GENOTYPE × AGE INTERACTION, AND THE INSULIN-LIKE GROWTH FACTOR I AXIS IN THE SAN ANTONIO FAMILY HEART STUDY: A STUDY IN HUMAN SENESCENCE

BY VINCENT PAUL DIEGO BA, University of Guam, 1995 MA, Binghamton University (SUNY), 2001

DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Anthropology in the Graduate School Binghamton University State University of New York 2005

ÂŠ 2005 by Vincent P. Diego. All rights reserved

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Accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Anthropology in the Graduate School of Binghamton University State University of New York 2005

April 15, 2005 Ralph M. Garruto, Department of Anthropology, Binghamton University Jean W. MacCluer, Department of Genetics, Southwest Foundation for Biomedical Research Michael A. Little, Department of Anthropology, Binghamton University John Blangero, Department of Genetics, Southwest Foundation for Biomedical Research John Relethford, Department of Anthropology, Binghamton University (Adjunct), Department of Anthropology, SUNY at Oneonta

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ABSTRACT This dissertation approaches human senescence from a statistical genetic perspective and works with data provided by the San Antonio Family Heart Study (SAFHS). It is shown how statistical genetics provides a logical foundation for traditional approaches to studying human senescence. For analytic tractability, the insulin-like growth factor I (IGF-I) axis is adopted as the main physiological system of interest. In theory, however, the statistical genetic approach used in this research can be applied to any physiological system. Working from within the statistical genetic framework, the basic model therein is improved upon and extended to include genotype × age interaction. Genotype × age interaction was found to be important in the overall behavior of the IGF-I axis in the SAFHS. The statistical genetic, biomedical and evolutionary implications of this finding are explored. The theory of genotype × age interaction is then extended to include mitochondrial effects, which are known to play important roles in human senescence. Lastly, the findings of this dissertation research are summarized.

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Dedicated to my parents, Frank Paulino Diego and Terrisita Leon Guerrero Taitague Diego, my siblings, Eileen Diego Meno, Michael Diego, Patrick Diego, Bernadette Diego Lujan, and Frank P. Diego, Jr., and their families. In His infinite wisdom and mercy, God knew I was weak, so He gave me a loving family.

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ACKNOWLEDGEMENTS My doctoral journey starts with Professor Gary M. Heathcote at the University of Guam. He introduced me to biological anthropology, nurtured my growth in said field, and then treated me early on like a full-fledged colleague of his. If it were not for his early, positive influence, I would not have pursued graduate studies in biological anthropology. And it was he who introduced me to Professor Ralph M. Garruto at Binghamton University (State University of New York), about whom I have more to say below. I thank Gary also for being a close friend of mine over the years, a sounding board whenever I needed one, an academic ally and promoter when I had none, a man in my academic corner whenever I felt dejected and down-trodden, and someone competent to share my academic ideas and dreams with, no matter how outlandish they might have been. Drs. Jane Underwood at the University of Arizona at Tucson and Alexander Kerr now at the University of Guam and my good friend Frank Camacho have similarly been there for me along the way and I am thankful to them for their warm friendship. So I came to Binghamton University (SUNY) to study with Dr. Ralph M. Garruto. What can I say about the man? To begin with, heâ€™s a great human being. Thatâ€™s what Dr. Jane Underwood said about him when I had inquired with her about his personality when I was a prospective student. Fortunately, I came to the same view on my own. Ralph always made sure that I pursued what I was interested in, not what he was interested in. Early in my first year in 1999 in the fall, I told him that I was interested in doing my dissertation research on the statistical genetics of the complex diseases associated with aging. It just so happened that the annual meeting for the physical anthropology and human biology societies were being held in San Antonio in the spring

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of 1999. Ralph took the time to meet with the folks at the Department of Genetics, Southwest Foundation for Biomedical Research (SFBR) at San Antonio and inquire with them about the possibility of training a student of his (me of course). I remember vividly our meeting in his office upon his return from San Antonio. He told me that he had spoken with some people at the SFBR and that they had expressed interest in taking on one of his students. He mentioned two names in particular, Drs. Jean W. MacCluer and John Blangero. I will talk more about the SFBR and these latter two individuals below, but for now I continue on with Ralph and my time at Binghamton. So I progressed through the masterâ€™s program in due time and graduated in 2001. Now, I am still embarrassed to report a certain height of absent-mindedness of mine, but it is a necessary part of my later story. I had completely forgotten to apply for financial support from the Department of Anthropology in the spring of 2000 for the following academic year. When this became known, Ralph was extremely upset and angry and I was feeling very down-trodden. While the first half of that summer immediately following was extremely trying, the second half held out hope. It just so happened that Jean was at this time looking for a pre-doctoral-level research assistant and that this person would be trained in statistical genetics, and have their pick of studies being carried out by the Department of Genetics. So when Jean had approached Ralph and Professor Michael A. Little about taking on a student of theirs, it came to pass that I took up the position and made my way to San Antonio. They say you reap what you sow, and I guess we were reaping what Ralph had sown in the spring of 1999. Before moving onto my San Antonio phase I should say a little more about Professor Little. I greatly appreciate Mikeâ€™s sincere interest in my academic development and his help in this regard. Mike served on my

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master’s committee and now also on my dissertation committee. I should also thank Professor John Relethford, who also served on my dissertation committee. Dr. Relethford is the only person I know who can lecture on Hardy-Weinberg Equilibrium and have the class cracking up most of the way through. It is such an honor for me to be able to say that these three distinguished biological anthropologists were on my dissertation committee. But it gets better! So I moved to San Antonio to learn statistical genetics under Drs. Jean W. MacCluer and John Blangero at the SFBR. I’ve been there ever since and am just now finished with my dissertation research, the spring of 2005. Jean is the nicest, sweetest scientist I know. She has always made me feel at home in the Department of Genetics and, perhaps more importantly, that my work was valuable and my thoughts were worth discussing. It’s important to realize that Jean is a highly-respected human geneticist and is the principal investigator (PI) and co-PI of several multi-million dollar research grants. Yet, she is always humble and unassuming in her conversations and always willing to listen to what you have to say. John Blangero taught me what I know in statistical genetics, which is still a little to be sure but much more than what I had coming in. Dr. Ravindranath Duggirala, who is a Scientist in the department, and myself call John “Maha Guru”, which is Indian for “Great Teacher”. The dude is straight-up brilliant and his knack for real-world problem-solving in statistical genetics never ceases to amaze me. Also, if there be any complaints on the mathematical nature of my dissertation research, the proper person to complain to is Dr. Blangero. Now there are the friends and family to thank. I’ll take my friends first. I am happy to thank my friends in Binghamton: David Hopwood, Nasser Malit, Bretton Giles,

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Marie Marley, Jen Bauder, Patrick Clarkin, Stephanie Rutledge, Bridgette Zavala, Felix Acuto, Alex Novgloski, Julia McCausland Gaines, Ralph Quam (Dr. P!), Tom Beasom, Tom Pearson, Laura Soloway, and Helene van Berge-Landry. You all made Binghamton a bearable and warm place for a Pacific Islander, even during the harsh winter. A special thanks goes out to David Hopwood for going the extra mile a number of times for me . . . David is from Canada (not Canadia!). So for Dave: Go Canada!! Iâ€™m also happy to thank my friends at San Antonio. At the SFBR, I would like to thank Nico Guoin, Prakash Nair, and the Population Genetics Office people, Linda Freeman-Shade, Amuche Ezeilo, Kent Polk, Debbie Newman (lifetime member) and Cheryl Reindl (honorary member). There are many others but we have a big department. I would also like to thank my two pool shooting friends who helped me to keep sane, Jonathan Camacho and Art Williams. At my church, Freedom Baptist Church, I would like to thank Preacher Lamb, and brothers V, Nacho, Thomas, Rob, Randy, Roy, Ben, Joseph, Sam, Rudy, Eakin, Henry, Miguel, Reggie, Nate and others for their fellowship in the Lord. I have to acknowledge the love and support that my family has given me throughout the years. I would like to thank my mom and dad for being wonderful, loving parents. I cannot thank them enough. They taught me the value of hard work and of humility and it is these traits in particular that have brought me to this point. I would also like to thank my brothers and sisters but especially my oldest sister Eileen, who has bore the brunt of my vacation visits. It was really important for me to see family once in a while as I was working on my dissertation. Lastly, I thank God, Jesus Christ, and the Holy Spirit. I was lost in darkness, and Jesus brought me back to live in the light of His righteousness. I offer my lifeâ€™s work as my humble service and in honor of the Lord.

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Table of Contents Title Page…………………………………………………………………………………..i Copyright Notice………………………………………………………………………….ii Signature Page……………………………………………………………………………iii Abstract…………………………………………………………………………………...iv Dedication………………………………………………………………………………....v Acknowledgements……………………………………………………………………….vi Table of Contents………………………………………………………………………….x List of Tables…………………………………………………………………………….xii List of Figures…………………………………………………………………………...xiii Chapter 1. Introduction…...………………………………………………………………1 Chapter 2. Background: Mathematical Biology of Senescence…..………………………6 Chapter 3. Background: Endocrinology of the IGF-I Axis in Relation to Senescence…32 Chapter 4. Background: The Study Population and Epidemiological Patterns…………44 Chapter 5. Methods I: Sampling Design, Pedigrees, and Phenotypes…………………..61 Chapter 6. Methods II: The Multivariate Mixed Effects Linear and Polygenic Models……………………………………………………………………………72 Methods II: Theory and Model of Genotype × Environment Interaction……......77 Chapter 7. Methods III: Likelihood Theory and Maximum Likelihood Estimation……...………………………………………………………………...93 Methods III: Hypotheses and Statistical Inference……………………………..101 Methods III: Power and Alternative Test Statistics………………………….…112 Chapter 8. Results: Statistical Behavior of the Phenotypes..…………………………..116 Results: Model Results………………………………………………………....116 Results: Power Analyses of the Genotype × Age Interaction Model…………..132 Chapter 9. Discussion: Statistical Genetic Finding……...……………………………..144 Discussion: Biomedical Ramifications…………………………………………153 Relation to Metabolism in Adulthood and the Metabolic Syndrome……………………………………………………….154 Ontogeny, Aging, and Neuroendocrine Cascades……………...159 Discussion: Evolutionary Ramifications……………………………………….166 Chapter 10. Conclusions: Caveats….……...…………………………………………..172 Conclusions: Prospectus………………………………………………………..172 Conclusions: Conclusions………………………………………………………175

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Appendix A: A Geometric Proof of the G × E Interaction Theorem…………………...178 Appendix B: Relation to the Gaussian and Ornstein-Uhlenbeck Stochastic Processes………………………………………………………………………..188 Appendix C: Derivation of the Elements in the Expected Fisher Information Matrix……….…………………………………………………………………..196 Appendix D: Geometry of the Likelihood Function……………………………………237 References………………………………………………………………………………240

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List of Tables •Table 1. Numbers of Relative Pairs in the SAFHS…………………………………….63 •Table 2. Descriptive Statistics of Raw Data……………………………………………67 •Table 3. Descriptive Statistics of Log-Transformed Data……………………………...67 •Table 4. Genome-wide expectations for alleles identical by descent (IBD)…………...75 •Table 5. Trait Heritabilities of the IGF-I Axis Components in the SAFHS…………..125 •Table 6. Models: Polygenic versus Genotype × Age Interaction……………………..125 •Table 7. Model Fitting for Log IGF-I under the Genotype × Age Interaction Model…………………………………………………………………………...126 •Table 8. Model Fitting for Log IGFBP-3 under the Genotype × Age Interaction Model…………………………………………………………………………...126 •Table 9. Model Fitting for Log Ratio3 under the Genotype × Age Interaction Model…………………………………………………………………………...127 •Table 10. Power Analyses: Parameter Sets…………………………………………...135

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List of Figures •Figure 1. Linear decline with age in physiological variables…………………………....8 •Figure 2. Brown and Forbes model of the mortality process with increasing age……..11 •Figure 3. Reliability Structures for (a) technical devices and (b) complex organisms...14 •Figure 4. First-order Taylor approximations…………………………………………...18 •Figure 5. Relation between fitness, measured by r, and the level of investement in somatic maintenance, s……..………………………………………………….30 •Figure 6. The main endocrine axes in aging and senescence…………………………..33 •Figure 7. IGF-I secretion pattern early in the human life span………………………...35 •Figure 8. Schematic description of the IGF-I axis and the two major sites of IGF-I secretion…………...……………………………..…………………………….36 •Figure 9. Endocrine, paracrine, and autocrine modes of IGF-I action…………………38 •Figure 10. The somatomedin hypotheses………………………………………………39 •Figure 11. Schematic of a gene expression network…………………………………...43 •Figure 12. Map of Bexar County in Texas……………………………………………..45 •Figure 13. Map of San Antonio in Bexar County……………………………………...46 •Figure 14. Rise of the Hispanic population in San Antonio, 1900-2000………………47 •Figure 15. Relative increase of the Hispanic population in Bexar County, Texas, 1990-2030…………….……………………………………………………..…48 •Figure 16. Relative standardized mortality ratios (RSMR) for total mentions of diabetes mellitus: Spanish surname and non-Hispanic whites age 30 and over by sex………...…….………………………………………………...50 •Figure 17. Change in T2D incidence in San Antonio, Texas………………………………..51 •Figure 18. T2D and CVD mortality in San Antonio…………………………………………..53 •Figure 19. Heart disease and T2D mortality in Bexar County, 2002…………………..55 •Figure 20. Schematic diagram of the epidemiologic transition………………………...56 •Figure 21. SAFHS recruitment area……………………………………………………62 •Figure 22. Schematic pedigree structure for the typical extended family unit in the SAFHS…………………………………………………………………………63 •Figure 23. Histograms of raw IGF-I and IGFBP-1 data………………………………..68 •Figure 24. Histograms of raw IGFBP-3 and Ratio3 data………………………………69 •Figure 25. Histograms of log transformed IGF-I and IGFBP-1 data…………………..70 •Figure 26. Histograms of log transformed IGFBP-3 and Ratio3 data………………….71 •Figure 27. G × E interaction variance under heteroscedasticity and homoscedasticity..84 •Figure 28. Graphical representation of exponential functions………………………..108 •Figure 29. One- and two-tailed tests on the assumption that maximum likelihood 2 estimates (MLEs) are normally distributed N μ = 0, σ = 1 …………………111 •Figure 30. Age-specific means and variances in IGF-I levels (ng/ml)………………..117 •Figure 31. IGF-I versus age and BMI………………………………………………...118 •Figure 32. Age-specific means and variances in IGFBP-1 levels (ng/ml)……………119 •Figure 33. IGFBP-1 versus age and BMI……………………………………………..120 •Figure 34. Age-specific means and variances in IGFBP-3 levels (ng/ml)……………121 •Figure 35. IGFBP-3 versus age and BMI……………………………………………..122 •Figure 36. Age-specific means and variances in Ratio3……………………………...123 •Figure 37. Ratio3 versus age and BMI………………………………………………..124

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•Figure 38. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans (variance functions)………………………………………………128 •Figure 39. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans: The same genetic correlation function is displayed on 2 scales…129 •Figure 40. Questionable genotype × age interaction for log IGFBP-3 in SAFHS Mexican Americans: Phenotypic, additive genetic and environmental variance functions…………………………………………………………….130 •Figure 41. Apparent genotype × age interaction due to significant heteroscedasticity in the environmental variance function for log Ratio3 in the environmental variance in SAFHS Mexican Americans……………………………………..131 •Figure 42. Age-distribution used for power analyses…………………………………134 •Figure 43. Power analyses: additive genetic variance……………………………...…136 •Figure 44. Power analyses: genetic correlation……………………………………….137 •Figure 45. Power analyses: environmental variance………………………………….138 •Figure 46. Additive genetic variance in fetal ultrasound morphometrics in the baboon, Papio hamadryas (spp.)……………………………………………..146 •Figure 47. Additive genetic variances in phenotypes associated with atherosclerosis………………………………………………………………...147 •Figure 48. Genetic parameters for atherosclerosis risk factors in the Framingham Heart Study…………………………………………………………………...148 •Figure 49. Power to detect G × E interaction by ANOVA……………………………150 •Figure 50. Power analysis of G × E interaction in samples of twin pairs…………….152 •Figure 51. Schematic diagram of changes in rank and scale along n segments of a continuous environment. I.…………………………………………………..157 •Figure 52. Schematic diagram of changes in rank and scale along n segments of a continuous environment. II…………………………………………………..163 •Figure 53. Additive genetic variance in ln mortality in Drosophila melanogaster…...170 •Figure A1. Schematic Representation of Vector Space in ℜ 2 ………………………..179 •Figure A2. Geometry of Heteroscedasticity I: The Degenerate Triangle in Vector Space………………………………………………………………………….184 •Figure A3. Geometry of Heteroscedasticity II: The Law of Pythagoras in Vector Space………………………………………………………………………….184 •Figure A4. Geometry of Heteroscedasticity III: The Degenerate Triangle in Vector Space………………………………………………………………………….185 •Figure A5. Geometry of Homoscedasticity I: The Zero Vector in Vector Space…….186 •Figure A6. Geometry of Homoscedasticity II: The Law of Pythagoras in Vector Space………………………………………………………………………….187 •Figure A7. Geometry of Homoscedasticity III: The Degenerate Triangle in Vector Space………………………………………………………………………….187 •Figure D1. Geometry of the Ln-Likelihood Function………………………………...238

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Chapter 1 Introduction Historically, biological anthropology has maintained a deep and abiding interest in the genetics of complex traits (MacCluer, 1993; Weiss, 1993, 1998a&b, 2000; Williams-Blangero and Blangero, 1993; Blangero, 1993; Rogers et al., 1999). Yet only as two decades ago have the analytical methods needed to study the genetics of complex traits in anthropological settings come to fruition. This is not a criticism of the field, but, rather, a reflection of the difficulties inherent in studying complex traits. Effecting a breach of the seemingly insurmountable difficulties has required nothing short of scientific revolutions in molecular and statistical genetics, and in mathematical and computational statistics. Now that these experimental and analytical methods have been developed, biological anthropology can examine anew its subordinate interests with respect to the larger category of complex traits. In essence, the objective of the present dissertation is to examine a traditional topic of interest, under the larger category of complex traits, from the perspective of modern statistical genetics. To fully understand the goals of this research, the developments just discussed need to be taken contrapuntally with other, intimately-related developments within the field of biological anthropology itself, which are specifically increasing interests in research on aging (Crews, 1993, 1997; Crews and Garruto, 1994) and in biomedical problems (Garruto et al., 1989, 1999; Little and Haas, 1989; Little and Garruto, 2000). For the purposes of this dissertation, one can combine all of these developments into one theme, namely the statistical genetics of human senescence.

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The approach of Blangero (1993), which is framed in terms of the statistical genetic theory of genotype × environment (G × E) interaction, is used to address human senescence. Because of the interest in senescence, this dissertation focuses on a specific class of G × E interaction, namely genotype × age interaction, where the age continuum has commonly been conceptualized as a special class of continuous environments (Hegele, 1992; Zerba and Sing, 1992; Zerba et al., 1996, 2000; Jaquish et al., 1997; Duggirala et al., 2000). For analytic tractability, the growth hormone/insulin-like growth factor I (GH/IGF-I) axis is used as a microcosm of the complex physiology of senescence. Thus, this dissertation is specifically on genotype × age interaction in the GH/IGF-I axis in relation to the biology of senescence. It is common to focus on the components of the GH/IGF-I axis involving just IGF-I and its binding proteins because it is difficult to get a useful measure of GH without requiring overnight stays on the part of study individuals (Neely and Rosenfeld, 1994). For this reason, the GH/IGF-I axis is hereon referred to as the IGF-I axis. This dissertation is also a small part of a comprehensive research project on the statistical genetics of cardiovascular disease (CVD), namely the San Antonio Family Heart Study (SAFHS). CVD is considered to be one of the major diseases of the metabolic syndrome (Reaven, 1988, 1993, 1995, 1999). Given that the overall metabolic dysfunction encompassed by the metabolic syndrome is known to be strongly age-related (Liese et al., 1998), it is perhaps safely assumed that the metabolic syndrome is one of the more complex manifestations of senescence. In other words, the metabolic syndrome is studied here from the perspective of the biology of senescence.

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Statement of the Problem This dissertation research will characterize the statistical genetics of the IGF-I axis in relation to age in the development of the metabolic syndrome in Mexican Americans of San Antonio participating in the SAFHS. Specific Aims and Hypotheses The statistical genetic characteristics of the IGF-I axis along the age continuum are analyzed under a sequence of models. Firstly, the simplest statistical genetic model, known as the polygenic model, is used to establish whether or not genetic factors are important in the phenotypic determination of the components of the IGF-I axis. If the heritability—which is taken as an indicator of genetic influence—of a given component is found to be significant, then that component will be analyzed further using the genotype × age interaction model. The null hypothesis under the theory of genotype × age interaction is that the gene expression network (GEN) underlying the IGF-I axis is insensitive to changes in age. That is, changes in age have no effect on the GEN of the IGF-I axis. Supposing that significant genotype × age interaction effects are found, it will be interesting to establish whether these arise from either of two sources of genotype × age interaction, which are variance heterogeneity and a genetic correlation coefficient significantly different from 1, or from the two sources acting jointly. The chapters to follow will develop the background knowledge and statistical genetic theory needed to better understand these aims. The aims, hypotheses, and statistical analyses used in this study are as follows: Specific Aim 1: It became apparent early on to the author, in carrying out this dissertation research, that the relation between statistical genetics on the one hand and more established approaches towards studying senescence on the other was not at all clear. Therefore, the first aim of this dissertation is to show how the statistical genetic approach

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is related to other approaches to studying senescence. This aim can be addressed by showing how the major theories of senescence can be unified and then by showing how statistical genetics provides both a foundation and extension of this unified structure. Specific Aim 2: To determine if the components of the IGF-I axis are significantly influenced by genetic factors. •Hypothesis 1: The heritabilities of the components of the IGF-I axis are significant. Specific Aim 3: To determine if the components of the IGF-I axis are influenced by age effects. •Hypothesis 2: The components of the IGF-I axis are more consistent with the genotype × age interaction model than with the simple, polygenic model. Specific Aim 4: To describe in terms of statistical genetic parameters how the behavior of the IGF-I axis is sensitive to the age continuum. This requires having found significant heritability and then significant genotype × age interaction effects. •Hypothesis 3: The additive genetic variance significantly changes with age. •Hypothesis 4: The genetic correlation coefficient is significantly different from 1. Outline of the Dissertation It is perhaps worthwhile to discuss the organizational structure of this dissertation. Important background concepts are introduced and developed in the first three ensuing chapters. The first background chapter covers the mathematical biology of senescence, which includes proximate and ultimate mathematical models of senescence. It will be pointed out in this chapter how major theories of senescence can be unified and how statistical genetics provides a foundation for the unified structure. The second background chapter delves into the physiology of senescence with a focus on the role played by the IGF-I axis. The third background chapter discusses the basic population biology for this study, including the study population and important epidemiological concepts. The ensuing chapters generally follow the traditional organization of methods, results, discussion, and conclusions, with a prospectus section being included in the

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conclusions chapter. The methods are discussed in a sequence of three chapters. The first of the methods chapters treats the more empirical aspects of the present work, which includes the sampling design, pedigree and relationship structures, demographics, and the phenotypes. The next two methods chapters follow the logical structure of statistical inference in that the first of these develops the statistical genetic models employed in this research and the next discusses the elegant machinery of likelihood-based statistical inference, which includes maximum likelihood estimation, hypothesis testing by recourse to the likelihood ratio test statistic, and statistical power calculations. As regards the chapter on statistical genetic models, it will be shown in that chapter how the basic model can be improved by allowing for G Ă— E interaction in general and genotype Ă— age interaction in particular. The next two chapters focus on the results and discussion. In the prospectus section of the conclusions chapter, an extension of the genotype Ă— age interaction model in relation to mitochondrial theories of senescence is developed. The conclusions of this dissertation research are then summarized in the section just following. It should be pointed out that there are four appendices that follow the main body of the text. These appendices at once enable a more coherent and flowing structure in the main body of the text and a forum for the discussion of concepts and the derivation of equations that are not immediately necessary for understanding the dissertation research. The appendices are introduced in the development of the main body of the text.

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Chapter 2 Background: Mathematical Biology of Senescence Senescence can be approached from diverse perspectives. Indeed, according to Medvedev (1990), there are more than 300 theories of aging and senescence. It will be argued in this chapter that these diverse perspectives can be unified and that statistical genetics offers a strong foundation for this unified structure. There are two categories of theories of senescence, which may be called proximate and ultimate explanations of senescence, after Mayr’s (1961) dichotomy of the explanation of biological phenomena (on a similar approach to senescence, cf. Finch and Rose, 1995; Masoro, 1999: ch. 5). One way to unify these two categories is to show explicitly how the proximate-level models can be used to build up, as it were, to the ultimate-level models. To unify the proximate and ultimate categories along these lines, two ideas are needed, which are Cannon’s (1929, 1939a) concept of physiological homeostasis and Simms’s (1942a) observation that the linear decline in homeostasis with increasing age can be related to the mortality risk observed in animal populations. Although the concept of homeostasis is original with Cannon (1929), a preferable definition is given by Shock (1977) as the systemic regulation of physiological functions such that organism-level integration is achieved in the face of a dynamic environment. Senescence is defined as the physiological deterioration associated with aging (Finch, 1990), which is brought about by the age-related decline in homeostasis. Canon (1939a&b, 1942) also originally proposed the view that senescence is characterized by an age-associated deterioration in the ability to maintain homeostasis against continual perturbations, extrinsic or intrinsic to the organism. Cannon’s view became a principle that was widely invoked in the fields

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of physiology (Simms, 1940, 1942a&b, 1946; Shock, 1952, 1961, 1969, 1974, 1977; Comfort, 1956, 1968; Kenney, 1982), and clinical science (Selye, 1946, 1950, 1951, 1955, 1956, 1970a&b, 1976; Kohn, 1963, 1978, 1982, 1985; Selye and Prioreschi, 1972). Indeed, Dilman (1981) proclaimed the above principle to be a biological law, “the law of deviation of homeostasis”. This principle is of importance because it suggests that aging individuals are increasingly predisposed to succumbing to perturbations in homeostasis (Strehler, 1977; Kohn, 1978). In fact, the interaction of stress and homeostasis in relation to disease and aging formed a central component of Selye’s theory of the “general adaptation syndrome” (Selye, 1946, 1950, 1951, 1955, 1956, 1970a&b, 1976; Selye and Prioreschi, 1972; for recent reformulations, see Frolkis, 1993; McEwen and Stellar, 1993; McEwen, 1998). Simms (1942a) made another advance when he suggested that the observation of a gradual or linear decline in homeostasis (Canon, 1939a; Simms, 1940, 1942a, 1946; Shock, 1955, 1961, 1977, 1985; Kohn, 1963, 1978, 1985; Fig. 1) can be logically related to the exponential mortality rate in animal populations. Simms’s (1942a) observation encouraged a number of theories of senescence relating the physiological characteristics of populations to their mortality rate, which is taken to be a proxy of the senescence rate (for historical reviews, see Strehler, 1959, 1977; Mildvan and Strehler, 1960; Kohn, 1978; Economos, 1982). As will be seen, linear decline in homeostasis or, more usually, an inversely proportional linear increase in physiological damage, thought to accrue under declining homeostasis, is often the critical assumption in proximate-level models that predict a fairly universal mortality pattern. This universal mortality pattern is known as the Gompertz and GompertzMakeham mortality functions (Gompertz, 1825; Olshansky and Carnes, 1997), which are

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Figure 1. Linear decline with age in physiological variables. All values were standardized against the value at 30 years of age and so percent remaining means deviation from that value. Source: Strehler (1959).

respectively given as: m(x ) = AeÎąx ,

Eq. 1

m(x ) = AeÎąx + E ,

Eq. 2

and

where A and Îą are constants to be determined by data, and E is a correction term that accounts for mortality due to sources extrinsic to the organism such as accidents and infectious diseases. There are several models that have derived the Gompertz on the basis of general assumptions and that can be applied to physiological systems in general. These are the models provided by Sacher and Trucco (Sacher, 1956, 1966, 1978; Sacher

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and Trucco, 1962; Trucco, 1963a&b), Brown and Forbes (1974a&b, 1975, 1976), Koltover (1982, 1983, 1992, 1996, 1997, 2004), Gavrilov and Gavrilova (2001, 2002a&b), van Leeuween et al. (2002), and Mangel and Bonsall (2004). Because the models of Sacher and Trucco and of Brown and Forbes are mathematically equivalent, a review of one of them will suffice. The Gavrilov and Gavrilova model is radically different and so this too will be reviewed. The models by Koltover, van Leeuween and colleagues, and Mangel and Bonsall are conceptualized in relation to oxidative stress. After making some introductory remarks on the roles of oxidative stress and mitochondrial dysfunction in senescence, these models will be discussed together. It is perhaps encouraging that very different perspectives lead to the same outcome. Brown and Forbes (1974a) developed a model that is mathematically equivalent to the model of Sacher and Trucco (see also extensions of the model in Brown and Forbes, 1974b, 1975, 1976). The assumptions of the Brown-Forbes model are: 1) The state of physiological injury that may lead to death, if severe enough, is inversely and linearly related to the decline in homeostasis. 2) The observed state of physiological injury y x at corresponding age, x , (satisfying assumption 1) may be taken as an

(

)

observation from a Gaussian random variable, Y , where Y ~ N μ, σ 2 , where by

convention upper case will be taken to denote the random variable, Y , and lower case, y x , denotes an observation on Y , μ is the mean of the normal distribution and σ 2 is the

variance. As such, observations on Y may be modeled as a linear regression on age: y x = μ + ε = α + β ⋅ x ; E (ε ) = 0 ,

Eq. 3

where ε is a random error term with expectation zero, and α and β are respectively the intercept and slope of the regression line of y x on x . 3) There is a theoretical cut-off

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level giving the absolute level of physiological injury that may be sustained before dying. This cut-off level is represented by a horizontal line that gives some constant high value of y for all x and that lies above and is approached from below by the regression line. Therefore, the cut-off level, denoted by y c , satisfies the constant function: yc = x ; ∀ x ∈ ℜ .

Eq. 4

Conceive now of a series of normal distribution curves, where each one is centered on a point on the regression line given by Equation 3. Since the regression line is approaching the horizontal line y c from below, the normal curves will come to have increasing area falling above y c . By the assumptions of the model, the area for any given y x falling above y c will give the probability of mortality, m(x ) , at the corresponding age, x (Fig. 2). Therefore, integration under the appropriate interval, from y c to + ∞ (in the y-axis), will give m(x ) . Now, Equation 3 allows m(x ) to be written as: m (x ) = ∫

⎧ 1 2⎫ exp⎨− 2 (y x − [α + β ⋅ x ] ) ⎬dy x . yc σ 2π ⎩ 2σ ⎭ ∞

1

Eq. 5

Generally, random variable Y may be expressed in terms of the standard normal distribution, N(μ = 0, σ 2 = 1), by way of the Z-transformation:

Z=

Y−μ . σ

Eq. 6

To prepare for use of the Z-transformation in the present context, define the distances: ε x = y c − (α + β ⋅ x ) ,

Eq. 7

ε0 = yc − α .

Eq. 8

and

10

m (x )

yc = x

y

y = α +β⋅x

x Figure 2. Brown and Forbes model of the mortality process with increasing age. Redrawn from Brown and Forbes (1974a).

To evaluate the integral in the interval [y c ,+∞ ) , we may now take the Z-transformed version of y c , i.e.,

εx σ

, and evaluate it under the standard normal, noting that here μ = 0

and σ 2 = 1 : ⎛ 1 ⎡ε ⎤2 ⎞ σ ⎧ 1 2⎫ exp⎨− y x ⎬dy x ≈ exp⎜ − ⎢ x ⎥ ⎟ . ⎜ 2⎣σ⎦ ⎟ 2π ε x 2π ⎩ 2 ⎭ ⎝ ⎠

1

∞

m (x ) = ∫ ε x σ

Eq. 9

where the solution is an approximation of the integral (from Feller, 1968: 166). On expressing Equation 7 in terms of Equation 8, the quadratic term in Equation 9 may be written as: 2

εx σ

2

=

(ε 0 − β ⋅ x ) 2 σ

2

ε 0 + 2ε 0 (− β ⋅ x ) + (− β ⋅ x ) 2

=

σ

2

2

.

Eq. 10

At large values of y x , the inequality:

11

−β⋅x σ

holds true such that

(− β ⋅ x ) 2 σ

2

<<

ε0 σ

,

Eq. 11

can be assumed in this case to make a negligible

contribution in the expansion of the quadratic term in Equation 10. Consequently, we arrive at: 2

εx σ

2

ε 0 + 2ε 0 (− β ⋅ x ) 2

≈

σ

2

,

Eq. 12

; ∀ β ⋅ x << ε 0 .

Eq. 13

and εx σ

≈

ε0 σ

−

2ε 0 ⋅ β ⋅ x

≈

σ

ε 0 − 2ε 0 ⋅ β ⋅ x σ

≈

ε0 σ

These approximations can be substituted into Equation 9 as follows: m (x ) ≈

⎛ 1 ⎡ ε 2 + 2ε 0 (− β ⋅ x ) ⎤ ⎞ exp⎜ − ⎢ 0 ⎥ ⎟⎟ 2 ⎜ 2π σ ⎥⎦ ⎠ ⎝ 2 ⎢⎣

σ ε0

Eq. 14 ≈

⎛ 1 ⎡ε ⎤ 2 ⎞ ⎛ ε ⋅β ⋅ x ⎞ exp⎜ − ⎢ 0 ⎥ ⎟ exp⎜⎜ 0 2 ⎟⎟ . ⎜ 2⎣σ⎦ ⎟ 2π ⎝ σ ⎠ ⎝ ⎠

σ ε0

At age x = 0 , define m(0) = m 0 and y(0) = y 0 . In analogy to Equation 9, an expression for the mortality function at age x = 0 is: ∞

m 0 = ∫ ε0 σ

⎛ 1 ⎡ε ⎤2 ⎞ σ ⎧ 1 2⎫ exp⎨− y 0 ⎬dy 0 ≈ exp⎜ − ⎢ 0 ⎥ ⎟ , ⎜ 2⎣σ⎦ ⎟ 2π ε 0 2π ⎩ 2 ⎭ ⎝ ⎠

1

Eq. 15

where we see that the first two terms in the right hand side of Equation 14 are in fact identical to m 0 . On putting α * =

β ⋅ ε0 σ

2

, m 0 = A , and appropriately substituting in

Equation 14, we now have:

12

m(x ) ≈ Ae

α* x

,

Eq. 16

which is the Gompertz mortality function. We now turn to the model developed by Gavrilov and Gavrilova (2001, 2002a&b), which is based on reliability theory. The application of reliability theory to the problem of aging in organisms was independently pioneered by several groups in the late seventies (Rosenberg et al., 1973; Skurnick and Kemeny, 1978, 1979; Abernethy, 1979; see also the work of Gavrilov and Gavrilova (cited in their 1991 book) in Russian publications in the late seventies). Reliability models sensu stricto of aging in organisms were developed by Abernethy (1979, 1998), Ďoubal (1982), Koltover (1982, 1983, 1992, 1996, 1997, 2004), Witten (1983, 1984a&b, 1985), Miller (1987, 1989), Ďoubal and Klemera (1989, 1990), Gavrilov and Gavrilova (1991, 2001, 2002a&b), and Izsák and Gavrilov (1995) (again, see the earlier work in Russian by Gavrilov and Gavrilova). The model developed by Gavrilov and Gavrilova (hereon G&G) is fairly general and may be taken as representative of the scope of reliability models (however, the Koltover model will also be reviewed shortly). Exposition of the G&G model requires some terminology from reliability theory regarding how systems are constructed (see Fig. 3). A serial or serially-constructed system is one that requires for its correct and continued operation that every single one of its components is correctly operating or functioning. Failure in one component results in system failure. This brings to mind the old saying that “A chain is only as strong as its weakest link”. In this case, component redundancy is irrelevant to system operation. A parallel or parallel-constructed system is one that requires that at least 1 out of n components are properly operating or functioning for its correct and continued operation. Thus, in this case, the probability that a system remains operational

13

Figure 3. Reliability Structures for (a) technical devices and (b) complex organisms. (a) Technical devices are serially connected between and within sub-systems. The large blocks (j = 5) represent sub-systems and the small blocks represent elements therein. (b) Organisms exhibit serial connections between subsystems (represented by the larger, m = 5 vertical rectangular blocks) that themselves exhibit parallel construction (represented by the smaller, k = 10 horizontal rectangular blocks). (a) and (b) also differ in the quality of elements. Organisms can sustain an initially high degree of defects (cross-marks) whereas technical devices start out with an initially low level of defects by design. Source: Gavrilov and Gavrilova (2001).

or alive is a function of component redundancy. Under the G&G model, multicellular organisms exhibit both types of construction in that organisms are serially constructed out of sub-systems (each one necessary for survival of the organism) but each sub-system can be described as being parallel constructed. However, as senescence takes its toll, the redundancy at the sub-system level becomes completely exhausted and the organism degenerates into a serially-constructed system at which point any new instance of damage is sufficient to cause system failure or death. Further, at this point, the mortality rate becomes constant; that is, a mortality plateau is produced. It will be useful at this point to briefly state some of the fundamental concepts common to demography and reliability

14

engineering, as these concepts will form a common underlying theoretical basis (see Cox, 1962; Gross and Clark, 1975; Elandt-Johnson and Johnson, 1980; Crowder et al., 1991). Let X be a random variable representing the lifetime of individuals. Then the lifetime cumulative distribution function is defined as: F(x ) = Pr (X ≤ x ) ,

Eq. 17

and the survivorship function, now denoted by s(x ) , is defined relative to F(x ) as: s(x ) = 1 − F(x ) = Pr (X > x ) .

Eq. 18

Now, the lifetime probability density function, f (x ) , as for all probability density functions, is found by taking the first derivative of the corresponding cumulative distribution function: f (x ) =

dF(x ) dx

=

d[1 − s(x )] dx

=

− ds(x ) dx

.

Eq. 19

The mortality function, m(x ) , is defined as:

m (x ) =

f (x )

⎡ ds(x ) ⎤ ⎡ 1 ⎤ − d ln s(x ) ⎡ ds(x ) ⎤ ⎡ 1 ⎤ = −⎢ . ⎥ = −⎢ ⎥⎢ ⎥ = ⎥⎢ s (x ) dx ⎣ dx ⎦ ⎣ s(x ) ⎦ ⎣ s(x ) ⎦ ⎣ dx ⎦

Eq. 20

If F(x ) is known or taken as given, then s(x ) , f (x ) , and m(x ) are determined by Equations 18-20. The G&G model starts by deriving the mortality rate for blocks that are parallel constructed out of k mutually substitutable elements, each described by a constant failure rate φ . The cumulative distribution function for block failure, Fb (k, φ, x ) , is assumed to be:

(

)

Fb (k , φ, x ) = 1 − e −φ x . k

Eq. 21

From the relations mentioned just above, we have for the survivorship function:

15

(

)

s b (k, φ, x ) = 1 − 1 − e −φ x , k

Eq. 22

for the probability density function:

( [

− d 1 − 1 − e −φ x f b (k , φ, x ) = dt

]

k

) = kφ ⋅ e

−φ x

(1 − e )

− φ x k −1

,

Eq. 23

and for the mortality function:

(

f (k, φ, x ) kφ ⋅ e −φ x 1 − e −φ x m b (k, φ, x ) = b = k s b (k , φ, x ) 1 − 1 − e −φ x

(

)

)

k −1

.

Eq. 24

For a serially constructed system comprised of j blocks made up of k elements, the mortality function of the system is found by simply summing the block mortality rates: j

m s (k, φ, x ) = ∑ m b (h ) = j ⋅ m b (k, φ, x ) =

(

jkφ ⋅ e −φ x 1 − e −φ x

(

1 − 1 − e −φ x

h =1

)

k

)

k −1

.

Eq. 25

Now consider the more realistic case for organisms wherein which blocks are comprised of mutually substitutable elements, each of which may be defective or functional. For the distribution of the number of functional elements, denoted by i , out of k total elements, G&G postulate a truncated Poisson distribution: ⎧0 ; ∀ i = 0, k + 1, k + 2, k + 3,..., ⎪ Pi = ⎨ −λ λi ; ∀ i = 1,2,3,..., k , ⎪ce i! ⎩

Eq. 26

where c=

1 −λ

1− e − e

−λ

∑ i=k +1 λi i! ∞

,

Eq. 27

where λ is the parameter of the Poisson distribution and c is a normalizing factor ensuring that the probabilities of all possible outcomes sum to unity:

16

k

∑ P = 1.

Eq. 28

i

i =1

The Poisson distribution is truncated at the left as stipulated to acknowledge the fact that organisms cannot survive with zero functional elements and is truncated at the right as stipulated because the number of functional elements cannot exceed the total number of elements. Note that the normalizing constant accounts for the cases when i = 0 and

i = k + 1, k + 2, k + 3,..., ∞ . In this case, the mortality rate for such a system is given by: j

k

k

h =1

i =1

i =1

m s (k , φ, x ) = ∑ m b (i, h ) = ∑ jPi m b (i ) = jce −λ ∑

λi m b (i ) , i!

Eq. 29

where the mortality rate of blocks with i initially functional elements, denoted by m b (i ) , is given by an expression analogous to Equation 24: m b (i ) =

(

iφ ⋅ e − φ x 1 − e − φ x

(

1 − 1 − e −φ x

)

i

)

i −1

.

Eq. 30

In view of Equation 30, Equation 29 may be rewritten as: m s (k, φ, x ) = φλjce e

−λ −φ x

∑ i =1

(

λi −1 1 − e −φ x

k

[

)

(i − 1)! 1 − (1 − e

i −1

)]

−φ x i

.

Eq. 31

The situation seems rather messy at this point. However, some approximations and simplifications are possible. For g(x ) = 1 − e

, where g (⋅) will be hereon referred to as

− φx

the function for which the Taylor approximation is to be applied, we have: g ′(x ) =

[

d1− e dx

− φx

] = − de

− φx

dx

= φe

The first-order Taylor approximation of g(x ) = 1 − e

− φx

− φx

.

Eq. 32

about the point (x − a ) at a = 0 is

given as:

17

g (x ) ≈ g (0 ) + g ′(0 )(x − a ) = 0 + φx = φx .

Eq. 33

It is important to note that since φ gives the failure rate, the approximation assumes that the failure rate is linear in x (see Fig. 4: Left Panel). Using Equation 33 in the numerator in Equation 31, we can write the following expression, which employs another first-order Taylor approximation justified below: m s (k, φ, x ) ≈ φλjce e −λ

− φx

k (λφx ) i−1 (λφx ) i−1 −λ = φλjce ∑ . ∑ − φx i =1 (e i =1 (i − 1) ! )(i − 1)! k

Eq. 34

On comparing Equations 31 and 34, it would seem that the critical part of this last

0.2

1.2

0.18

1.15

0.16

1.1 0.14

1.05 f(x)

f(x)

0.12 0.1 0.08

1 0.95

0.06

0.9 0.04

0.85

0.02

0.8

0 20

30

40

50

60

70

20

80

30

40

50

60

70

80

Age (years)

Age (years)

Figure 4. First-order Taylor approximations. Left Panel: the function g (x ) = φx (solid

line) is a good approximation of the function g(x ) = 1 − e

(

φx << 1 , the function g(x ) = ⎡1 − 1 − e ⎢⎣ − φx (diamonds). function g(x ) = e

−φ x

− φx

(diamonds). Right Panel: for

) ⎤⎥⎦ (solid line) is well approximated by the i

18

approximation appearing in the denominator is:

(

)

⎡1 − 1 − e − φ x i ⎤ ≈ exp(− φx ) . ⎢⎣ ⎥⎦

Eq. 35

Equation 35 is justified as follows. The first derivative of the left hand side of Equation 35 is:

(

d ⎡1 − 1 − e ⎢ g ′(x ) = ⎣ dx

−φ x

) ⎤⎥⎦ i

(

− d⎡ 1 − e ⎢⎣ = dx

−φ x

) ⎤⎥⎦ i

(

−φ x = ⎡− i 1 − e ⎢⎣

)

i −1

(

⎤ d 1− e ⎥⎦ dx

−φ x

) Eq. 36

(

= ⎡− i 1 − e ⎢⎣

−φ x

)

i −1

−φ x

⎤ − de ⎥⎦ dx

= −φie

−φ x

(1 − e ) −φ x

i −1

.

For a = 0 , the first-order Taylor approximation about (x − a ) of the left hand side of Equation 35 is: g (x ) ≈ g(0 ) + g ′(0 )(x − a ) = 1 + 0 = 1 .

Eq. 37

The first derivative of the right hand side of Equation 35 is: g ′(x ) =

de

− φx

dx

= −φe

− φx

.

Eq. 38

For a = 0 , the first-order Taylor approximation about (x − a ) of the right hand side of Equation 35 is: g (x ) ≈ g (0 ) + g ′(0 )(x − a ) = 1 − φx .

Eq. 39

Therefore, for φx << 1 , the approximation is legitimate (Fig. 4: Right Panel). The summation in the right most term in Equation 34 can be expressed as the difference of infinite series:

(λφx )i−1 ∞ (λφx )i−1 ∞ (λφx )i−1 =∑ −∑ , ∑ i =1 (i − 1) ! i =1 (i − 1) ! i = k +1 (i − 1) ! k

Eq. 40

19

where the first term on the right hand side of Equation 40 is the power series definition of the exponential function, which in general is given as: x

e = 1+ x +

x2 2!

+

x3 3!

∞

xi

i =0

i!

+ ... = ∑

∞

x i −1

i =1

(i − 1)!

=∑

.

Eq. 41

Thus, Equation 34 may be rewritten as follows: m s (k, φ, x ) ≈ φλjce

−λ

(λφx )i−1 ⎡ αx ∞ (αx )i−1 ⎤ = A ⎢e − ∑ ⎥, ∑ i =1 (i − 1) ! i = k +1 (i − 1) ! ⎥ ⎢⎣ ⎦ k

Eq. 42

where A = φλjce − λ , and α = φλ . On noting the limit:

(αx )i−1 lim ∑ = 0, k →∞ x →0 i = k +1 (i − 1) !

Eq. 43

m s (k, φ, x ) ≈ Aeαx ,

Eq. 44

∞

we find that:

which is the Gompertz mortality function again. However, this holds to the extent that the approximation holds, which in turn is valid early in the lifespan (Fig. 4). Later in the life span, the system degenerates to a serially-constructed system, in which case the number of functional elements given by k approaches 1. Note also that the postulation of a Poisson distribution for the functional elements is no longer necessary. Therefore, for k ≈ 1 , we have from Equation 25: m s (k , φ, x ) =

(1 − e ) 1 − (1 − e )

jkφ ⋅ e

−φ x

− φ x k −1

−φ x k

≈

jφ ⋅ e

−φ x

(1 − e )

1−1+ e

−φ x 0

−φ x

= jφ .

Eq. 45

Thus, the phenomenon of the mortality plateau is observed. Oxidative stress and mitochondrial dysfunction are increasingly thought to play major roles in senescence (Wallace, 1992a&b, 1995, 1999; Ames et al., 1993; Shigenaga

20

et al., 1994; Sohal and Weindruch, 1996; Beckman and Ames, 1998; Lenaz, 1998; Wei, 1998; Wei et al., 1998; Ashok and Ali, 1999; Cortopassi and Wong, 1999; Finkel and Holbrook, 2000; Grune and Davies, 2001; Van Remmen and Richardson, 2001; Lenaz et al., 2002; Reis, 2003; Sastre et al., 2003; Singh et al., 2003; Barja, 2004). The two processes of oxidative stress and mitochondrial dysfunction are logically connected because mitochondria are by far the predominant source of reactive oxygen species (ROS), which cause oxidative stress (Cadenas and Davies, 2000; Grune and Davies, 2001; Sastre et al., 2003; Singh et al., 2003; Turrens, 2003), although there are other sources of ROS. The modern view ultimately derives from Harmanâ€™s (1956) original â€œfree radical theoryâ€? of senescence, which Harman (1972, 1983) himself first extended to also incorporate mitochondrial effects (see reviews in Harman, 1981, 1991, 1992, 2001). There are at least two differential equation models (van Leeuween et al., 2002; Mangel and Bonsall, 2004) and a reliability model (Koltover, 1982, 1983, 1992, 1996, 1997, 2004) that can recover the Gompertz mortality function on the basis of general assumptions at the proximate level and in terms of oxidative stress. Of the two differential equation models, only the model developed by van Leeuwen et al. (2002) admits a straightforward analytic solution whereas the model by Mangel and Bonsall (2004) is a little more complicated and must be solved numerically. Even more encouraging, both the van Leeuwen et al. (2002) and Koltover (1982, 1983, 1992, 1996, 1997, 2004) models assume linearity in the oxidative damage accruing with age. It will be instructive to review the van Leeuwen et al. (2002) and Koltover (1982, 1983, 1992, 1996, 1997, 2004) models. For mortality risk, van Leeuween et al. (2002) propose the following model:

21

m(t ) = β

D( t ) V (t )

,

Eq. 46

where D(t ) is the amount of oxidative damage, V(t ) is the structural volume, and β is the damage-specific killing rate. As will be seen, V(t ) need not interest us here. However, it should be noted that the model of van Leeuwen et al. (2002) was developed within the framework of what has been called a “Dynamic Energy Budget” (DEB) approach (for reviews, see Kooijman, 2001; Lika and Kooijman, 2003). It is through V(t ) that the model of van Leeuween et al. (2002) is coupled to the DEB approach.

Having stated this, the present focus is to derive an expression for D(t ) . The model makes four assumptions. The first assumption is that the ROS-generation rate is proportional to the catabolic rate: J + (t ) = α(t )C(t ) ,

Eq. 47

where J + (t ) is the ROS-production rate, α(t ) is the amount of ROS produced per utilized reserve unit, and C(t ) is the catabolic rate, which itself satisfies: C(t ) = c(t )e max . ,

Eq. 48

where c(t ) is the scaled catabolic rate taken as a product with the maximum energy reserve density, e max . . The second assumption is that ROS reactivity is effectively instantaneous following ROS production such that the ROS-generation rate translates immediately into the ROS-reaction rate. ROS reactivity, however, is reduced or eliminated by antioxidant defenses at rate J − (t ) . Therefore, the total ROS-reaction rate, denoted by J r (t ) , is given by: J r (t ) = J + (t ) − J − (t ) = γJ + (t ) ,

Eq. 49

22

where γJ + (t ) gives the fraction of ROS actually reacting. The third assumption is that the rate of oxidative damage is a linear combination of the fraction of ROS-reactions actually inducing damage, given by zJ r (t ) , the amplification to the oxidative damage rate due to cellular and intracellular damage, occurring at rate x (t ) , and the repair rate, y(t ) . Therefore, on suppressing the function notation, a preliminary differential equation is given as: dD dt

= zJ r + xD − yD .

Eq. 50

The fourth assumption is that α is a linear function in D. Here once again is the crucial assumption of linearity in physiological damage or its inverse. As regards oxidative stress, this assumption appears to be supported at least in humans (Jones et al., 2002; Junqueira et al., 2004) and rats (Driver et al., 2000). The assumption is formulated as: J + = [α 0 + α1D]C .

Eq. 51

Using Equations 49 and 51, Equation 50 may be written as:

dD dt

= zγα 0 C + [zγα1C + (x − y )]D .

Eq. 52

Define new, compound parameters ψ = (x − y ) , φ = α 1 zγe max . , and ϕ = α 0 zγe max . . Using these definitions and Equation 48, Equation 52 becomes: dD dt

= [ψ + φc]D + ϕc .

Eq. 53

On supposing that c = c* , a constant, and ϕ = 0 , Equation 53 becomes: dD dt

= [ψ + φc* ]D ,

Eq. 54

which is a separable differential equation and is solved as follows:

23

D′

∫ D = ∫ [ψ + φc ]dt ⇒ *

ln D = [ψ + φc* ]t + κ1 ⇒

Eq. 55

D = D 0 exp([ψ + φc* ]t ) ; ∀ D 0 = e

κ1

where κ 1 is a constant of integration. On the assumption that V(t ) is constant and given by V* , use of Equation 55 in Equation 46 gives: m(t ) = β

D0 V*

exp([ψ + φc* ]t ) = Ae * , α t

which is the Gompertz mortality function, and where A = β

Eq. 56 D0 V*

, and α * = [ψ + φc * ] .

The reliability approach discussed earlier has also been conceptualized in terms of oxidative stress (Koltover, 1982, 1983, 1992, 1996, 1997, 2004). Koltover (1992) noted the necessity of relating the linear increase in oxidative damage to the Gompertz mortality function (on the linear increase in oxidative damage, see also Driver et al., 2000; Jones et al., 2002; Junqueira et al., 2004). Although oxidative damage occurs linearly, the distribution of damaged structures that are critical to survival is what matters most under Koltover’s approach (this argument goes back to Simms, 1942a; the argument is reiterated explicitly in Sacher and Trucco, 1962; Brown and Forbes, 1974a). To derive the Gompertz mortality function in terms of oxidative damage, Koltover developed the following model. Koltover postulates the existence of Q critical structures, each one essential for life. Therefore, the organism is conceptualized as being serially constructed out of Q critical systems. Koltover motivates the model by considering the jth critical system, where j = 1,2, K , Q . Define m j as the number of defective elements due to

24

oxidative damage in the jth critical system and m c as a critical threshold in the number of defects due to oxidative damage that the jth system can sustain. By these definitions, the difference (m c − m j ) can be seen to be a safety margin defined on the interval: 0 ≤ m j ≤ m c . For simplicity, assume that m c is the same for all j∈ Q . Now imagine a process in which m j accumulates in time so that in all that follows m j (t ) ≡ m j (that is, m j is now a function of time). For the jth system, the time of failure-free functioning, denoted by τ j , is assumed to be proportional to the safety margin, and is given as: τ j = b(m c − m j ) ,

Eq. 57

where b is a constant of proportionality. In general, the time of failure-free functioning is given as: t = b(m c − m ) ; ∀ j ∈ Q ,

Eq. 58

which of course implies that: m = mc −

t b

.

Eq. 59

On supposing m j to be a random variable, the Palm-Khintchine Theorem (Khintchine, 1969: ch. 5; Koltover, 1982) suggests that the exponential distribution will suffice as the probability law governing m j . The idea of a critical threshold given by m c , however, requires a truncated exponential distribution; that is, an exponential distribution that is truncated at m c . From these considerations, Koltover (1997; and implicitly in his related works) suggested the following density distribution function: f (m ) =

a exp(− am )

1 − exp(− am c )

; ∀ 0 < m < mc ,

Eq. 60

25

where a is a parameter of the exponential distribution. From the relation between the density and cumulative distribution functions (see Eq. 19), we have for the cumulative distribution function:

∫

mc m

t = mc ⎡ exp(− at ) ⎤ exp(− am c ) − exp(− am ) dt = ⎢− . = ⎥ 1 − exp(− am c ) exp(− am c ) − 1 ⎣⎢ 1 − exp(− am c ) ⎦⎥ t = m a exp(− at )

Eq. 61

Denote the above cumulative distribution function by G j (t ) . Writing the second term in the numerator in G j (t ) in terms of m c (Eq. 59), we have: G j (t ) =

=

exp(− am c ) − exp(− a [m c − t b] ) exp(− am c ) − 1

exp(− am c )[1 − exp(at b )] exp(− am c ) − 1

=

exp(− am c ) − exp(− am c ) exp(at b ) exp(− am c ) − 1

Eq. 62 =

1 − exp(at b )

exp(am c )[exp(− am c ) − 1]

=

exp(at b ) − 1

exp(am c ) − 1

.

From the relation between the failure cumulative distribution and survivorship functions (see Eq. 18), and the assumption that m c is the same for all j∈ Q , the following survivorship function for individuals is derived as: Q

[

]

s(t ) = ∏ 1 − G j (t ) = [1 − G (t )] . j=1

Q

Eq. 63

Koltover suggested the following approximation for the survivorship function:

[1 − G (t )] Q ≈ exp[− QG(t )] .

Eq. 64

Since exp[− QG (t )] = (exp[− G (t )] ) , it is sufficient show that: Q

[1 − G (t )] ≈ exp[− G(t )] .

Eq. 65

Starting with the left hand side of Equation 65, the first derivative is:

26

g ′(t ) =

d[1 − G (t )] dt

=−

⎤ d[exp(at b ) − 1] d ⎡ exp(at b ) − 1 ⎤ ⎡ −1 ⎢ ⎥=⎢ ⎥ dt ⎣⎢ exp(am c ) − 1⎦⎥ ⎣⎢ exp(am c ) − 1⎦⎥ dt

Eq. 66 =−

a exp(at b )

b[exp(am c ) − 1]

.

At x = 0 , the first-order Taylor approximation about the point (t − x ) is: g(t ) ≈ g(0) + g ′(0)(t − x ) = 1 −

at b[exp(am c ) − 1]

.

Eq. 67

For the right hand side of Equation 65, the first derivative is:

g ′(t ) =

d exp[− G (t )] dt

=

d ⎧⎪ ⎡ exp(at b ) − 1 ⎤ ⎫⎪ ⎥⎬ ⎨exp ⎢− dt ⎪⎩ ⎢⎣ exp(am c ) − 1⎥⎦ ⎪⎭ Eq. 68

=−

a exp(at b )

⎡ exp(at b ) − 1 ⎤ exp ⎢− ⎥ . b[exp(am c ) − 1] ⎣⎢ exp(am c ) − 1⎦⎥

Therefore, the first-order Taylor approximation about the point (t − x ) at x = 0 is: g(t ) ≈ g(0) + g ′(0)(t − x ) = 1 −

at b[exp(am c ) − 1]

.

Eq. 69

We have just seen that the first-order Taylor approximations are identical. Hence, Koltover’s approximation can be said to hold true up to first order. On writing α =

and β =

Q exp(am c ) − 1

a b

, the survivorship function becomes: s(t ) ≈ exp[− QG (t )] = exp{− β[exp(αt ) − 1] } .

Eq. 70

Finally, from the relation between the mortality and survivorship functions, we find: m (t ) =

− d ln s(t ) dt

≈

− d[− β exp(αt ) + β] dt

αt

= Ae ,

Eq. 71

27

which is the Gompertz mortality function yet again for A = αβ . Koltover (1982, 1983, 1992, 1996, 2004; see also Koltover et al., 1993) empirically tested this model against data on oxidative damage available in the literature and found an excellent fit between data and predictions under the model. Given a Gompertzian mortality function, ultimate-level models can easily explain the evolution of senescence. The following discussion is a selective account of evolutionary approaches to senescence (see Rose, 1991 for a comprehensive account). The deterioration in homeostasis can be understood in ecological evolutionary terms using the disposable soma (DS) theory of the evolution of senescence. The DS theory was developed by T. B. L. Kirkwood and colleagues (Kirkwood, 1977, 1981, 1987, 1990, 1996, 1997, 2002; Kirkwood and Holiday, 1979, 1986; Kirkwood and Cremer, 1982; Kirkwood and Rose, 1991), and is predicated on the life history tradeoff in the allocation of resources to reproduction and to growth and maintenance (Perrin and Sibly, 1993; Zera and Harshman, 2001). It is important to note that Kowald and Kirkwood (1994, 1996, 2000; see also Kirkwood and Kowald, 1997) have begun to show how the DS model can be connected with the cellular-level processes of oxidative stress and mitochondrial dysfunction. Kirkwood and Rose (1991) developed an elegant mathematical model of the DS theory (cf. similar models in Kirkwood and Holliday, 1986; Kirkwood, 1990). The DS model starts with the Euler-Lotka Equation:

∫

∞ 0

L(x , s ) ⋅ M (x , s ) ⋅ e − rx dx = 1 ,

Eq. 72

( )

where L x , s and M (x , s ) are respectively survivorship and fecundity functions of age, denoted by x, and of the level of investment in somatic maintenance, denoted by s, and r is the intrinsic rate of increase (note that the notation here follows Kirkwood and Rose for

28

( )

investment in somatic maintenance). Once L x , s and M (x , s ) are specified, r can be solved for by standard methods (Charlesworth, 1994a). Note that survivorship and mortality have the following relation:

()

()

L x = exp ⎡⎢− ∫ m x dx ⎤⎥ . ⎣ ⎦

Eq. 73

Using the Gompertz-Makeham in Equation 47, and integrating across the interval from the age at which reproduction begins, denoted by a, to x, we have:

(

)

x x x αt αt L(x ) = exp ⎡− ∫ Ae + E dt ⎤ = exp ⎡− ∫ Ae dt − ∫ Edt ⎤ ⎢⎣ a ⎥⎦ ⎢⎣ a ⎥⎦ a

Eq. 74 ⎡ A αt = exp ⎢− e ⎢⎣ α

( )

t =x t =a

−E⋅t

⎤ ⎡ A α x αa ⎤ ⎥ = exp ⎢− e − e − E(x − a )⎥ . t =a ⎥ ⎣ α ⎦ ⎦

(

t =x

)

Assuming that the total juvenile mortality is given by V, the DS model specifies the adult survivorship function as:

(

)

⎡ A αx αa ⎤ L(x , s ) = (1 − V ) exp ⎢− e − e − E(x − a )⎥ . ⎣ α ⎦

Eq. 75

To specify the fecundity function, the DS model assumes that fecundity declines like a survivorship function following the Gompertz mortality function:

(

⎡ A αx α a M (x, s ) = h exp ⎢− e − e ⎣ α

)⎤⎥ , ⎦

Eq. 76

where h is the reproduction rate. The DS model requires that the parameters α and h are given by increasing functions in s, and that the parameter a is given by a decreasing function in s. Candidate forms for the α , h, and a functions are given respectively as: ⎧ ⎛ s′ ⎞ ⎟ ; ∀ s < s′ ⎪α 0 ⎜ , α = ⎨ ⎜⎝ s − 1 ⎟⎠ ⎪0 ; ∀ s ≥ s′ ⎩

Eq. 77

29

( )

h = h max 1 − s ,

Eq. 78

and a=

a min

(1 − s)

,

Eq. 79

where the region from s ′ to 1 defines a non-senescence region. The DS model shows that the optimal level of investment in somatic maintenance is lower than the amount of investment required to be in the non-senescence region (Fig. 5). Taking r as a measure of fitness, the DS model also shows that senescence is a consequence of the optimal life history strategy (cf. more sophisticated models by Abrams and Ludwig, 1995; Cichoń, 1997; Cichoń and Kozłowski, 2000; Shanley and Kirkwood, 2000; Mangel, 2001;

intrinsic rate of increase, r

Novoseltev et al., 2002). Since the amount of investment in somatic maintenance is less

dr ds

0

=0

s* s’ investment in somatic maintenance, s

1

Figure 5. Relation between fitness, measured by r, and the level of investement in somatic maintenance, s. The optimal amount of investment corresponding to the maximum fitness (maximum intrinsic rate of increase, r) is denoted by s*. Redrawn from Kirkwood and Rose (1991). 30

than what is required for the non-senescence phenotype (i.e., nearly-perfect to perfect fidelity in somatic maintenance), it follows that the soma would accumulate defects with increasing age and that homeostasis would progressively deteriorate. To recapitulate, senescence can be understood from unified proximate and ultimate perspectives. Recall that the DS model assumes the Gompertz mortality function and that the derivations of the Gompertz mortality function assume linearity in either homeostatic decline or an inversely proportional increase in the damage or injury accruing thereto. It has been remarked that linearity appears in so many biological processes because the linear terms of their respective Taylor approximations tend to dominate the overall behavior (Starmer and Starmer, 2002). Economos (1982) argued that the linear decline is a frame of mind. Similarly, Finch (1990) holds such a pattern to be “untrue” (see Finch, 1990: 155). However, in a review of 469 studies, Sehl and Yates (2001: B200) noted “We did encounter some cases of curvilinear loss. However, the linear term in most polynomial fits carried most of the weight.” Thus, the general argument of Starmer and Starmer (2002) appears to empirically validated, at least for the case of senescence. This observation of linearity has significant ramifications for this dissertation because it implies that the linear model derived from Fisher (1918), on which all of contemporary statistical genetics is predicated, is a sufficient basis for the statistical genetic investigation of processes that are fundamental to senescence. It must be pointed out, however, that the statistical genetics approach, more than providing a foundation, also makes a valuable extension to traditional approaches to studying senescence by accounting for genetic variation among individuals of a given population.

31

Chapter 3 Background: Endocrinology of the IGF-I Axis in Relation to Senescence This chapter is an extension of the discussion of senescence in the previous chapter, but with a focus on the physiological approach to senescence and on the role played therein by the IGF-I axis. The physiological basis of the statistical genetic hypothesis to be tested, which was briefly mentioned in the introduction, is discussed in this chapter as well. From the corpus of work on the physiology and clinical biology of senescence, we know that the deterioration in homeostasis is causally related to the development of agerelated pathology and disease (Strehler, 1977; Dilman, 1981, 1992 1994; Kohn, 1978, 1982; Kenney, 1982). Since the pathophysiology associated with senescence is exceedingly complex, one can take the reductionism route. A major undertaking in this direction is provided by the neuroendocrine theory of senescence, which has been elaborated by Finch (1975, 1976, 1977, 1979, 1987, 1988, 1990, 1993; Finch and Landfield, 1985) among others (see also Frolkis, 1966, 1968, 1972, 1976, 1981; Dilman, 1971, 1976, 1979, 1981, 1984, 1986, 1992 1994; Everitt, 1973, 1976a&b; 1980a&b; Dilman and Anisimov, 1979; Dilman and Berstein, 1979; Dilman et al., 1979a&b, 1986). According to Finchâ€™s theory, senescence involves neuroendocrine cascades that are dysfunctional, late-life occurrences of the same physiological control systems responsible for maintaining homeostasis in earlier ontogeny. Under this view, the neuroendocrine cascades may be seen as inducers of pathology or as inefficacious mechanisms for restoring homeostasis. The neuroendocrine cascades refer to the cascading interactions of the two main effector arms of the central nervous system (CNS) that are responsible

32

for the maintenance of homeostasis, which are the autonomic nervous system (ANS) and endocrine arms, hence the name â€œneuroendocrineâ€?. The neuroendocrine cascades theory of senescence will be taken as the general physiological foundation for the current approach. One can focus further still on one of the three main endocrine axes involved in aging processes, which are the IGF-I, sex hormone, and the hypothalamic-pituitaryadrenal (HPA) axes (Fig. 6). With a view towards understanding senescence, the IGF-I axis across the age continuum will be taken as the system of study. Indeed, Finch and colleagues suggest that a focus on the IGF-I axis in relation to senescence may well be profitable (Finch and Ruvkun, 2001; Longo and Finch, 2002, 2003).

Figure 6. The main endocrine axes in aging and senescence. Left: The IGF-I axis. Middle: The Sex hormone axis. Right: The HPA axis. See text. Source: Lamberts et al. (1997). 33

The IGF-I axis—a complex network of hormones, binding proteins, proteases and receptors (Sara and Hall, 1990; Werner et al., 1994; Jones and Clemmons, 1995; CollettSolberg and Cohen, 1996)—is an important regulator of prenatal development (Gluckman, 1986; Gluckman and Pinal, 2003), postnatal growth (Daughaday, 2000; Lupu et al., 2001), aging processes (Barbieri et al., 2003; Tatar et al., 2003) and metabolism (Liu and Barrett, 2002; Murphy, 2003). Moreover, the IGF-I axis plays critical roles in osteoporosis (Geusens and Boonen, 2002; Žofková, 2003), sarcopenia and muscle atrophy (Borst and Lowenthal, 1997; Grounds, 2002), a number of cancers (LeRoith and Roberts, 2003; Fürstenberger and Senn, 2003), a number of neurodegenerative disorders (Gasparini and Xu, 2003; Trejo et al., 2004) and the four components of the metabolic syndrome, namely T2D, CVD, hypertension and obesity (Raines and Ross, 1995, 1996; Sowers and Epstein, 1995; Froesch, 1997; Bayes-Genis et al., 2000; Maccario et al., 2000; Hausman et al., 2001; Frystyk et al., 2002; Holt et al., 2003). Thus, a study of the IGF-I axis leads naturally to the more general concern of senescence. A fund of studies on a wide range of human populations have established that the pattern of IGF-I secretion follows a rise from low levels during early postnatal growth to maximal levels at puberty, declines shortly thereafter and culminates at relatively lower levels at older ages (Hall et al., 1980, 1981; Bala et al., 1981; Luna et al., 1983; Rosenfield et al., 1983; Hall and Sara, 1984; Furlanetto and Carra, 1986; Cara et al., 1987; Savage et al., 1992; Argente et al., 1993; Hesse et al., 1994; Juul et al., 1994, 1995; Olivié et al., 1995; Yamada et al., 1998; Kawai et al., 1999; Barrios et al., 2000; Löqvist et al., 2001; Low et al., 2001; reviewed in Juul, 2003). Data provided by Diagnostic Systems Laboratory (DSL) for 1700 boys and 1700 girls from 3 to 17 years of age are

34

plotted in Figure 7 (the data may be obtained from their web page at the following URL: http://www.dslabs.com). Figure 7 shows the archetypical secretion pattern up until shortly after puberty. Data from the SAFHS will demonstrate the continued decline at older ages (reported below). Generally, females achieve their peak IGF-I secretion height before males at puberty, which is consistent with general patterns of pubertal growth (Tanner, 1978; Bogin, 1999). The general features of this IGF-I secretion pattern over the life span has been documented in baboons (Copeland et al., 1981, 1982; Crawford and Handelsman, 1996; Crawford et al., 1997), chimpanzees (Copeland et al., 1985), rhesus macaques (Liu et al., 1991; Styne, 1991) and gibbons (Suzuki et al., 2003).

700

Mean IGF-I (ng/ml)

600 500 400 300 200 100 0 3

5

7

9

11

13

15

17

Age (years) Boys

Girls

Sex-Averaged

Figure 7. IGF-I secretion pattern early in the human life span. Note that girls typically achieve their peak secretion height earlier than boys. Data are from Diagnostic Systems Laboratories for 1700 boys and 1700 girls from 3 to 17 years of age.

35

The liver is by far the main source of systemic IGF-I and accounts for around 80% of the total IGF-I pool in circulation (Sara and Hall, 1990; Jones and Clemmons, 1995). In the circulatory system, IGF-I may form several complexes with its binding proteins (IGFBPs), of which six are known, designated as IGFBP-1 to IGFBP-6, and an acid labile sub-unit (ALS) (Fig. 8; Rechler, 1993; Clemmons, 1999; Baxter, 2000). Of the IGFBPs, IGFBP-1 and IGFBP-3 are considered to be the most important in determining the availability of free IGF-I to tissues (Clemmons, 1999; Baxter, 2000). The system is more complicated than depicted in Figure 8 because there are also proteases and phosphorylating proteins that modulate IGFBP activity (Coverly and Baxter, 1997; Bunn and Fowlkes, 2003). All of these proteins constitute a complex

Figure 8. Schematic description of the IGF-I axis and the two major sites of IGF-I secretion. GHRH – GH release hormone; SS – Somatostatin; GH-R – GH receptor; IGF1 = IGF-I; IGF-1R – IGF-I receptor (see text). Source: Carter et al. (2002a). 36

system operating under dynamic biochemical equilibria that modulate the tissue-level availability of free IGF-I. Liver secretion of IGF-I is stimulated mainly by growth hormone (GH), which is secreted by the somatotrophs of the anterior pituitary (Corpas et al., 1993; Giustina and Veldhuis, 1998; Müller et al., 1999). It should be noted that insulin and nutritional factors are also important in stimulating liver secretion of IGF-I (Clemmons and Underwood, 1991; Thissen et al., 1994; Jones and Clemmons, 1995; Ketelslegers et al., 1995). Shortly after sexual maturation, the decline in circulating IGFI is mediated foremost by negative feedback regulation of GH secretion by two wellknown pathways: 1) the short-loop pathway, which refers to the action of IGF-I directly at the somatotrophs and 2) the long-loop pathway, which refers to the actions of IGF-I at the hypothalamus, namely down-regulation of GH release hormone and up-regulation of somatostatin, which are respectively positive and negative regulators of somatotroph secretion of GH (Corpas et al., 1993; Giustina and Veldhuis, 1998; Müller et al., 1999). IGF-I is also expressed and regulated in virtually all other tissue types, as first demonstrated by the work of D’Ercole and colleagues on the tissue distribution of IGF-I synthesis in the human fetus and the rat (D’Ercole et al., 1980a&b, 1984; D’Ercole and Underwood, 1981, 1986; Van Wyk et al., 1981; Underwood et al., 1984, 1986; D’Ercole, 1996). Similarly, work by Isaksson and colleagues on the endocrinology of bone growth suggested that GH promotes the local expression and regulation of IGF-I in bone tissue (Isaksson et al., 1982, 1985, 1987, 2000; Ohlsson et al., 1998, 1999). These and similar such findings established the concept that the IGF-I axis has an autocrine/paracrine mode of action in addition to its classical endocrine mode (i.e., via liver-secreted IGF-I) (Fig. 9; Underwood et al., 1986; Holly and Waas, 1989; Chatelain et al., 1991). The original

37

Pituitary

Nutrition & Other Factors

GH Liver

Circulation

IGF-I IGF-I IGFBP-3 ALS

IGF-I

Endocrine Paracrine

IGF-I Autocrine

Figure 9. Endocrine, paracrine, and autocrine modes of IGF-I action. Courtesy of Dr. A. J. Dâ€™Ercole.

somatomedin hypothesis (Fig. 10a; Salmon and Daughaday, 1957; Daughaday and Garland, 1972; Daughaday et al., 1972), which claimed that GH exerts its effects solely through the mediating actions of liver-produced IGF-I (originally named sulfation factor and then somatomedin by Daughaday and colleagues), was accordingly revised to acknowledge the ubiquitous autocrine/paracrine mode of action. Under the revised somatomedin hypothesis, it was still maintained that the predominant effects of GH arise through the endocrine mode (Fig. 10b; Daughaday and Rotwein, 1989; Daughaday, 1989, 1997, 2000; Spagnoli and Rosenfeld, 1996; Salmon and Burkhalter, 1997). The revised somatomedin hypothesis has come under scrutiny because it appears from work on transgenic mice that the endocrine mode is not at all essential for normal 38

Figure 10. The somatomedin hypotheses. (a) The original somatomedin hypothesis. (b) The revised somatomedin hypothesis. (c) The current somatomedin hypothesis. Igf1-/- – Transgenic IGF-I double-negative mutant mice that are unable to synthesize liver IGF-I following fetal development. Such mice can be used to study the effects of postnatal ablation in liver secretion of IGF-I. All other abbreviations are as mentioned previously. See text for discussion. Source: LeRoith et al. (2001a).

growth whereas the autocrine/paracrine mode is both sufficient and necessary to this end (Fig. 10c; Liu and LeRoith, 1999; Sjögren et al., 1999, 2002a-c; Yakar et al., 1999, 2000; Ohlsson et al., 2000a&b; Liu et al., 2000; Butler and LeRoith, 2001a&b; Isaksson et al., 2001a&b; LeRoith et al, 2001a&b; Butler et al., 2002). Still, the data in favor of the concept that the endocrine mode is important in somatic growth is compelling, such as the clinical observations that reduced growth is incurred under systemic IGF-I deficiency and/or resistance (Spagnoli and Rosenfeld, 1996; Hintz, 1999; Laron, 1999, 2002; Zapf and Froesch, 1999; Daughaday, 2000; López-Bermejo et al., 2000; Camacho-Hübner and

39

Savage, 2001; Reiter and Rosenfeld, 2003; Rosenfeld, 2003) and in disparate human pygmy populations (Merimee et al., 1981, 1982; Jain et al., 1998; Clavano-Harding et al., 1999; Dávila et al., 2002). Further, the transgenic mouse studies are subject to several ambiguities of interpretation, with the consequence that they cannot clearly reject an important role for endocrine IGF-I in somatic growth (D’Ercole and Calikoglu, 2001; Robson et al., 2002; van der Eerden et al., 2003). Further still, a recent transgenic mouse study along the lines of somatic growth regulation has found that there appears to be a critical threshold-level for circulating IGF-I below which longitudinal bone growth and bone density are severely affected (Yakar et al., 2002a&b; Yakar and Rosen, 2003). In an earlier review of the above debate by Gluckman et al. (1991) it was thought that the main role of endocrine IGF-I was in the regulation of whole-body protein metabolism. In this regard, it is noteworthy that studies using isotopic tracer infusions of the essential amino acid leucine as a marker of whole-body protein metabolic activity have found that IGF-I promotes the protein anabolism typical of pubertal growth (Arslanian and Kalhan, 1996; Mauras et al., 1996; Mauras, 1999). These results are consistent, moreover, with the well-known anabolic effects of GH and IGF-I in skeletal muscle metabolism (Fryburg, 1994; Florini et al., 1995, 1996; Fryburg and Barrett, 1995; Liu and Barrett, 2002; Rennie et al., 2004). Further, other transgenic mouse studies have also demonstrated that liver-derived IGF-I plays an important role in the metabolic regulation of carbohydrate and lipids (Fernández et al., 2001; Sjögren et al., 2001; Wallenius et al., 2001; Yakar et al., 2001, 2002b; 2004; Haluzik et al., 2003; Clemmons, 2004). This said, it should be recalled that IGF-I had long been thought to be a regulator of at least glucose homeostasis due largely to the seminal work of Froesch and colleagues

40

(Froesch et al., 1963, 1966, 1967). These earlier studies are relevant to the current debate because they were carried out on what would later be identified as IGF-I extracts from human serum (Froesch et al., 1985, 1996a). Because liver-derived IGF-I constitutes the vast majority of the IGF-I pool in circulation (Sara and Hall, 1990; Jones and Clemmons, 1995), the early studies are consistent with an important endocrine mode of action. One balanced conceptual model that has developed out of this lively debate is that both modes of action of the IGF-I axis are important in somatic growth regulation and their relative importance will vary according to developmental stage and tissue-type (D’Ercole and Calikoglu, 2001; see also D’Ercole and Underwood, 1986; D’Ercole, 1996 for an earlier version of this model). D’Ercole and Calikoglu (2001) postulated that the autocrine/paracrine mode predominates during early fetal development and the endocrine mode becomes increasingly important over the course of postnatal growth. In statistical genetic studies on the mouse, it has been demonstrated that there are at least two different gene systems controlling the overall dynamics of growth, one operative early in ontogeny and the other later (Cheverud et al., 1996; Atchley and Zhu, 1997; Vaughn et al., 1999). Cheverud et al. (1996) hypothesized that the genetic system controlling late growth was related to IGF-I and suggested that their results are consistent with the model proposed by D’Ercole and Underwood (1986). More recently, using GH-deficient lit/lit mutant mice and IGF-I knockout mice, Mohan et al. (2003) demonstrated that GH-independent mechanisms controlled prepubertal bone growth whereas GH-dependent IGF-I was largely responsible for pubertal bone growth, which is consistent with the 2-phase model. If one may paraphrase the conclusions of D’Ercole and Calikoglu (2001) in which they proposed a subtle extension of the above 2-phase model and integrate these with

41

general tenets of physiological ecology, we come now to a development-oriented 3-phase model for the behavior of the IGF-I axis according to which: 1) the autocrine/paracrine mode predominates during late fetal development; 2) the endocrine mode becomes increasingly important for somatic growth and is maximally important for the pubertal growth spurt; and 3) the endocrine mode undergoes a transition from being a regulator of somatic growth to being a regulator of metabolism and somatic maintenance over the course of adulthood. Phase 3 at once has the potential to resolve the debate regarding the “true” role of endocrine IGF-I and highlights the likely role that the IGF-I axis might play in the physiological mechanisms underlying the well-known life history tradeoffs obtaining amongst growth, reproduction and somatic maintenance (on the theory of life history tradeoffs in relation to senescence, see Kirkwood and Rose, 1991; Partridge and Barton, 1993; Abrams and Ludwig, 1995; Cichoń, 1997; Cichoń and Kozłowski, 2000). The observation that the IGF-I axis is important throughout all the major stages of the life span for a diverse array of physiological phenomena would seem to indicate the existence of a dynamic gene expression network (GEN) (sensu Wyrick and Young, 2002; see Fig. 11) reflecting the behavior of the IGF-I axis. This is to be expected from the tenet of endocrinology that hormones initiate signal transduction networks that, in turn, modulate the behavior of a GEN. If the 3-phase model of the behavior of the IGF-I axis has any credence, then the predicted shifts should be reflected by the behavior of the IGFI axis GEN translated along the age continuum. Here, with data from the SAFHS for the relevant age range, the statistical genetics of the third phase will be addressed. A simplified genetic model of the IGF-I axis may be postulated. Under this model, the IGF-I axis interacts with an underlying GEN, and the genes of the GEN exhibit

42

pleiotropy and variation in age-specific effects (cf. Cheverud et al., 1996). Minimally, this theoretical model consists of two testable hypotheses: 1) The IGF-I axis GEN is pleiotropic. 2) The components of the IGF-I axis GEN exhibit age-specific effects. Both of these hypotheses can be rigorously addressed using current statistical genetic models. This dissertation will focus on the second hypothesis.

Figure 11. Schematic of a gene expression network. The IGF-I axis can be thought of as occupying the arrow connecting an environmental stimulus to a signal transduction network, with control over a number of transcriptional activators. The transcriptional activators in turn control the expression of a number of genes, either in one-to-one fashion as in the left set of examples or in multiple-to-one fashion as in the middle and right set of examples. Source: Wyrick and Young (2002).

43

Chapter 4 Background: The Study Population and Epidemiological Patterns The study population is derived from the San Antonio Family Heart Study (SAFHS). The demographic and epidemiological foundations of the SAFHS will be reviewed. The SAFHS is comprised of Mexican Americans recruited from low-income barrios in San Antonio, Texas. San Antonio is the largest city of Bexar County, Texas, and Bexar County is itself situated in southcentral Texas (Figs. 12 and 13). San Antonio is currently the second largest city in Texas (after Houston and Dallas) and the ninth largest in the United States (U.S.) (cf. Fehrenbach, 2002). Arreola (2002: 131) points out that â€œamong large cities, San Antonio is the urban area with the highest proportion of Mexican Americans in the country; the Hispanic subgroup was 59 percent of the city in 2000.â€? According to statistics from the San Antonio Metropolitan Health District (SAMHD), the San Antonio population grew from 1,185,394 residents in 1990 to 1,392,931 residents in 2000 at a rate of about 17.5% (SAMHD, 2000). An important point for the current study is that most of the San Antonio population growth is attributable to the 29% growth in the Hispanic population, which, in turn, was due to both a relatively high Hispanic birth rate and a high Mexican immigration rate (SAMHD, 2000). The rise of those of Mexican ancestry in San Antonio from 1900 to 2000 is depicted in Figure 14. This agrees roughly with the projected change in the ethnic composition in Bexar County from 1990 to 2030 (Fig. 15). At both the city and county level, Hispanics were the majority population by the year 2000. According to Figure 15, this pattern in the dominance of Hispanics in the contribution to the total population looks to be increasing up until at least the year 2030.

44

Figure 12. Map of Bexar County in Texas. Blow-up at bottom left shows Bexar County surrounded by seven counties.

45

Figure 13. Map of San Antonio in Bexar County. Yellow indicates the inner cityâ€” encircled by Loop 410â€”and green indicates the city limits.

46

Population x 1000

San Antonio Population 1400 1200 1000 800 600 400 200 0 1900

1920

1940

1960

1980

2000

Decades Hispanic

Total

San Antonio Hispanic Population 70

% Hispanic

60 50 40 30 20 10 0 1900

1920

1940

1960

1980

2000

Decades Figure 14. Rise of the Hispanic population in San Antonio, 1900-2000. Top Panel: Population growth for the total and Hispanic population in San Antonio. Botom Panel: Percentage of total population growth due to Hispanic population growth. Data are from Arreola (2002: Table 7.3, p. 145).

47

Population x 1000

Bexar County Population 1400 1200 1000 800 600 400 200 0 1990

2000

2010

2020

2030

Decades Hispanic

NH White

Black

Other

Bexar County Hispanic Population

% Hispanic

65 60 55 50 45 1990

2000

2010

2020

2030

Decades Figure 15. Relative increase of the Hispanic population in Bexar County, Texas, 19902030. Data and projections are from the U.S. Census and Texas State Data Center as reported in SAMHD (2000).

48

The main goal of the SAFHS is to discover the genetic determinants of atherosclerosis Mexican Americans, focusing on the Mexican American population of San Antonio. Atherosclerosis is perhaps the most important cause of mortality under the more general category of cardiovascular disease (CVD). CVD in turn is one of the major diseases of the metabolic syndrome. A brief discussion of the historical and current epidemiology of the metabolic syndrome in San Antonio will show the SAFHS to be a logical step in addressing these problems. All of the metabolic syndrome components, namely CVD, type 2 diabetes (T2D), obesity and hypertension, are classic diseases of modernization. Their etiologies, known to be physiologically related (Reaven, 1988, 1993, 1995, 1999), involve environmental effects associated with modernization and a poorly understood genetic predisposition (Zimmet and Thomas, 2003; for representative work particularly on the Mexican American population of San Antonio, see Diehl and Stern, 1989; Stern and Haffner, 1990; Stern et al., 1991, 1992; Mitchell et al., 1996a&b, 1999; MacCluer et al., 1999; Hixson and Blangero, 2000). Early features of the metabolic syndrome involve deranged carbohydrate and lipid metabolism, which promote progression to T2D and obesity (Haffner et al., 1992; Liese et al., 1997, 1998; Reaven, 1999). Moreover, T2D is one of the more important predictors of CVD and type 2 diabetics are at higher risks for CVD morbidity and mortality relative to nondiabetics (Laakso and Lehto, 1997; Howard and Magee, 2000; Laakso, 2001; Resnick and Howard, 2002; Laakso and Kuusisto, 2003; Nesto, 2003, 2004). The fact that T2D has an earlier age of onset than CVD will be important below.

49

Judging from the historical epidemiology of T2D, the current problems associated with the metabolic syndrome in Mexican Americans of San Antonio appear to have started shortly after 1940 (Fig. 16; Ellis, 1962; Carey et al., 1992; Bradshaw et al., 1995). For simplicity, it is here assumed that the historical works referred to herein were speaking of what we today recognize as T2D, which is justifiable because the criterion of diabetes mellitus in adults was often met or implied in these reports. In one of the earliest studies focusing on Mexican American mortality in Bexar County, Ellis (1962) reported that Spanish-surname men and women had T2D mortality rates of 16.97 and 22.27

8 7

RSMR

6 5 4 3 2 1 0 1

2

3

4

5

Decades Spanish surname female Non-Hispanic white female

Spanish surname male Non-Hispanic white male

Figure 16. Relative standardized mortality ratios (RSMR) for total mentions of diabetes mellitus: Spanish surname and non-Hispanic whites age 30 and over by sex. Data are from multiple-cause-of-death records for Bexar County, Texas, 1935-1944 to 1975-84. Decades 1-5 correspond to 1935-1944, 1945-1954, 1955-1964, 1965-1974, and 19751984, respectively. Modified from Bradshaw et al. (1995).

50

whereas their other-white counterparts had rates of 10.75 and 9.02, respectively. The differentials in T2D mortality in Mexican Americans and non-Hispanic whites appear to have started shortly after 1940 for females and shortly after 1950 for males. These differentials would increase with time (Fig. 16; Carey et al., 1992; Bradshaw et al., 1995). The negative trend continued unabated, as evidenced by reports of increased T2D incidence from the late 1970s to the late 1980s (Haffner et al., 1991, 1992; Fig. 17). These results are from earlier phases of the San Antonio Heart Study (SAHS) (not to be confused with the SAFHS). In a more recent phase of the SAHS, Burke et al. (1999) found that the incidence of T2D for 7- to 8-year follow-up examinations carried out from 1987 to 1996 for cohorts enrolled from 1979 to 1988 increased in both Mexican Americans and non-Hispanic whites (Fig. 17).

Incidence, %

T2D in San Antonio, 1979-1996 20 18 16 14 12 10 8 6 4 2 0

15.7

9.4 5.7 2.6

1979-1988

1987-1996

Cohorts MA

CA

Figure 17. Change in T2D incidence in San Antonio, Texas. Data are T2D incidence (%) by ethnicity, Mexican American (MA) and Caucasian American (CA). Data were reported in Burke et al. (1999).

51

One related and potentially contentious issue needs to be considered in some detail. It was once widely held that Mexican Americans have lower CVD mortality than non-Hispanic whites, which is somewhat paradoxical given that Mexican Americans have higher T2D mortality and/or morbidity indices and lower socioeconomic status (SES) indices relative to non-Hispanic whites (Ellis, 1962; Kautz, 1982; Bradshaw et al., 1985; Castro et al., 1985; Markides and Coreil, 1986; Diehl and Stern, 1989; Rosenwaike and Bradshaw, 1989; Stern and Haffner, 1990; Bradshaw and Liese, 1991; Bradshaw and Frisbie, 1992; Carey et al., 1992; Mitchell et al., 1992; Stern, 1993). This pattern has been called the Hispanic Paradox (Hunt et al., 2002, 2003). The Hispanic Paradox is really more complex than can be adequately described here (for more comprehensive treatments, see Franzini et al., 2001; Palloni and Morenoff, 2001; Morales et al., 2002). As can be seen in Figure 18, Mexican Americans of San Antonio had advantages over or were comparable to Caucasian Americans of San Antonio in CVD mortality overall while the situation is reversed with respect to T2D. Stern and Wei (1999) argued that the pattern is spurious because it is based on vital statistics and these data tend to underestimate deaths in minority segments of the population (see also Wei et al., 1996). Their analyses of risk factor distributions derived from the SAHS indicate that Mexican Americans have higher CVD mortality than non-Hispanic whites. More recent work from the SAHS by Hunt et al. (2002, 2003) hasconfirmed these findings and the argument that vital statistics data underestimate mortality in minorities was reiterated. However, Espino et al. (1994), from analyses of Bexar County death certificates, found that elderly Mexican Americans had higher mortality risks of T2D and CVD than their non-Hispanic white counterparts. Thus, while the pattern uncovered by Espino et al.

52

Acute Myocardial Infarction

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

1.2 1

SMR

SMR

T2D

0.8 0.6 0.4 0.2 0

1940

1950

1960

1970

1940

1980

1950

Decades CAM

CAF

MAM

MAF

MIM

1960

1970

1980

Decades CAM

MIF

Chronic Ischemia

CAF

MAM

MAF

MIM

MIF

Other Circulatory 2

1.4 1.2

1.5

SMR

SMR

1 0.8 0.6 0.4

1 0.5

0.2 0

0 1940

1950

1960

1970

1980

1940

1950

CAM

CAF

MAM

MAF

MIM

1960

1970

1980

Decades

Decades MIF

CAM

CAF

MAM

MAF

MIM

MIF

Figure 18. T2D and CVD mortality in San Antonio. The last letters in the legend stand for male (M) or female (F). CA and MA stand for Caucasian American and Mexican American, respectively. MI stands for Mexican immigrant. Data are for standardized mortality ratios (SMR) for total mentions of the cause of mortality, where the values for CAM and CAF over the decade beginning at the year 1960 are taken as the standards. Data are from Carey et al. (1992).

(1994) is consistent with the current reports from the SAHS, the argument that the bias in vital statistics data gives rise to the Hispanic Paradox is not entirely accurate. Indeed, the most recent community health review by the SAMHD (2002) reported, on the basis of vital statistics data, that Hispanics have higher T2D and CVD mortality rates than nonHispanic whites, although the differential in heart disease mortality may not be statistically significant until the oldest age group (Fig. 19). Similar findings were reported from the Corpus Christi Heart Project in which higher CVD mortality rates in Mexican Americans relative to non-Hispanic whites were also observed (Goff et al., 1994; Pandey et al., 2001). Thus, the Hispanic Paradox would seem to be falsified for the last decade in San Antonio (and perhaps Corpus Christi). It is perhaps significant to note that all of the studies rejecting the Hispanic Paradox in San Antonio appeared relatively recently whereas the studies suggesting the existence of a Hispanic Paradox all appeared earlier. This observation suggests an alternative explanation that is explored below. The problems with vital statistics data notwithstanding, it is possible that the Hispanic Paradox was a real phenomenon in the past that was brought about by heterogeneity in the position along the epidemiologic transition (see below) occupied by the Mexican immigrant population, and the Mexican and Caucasian American populations. Before proceeding with this argument, some tenets of the epidemiology of modernization need to be discussed. There are generally two versions of biomedical studies termed natural experimental models (sensu Garruto et al., 1989, 1999) that seek to identify the health effects of modernization and the respective etiologies of the health problems associated with modernization. A caveat to be taken with the gross generalization to follow is that

54

Heart Disease Mortality

Mortality Rate (per 1,000,000)

12000 10000 8000 6000 4000 2000 0 0-19

20-29 30-39 40-49 50-59 60-69 70-79

Age Intervals MA

CA

T2D Mortality 4000

Mortality Rate (per 1,000,000)

3500 3000 2500 2000 1500 1000 500 0 0-19

20-29 30-39 40-49 50-59 60-69 70-79

Age Intervals MA

CA

Figure 19. Heart disease and T2D mortality in Bexar County, 2002. Top Panel: Mortality due to heart disease. Bottome Panel: Mortality due to T2D. Data were compiled and reported by SAMHD (2002).

55

the two versions are not to be treated typologically, but rather, as ends of a populational continuum. Under what might be called the in situ modernization model, a particular community undergoes modernization and as the result of such change there occurs an epidemiologic transition (Omran, 1971) from an epidemiological profile dominated by infectious disease to one dominated by non-infectious disease (Fig. 20). The epidemiologic transition was originally conceived with respect to developed Western nations. Therefore, the rate at which such nations progressed through the three stages of the epidemiologic transition (see Fig. 20) is taken as the standard case strictly for

Stage I

Stage II

Infectious mortality

Stage III

Noninfectious mortality

Modernization Figure 20. Schematic diagram of the epidemiologic transition. As modernization increases, infectious mortality decreases while noninfectious mortality increases. At Stage I, the “age of pestilence”, infectious mortality accounts for most of the population mortality. At Stage II, the “transition stage”, infectious mortality decreases while noninfectious mortality increases. At Stage III, the “age of chronic degenerative diseases”, noninfectious mortality now accounts for most of the population mortality.

56

comparative purposes. Thus, it is commonplace to speak of a relatively rapid or delayed epidemiologic transition for other nations or populations. Classic examples of the in situ modernization model are the Republic of Nauru (Zimmet, 1978, 1979) and the United States Territory of American Samoa (Baker et al., 1986), both of which are considered to be instances of rapid modernization and, hence, rapid progression through the three stages of the epidemiologic transition. Under what might be called the migration model, there arises a migration link from a lesser-developed nation or community to a moredeveloped nation or community. The pattern may be international, rural-to-urban or some mix of both and the prevailing commonality is that large numbers of people radically transform their environment by moving from one place to another (for illustrative examples from the Pacific, see Garruto, 1990). Migration from various sources in Mexico to various communities in the United States is a well-known phenomenon. Not surprisingly, migration effects have figured prominently in studies of T2D and CVD mortality and morbidity in the Hispanic population (to include both those born in the U.S. and in Mexico) of San Antonio (Rosenwaike and Bradshaw, 1989; Bradshaw and Frisbie, 1992; Carey et al., 1992; Wei et al., 1996; Stern and Wei, 1999; Hunt et al., 2002). The situation of the Mexican Americans of San Antonio would seem to fall somewhere in between the in situ modernization and migration models. In this population, two forces are inextricably entangled in their effects; migration from Mexico and rural areas of south Texas and in situ modernization exacerbated by the socioeconomic stratification and inequality â€œendemicâ€? to large metropolitan areas in the U.S. (Sen, 1993). That in situ modernization (with all its unintended problems) plays a

57

role in the causal structure of the metabolic syndrome in San Antonio is suggested by observations that T2D prevalence in Mexican American participants in the SAHS is inversely related to socioeconomic status (Hazuda et al., 1988; Mitchell and Stern, 1992). The mechanism leading to the previous relation presumably involves the increase in relative deprivation and the genesis of socioeconomic gradients in health (Marmot, 1994; Williams and Collins, 1995; Daniels et al., 1999; Nguyen and Peschard, 2003) that are known to occur under the epidemiologic transition (Wilkinson, 1994). Similarly, that migration plays a role is suggested by comparisons of T2D prevalence in Mexican American participants in SAHS and foreign-born Mexican Americans: T2D prevalence in Mexican American participants in the SAHS is inversely related to acculturation status independent of socioeconomic status (Hazuda et al., 1988; Stern and Haffner, 1990), whereas foreign-born Mexican Americans tend to have relatively lower prevalences of T2D and CVD than U.S.-born Mexican Americans in San Antonio (Bradshaw and Frisbie, 1992; Wei et al., 1996; Hunt et al., 2002). The latter observation is consistent with reports in the literature that foreign-born Mexican Americans have lower CVD mortality in Texas (Rosenwaike and Bradshaw, 1989). Studies by Sundquist and Winkelby (1999, 2000) on data from the National Health and Nutrition Examination Survey III (NHANES III) came to similar conclusions at the national level. Sundquist and Winkelby (1999, 2000) divided the Mexican American group from NHANES III into three sub-groups roughly reflective of migration and acculturation status: 1) Mexicoborn, 2) U.S.-born English-speaking, and 3) U.S.-born Spanish-speaking. These subgroups were compared against each other and against non-Hispanic whites for a number of CVD risk factors (e.g. BMI and T2D). Sundquist and Winkelby (1999, 2000) found

58

that overall, Mexican Americans are at higher risk for CVD than non-Hispanic whites. In comparisons among the Mexican American sub-groups, they also found that the U.S.born Spanish-speaking individuals were at significantly greatest risk for CVD. Taken together, these observations suggest that as Mexican immigrants become assimilated into American society, the concomitant changes in environment exact increases in the risk of T2D and CVD. Now, migration is a complex sociocultural phenomenon (for Mexican immigration to the U.S., see Massey, 1986; Massey and Espa単a, 1987; Durand and Massey, 1992). Based on studies of Mexican immigration into the U.S., it appears that the demographic structure of the migrant flow to the U.S. changes in accordance with a three phase model of migration (Massey, 1986). In the first phase, the migrant flow is comprised predominantly of young male adults. Inevitably, these young male adults become well-adapted to their foreign setting, thus setting up the next two phases. From the transition to the settlement phases, women and children, who are the families of the young male adults, become part of the migrant flow. The important point here is that the demographic structure of the receiving Hispanic populations in the U.S. would be accordingly affected. Given that Bradshaw and Frisbie (1992) have demonstrated that Mexicans were in stage II of the epidemiologic transition relative to Caucasian Americans, it follows that the Hispanic population of San Antonio would have characteristics that are intermediate between stages II and III of the epidemiologic transition for most of the last century. Bradshaw and Frisbie (1992) did in fact find that Mexican Americans of San Antonio are intermediate between Caucasian Americans and Mexicans. An immediate corollary of this line of thinking is that only after the Mexican

59

American population becomes demographically agedâ€”due to increasing maturation of the demographic structure of the migrant stream as well as to in situ demographic agingâ€”do we begin to observe ethnic group differentials in CVD mortality that are consistent with expectations. Under this scenario, the differential in CVD mortality arises simply because proportionately more and more Mexican Americans are now living to the age of onset for CVD. This scenario is consistent with the facts that Hispanics have always had higher T2D mortality than non-Hispanic Whites and that T2D has an earlier age of onset than CVD. San Antonio has been the venue of a number of informative epidemiological studies on the etiology of the metabolic syndrome (e.g., Hazuda et al., 1988; Stern et al., 1991, 1992; Wei et al., 1996). This epidemiological work has established the unquestionable importance of environmental factors. Besides socioeconomic status and acculturation status, dietary behaviors related to fat and sugar intake patterns have also been implicated as contributing risk factors to obesity and T2D (Stern and Haffner, 1990). In stark contradiction to our knowledge of the role of environmental factors, very little is known about the genetic factors that may either predispose individuals to or be in some way protective against the metabolic syndrome. The SAFHS seeks to redress the dearth of knowledge on the role of genetic factors in the metabolic syndrome. This dissertation is only one small part of this large-scale research enterprise.

60

Chapter 5 Methods I: Sampling Design, Pedigrees, and Phenotypes The SAFHS is â€œthe first comprehensive genetic epidemiologic study of atherosclerosis and its correlates in Mexican Americans (Dr. J. W. MacCluer, personal communication)â€?. The SAFHS is a research enterprise jointly carried out by the Department of Genetics at the Southwest Foundation for Biomedical Research (SFBR), San Antonio, Texas, and the School of Medicine at the University of Texas at San Antonio Health Science Center. The findings of this research are reviewed in Mitchell et al. (1996a&b, 1999) and MacCluer et al. (1999). Two phases have been completed so far, designated as SAFHS1 and SAFHS2, and a third phase began in 2002. The current study focuses on SAFHS1, but the SAFHS abbreviation will be used in the ensuing. In general, genetic epidemiology studies require data on: 1) pedigree structure, 2) phenotypes, 3) covariates and 4) genotypes. A description of the data is given just below. The analytical methods are described in detail in the next two chapters. Detailed descriptions of the study design and protocols are reported in Mitchell et al., (1996a) and MacCluer et al. (1999). The current study focused on carrying out quantitative genetic analyses (Lange, 1997; Thomas, 2004) as opposed to linkage analyses. Consequently, genotype data were not required, as quantitative genetic analyses minimally require pedigree structure and trait data. Participants in the SAFHS were recruited from low-income barrios of greater than 90% Mexican American residency. These barrios were identified by reference to published socioeconomic and demographic profiles of the neighborhoods of San Antonio. The distribution of the sample population in San Antonio is shown in Figure 21.

61

IH 10 SAFHS population IH 90

Loop 410

IH 35

Figure 21. SAFHS recruitment area. The inner city of San Antonio is roughly encircled by Loop 410 (in yellow). The SAFHS population is located in the gridded area. The right boundary is formed by Interstate Highways (IH) 10 and 35.

Sampling Design, Pedigree Structure and Basic Demographics Probands for the SAFHS were chosen from among individuals of 40 to 60 years of age who reside in identified low-income barrios. Extended families were identified through probands chosen because they have at least six living, first-degree relatives (i.e., siblings and/or age-eligible offspring) (Fig. 22). The same set of relatives of the probandâ€™s spouse were also recruited. The subset of the SAFHS for the current study consists of 1,047 participants from 48 families. The mean pedigree size is 29 individuals per family and the pedigree size ranges from 3 to 76 individuals per family. The numbers of relationship types are reported in Table 1. The mean age is 39.5 years and the range is from 15.5 to 94.2 years of age. There are 404 males and 643 females.

62

S an Anto nio Family He art S tudy

Figure 22. Schematic pedigree structure for the typical extended family unit in the SAFHS. The arrow indicates the proband. First-, second- and third-degree relatives are in turqoise, yellow and red, respectively. Courtesy of Dr. J. W. MacCluer.

Table 1. Numbers of Relative Pairs in the SAFHS Relationship Type Number Parent-offspring 1788 Sibs 1337 Half-sibs 186 Grandparent-grandchild 1598 Great grandparent-grandchild 784 Avuncular 2686 Grand avuncular 978 Half avuncular 431 First cousins 2738 First cousins once removed 2633 Second cousins 672 Other 1172 Total Relative Pairs 17003

63

Phenotypes All hormonal phenotypes were measured in the physiology laboratory of Dr. John Blangero at the SFBR Department of Genetics. Circulating levels of IGF-I (ng/ml) were measured using an IGF-I immunoradiometric assay (IRA) kit (Nichols Institute Diagnostics, San Juan Capistrano CA). IGFBP-1 and IGFBP-3 levels (ng/ml) were measured using IRA kits specific to the binding protein (Diagnostic Systems Laboratories, Inc.). It is commonplace in the literature to also analyze the molar ratio of IGF-I to IGFBP-3 (because IGFBP-3 is the main binding protein in circulation; Juul et al., 1994, 1995). To compute the molar ratio, the molar masses of 7,649 daltons for IGFI and 28.5 kilodaltons for IGFBP-3 were used (Jones and Clemmons, 1995). The resultant trait is referred to as Ratio3 for brevity. Body mass index (BMI) is commonly used as a covariate (see below) for traits related to growth and metabolism. BMI was computed as the ratio of weight (Kg) to height squared (m2), where weight and height measurements were taken during the participantâ€™s clinic visit. Covariates The covariates data were obtained from participant responses to the questionnaires and interviews. These data are on age, sex, medical history, reproductive history, smoking habits, dietary habits (based on a food frequency questionnaire), alcohol consumption, physical activity levels (based on a modified Stanford 7-Day Physical Activity Recall Instrument), and acculturation and socioeconomic status. Also, any of the phenotypes may serve as covariates. The covariates in all the models were screened for significance. The significant covariates in all the models were some combination of age, sex, age2, sex Ă— age, and BMI.

64

Descriptive Statistics, Transformations, and Treatment of Outliers Generally, the raw data were significantly kurtotic and skewed and thus in violation of the assumptions of multivariate normality and additivity (Table 2; Figs. 2324; see the following chapter on these assumptions). Beaty et al. (1985) demonstrated that significant kurtosis has an adverse influence on downstream statistical inferences derived from variance components models with more than two variance components, which is the case for the genotype Ă— age interaction model. Inducing univariate normality is a reasonable first step towards satisfying multivariate normality (Looney, 1995). To this end, two remedies were sequentially employed. For all analyses, the data were first subjected to a logarithmic transformation, which, as Wright (1968: ch.10-11) has shown, is often sufficient to achieve normality and, as Freeman (1985) noted, is also sufficient to induce additivity (see also just below). Following logarithmic transformation, outliers were removed at Âą 4 standard deviations from their respective means (cf. the recommendations by Freeman, 1985). Few outliers were removed for all traits (< 0.5 % of their respective total sample sizes). These remedies rendered the derived traits sufficiently normally distributed as confirmed each time by inspection of the resultant distributional properties (Table 3; Figs. 25-26). It should be noted that using a transformation to conform to the assumptions of normality and additivity, as opposed to seeking a variance-stabilizing transformation that would induce constant variance, is in no way inconsistent with modeling (co)variance heterogeneity (see the next chapter). The assumptions of normality, additivity, and constant variance involve separate but related issues, as has been clearly delineated in seminal works on the use of transformations in data analysis (Bartlett and Kendall, 1946;

65

Bartlett, 1947; Tukey, 1957; Box and Cox, 1964), although transformations may often simultaneously achieve a close approximation to all three assumptions (Bartlett, 1947; Tukey, 1957; but see Sampford, 1964). As Box and Cox (1964) pointed out in the reply section of their article, the assumption of an underlying distribution is the logical starting point for any parametric analysis. Subsequent to this observation, and in the context of interaction analyses, whether or not one seeks transformations specifically to conform to the assumptions of additivity and constant variance will largely depend on one’s definition of statistical interaction. Thus, when Cox (1984) defined interaction as “inconstancy of variance”, he suggested that as a first-check a variance-stabilizing transformation should be employed, if such exists, and, similarly, when Freeman (1985) defined interaction as nonadditivity, he suggested, again as a first-check, that a transformation that induces additivity, if such exists, should be employed. The assumptions of normality and additivity are maintained in the present analyses, but, for the reasons detailed in the following chapter, (co)variance heterogeneity will be modeled. This procedure of modeling G × E interaction is consistent with standard practice in statistical modeling in that it represents a slightly more complex model that is firmly predicated on a simpler model. The idea here is to incrementally increase agreement with reality. Further, given that the logarithmic transformation may often induce close agreement with the above three assumptions, the present search for interactions is rather conservative.

66

Table 2. Descriptive Statistics of Raw Data Log Trait IGF-I

Mean

Variance

Kurtosis

Skewness

N

147.605209

14022.30

8.92850

2.29894

1001

978.57114 13.82813

2.56707

955

IGFBP-1

34.36576

IGFBP-3

3289.36581

3042352

8.33261

2.17906

1005

0.000501

0.000163

931.3702 30.49015

935

Ratio3

Table 3. Descriptive Statistics of Log-Transformed Data Log Trait

Mean

Variance Kurtosis Skewness

N

IGF-I

2.04932

0.11208

0.81848

-0.38833

1001

IGFBP-1

1.37076

0.16360

0.23321

-0.43841

955

IGFBP-3

3.46626

0.04382

0.42641

-0.00125

1005

Ratio3

-5.35965

1.25415

0.78329

-0.14718

935

67

Raw IGF-I Histogram 250

Frequency

200 150 100 50 0 5.68

401.1087097

796.5374194

Measurement

Raw IGFBP-1 Histogram 250

Frequency

200 150 100 50 0 0.612

128.2853333

255.9586667

Measurement

Figure 23. Histograms of raw IGF-I and IGFBP-1 data.

68

Frequency

Raw IGFBP-3 Histogram 200 180 160 140 120 100 80 60 40 20 0 662.0945301

6036.487771

11410.88101

Measurement

Frequency

Raw Ratio3 Histogram 1000 900 800 700 600 500 400 300 200 100 0 4.02708E-10

0.143242348

0.286484696

Measurement

Figure 24. Histograms of raw IGFBP-3 and Ratio3 data.

69

Log IGF-I Histogram 120

Frequency

100 80 60 40 20 0 0.75435

1.568669355

2.38298871

Measurement

Frequency

Log IGFBP-1 Histogram 100 90 80 70 60 50 40 30 20 10 0 -0.21325

0.797224667

1.807699333

Measurement Figure 25. Histograms of log transformed IGF-I and IGFBP-1 data.

70

Log IGFBP-3 Histogram 120

Frequency

100 80 60 40 20 0 2.82092

3.309873548

3.798827097

Measurement

Log Ratio3 Histogram 140 120 Frequency

100 80 60 40 20 0 -9.39501

-6.099846333 -2.804682667 Measurement

Figure 26. Histograms of log transformed IGFBP-3 and Ratio3 data.

71

Chapter 6 Methods II: The Multivariate Mixed Effects Linear and Polygenic Models As discussed in chapter 2, a linear model of physiological function or of physiological damage along the life span is important in a number of the more general theories of senescence at the proximate level. The statistical models to be employed in this dissertation research are a special class of the multivariate linear model known as variance components models (Searle et al., 1992; Hopper, 1993; Schork, 1993), which ultimately derive from Fisher (1918). This chapter is divided into two sections. The first section discusses the basic statistical genetic model whereas the second section discusses extensions of the basic model to incorporate G × E interaction. The variance components approach assumes that a phenotype vector of individuals in a given pedigree, denoted by y , follows a multivariate normal (MVN) distribution. In matrix notation, the MVN is given as: f (y ) = (2 π )

−N 2

Σ

−1 2

1 ′ exp ⎡⎢− (y − μ ) Σ −1 (y − μ )⎤⎥ . ⎣ 2 ⎦

Eq. 80

Stylized for the present context, the parameters μ and Σ are respectively the vector of phenotype means and the variance-covariance matrix (usually just referred to as the covariance matrix) of the phenotype variances and covariances for a single pedigree, π is the mathematical constant (= 3.14…), N is the number of individuals in the pedigree, Σ is the determinant of Σ , the prime indicates matrix or vector transpose, and Σ −1 is the inverse matrix of Σ . For the MVN in general, we have: E[y ] = μ . A better model of E[y ] that takes into account the effects of covariates is formulated as follows: E[y ] = Xβ ,

Eq. 81

72

where X is an incidence matrix augmented by a column of 1’s, and β is a vector of the grand trait mean, μ (so that β 0 = μ ), and covariate effects, β1 ,..., β n (Searle et al., 1992). On conceiving of the phenotype as being additively determined by random genetic and environmental effects (i.e., the additivity assumption), the phenotype vector

y may be expressed in terms of a multivariate mixed effects linear model: y = Xβ + g + e ,

Eq. 82

where g is a vector of random genetic effects, and e is a vector of random environmental effects. Moreover, g and e are distributed as mutually independent MVNs, with E[g ] = E[e] = 0 , where 0 is the null vector (Searle et al., 1992). By these assumptions, Equation 82 implies that the variability in y , given by the covariance matrix Σ , is related to the variances in the random effects. Hence, our interest should now lie in modeling the components of Σ , under the model of Equation 82. Equation 82 is called a mixed effects model because it is comprised of fixed effects in the component modeled by “ Xβ ” and of random effects in the component modeled by “ g + e ”. Searle et al. (1992) discuss the history and meanings of fixed, random, and mixed effects models (see also Rao, 1997). Equation 82 has a complementary interpretation in that it is understood to be a probabilistic model of the population-level behavior of a measurement of interest, which in the context of statistical genetics is usually a phenotype. As a probabilistic model, it is comprised of deterministic and stochastic components (see the discussion by Wackerly et al., 1996: 476-479). In the present case, the deterministic component of the model is given by Equation 81, which appears in the right hand side of Equation 82. Equation 81 is deterministic in that it determines the values to be assigned in a precise, specified manner and, by itself, does 73

not allow for random error or stochasticity. To this extent, the function specifies law-like behavior. However, it has long been known that population phenomena cannot be fully explained by a deterministic function. That is, there is always stochastic behavior about the law-like process implied by a candidate deterministic model. Accounting for this stochastic behavior constitutes the second component of all probabilistic models. It would be left to Fisher (1918) to develop a measure of the stochasticity about the deterministic function. This measure he called the variance: It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance of the normal population to which it refers, and we may now ascribe to the constituent causes fractions or percentages of the total variance which they together produce (emphasis mine). Fisher (1918: 399) Moreover, Fisher (1918) is credited with the development of the linear model (see Searle et al., 1992; Rao, 1997) and, as can be perceived from the quote, his main interest at least in that paper lay in modeling the variance components. On assuming that dominance and epistatic genetic effects are negligible, pair-wise comparisons of individual phenotype, denoted by y (the scalar values of y ), define the elements of the covariance matrix as (Lange et al., 1976; Lange, 1997): ⎧⎪σ 2y = σ g2 + σ e2 ; ∀ x = z, 2φ xz = 1, δ xz = 1, Cov( y x , y z ) = 2φ xz σ + σ δ xz = ⎨ ⎪⎩2φ xz σ g2 ; ∀ x ≠ z, δ xz = 0, 2 g

2 e

Eq. 83

where x and z index individuals, 2φ xz gives the expected coefficient of relationship n

(where φ xz is defined below), σ g2 = ∑ σ gj2 is the additive genetic variance summed across j=1

n loci, σ e2 is the environmental variance, and δ xz is defined as 1 when individuals x and z are the same and 0 otherwise. For pedigrees, the matrix formulation of Equation 83 is:

74

Σ = 2Φσ g2 + Iσ e2 ,

Eq. 84

where Φ is the kinship matrix of the pedigree and I is the identity matrix. From Feller (1957: 215-216, 221-222), random variables are mutually independent if and only if: Cov(g, e ) = 0 ,

Eq. 85

whence the use of δ xz and I in the scalar and matrix formulations, respectively. The model of Equations 83 and 84 will be referred to as the polygenic model, under the scalar and matrix formulations, respectively. For pair-wise comparisons of any two relatives in a pedigree, the expected kinship coefficient over the genome (Malécot, 1969), denoted by φ xz , is defined as:

φ xz =

1 2

E[(κ1 j 2 + κ 2 j )] ,

Eq. 86

where the κ ij are coefficients giving the jth locus-specific probability that a given pair of relatives share i alleles identical by descent (IBD) (Cotterman, 1940). Examples of genome-wide expected probabilities for sharing 0, 1, and 2 alleles IBD, denoted by κ 0 , κ1 , and κ 2 , respectively, for typical pair-wise relationships in an extended pedigree are

presented in Table 4 (note that the subscript j has been dropped). For all loci, the κ i must satisfy the following restriction: 2

∑κ

i

= 1,

Eq. 87

i =0

which states that the allele sharing probabilities sum to 1. Discussions of the theory underlying the computation of the elements in a kinship matrix, Φ , for a given pedigree can be found in Thompson (1986, 2000), Lange (1997), and Thomas (2004).

75

Table 4. Genome-wide expectations for alleles identical by descent (IBD) Pair-Wise Relationship κ0 κ1 κ2 φ xz

MZ twins Parent-offspring Full-sib Half-sib-+-first-cousin Half-sib Grandparent-grandchild Avuncular First-cousin Half-avuncular Half-first-cousins Unrelated

0 0 0.25 0.375 0.5 0.5 0.5 0.75 0.75 0.875 1

0 1 0.5 0.5 0.5 0.5 0.5 0.25 0.25 0.125 0

1 0 0.25 0.125 0 0 0 0 0 0 0

0.5 0.25 0.25 0.1875 0.125 0.125 0.125 0.0625 0.0625 0.003125 0

It cannot be overemphasized that the underlying assumption of multivariate normality is justified (see the discussion in Lynch and Walsh, 1998: 26-27). The fundamental importance of the MVN is a direct consequence of the much-celebrated Central Limit Theorem. From Cramér (1946: 213-218), the Central Limit Theorem holds that the sum: ξ = ξ1 + ξ 2 + ... + ξ n ,

Eq. 88

of n independent random variables, denoted by ξi (i = 1,2,..., n ) , is approximately distributed as a normal distribution as n becomes large and the approximation becomes increasingly better as n → ∞ . The theorem has been proven to hold in regard to the MVN in general (Cramér, 1946: 316-317; 1970: ch. 10; Feller, 1957: 252-259). Under restrictive conditions, Lange (1978) proved that the MVN Central Limit Theorem holds for quantitative traits that are distributed in human pedigrees. The Central Limit Theorem can be seen as being related to another important but perhaps lesser-known theorem on the addition of independent, normally distributed random variables, which

76

can be called the Addition Theorem (sensu Cramér, 1946: 212-213, 1970: chs. 5-6). The Addition Theorem holds that the sum: η = η1 + η2 + ... ,

Eq. 89

of any number of normally distributed random variables, denoted by ηi (i = 1,2,...) , is itself normally distributed (Cramér, 1946: 212). Note that the Addition Theorem holds for any number of normally distributed random variables whereas the Central Limit Theorem requires n to become large, which implies that the normal approximation may not hold for small n. The Addition Theorem is important not merely for the distinction just made but also because of its implications. In particular, Cramér (1946: 213) noted that the Addition Theorem implies that linear functions of normally distributed random variables are also normally distributed and, conversely, that if a linear function of random variables is normally distributed, then its components are also normally distributed. It was further noted by Cramér (1946: 316; cf. 1970: ch. 10) that the Addition Theorem holds for the MVN as well. Taken together, these two theorems put the multivariate mixed linear and polygenic models on strong theoretical grounds. Firstly, the Central Limit Theorem underwrites the fundamental assumption that phenotypes are MVN distributed. Secondly, the Addition Theorem underwrites the notion that MVN phenotypes may be expressed as a linear function, where its components are also MVN. Methods II: Theory and Model of Genotype × Environment Interaction

It will be convenient to review the mathematical definitions and relations of the terms variance, standard deviation, covariance, and correlation coefficient because the genotype × environment (G × E) interaction model is most easily derived from said definitions. The following discussion will be based on material that can be found in most

77

textbooks on statistics and probability. Highly recommended sources include Cramér (1946), Feller (1957), Parzen (1960) and Anderson (1984). Wackerly et al. (1996) and Ross (2003) provide more current treatments. The following definitions will be made with respect to the random variables Y and Z. The definition of the variance of Y is:

[

] [

]

Var(Y ) ≡ σ 2Y = E (Y − E[Y ]) = E (Y − μ Y ) = E[Y 2 − 2Yμ Y + μ 2Y ] 2

2

= E[Y ] − 2E[Y ]μ Y + μ = E[Y 2 ] − 2μ 2Y + μ 2Y 2

2 Y

Eq. 90

= E[Y 2 ] − μ 2Y = E[Y 2 ] − (E[Y ]) . 2

The definition of the covariance of Y and Z is: Cov(Y, Z) ≡ σ Y , Z = E[(Y − E[Y ])(Z − E[Z])]

= E[(YZ) − Y E[Z] − E[Y ] Z + E[Y ] E[Z]] = E[(YZ) − Yμ Z − μ Y Z + μ Y μ Z ]

Eq. 91

= E[YZ] − E[Y ] μ Z − μ Y E[Z] + μ Y μ Z

= E[YZ] − μ Y μ Z − μ Y μ Z + μ Y μ Z

= E[YZ] − μ Y μ Z = E[YZ] − E[Y ] E[Z] .

A useful identity follows from these definitions. For the case of random variable Y say: Cov(Y, Y ) = E[(Y − E[Y ])(Y − E[Y ])]

[

]

= E (Y − E[Y ]) = Var(Y ) . 2

Eq. 92

That is, the covariance of a random variable with itself is simply the variance. The standard deviation is defined as the positive square root of the variance: σ 2Y = + σ Y .

Eq. 93

The correlation coefficient is defined as: ρY,Z =

σY,Z σ 2Y σ 2Z

=

σ Y,Z σ 2Y σ 2Z

=

σ Y,Z σYσZ

,

Eq. 94

from whence, we obtain the relations:

78

ρY ,Y =

ρY,Z =

σ Y ,Y

=

σYσY

σ Y,Z

=

σYσZ

σ 2Y σ 2Y

= 1,

σ Z,Y σ Zσ Y

Eq. 95

= ρ Z,Y ,

Eq. 96

and an alternative expression for the covariance of Y and Z: σ Y,Z = ρY,Zσ Y σ Z .

Eq. 97

Further, the correlation coefficient takes values in the closed interval [− 1,+1] . That is, − 1 ≤ ρ Y , Z ≤ +1 .

Eq. 98

With slight modifications to Anderson’s (1984: 22) definition of a bivariate normal covariance matrix, the above relations can be compactly illustrated as follows: ⎡ρ Y ,Y σ Y σ Y Σ=⎢ ⎢ ⎢⎣ ρ Z,Y σ Z σ Y

ρ Y , Z σ Y σ Z ⎤ ⎡ σ 2Y ⎥=⎢ ⎥ ⎢ ρ Z, Z σ Z σ Z ⎥⎦ ⎢⎣σ Z,Y

σY,Z ⎤ ρ Y ,Y = ρ Z, Z = 1 ⎥;∀ , ⎥ 2 ρ Y , Z = ρ Z,Y σ Z ⎥⎦

Eq. 99

where the modifications only involve being explicit about the correlation coefficients. Finally, there is a special relation, derivable from the above definitions, on the variance of the difference of two random variables: Var(Y − Z) = E{[(Y − Z ) − E(Y − Z)]

2

}

= E{(Y − Z) − 2(Y − Z)E(Y − Z) + [E(Y − Z)] }, 2

⇓

2

⇓⇓

Eq. 100a

⇓⇓⇓

where ⇓ , ⇓⇓ , and ⇓⇓⇓ indicate terms that will be taken separately.

⇓

[

]

E (Y − Z ) = E[(Y − Z )(Y − Z )] = E[Y 2 − 2YZ − Z 2 ] 2

= E[Y 2 ] − 2E[YZ] + E[Z 2 ] ,

⇓⇓

79

− 2E{(Y − Z)E[Y − Z]} = −2E{(Y − Z)[E(Y ) − E(Z)]} = −2E[(Y − Z)(μ Y − μ Z )]

= −2E[Yμ Y − Yμ Z − Zμ Y + Zμ Z ]

= −2E[Y ]μ Y + 2E[Y ]μ Z + 2E[Z]μ Y − 2E[Z]μ Z = −2μ 2Y + 4μ Y μ Z − 2μ 2Z ,

⇓⇓⇓ E[E(Y − Z)E(Y − Z)] = E{[E(Y ) − E(Z )] [E(Y ) − E(Z)]} = E[(μ Y − μ Z )(μ Y − μ Z )] = E[μ 2Y − 2μ Y μ Z + μ 2Z ] = μ 2Y − 2μ Y μ Z + μ 2Z ,

where summing the terms under ⇓⇓ and ⇓⇓⇓ yields the quantity: − μ 2Y + 2μ Y μ Z − μ 2Z , to which we add the terms under ⇓ to complete the main expectation:

[E(Y )− μ ]+ [E(Z )− μ ]− 2[E(YZ) − μ μ ] 2

↓

Var (Y )

2 Y

2

2 Z

↓

+ Var (Z)

Y

Z

↓

− 2Cov(Y, Z) .

We have just shown the following result: Var(Y − Z) = Var(Y ) + Var(Z ) − 2Cov(Y, Z )

= σ 2Y + σ 2Z − 2ρ Y , Z σ Y σ Z ,

Eq. 100b

With the previous section and the above discussion, we now have all that is needed to proceed with the mathematical theory of G × E interaction. A variance component model designed to detect G × E interaction can be used to address the potentially dynamic gene expression network (GEN) that is reflective of the behavior of the IGF-I axis along the age continuum. The foundations for this approach trace back to Haldane’s (1946) early ideas on the importance of G × E interaction for the determination of quantitative phenotypes, and to Falconer’s (1952) idea of treating trait states in different environments as different traits. An operational definition of G × E 80

interaction may be taken as the environmental dependency or sensitivity of genotype expression in the process of phenotype determination (Haldane, 1946; Falconer, 1952, 1960a&b, 1989, 1990; Lynch and Walsh, 1998). In this connection, it is common to speak of the trait response to a change in environment (Falconer, 1989; Lynch and Walsh, 1998). To motivate the theory, consider the simplest case of a trait measured in two different environments. In this case, the additive genetic variance in trait response to the change in environment, denoted as σ g2Δ , can be written as (Robertson, 1959; Blangero, 1993; Wu, 1998):

σ g2Δ

⎧σ g21 + σ g2 2 − 2ρG σ g1σ g 2 ; ∀ σ g21 ≠ σ g2 2 , ⎪ =⎨ ⎪2σ 2 (1 − ρ ) ; ∀ σ 2 = σ 2 = σ 2 , G g1 g2 g ⎩ g

Eq. 101

where σ g21 and σ g2 2 are the additive genetic variances of the trait in environments 1 and 2, and ρG is the genetic correlation of the traits between environments. Incidentally, Robertson (1959) derived his version of Equation 101 by taking expectations as in Equation 100a&b to get a slightly more complicated equation, which includes Equation 101. However, if we start with Falconer’s (1952) idea of treating the trait states in different environments as different traits, then Equation 101 is seen to be merely the statistical genetic version of Equation 100a&b. That is, on treating trait states in two different environments as two different random variables, the above formulation follows directly from the definition of the variance of the difference of two random variables. There is no G × E interaction when σ g2Δ = 0 (Robertson, 1959; Blangero, 1993). Nonzero G × E interaction is comprised of two components, a component due to heteroscedasticity (also known as variance heterogeneity or unstable variance) and

81

another component due to the genetic correlation (Robertson, 1959; Dickerson, 1962; Yamada, 1962; Eisen and Saxton, 1983; Yamada et al., 1988; Falconer, 1990; Itoh and Yamada, 1990; Blangero, 1993; Wu, 1998). A useful theorem that makes the preceding statements a little more rigorous will now be proven. Equation 101 specifies two main outcomes, one holding under heteroscedasticity ( σ g21 ≠ σ g2 2 ) and the other under homoscedasticity ( σ g21 = σ g2 2 = σ g2 ). These outcomes each give in turn yet three more, general outcomes, under conditions specified for the genetic correlation that are representative of full positive or negative correlation and of zero correlation. Assuming σ g21 ≠ σ g2 2 , Equation 101 gives:

σ g2Δ = σ g21 + σ g2 2 − 2ρG σ g1σ g 2

⎧σ g21 + σ g2 2 − 2σ g1σ g 2 ; ∀ ρG = 1. ⎪ ⎪ ⎪ ; ∀ ρG = 0. = ⎨σ g21 + σ g2 2 ⎪ ⎪ ⎪σ g21 + σ g2 2 + 2σ g1σ g 2 ; ∀ ρG = −1. ⎩

Eq. 102

Assuming σ g21 = σ g2 2 = σ g2 , Equation 101 gives:

σ g2Δ

⎧0 ; ∀ ρG = 1. ⎪ ⎪ ⎪ 2 = 2σ g (1 − ρG ) = ⎨2σ g2 ; ∀ ρG = 0. ⎪ ⎪ ⎪4σ g2 ; ∀ ρG = −1. ⎩

Eq. 103

Let nonzero G × E interaction be defined as σ g2Δ ≠ 0 . The cases will be discussed in relation to the conditions giving rise to σ g2Δ ≠ 0 , and will be taken in descending order from top to bottom for Equation 102 and then for Equation 103. With little loss in 2

2

2

generality, it will be assumed that σ g1 , σ g 2 , and σ g are non-zero for we shall never be 82

interested in traits that show no variation. For the top case of Equation 102, setting σ g2Δ = 0 gives a quadratic equation in the variables, namely:

0 = σ g21 − 2σ g1σ g 2 + σ g2 2 = (σ g1 − σ g 2 )(σ g1 − σ g 2 ) .

Eq. 104

⇒ σ g1 = σ g 2

However, σ g21 ≠ σ g2 2 by assumption and so Equation 104 amounts to a contradiction. Therefore, for the top most case, σ g2Δ ≠ 0 even when ρG = 1 . The middle case of Equation 102 arises for completely uncorrelated random variables. To preclude confusion, recall from Feller (1957: 215-216, 221-222) that independence of random variables implies 0 covariance and, of course, 0 correlation. However, the converse, as noted by Feller (1957: 222), is not true. That is, 0 correlation can say nothing about whether or not the random variables of interest are independent. At any rate, the important point here is that σ g2Δ ≠ 0 by the definition of the variance as the expected squared deviations from the mean (Eq. 90). For the bottom case of Equation 102, that σ g2Δ ≠ 0 follows immediately from the definitions of the variance and standard deviation

(Eqs. 90 and 93), for a sum of positive terms is itself positive. Equation 103 is easier to interpret. As ρG goes from +1 to –1, σ g2Δ goes from 0 to 4σ g2 . Clearly, σ g2Δ = 0 for ρG = 1 . Therefore, whenever ρG < 1 , we can conclude that σ g2Δ ≠ 0 (not including the

trivial case for σ g2 = 0 ). The following theorem has just been proven. There is no G × E interaction, i.e., σ g2Δ = 0 , if and only if σ g21 = σ g2 2 = σ g2 and ρG = 1 are simultaneously satisfied. In all other non-trivial cases, σ g2Δ ≠ 0 . The condition of σ g2Δ = 0 is now to be

83

understood as a null hypothesis for G × E interaction. It follows from the theorem that it is sufficient to reject σ g21 = σ g2 2 = σ g2 , ρG = 1 , or both in order to reject the null hypothesis that σ g2Δ = 0 . That is, by the simultaneity condition as required under the theorem, rejection of just one of the stated conditions amounts to a rejection of σ g2Δ = 0 . An alternative proof of the theorem is provided in Appendix A. Moreover, it can be seen that under homoscedasticity or heteroscedasticity, as the correlation continuum is traversed from complete positive correlation to zero correlation to complete negative correlation, the magnitude of σ g2Δ increases monotonically to its maximum (Fig. 27).

6 5

Variance

4 3 2 1 0 0

0.5

1 1.5 Genetic Correlation + 1

heteroscedasticity

2

homoscedasticity

Figure 27. G × E interaction variance under heteroscedasticity and homoscedasticity. The interaction variances were computed from Equations 102 and 103 (see text). Under heteroscedasticity, the trait variances in environments 1 and 2 were assigned the values 1 and 2, respectively. Under homoscedasticity, the trait variance was assigned the value 1. The lower limit of G × E interaction is clearly set by the homoscedasticity case whereas the upper limit is dependent on the magnitude of heteroscedasticity. 84

The theorem on G × E interaction can be generalized to multiple environments in a straightforward manner using matrix algebra. Let there be n environments with n corresponding trait states. The additive genetic covariance matrix for the n trait states, of dimensions n × n , is given as: ρ1, 2 σ1σ 2 σ12 ⎡ ⎢ ⎢ ⎢ ρ 2,1σ 2 σ1 σ 22 ⎢ ⎢ M M G=⎢ ⎢ ⎢ ⎢ρ n −1,1σ n −1σ1 ρ n −1, 2 σ n −1σ 2 ⎢ ⎢ ⎢⎣ ρ n ,1σ n σ1 ρn , 2σn σ 2

L ρ1,n −1σ1σ n −1 L ρ 2,n −1σ 2 σ n −1 O

M

L

σ 2n −1

L ρ n ,n −1σ n σ n −1

ρ1,n σ1σ n ⎤ ⎥ ⎥ ρ 2,n σ 2 σ n ⎥ ⎥ ⎥ ⎥, M ⎥ ⎥ ρ n −1,n σ n −1σ n ⎥ ⎥ ⎥ 2 ⎥⎦ σn

Eq. 105

where the elements are understood to be genetic parameters and subscripts indicate the environment. By the above theorem, G × E interaction can be evaluated for the set of hypotheses on the additive genetic variances across environments: σi2 = σ 2j ; ∀ i, j, . . . , n ,

and for the set of hypotheses on the genetic correlations across environments: ρi , j = 1 ; ∀ i, j, . . . , n .

Blangero and colleagues (Blangero et al., 1987, 1988, 1989, 1990a&b; Blangero and Konigsberg, 1991; Blangero, 1993) developed a similar model for the detection of G × E interaction under a complex segregation analysis approach. The problem, however, is that there is an “explosion” in the number of parameters that need to be estimated (Meyer and Hill, 1997; Pletcher and Geyer, 1999; Meyer, 2001). For a covariance matrix of

n × n dimensions, the number of parameters, denoted by N θ , is given as:

85

Nθ =

n (n + 1) 2

.

Eq. 106

For 5 environments say, there are 15 parameters in the additive genetic covariance matrix alone. For the polygenic model, this is in addition to the number of environmental variance parameters, which is given by n. Further, for the full multivariate linear mixed model, the sum N θ + n is added to the numbers of parameters for the environmentspecific means, which is also given by n, and for the environment-specific covariate effects estimates, which is given by n times the number of covariates. The simplest full model with no covariates would still give 25 parameters in all to be estimated. At this level of model complexity, serious problems arise in maximum likelihood estimation and the sampling variances of the parameter estimates tend to be prohibitively large (Searle et al., 1992; Meyer and Hill, 1997; Pletcher and Geyer, 1999; Meyer, 2001). Clearly, another approach is needed that can circumvent the problems arising from a model overburdened in parameters. Towards this end, σ g2 and ρG can be modeled as functions of the environment of interest, provided the environment is continuous (Blangero, 1993; Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002). This amounts to a generalization of the above ideas on G × E interaction from discrete to continuous environments (Kirkpatrick et al., 1994). For the present study, the continuous environment of interest is the age continuum. To model the null hypothesis of σ g2Δ = 0 , σ g2 and ρ G are parameterized as continuous functions of age:

[

]

σ g = exp α g + γ g (p − age ) ; ∀ p ∈ T = age min . K age max . ; T ⊂ ℜ ;

Eq. 107

ρG = exp(− λ p − q ) ; ∀ p, q ∈ T ,

Eq. 108

2

86

where σ g2 is modeled as an exponential function to ensure a positive variance (Blangero, 1993; Pletcher and Geyer, 1999), p denotes the age of an individual which belongs to the index set of ages, denoted by T , T ranges from the minimum age, age min . , to the

maximum age, age max . , in the sample population and is a positive, finite subset of the real line, and the average age is that for the sample population; ρG is modeled as the correlation function of an Ornstein-Uhlenbeck stochastic process (see Appendix B), p and q are any two ages in T and α , γ , and λ are parameters to be estimated. On taking the natural logarithm of the additive genetic variance function, we will have a log-linear function in the variance: ln σ g2 = α g + γ g (p − age ) ,

Eq. 109

which is just the equation of a line on the logarithmic scale. Thus, genotype × age interaction obtains for a nonzero slope on the logarithmic scale of the additive genetic variance function; that is, for γ g ≠ 0 . Similarly, genotype × age interaction obtains for λ ≠ 0 in the genetic correlation function, where the null hypothesis is satisfied for λ = 0

because e 0 = 1 . Taking the natural logarithm of the genetic correlation function also gives the equation of a line on the logarithmic scale: ln ρG = −λ p − q .

Eq. 110

As for the additive genetic variance function, genotype × age interaction obtains for a non-zero slope on the logarithmic scale of the genetic correlation. The environmental variance component of the response is modeled in similar fashion to σ g2 but there can be

87

no corresponding environmental correlation term because of the assumption that g and e are distributed as mutually independent MVNs. There is one more component needed to build the full genotype × age interaction model. To allow for a covariance formulation, and in keeping with the definition of the standard deviation (Eq. 93), let: 2

{ [

(

σ υx = σ υx = exp α υ + γ υ p x − age

)] }

1 2

; ∀ υ = g, e ,

Eq. 111

where this formulation holds for any individual, x , in the sample, but when taking covariances this can be indicated with individual specific subscripts, as in x and z for the generic case. The full genotype × age interaction model is a decomposition of the total phenotypic variance similar to Equation 83 and so the variance and covariance components are similarly subscripted. Taking Equations 107, 108, and 111 together and recalling the fundamental relations detailed earlier in this section, the phenotypic covariance may be written as: 1

1

Cov( y x , y z ) = 2φ xz {exp(− λ p x − q z )} { exp[α g + γ g (p x − age)] } 2 { exp[α g + γ g (q z − age)] } 2 1

1

+ δ xz { exp[α e + γ e (p x − age )] } 2 { exp[α e + γ e (q z − age )] } 2 , Eq. 112 where all the previous definitions hold. Note that because variances can always be expressed in terms of covariances and covariances can be defined in terms of the correlation coefficient and standard deviations, we effectively inherit a flexible means for formulating a variance/covariance relation. This is a cross-sectional model that applies generally to three types of pairwise comparisons of individuals. In one type, let x = z while p = q . Equation 112 gives the variances in this situation, in accord with the polygenic model. In a second type, it may be such that x ≠ z while p = q , and, in a third 88

type, it may be such that x ≠ z while p ≠ q . Note that none of these three types are longitudinal comparisons, which would be the case where x = z while p ≠ q (i.e., the same individual is measured at different ages). In the former two cases, where p = q , the genetic correlation function, written as a function of age differences, cannot play a role in genotype × age interaction because for this case the function equals 1. For the case where different individuals of different ages are compared ( x ≠ z while p ≠ q ), the variance and genetic correlation functions can both contribute to potential genotype × age interaction. Thus, an optimal data set for the discovery of genotype × age interaction under the above approach will have large extended pedigrees—this is because the genetic covariance is still also a function of relatedness—whose constituents are of widely varying ages. Taking all of these considerations together, we can rewrite Equation 112 to explicitly cover the three types of conditions just discussed as follows:

[

)]

(

[

)]

(

⎧exp α g + γ g p x − age + exp α e + γ e p x − age ; ⎪ ⎪ ∀ x = z,2φ xz = 1, δ xz = 1 . ⎪ ⎪ Cov( y x , y z ) = ⎨2φ {exp(− λ p − q )}× x z ⎪ xz 1 ⎪ 2 exp α + γ q − age g g z ⎪ exp α g + γ g p x − age ⎪ ∀ x ≠ z, δ = 0 . ⎩ xz

{ [

(

)] } { [

(

Eq. 113

)] }

1 2

;

The bottom form on the right hand side covers both cases where different individuals are of the same age or of different ages. Note that the assumption that g and e are distributed as mutually independent MVNs is still in operation (i.e., there is no environmental covariance term). Using the properties of the exponential function, Equation 113 can be written so that the genetic components are represented in one exponential function for the bottom form.

89

⎧ ⎪exp α + γ p − age + exp α + γ p − age ; g g x e e x ⎪ ⎪ ∀ x = z,2φ xz = 1, δ xz = 1 . ⎪ Cov( y x , y z ) = ⎨ ⎪ γ ⎪2φ xz exp ⎡α g + g p x + q z − 2age − λ p x − q z ⎤ ; ⎢ ⎥ ⎪ 2 ⎣ ⎦ ⎪ ⎩ ∀ x ≠ z, δ xz = 0 .

[

)]

(

[

(

(

)]

Eq. 114

)

Equations 112-114 are completely analogous to Equation 83. To begin to write the matrix model, we may use the equivalence relations regarding age and individual identity to determine the elements of the matrix specifying the two genetic outcomes in Equation 114 (sensu Lange, 1986). It is significant that the equivalence relations regarding age and individual identity specify mutually exclusive conditions that exhaust all possibilities in a cross-sectional design. Moreover, because there is only one outcome with respect to the environmental component, it is as if the variance component, σ e2 , is merely reparameterized (in fact, all the variance components are reparameterized). Let there be a new matrix, A = {a ij }. The elements in this new matrix are specified as follows: ⎧ ⎪ ⎪exp α g + γ g (p i − age ) ; ∀ i = j . ⎪ a ij = ⎨ ⎪ γ ⎪exp ⎡α + g (p + q − 2age ) − λ p − q ⎤ ; ∀ i ≠ j . ⎢ g i j i j ⎥ ⎪⎩ ⎣ 2 ⎦

[

]

Eq. 115

Also, let there be a diagonal matrix B = {b ij }, where the diagonal elements are given by:

[

(

)]

exp α e + γ e p x − age . All together, the matrix formulation for the genotype × age

interaction model may be given as follows:

90

Σ = 2Φ o A + B ,

Eq. 116

where o is the Hadamard product operator (Horn and Johnson, 1991: ch. 5). Equation 116 is completely analogous to Equation 84. That the (co)variance is being modeled as a function of some environmental variable of interest represents a departure from traditional quantitative genetics, as is now explained and justified. In their comprehensive discussion of G × E interaction, Lynch and Walsh (1998: 663) noted that G × E interaction may exist even when ρG = 1 , but then they suggested that a variance-stabilizing transformation would remove such effects. However, Bulmer (1980: 25) and Falconer (1989: 296) both pointed out that such transformations may not always be successful at removing interaction effects. It is notable in this regard that D. S. Falconer, the founder of the genetic correlation approach, emphasizes the genetic correlation in diagnosing G × E interaction in all editions of his widely-used textbook on quantitative genetics (e.g., Falconer, 1989: 322-326; but see Falconer (1990) for a treatment of variances) and that Robertson (1959), who originally derived Equation 101, ultimately deferred to Falconer’s method. Intuitively, not accounting for variance heterogeneity in a model of G × E interaction, when it is known a priori to have an effect, leads to biased estimates of the genetic correlation and,

consequently, to necessary corrections for this bias (Robertson, 1959; Eisen and Saxton, 1983; Fernando et al., 1984; Yamada et al., 1988; Itoh and Yamada, 1990; Dutilleul and Potvin, 1995). However, statisticians and some statistical geneticists have pointed out that modeling (co)variance heterogeneity (Carroll and Rupert, 1982, 1988; Aitkin, 1987; Davidian and Carroll, 1987; Blangero, 1993; Verbyla, 1993; Denis et al., 1997; Frensham et al., 1997; Carroll, 2003), as is being done under the genotype × age interaction model,

91

is in many cases more desirable and powerful than the traditional approach of seeking a variance-stabilizing transformation or a correction to this effect (Bartlett and Kendall, 1946; Bartlett, 1947; Box and Cox, 1964; Cox, 1984). Further, both the variance and correlation functions can be shown to have a rigorous mathematical foundation in the theory of stationary Gaussian stochastic processes (Kirkpatrick and Heckman, 1989; Kirkpatrick and Lofsvold, 1989; Kirkpatrick et al., 1990, 1994; Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002; Appendix B). Thus, under the model espoused here, equal weight is accorded to the variance and genetic correlation functions in the search for genotype × age interaction (but their interpretations will be different).

92

Chapter 7 Methods III: Likelihood Theory and Maximum Likelihood Estimation As mentioned in the first section on models in the preceding chapter, Fisher is credited with the development of the linear model in general and variance components models in particular. It is remarkable that he (Fisher, 1912, 1922, 1925, 1934a&b, 1935, 1990) is also credited with the development of the theory of likelihood and maximum likelihood estimation (see Edwards, 1992). These concepts will be of prime importance in the three sections of this chapter on estimation, inference, and power. The statistical genetics software SOLAR (Almasy and Blangero, 1998) was used for all model analyses. In particular, SOLAR employs standard numerical computation algorithms to compute: 1) the ln-likelihood of a statistical model, 2) the maximum likelihood estimates of parameters under a model, and 3) the standard errors of the maximum likelihood estimates. These will be referred to as goals. The underlying theory is reviewed herein. It should be held in mind that for all three goals, the end result is a scalar. Thus, SOLAR can be thought of as a fancy calculator that is used to compute the above scalar values. The three goals will be taken in turn. In general, the likelihood function for a population sample comprised of z pedigrees, given a single-parameter, probability model is given as: z

L(θ data ) = c∏ f i ( y ) ; ∀ c > 0 ,

Eq. 117

i =1

where θ and f ( y ) denote the parameter and model, respectively; the likelihood of θ conditional on the data is given by a multiplicative function of f ( y ) , which holds up to a multiplicative constant, c > 0 ; and multiplication is carried out across pedigrees, the

93

constituents of which have measurements, y , that are distributed within pedigrees according to f ( y ) . Notice that on taking logarithms we will have, by a property of

logarithms, an additive function: z

log L(θ data ) = log c + ∑ log f i ( y ) ; ∀ c > 0 ,

Eq. 118

i =1

where we may now more conveniently sum across pedigrees to obtain the sample log likelihood. Keeping these general points in mind, we may now take the case for say a single pedigree specifically in regard to the multivariate mixed linear and polygenic models. Under the assumption that the trait of interest is MVN within pedigrees, and using Equations 82-84 as an example that may be generalized to more complex models, the likelihood function for a single pedigree of N individuals is given as: ⎧ ⎡ 1 ⎤⎫ −N 2 −1 2 L(β, σ g2 , σ e2 y, X ) = c ⋅ f (y ) = c ⎨(2π) Σ exp ⎢− Δ′Σ −1Δ⎥ ⎬ ; ∀ c > 0 , ⎣ 2 ⎦⎭ ⎩

Eq. 119

where the parameters under the multivariate mixed linear and polygenic models, namely, β , σ g2 , and σ e2 , are expressed as the hypothesis that they are proportional to the MVN up

to an arbitrary, multiplicative constant, c > 0 , conditional on the vector of trait values, y , and the covariates matrix, X , and Δ = y − E[y ] = y − Xβ (Blangero et al., 2001). Traditionally, it is assumed that c = 1 (Rohatgi, 1984). Taking natural logarithms yields: ln L(β, σ g2 , σ e2 y, X ) = −

1 2

[N ln(2π) + ln Σ + Δ′Σ Δ] . −1

Eq. 120

The ln-likelihood for the population sample comprised of z pedigrees is then computed by the following additive function: ln L z (β, σ g2 , σ e2 y, X ) = −

1

z

∑ [N ln(2π) + ln Σ 2 i

i =1

i

+ Δ′i Σ i−1Δ i ] .

Eq. 121

94

The right hand sides of Equations 120 and 121 each have three scalar terms in the brackets. The latter two of these terms are perhaps not so clearly seen as scalars. The second term involves the determinant of a matrix, which is always a scalar. As for the last term, a row vector post-multiplied by a matrix gives a row vector still, which when post-multiplied by a column vector gives a scalar. Parameter estimation is carried out under standard maximum likelihood estimation procedures (Lange, 1997; Lynch and Walsh, 1998; Thompson, 2000; Thomas, 2004). Let θ = [β, σ g2 , σ e2 ] ′ denote a parameter vector. Maximum likelihood estimation gives the parameter estimates in θ that make the ln-likelihood function (Equations 120 or 121) a maximum. To this end, there are multivariable generalizations of techniques in univariable calculus for the identification of local maxima and minima, which are the first and second derivative tests. According to the univariable method, a local maximum exists where the function of interest, evaluated at the first derivative set equal to 0, is concave down, which obtains only when the second derivative is negative in sign. For the multivariable case, the first requirement is that the vector of first partial derivatives, called the score vector and denoted by S(θˆ ) , equals 0 (Lange, 1997; Magnus and Neudecker, 1999). On simplifying the notation for the ln-likelihood function and taking conditionality as understood, we require that:

()

⎡ ∂ ln L θˆ S θˆ = ⎢ , ˆ ∂ β ⎣

()

()

∂ ln L θˆ , ∂σˆ g2

′ ∂ ln L θˆ ⎤ ⎥ = 0, ∂σˆ e2 ⎦

()

Eq. 122

where estimates are indicated by a carat. The second requirement involves the Hessian matrix, denoted by H , which is defined as the matrix of second partial derivatives

95

evaluated at S(θˆ ) = 0 , and, for a n ×1 column vector, is of dimensions n × n (Magnus and Neudecker, 1999). That is:

H=

∂ 2 ln L(θˆ ) ∂θˆ ∂θˆ ′

S(θˆ )= 0

⎡ ∂ 2 ln L(θˆ ) ⎢ ⎢ ∂βˆ ′∂βˆ ⎢ ⎢ ∂ 2 ln L(θˆ ) =⎢ ⎢ ∂βˆ ∂σˆ g2 ⎢ ⎢ 2 ⎢ ∂ ln L(θˆ ) ⎢ ˆ 2 ⎣ ∂β∂σˆ e

∂ 2 ln L(θˆ ) ∂σˆ g2 ∂βˆ ∂ 2 ln L(θˆ ) ∂σˆ g2 ∂σˆ g2 ∂ 2 ln L(θˆ ) ∂σˆ g2 ∂σˆ e2

∂ 2 ln L(θˆ ) ⎤ ⎥ ∂σˆ e2 ∂βˆ ⎥ ⎥ 2 ∂ ln L(θˆ ) ⎥ ⎥. ∂σˆ e2 ∂σˆ g2 ⎥ ⎥ ⎥ ∂ 2 ln L(θˆ ) ⎥ ⎥ ∂σˆ e2 ∂σˆ e2 ⎦

Eq. 123

In general, for a multivariable function f ( x , y ) , a theorem from differential calculus holds true, provided f ( x , y ) is continuous and differentiable (Horn and Johnson, 1985: 167, 392; Widder, 1989: 52-53; Magnus and Neudecker, 1999: 105-106): ∂ 2f (x, y ) ∂x∂y

=

∂ 2f (x, y ) ∂y∂x

.

Eq. 124

The theorem states that the order in which the second partial derivatives of f ( x , y ) are obtained (that is, on differentiating with respect to x and then y or vice versa) is inconsequential for the two partial derivatives are equal. The theorem generalizes to all multivariable functions, f ( x , y,...) , and applies to all second partial derivatives. Therefore, the Hessian matrix obtained from a multivariable scalar function, as in the lnlikelihood function, is always a symmetric matrix by the theorem since the off-diagonals are correspondingly equal. That is, for the matrix {f ij }, f ij = f ji for all i and j . This point

will become relevant below. The second requirement for ln L(θˆ ) to be a local maximum is that H is negative definite, which is defined just below. That is, if H is negative definite, then ln L(θˆ ) is taken to be a local maximum and the values in θˆ are taken to be

96

the maximum likelihood estimates (MLEs) (Magnus and Neudecker, 1999). Given a matrix F and a column vector x , F is negative definite if its corresponding quadratic form, x′Fx , is negative definite, which holds for (Horn and Johnson, 1985: 396-397):

x′Fx < 0 ; ∀ x ≠ 0 ,

Eq. 125

where the end-result is always a scalar quadratic function in the elements in x (recall that a row vector post-multiplied by a matrix gives a row vector which is post-multiplied by a column vector to give a scalar). To check if H is in fact negative definite, we can use the second-order Taylor expansion of ln L(θˆ ) about some nearby point in the parameter ~ space, say θ , to obtain (cf. Horn and Johnson, 1985: 391-392; Stengel, 1994: 33-34): ~ ~′ ln L(θˆ ) = ln L(θ ) + (θˆ − θ ) S(θˆ )

~ θˆ = θ

+

1 ˆ ~′ ˆ ~ (θ − θ ) H(θ − θ ) . 2

Eq. 126

~′ ~ ~′ ~ If (θˆ − θ ) H (θˆ − θ ) < 0 , where (θˆ − θ ) H (θˆ − θ ) is the quadratic form, then H is negative definite. There are other methods to determine if ln L(θˆ ) is a maximum that require S(θˆ ) and H (Tracy and Dwyer, 1969; Magnus and Neudecker, 1999) but the above method is sufficient to illustrate the principles involved. Thus far, the principles underlying the likelihood function and maximum likelihood estimation have been discussed. The computation of the standard errors of the parameter estimates may now be addressed. These are derived from the sampling covariance matrix of the parameter estimates, which in turn is derived from the Fisher information matrix. The expected Fisher information matrix, denoted by FI , is found by taking the negative of the expectation of H (Lehmann, 1983: 126; Edwards, 1992: 146; Searle et al., 1992: 472-474; White, 1994: 94; Shao, 1999: 136): FI = −E[H ] .

Eq. 127

97

Similarly, the observed Fisher information matrix, denoted by Fri , is the negative of H (Efron and Hinkley, 1978). The reason for taking the negative of E[H ] or H is that we are working towards the sampling covariance matrix for θˆ . Therefore, since a proper covariance matrix has to be positive semidefinite, one achieves this by simply taking the negative of E[H ] or H . Similar to the definition of a negative definite matrix, F is positive semidefinite if (Horn and Johnson, 1985: 396-397):

x′Fx ≥ 0 ; ∀ x ∈ ℜ n .

Eq. 128

That we are working towards the covariance matrix for θˆ also explains the relevance of the point made above that H is a symmetric matrix, for a proper covariance matrix, in addition to being positive semidefinite, must also be symmetric such that σ i , j = σ j,i for all i ≠ j (Magnus and Neudecker, 1999: 246). Efron and Hinkley (1978) argued for using Fri in favor of FI in statistical inference (see also Skovgaard, 1985; Lindsay and Li, 1997). However, Huzurbazar (1949) showed that for fairly simple likelihood functions FI and Fri are in fact identical (see also Edwards, 1992: 150-151).

The elements of S(θˆ ) and FI under the polygenic model have been derived and are reported in Blangero et al. (2001). They are reported here with some slight modifications to notation. Under the polygenic model, the elements of S(θˆ ) are given by: ′ ′ ∂ ln L θˆ ⎛⎜ ∂βˆ ⎞⎟ ′Σ −1Δ = (ei(n ) ) X′Σ −1Δ ; i = 0, 1, . . . , n , X =⎜ ˆ ⎟ ∂βˆ i ⎝ ∂β i ⎠

()

( ) = Tr (Σ Φ)+ Δ Σ ′

∂ ln L θˆ ∂σˆ

2 g

−1

−1

ΦΣ −1Δ ,

Eq. 129

Eq. 130

98

( ) = − 1 Tr (Σ ) − 1 Δ′Σ 2 2

∂ ln L θˆ ∂σˆ

−1

−1

2 e

−1

Σ Δ,

Eq. 131

where e i(n ) is an elementary n ×1 column vector, with a “1” at the ith position and a “0” at all other positions, and where the trace operator, Tr (⋅) , is defined below. For large samples, the MLEs are themselves MVN distributed, and the expected covariance of the effects in β and the variance components is 0 (Tracy and Dwyer, 1969; Cox and Reid, 1987; Lange, 1997; Blangero et al., 2001; McCulloch and Searle, 2001):

( )⎞⎟⎛⎜ ∂ ln L(θˆ )⎞⎟⎤⎥ = E ⎡⎢− ∂

⎡⎛ ∂ ln L θˆ E ⎢⎜⎜ ⎢⎣⎝ ∂βˆ i

⎟⎜ ∂σ ⎠⎝

2 υ

⎟ ⎠⎥⎦

⎢⎣

()

ln L θˆ ⎤ = 0 ; ∀ i = 0, 1, . . . , n ; υ = g, e . 2 ⎥ ∂βˆ i ∂σ υ ⎥⎦ 2

Eq. 132

Recall that Δ = y − E[y ] = y − Xβ . On taking its expectation, we have: E[Δ] = E[y − Xβ] = E[y ] − E[Xβ] = E[y ] − E[y ] = 0 nx1 .

Eq. 133

Therefore, after evaluating the second partial derivatives, all terms involving E[Δ] vanish. All together, we therefore have:

′

( )⎞⎟ = ⎛⎜ ∂βˆ ⎞⎟ X′Σ

⎛ ∂ 2 ln L θˆ E⎜ − ⎜ ∂βˆ ∂βˆ i j ⎝

⎟ ⎜ ∂βˆ ⎟ ⎠ ⎝ j⎠

−1

X

′ ∂βˆ = e (jn ) X′Σ −1Xei(n ) ∀ i, j, . . . , n . ∂βˆ

( )

Eq. 134

i

( )⎞⎟ = 0 ; ∀ i,..., n ; υ = g, e .

Eq. 135

( )⎞⎟ = 2Tr (Σ

ΦΣ −1Φ .

Eq. 136

)

Eq. 137

⎛ ∂ 2 ln L θˆ E⎜ − ⎜ ∂βˆ ∂σˆ 2 υ i ⎝

⎛ ∂ 2 ln L θˆ E⎜ − ⎜ ∂σˆ 2 ∂σˆ 2 g g ⎝

⎟ ⎠

⎟ ⎠

( )⎞⎟ = 1 Tr (Σ

⎛ ∂ 2 ln L θˆ E⎜ − ⎜ ∂σˆ 2 ∂σˆ 2 e e ⎝

⎟ ⎠

2

)

−1

−1

Σ −1 .

99

( )⎞⎟ = Tr (Σ

⎛ ∂ 2 ln L θˆ E⎜ − ⎜ ∂σˆ 2 ∂σˆ 2 g e ⎝

⎟ ⎠

−1

)

ΦΣ −1 .

Eq. 138

Equations 129-131 and 134-138 are all scalar-valued functions. The trace of a matrix, Tr (⋅) , is a special summation operator, which sums the diagonal elements of a matrix.

The outcomes at Equations 129 and 134 and the right most terms of Equations 135 and 136 are ultimately instances of a quadratic form, which we have seen to be a scalar function. Equations 134-138 fully specify the elements in FI under the polygenic model. It will be shown in Appendix C how the elements in the score vector and the expected Fisher information matrix are derived once the ln-likelihood function is known. Inversion of FI gives the covariance matrix for θˆ , denoted by Σ θˆ (Lehmann, 1983: 427-430; Edwards, 1992: 159; Searle et al., 1992: 472-474; White, 1994: 94-95): FI−1 = Σ θˆ ,

Eq. 139

which can be used to give the standard errors of the parameter estimates in θˆ . A geometric interpretation of the relation between FI and Σ θˆ is provided in Appendix D. On writing Σ θˆ in partitioned form (after Lange, 1997), we have:

⎡ ⎡ σμ2ˆ ⎢⎢ ⎢ ⎢σμˆ βˆ 1 ⎢ ⎢σ ⎢ ⎢ μˆ βˆ 2 ⎢ ⎢σ ˆ Σ θˆ = ⎢ ⎢ μˆ β3 ⎢ ⎢σμˆ βˆ 4 ⎢ ⎢σ ˆ ⎢ ⎣ μˆ β5 ⎢ ⎢ ⎣

σβˆ 1μˆ

σβˆ 2μˆ

σβˆ 3μˆ

σβˆ 4μˆ

2 βˆ 1

σβˆ 2βˆ 1

σβˆ 3βˆ 1

σβˆ 4βˆ 1

σβˆ 1βˆ 2

σβ2ˆ 2

σβˆ 3βˆ 2

σβˆ 4βˆ 2

σβˆ 1βˆ 3

σβˆ 2βˆ 3

σβ2ˆ 3

σβˆ 4βˆ 3

σ

2 βˆ 4

σβˆ 1βˆ 4

σβˆ 2βˆ 4

σβˆ 3βˆ 4

σ

σβˆ 1βˆ 5

σβˆ 2βˆ 5

σβˆ 3βˆ 5

σβˆ 4βˆ 5

⎡0 0 0 0 0 0 ⎤ ⎢0 0 0 0 0 0 ⎥ ⎣ ⎦

σβˆ 5μˆ ⎤ ⎥ σβˆ 5βˆ 1 ⎥ σβˆ 5βˆ 2 ⎥ ⎥ σβˆ 5βˆ 3 ⎥ ⎥ σβˆ 5βˆ 4 ⎥ σβ2ˆ 5 ⎥⎦

⎡0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣0 ⎡ σ 2Vg ⎢ ⎣σ Ve,Vg

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡Μ 6×6 ⎥=⎢ ⎥ ⎣ 0 2×6 ⎥ ⎥ σ Vg ,Ve ⎤ ⎥ ⎥ σ 2Ve ⎦ ⎥⎦

0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎦

0 6×2 ⎤ , Ω 2×2 ⎥⎦

Eq. 140

100

where the sampling variances in the parameter estimates (including 5 covariates as in a typical analysis) under the polygenic model lie along the diagonals of the block matrices M 6×6 and Ω 2×2 ( Vg = σˆ g2 and Ve = σˆ e2 ). In standard matrix form, Σ θˆ is a proper covariance matrix in that it is symmetric—inherited from the Hessian—and positive semidefinite. The standard errors, denoted by SE, of the parameter estimates are then obtained by taking the square roots of the sampling variances along the diagonal to give ± SE . Under general regularity conditions—e.g., that the likelihood function is at least

twice differentiable—(for a full listing, see Cramér, 1946: 478-479), we have the following theorem on the second-order efficiency of parameter estimation. For an unbiased estimator of parameters in a parameter vector, denoted by ~ θ , the Cramér-Rao Inequality is given as (Stuart and Ord, 1991: 615-616; Shao, 1999: 251; named after Rao, 1945,1947; Cramér, 1946: 478-482):

( )⎞⎟⎤⎥

~ ⎡ ⎛ ∂ 2 ln L θˆ Var θ ≥ ⎢− E⎜⎜ ⎢⎣ ⎝ ∂θˆ ∂θˆ ′

()

−1

⎟ , ⎠⎥⎦

Eq. 141

which states that the variance of ~ θ can be no less than the inverse of the Fisher information matrix for a parameter estimate vector, θˆ . Note that the right hand side of the Cramér-Rao Inequality is Σ θˆ . This Cramér-Rao lower bound means that the maximum likelihood estimates are the best estimates. Methods III: Hypotheses and Statistical Inference Under the polygenic model, the genetic hypothesis of interest is that the heritability of a trait is significant. The heritability, denoted by h 2 , is defined as (Falconer, 1989):

101

h 2 = σ g2 σ 2p ,

Eq. 142

which is simply the ratio of the additive genetic variance to the phenotypic variance. Thus, the statistical null hypothesis under the polygenic model is that: σ g2 = 0 .

Rejection of σ g2 = 0 is taken as evidence of significant heritability. Under the genotype × age interaction model, the genetic hypotheses for no genotype × age interaction are that the variance is homoscedastic across the age continuum and the genetic correlation equals 1 across any age increment. For the example of two environments, these respectively hold that: σ g21 = σ g2 2 = σ g2 ,

and ρG = 1 , which correspond respectively, for the more general, continuous case, to the statistical null hypotheses that: γg = 0 , and

λ =0. By the arguments given in the preceding chapter, rejection of γ g = 0 or λ = 0 or both is taken as evidence of significant genotype × age interaction. On finding the maximum likelihood estimates, inferences are then made by consideration of the likelihood ratio statistic (Wilks, 1938; Wald, 1943):

102

( ) ( )

⎡ L θˆ N ⎤ Λ = −2 ln ⎢ ⎥ = −2 ln L θˆ N − ln L θˆ A , ˆ ⎣ L θA ⎦

[ ( )

( )]

Eq. 143

where the null hypothesis, H N (parameter constrained to 0, θˆ N ), is compared to the alternative hypothesis, H A (parameter estimated, θˆ A ). It should be pointed out that the likelihood ratio test and similar such tests (see below) developed out of the NeymanPearson school of thought (Neyman and Pearson, 1928a&b, 1933; Lehmann, 1950, 1959). Classically, Λ is distributed as a central chi-square random variable, denoted by χ 2 , with degrees of freedom (d.f.) equal to the difference in the number of parameters under the null (or restricted) and alternative (or general) hypotheses (for an excellent exposition of the d.f. concept in relation to Λ and in general, see Good, 1967, 1973). If the null hypothesis lies on a boundary of the admissible parameter space, the asymptotic distribution of Λ is given by a mixture of χ ν2 random variables, where ν denotes the d.f. and where the mixture may include ν = 0 (Chernoff, 1954; Miller, 1977; Self and Liang, 1987). Let the p-value obtained for Λ , evaluated as a χ ν2 with the appropriate d.f. or as a mixture thereof, be denoted by p(Λ ) . Then, significance is achieved for: p(Λ ) ≤ α , where we may take α = 0.05 to be our nominal significance level (White, 1994: 178). The appropriate mixture of χ ν2 random variables will be derived below (for a rigorous treatment of the derivation of the appropriate mixture of χ ν2 random variables under related models, see Shapiro, 1985, 1988). Before this is done, however, the concept of nested model analyses needs to be introduced. The comparisons of the full polygenic and genotype × age interaction models with their constrained alternatives are examples of nested model analyses, where the

103

appropriate d.f. of the χ 2 -tests are dictated by the difference in parameters (Thomas, 2004). For the present context, define standard conditions as cases where the null hypothesis is not on a boundary of the admissible parameter space. This is only one of several criteria, all termed regularity conditions, that enable a rigorous derivation of the distribution of Λ (see Chernoff, 1954; Cox and Hinkley, 1974: 281; for a recent discussion of what these are, see Cheng and Traylor, 1995: Sect. 2). Under standard conditions, it may happen that parameters are significant by themselves, as would be indicated under their respective 1-d.f. χ 2 -tests, or that parameters are significant only when considered jointly, as can be determined by carrying out their respective 1-d.f. χ 2 tests and 2-d.f. χ 2 -tests. Consider the scenario for a hypothetical 3-parameter model with parameters a, b, and c say. We can evaluate whether a, b, and c are significant when considered singly by carrying out their 1-d.f. χ 2 -tests. However, it may happen that none of these turn out to be significant when considered singly. At this point, we can still carry out 2-d.f. χ 2 -tests to evaluate the possibility that parameters need to be considered jointly in order to uncover their significance. Thus, we can constrain say parameters a and b, compare this model to the full 3-parameter model for a 2-d.f. χ 2 -test, find that the p-value indicates significance and conclude that parameters a and b are important only when considered jointly. It turns out that this example for a hypothetical 3-parameter model is a good description of the 5-parameter genotype × age interaction model for the variance components. There are 4 parameters (intercept and slope parameters on the logarithmic scale) for the additive genetic and environmental variance functions plus 1 parameter for the genetic correlation function. Whereas constraining the intercept

104

parameters ( α g and α e ) while allowing the slope parameters ( γ g and γ e ) to be estimated is nonsensical, the reverse scenario of constraining the slope parameters while “floating” the intercept parameters is plausible. Moreover, upon demonstrating that the polygenic model is significantly better than the so-called sporadic model (the model in which the phenotypic variance is not decomposed), it is no longer necessary to assess the possible significance of the intercept parameters. In fact, floating the intercept parameters while constraining the other 3 parameters produces a model with the exact same ln-likelihood as the polygenic model (analyses not shown). This merely reflects the principle that likelihoods (and ln-likelihoods) for models with continuous parameters are invariant under reparameterization (Edwards, 1992: 28). On reparameterizing the polygenic model in terms of the the genotype × age interaction model, we will have: ⎧σ 2y = exp(α g ) + exp(α e ) ; ⎪⎪ Cov( y x , y z ) = 2φ xz exp(α g ) + δ xz exp(α e ) = ⎨ ⎪2φ exp(α ) , g ⎪⎩ xz

Eq. 144

where the previous definitions hold. Accordingly, for cases where the genotype × age interaction model is significantly better than the polygenic model, we effectively have a 3-parameter model in terms of the kinds of χ 2 -tests that are plausible. That is, we can ask whether γ g , γ e , or λ (for the genetic correlation function) are significant when considered singly or jointly. All together, we can carry out 3 1-d.f. χ 2 -tests for each parameter considered singly and 3 2-d.f. χ 2 -tests for the possible permutations. Again, this discussion holds for standard conditions. The situation is more complicated when the null hypothesis is in fact on a boundary of the admissible parameter space.

105

The exact mixture of χ ν2 random variables or a conservative approximation thereof for the cases to be considered under an analysis of the genotype × age interaction model can now be derived. There are three cases that need to be considered. These cases are for the appropriate mixtures when comparing: 1) the polygenic model to the genotype × age interaction model, 2) the genotype × age interaction model with one parameter constrained to 0 to the full genotype × age interaction model, and 3) the genotype × age interaction model with two parameters constrained to 0 to the full genotype × age interaction model. These cases will be taken in order. It should be noted, however, that the traditional criterion (i.e., difference in parameters) is conservative (Stram and Lee, 1994, 1995; Almasy et al., 2001). On finding significant heritability under the polygenic model, the intercept parameters α g and α e of the variance functions under the genotype × age interaction model may be dropped from further consideration because they can be thought of as 2

2

reparameterized versions of σ g and σ e (Eq. 144). Now, the slope parameters γ g and γ e of the variance functions may take values in the interval (− ∞, ∞ ) ; i.e., any point on the real line ℜ . This can be demonstrated with the following inequality:

[

( )] exp[γ (p − age )] > 0

exp α + γ p i − age > 0 ⇒e ⇒

α

i

e e

⇒e

γ ⋅p i

γ ⋅age

γ ⋅p i

>0

Eq. 145

>0 .

Because of the restriction of individual age p i to the index set T, which is a positive, finite subset of the real line ( p ∈ T = age min . K age max . ; T ⊂ ℜ ), without loss in generality,

106

γ

age p i can be assumed to be 1. Whereas e always maintains positivity, γ can take any value in the interval (− ∞, ∞ ) and the inequality will always hold (Fig. 28). Therefore, the null hypothesis cannot lie on a boundary of the parameter space because the range of admissible values for the slope parameters is unbounded. From this fact, it is inferred 2

that γ g and γ e each give rise to a χ1 random variable (this satisfies the standard condition as defined above). By contrast, the null hypothesis with respect to the genetic correlation function, which is λ = 0 , does in fact lie on the boundary of the genetic 0

correlation function because ρ G = e = 1 (Fig. 28). Therefore, by arguments first developed by Chernoff (1954) and reiterated by Self and Liang (1987), λ gives rise to ⎛1 2 1 2⎞ 2 the mixture ⎜ χ 0 + χ1 ⎟ . Let χ M 2 ⎠ ⎝2

denote the appropriate mixture of χ ν2 random

⋅

variables. Note that the d.f.’s are additive with respect to independent χ ν2 random variables and that the weighting frequencies of the χ ν2 random variables must sum to 1 (Shapiro, 1985, 1988). On comparing the polygenic model to the full genotype × age interaction model, we find that Λ is distributed as follows: 2

χM

γ g ,λ , γ e

⎛1 2 1 2⎞ = ⎜ χ 0 + χ1 ⎟ 2 ⎠ ⎝2

(

2

2

+ χ1 + χ1 λ

)

γ g ,γ e

Eq. 146 ⎛1 2 1 2⎞ = ⎜ χ 0 + χ1 ⎟ 2 ⎠ ⎝2

2

+ χ2 λ

γ g ,γ e

⎛1 2 1 2⎞ = ⎜ χ 2 + χ3 ⎟ 2 ⎠ ⎝2

. γ g ,λ , γ e

2

2

Thus, Λ is approximately distributed as a 50:50 mixture of χ 2 and χ 3 random variables and p(Λ ) is determined accordingly. That this is an approximation to the exact distribution is demanded by the fact that γ g and λ are non-independent and so their

107

exp(x) 8 7 f(x) = exp(x)

6 5 4 3 2 1 0 -4

-2

0

2

x

f(x) = exp(-x)

exp(-x) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

x

Figure 28. Graphical representation of exponential functions. Top panel: values of x can be any value in the interval (− ∞, ∞ ) , whereas f (x ) = exp( x ) maintains positivity for all x ∈ ℜ . Bottom panel: the exponential decay function is restricted to the closed interval [0,1] , whereas x now takes values, under the restriction of f (x ) = exp(− x ) to [0,1] , in the half-open interval [0, ∞ ) ; that is, x now has a boundary to the left at x = 0 because 0

e = 1 , but is unbounded to the right (values approaching infinity are legitimate).

108

mixture of χ ν2 random variables is not simply additive (but note that both γ g and λ are independent in respect to γ e ). Indeed, the exact mixture would have to somehow account for the covariance of γ g and λ . However, one can argue that this is a conservative approximation (cf. Stram and Lee, 1994, 1995; Almasy et al., 2001). The second case was indirectly discussed just above. For either of the slope 2

parameters of the variance functions, the exact distribution for Λ is given as a χ1

random variable. For the situation where only the genetic correlation parameter λ is constrained to 0, the exact distribution for Λ is given by: 2

χM

λ

⎛1 2 1 2⎞ = ⎜ χ 0 + χ1 ⎟ 2 ⎠ ⎝2

.

Eq. 147

λ

Suppose now it is desired to ascertain whether parameters are significant when considered jointly. For the situation where the slope parameters of the variance functions 2

are jointly constrained to 0, the appropriate distribution for Λ is given as a sum of χ1 2

random variables, which is just given as a χ 2 random variable. For the situation where either of the slope parameters of the variance functions and the genetic correlation parameter λ are jointly constrained to 0, we have the following mixtures: 2

χM

2

γ g ,λ

2

χM

λ ,γ e

= χ1

γg

⎛1 2 1 2⎞ + ⎜ χ 0 + χ1 ⎟ 2 ⎠ ⎝2

⎛1 2 1 2⎞ = ⎜ χ 0 + χ1 ⎟ 2 ⎠ ⎝2

λ

2

+ χ1 λ

γe

⎛1 2 1 2⎞ = ⎜ χ1 + χ 2 ⎟ 2 ⎠ ⎝2

γ g ,λ

⎛1 2 1 2⎞ = ⎜ χ1 + χ 2 ⎟ 2 ⎠ ⎝2

λ,γ e

;

Eq. 148

,

Eq. 149

where, by the above arguments, the first is a conservative approximation and the second is exact.

109

The preceding theory derives from the classical result that Λ is distributed as a central chi-square χ ν2 . This is all enabled by a more fundamental result of mathematical statistics, which is the fact that maximum likelihood estimates (MLEs) are asymptotically normally distributed (Cramér, 1946: ch. 33; Cox and Hinkley, 1974: ch. 9; Stuart and Ord, 1991: ch. 18). From this earlier result on the asympototic normality of MLEs, we have conservative “tests” for one-tailed and two-tailed hypotheses (Fig. 29). For a onetailed test and a significance level at 0.05, MLEs should be greater than roughly 2 times their standard error (Fig. 29: Top panel). For a two-tailed test and a significance level at 0.05, MLEs should be greater than roughly 2.35 times their standard error (Fig. 29: Bottom panel). By the arguments given earlier on admissible parameter values, testing for λ = 0 corresponds to a one-tailed test. Therefore, the MLE of λ should be greater than 2 times its standard error. Similarly, testing either γ g = 0 or γ e = 0 corresponds to a two-tailed test each time, and so their respective MLEs should be greater than 2.35 times their respective standard errors.

110

One-tailed Test

~2 SD

0 -4

-3

-2

-1

0

1

2

3

4

2

3

4

Two-tailed Test

~2.35 SD

~2.35 SD

0 -4

-3

-2

-1

0

1

Figure 29. One- and two-tailed tests on the assumption that maximum likelihood 2 estimates (MLEs) are normally distributed N Îź = 0, Ďƒ = 1 . Top panel: one-tailed test of the hypothesis that a MLE is greater than zero. Bottom panel: two-tailed test of the hypothesis that a MLE is nonzero.

(

)

111

Methods III: Power and Alternative Test Statistics In mathematical statistics, the central chi-square χ ν2 is known to be a special case of the more general noncentral chi-square, denoted by χ′2 (ν, ζ ) , where the two parameters of its distribution are the d.f. given by ν and the noncentrality parameter, denoted by ζ (Johnson et al., 1995). Indeed, from Johnson et al. (1995), the central χ ν2 can be written in terms of the noncentral χ′2 (ν, ζ ) as χ′2 (ν, ζ = 0) = χ′2 (ν,0) . Given this relation, it is perhaps not too unexpected that the likelihood ratio statistic Λ , which is distributed as a central χ ν2 , can be understood in terms of the noncentral χ′2 (ν, ζ ) . This vague intuition was given rigorous form by Wald (1943), who showed that Λ is asymptotically distributed as a noncentral χ′2 (ν, ζ ) , with noncentrality parameter given by (see also Anderson, 1984: 75-77; Stuart and Ord, 1991: §§ 23.4-23.8; Williams and Blangero, 1999a&b; Blangero et al., 2001): ′ ζ = θˆ A − θˆ N FI θˆ A − θˆ N .

(

) (

)

Eq. 150

Power is strictly defined as the probability that a test will correctly reject a false null hypothesis (Blangero et al., 2001). An expression for the power of the likelihood ratio test, denoted by P(Λ ) , is given by the integral across the region of the noncentral

χ′2 (ν, ζ ) distribution with a lower limit of integration set by the 100(1 − α ) percentage point of the central χ ν2 distribution, where this lower limit is denoted by χ′α2 (ν,0) (Stuart and Ord, 1991: §§ 23.4-23.8; Williams and Blangero, 1999a&b; Blangero et al., 2001):

P (Λ ) = ∫

∞ 2 χ′α

dχ′ (ν, ζ ) . 2

(ν ,0 )

Eq. 151

112

The upper limit of integration (at ∞ ) in Equation 151 does not represent a computational problem by virtue of the fact that:

∫

∞ 0

dχ ′ (ν, ζ ) = 1 . 2

Eq. 152

Therefore, a more convenient form for numerical integration is given as: P (Λ ) = ∫

∞ 2 χ′α

∞

dχ′ (ν, ζ ) = ∫ dχ′ (ν, ζ ) − ∫ 2

(ν ,0 )

2

0

2

χ′α ( ν , 0 ) 0

dχ′ (ν, ζ ) 2

Eq. 153 = 1− ∫

2 χ′α

0

(ν,0 )

dχ′ (ν, ζ ) . 2

In the Appendix C, the elements in FI will be derived. As will be seen there, the elements in FI involve computationally-intensive matrix equations. The GaussTM software package (Aptech Systems, Inc.) will be used for these analyses. By Equation 150, ζ can be easily determined if we know FI . Equation 153 is then evaluated with respect to the noncentral χ′2 (ν, ζ ) distribution also using the GaussTM software. For completeness, the noncentral χ′2 (ν, ζ ) distribution is given here following Johnson et al. (1995). The central χ ν2 distribution is given first as: 1 exp[− x 2] ⎛ x ⎞ p ν , ζ = 0 (x ) = p ν , 0 (x ) = ⎜ ⎟ 2 Γ(ν 2 ) ⎝ 2 ⎠

(ν 2 )−1

;∀ x > 0,

Eq. 154

where Γ(⋅) is the gamma function, defined as: ∞

Γ(ν 2 ) = ∫ exp[− u ] 0

(ν 2 )−1

du ; ∀ u ∈ ℜ .

Eq. 155

The noncentral χ′2 (ν, ζ ) distribution is then given as: exp[− ζ 2]⎛ ζ ⎞ ⎜ ⎟ p ν + 2 n ,0 (x ) ; ∀ x > 0, ζ ≥ 0 . n! ⎝ 2⎠ n =0 ∞

p ν , ζ (x ) = ∑

n

Eq. 156

113

Two alternatives to the likelihood ratio statistic Λ are the Wald-type statistic (after Wald, 1943), denoted by W , and Rao’s score statistic (Rao, 1948), denoted by R s (following Bera and Bilias, 2001). Rao’s score statistic is also known in the econometrics literature as the Lagrange multiplier statistic (after Aitchison and Silvey, 1958, 1960; Silvey, 1959). The three statistics provide asymptotically optimal tests (Moran, 1970; Peers, 1971; Cox and Hinkley, 1974: ch.9; Buse, 1982; Engle, 1984; Rayner, 1997; Shao, 1999: 386-387; Greene, 2003: ch. 17). Moreover, the three statistics are equivalent asymptotically and are distributed as a χ ν2 random variable according to the theory just reviewed (Cox and Hinkley, 1974: ch.9; Buse, 1982; Engle, 1984; Rayner, 1997; Shao, 1999: 386-387; Blangero et al., 2001; Greene, 2003: ch. 17). The Wald-type statistic W is given as: W = θ′FIθ = θ′Σ θ−1θ .

Eq. 157

For a single scalar parameter, θ i say, this expression reduces to: W=

(θi ) 2 . σ θ2

Eq. 158

i

Rao’s score statistic R s is given as: ′ ′ R s = S(θ ) Σ θS(θ ) = S(θ ) FI−1S(θ ) .

Eq. 159

For the scalar parameter case, this expression reduces to: R s = S(θ i ) σ θ2i . 2

Eq. 160

These statistics utilize different features of the curvature about the maximum likelihood estimates to allow inferences to be made about the likelihood ratio of a null versus an alternative hypothesis (Buse, 1982; Engle, 1984; Greene, 2003: ch. 17).

114

Although the sources of geometrical information (cf. Fig. D1 in Appendix D) underlying these statistics are different, the fact that they are descriptions of the same ln-likelihood topography about the maximum suggests equivalence in large samples. Further, the three statistics impart to researchers the flexibility of using the most feasible statistical test given their research design (Shao, 1999: 387; Blangero et al., 2001). For example, R s is the least computationally-intensive because it requires estimation only under the null hypothesis, whereas Î› is the most intensive because it requires estimation under both the null and alternative hypotheses. It turns out, however, that W is the easiest to compute for preliminary investigations of the statistical power properties of the genotype Ă— age interaction model.

115

Chapter 8 Results Statistical Behavior of the Phenotypes Consistent with the studies reviewed in the background section on senescence and the IGF-I axis, circulating IGF-I levels exhibit a progressive decline starting after adolescence and plateaus in late adulthood (Figs. 30 and 31). In contrast, IGFBP-1 levels seem to rise at advanced ages (Figs. 32 and 33). Similar to the IGF-I pattern, circulating IGFBP-3 levels (Figs. 34 and 35) and Ratio3 (Figs. 36 and 37) exhibit declines from post-adolescence to late adulthood. Model Results Heritabilities for log IGF-I, log IGFBP-1, log IGFBP-3, and log Ratio3 are reported in Table 5. All the traits are significantly heritable. The genotype × age interaction model is significantly better than the polygenic model for log IGF-I, log IGFBP-3, and log Ratio3, but not for log IGFBP-1 (Table 6). At this point, all that can be said is that the genotype × age interaction model is more supported by the data than the polygenic model for the traits just mentioned. In order to answer the question of whether or not genotype × age interaction as strictly defined in chapter 6 is important, the full genotype × age interaction model was compared to its various constrained alternatives for log IGF-I, log IGFBP-3, and log Ratio3 (Tables 7-9, respectively). Three values are reported in these tables, the maximum likelihood parameter estimates, their standard errors, and the p-values under the appropriate tests (1 d.f., 2 d.f., or their equivalents). As discussed in chapter 7, the MLEs for γ g and γ e should be greater than 2.35 times their standard error, which correspond to a conservative two-tailed significance test each time,

116

400

Mean IGF-I Levels

350 300 250 200 150 100 50 0 >15-20 >20-25 >25-30 >30-40 >40-50

>50

Age Intervals

Variance in IGF-I Levels

40000 35000 30000 25000 20000 15000 10000 5000 0 >15-20 >20-25 >25-30 >30-40 >40-50

>50

Age Intervals

Figure 30. Age-specific means and variances in IGF-I levels (ng/ml).

117

12

Log IGF-I

10 8 6 4 2 0 10

20

30

40

50

60

70

80

90

Age (years)

12

Log IGF-I

10 8 6 4 2 0 10

20

30

40

50

60

Body Mass Index

Figure 31. IGF-I versus age and BMI. Top panel: log IGF-I versus age. Bottom panel: log IGF-I versus BMI.

118

50 Mean IGFBP-1 Levels

45 40 35 30 25 20 15 10 5 0 >15-20

>20-25

>25-30

>30-40

>40-50

>50

Age Intervals (years)

Variance in IGFBP-1 Levels

1400 1200 1000 800 600 400 200 0 >15-20 >20-25 >25-30 >30-40 >40-50

>50

Age Intervals (years)

Figure 32. Age-specific means and variances in IGFBP-1 levels (ng/ml).

119

3

Log IGFBP-1

2.5 2 1.5 1 0.5 0 0

20

40

60

80

100

Age (years)

300

Log IGFBP-1

250 200 150 100 50 0 0

20

40

60

80

Body Mass Index

Figure 33. IGFBP-1 versus age and BMI. Top panel: Log IGFBP-1 versus age. Bottom panel: Log IGFBP-1 versus BMI.

120

5000

Mean IGFBP3 Levels

4500 4000 3500 3000 2500 2000 1500 1000 500 0 >15-20 >20-25 >25-30 >30-40 >40-50

>50

Age Intervals

Variance in IGFBP3 Levels

7000000 6000000 5000000 4000000 3000000 2000000 1000000 0 >15-20 >20-25 >25-30 >30-40 >40-50 Age Intervals

>50

Figure 34. Age-specific means and variances in IGFBP-3 levels (ng/ml).

121

24

Log IGFBP-3

22 20 18 16 14 12 10 20 30 40 50 60 70 80 90 100 Age (years)

24

Log IGFBP-3

22 20 18 16 14 12 10

20

30

40

50

60

Body Mass Index

Figure 35. IGFBP-3 versus age and BMI. Top panel: Log IGFBP-3 versus age. Bottom panel: Log IGFBP-3 versus BMI.

122

0.12

Mean Ratio3 Levels

0.1 0.08 0.06 0.04 0.02 0 >15-20 >20-25 >25-30 >30-40 >40-50

>50

Age Intervals (years)

Variance in Ratio3 Levels

0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0 >15-20 >20-25 >25-30 >30-40 >40-50

>50

Age Intervals (years)

Figure 36. Age-specific means and variances in Ratio3.

123

0

20

40

60

80

100

50

60

0 -2

Log Ratio3

-4 -6 -8 -10 -12 -14 Age (years)

10

20

30

40

0 -2

Log Ratio3

-4 -6 -8 -10 -12 -14 Body Mass Index Figure 37. Ratio3 versus age and BMI. Top panel: Log Ratio3 versus age. Bottom panel: Log Ratio3 versus BMI.

124

Table 5. Trait Heritabilities of the IGF-I Axis Components in the SAFHS Trait

Covariates‡

Heritability (± SE)

N

Log IGF-I

0.28** (0.07)

age, age2, sex × age, and BMI

681

Log IGFBP-1

0.27** (0.07)

age, sex, and BMI

678

Log IGFBP-3

0.31** (0.07)

age, sex, and BMI

699

Log Ratio3

0.26** (0.07)

age, sex, age2, sex × age, and BMI

667

* p-value < 0.01 **p-value < 0.001 ‡ Screened for significance

Table 6. Models: Polygenic versus Genotype × Age Interaction Trait Log IGF-I

Ln-likelihood Polygenic Genotype × age -299.7655 -289.7816

Λ 19.96774

Log IGFBP-1

-304.4637

-303.8422

1.24299

0.63993

Log IGFBP-3

-335.7971

-330.7277

10.13884

0.01190

Log Ratio3

-315.0087

-311.1454

7.72652

0.03650

P(Λ ) ‡ at χ M 2

†

γ g ,λ , γ e

0.00011

Λ = −2[ln L(H 0 ) − ln L(H A )] , where H 0 and H A are the null (or restricted) and alternative (or general) hypotheses, respectively (see Equation 143 and the supporting text in chapter 7 of this dissertation). 2 ‡ P(Λ ) is the p-value obtained by evaluating Λ at χ M γ g ,λ , γ e . See Equation 146 and the †

supporting text in chapter 7 of this dissertation.

125

Table 7. Model Fitting for Log IGF-I under the Genotype × Age Interaction Model

Model Parameters

G E N

E N V

Maximum Likelihood Estimates

± Standard Error

P(Λ ) ‡ for

1 d.f. test or equivalent

2 d.f. test or equivalant

αg

-0.16339

0.06033

NN

NN

γg

0.01857

0.00399

1.15E-06

NN

λ

0.33675

0.13786

0.16417

1.08E-31

αe

-24.15996

11.93879

NN

NN

γe

-1.10072

0.55613

0.00204

NN

‡

see chapter 7 (pp. 104-109) for explanation. NN – Not necessary (see chapter 6, pp. 104-106). E denotes exponentiation (base 10).

Table 8. Model Fitting for Log IGFBP-3 under the Genotype × Age Interaction Model

αg

± Standard Error 0.25811

γg

0.01665

0.01167

0.17765

0.03620

λ

0.01862

0.01284

0.04000

NN

αe

-0.55627

0.17305

NN

NN

γe

0.00257

0.00901

0.78185

0.03620

Model Parameters

G E N

E N V

P(Λ ) for 1 d.f. test or 2 d.f. test or equivalent equivalent NN NN

Maximum Likelihood Estimates -0.88715

126

Table 9. Model Fitting for Log Ratio3 under the Genotype × Age Interaction Model

αg

± Standard Error 0.66491

γg

-0.08033

0.03676

0.00797

NN

λ

0.00000

0.07445‡

0.50000

0.01322

αe

-0.39432

0.11337

NN

NN

γe

0.01867

0.00629

0.01060

NN

Model Parameters

G E N

E N V ‡

P(Λ ) for 1 d.f. test or 2 d.f. test or equivalent equivalent NN NN

Maximum Likelihood Estimates -1.90485

– Computed by the method of “gridding” in SOLAR

and the MLE for λ should be greater than 2 times its standard error, which corresponds to a one-tailed significance test. For log IGF-I, Table 7 reveals that the null hypotheses as regards γ g and λ are significantly rejected. Therefore, there is significant genotype × age interaction for log IGF-I. The elements of this inference are illustrated in Figures 38 and 39. While the likelihood ratio test indicates significance in respect to γ e for log IGFI, the conservative requirement that the γ e estimate be greater than 2.35 times its standard error urges caution. However, since the environmental variance is decreasing ( γ e is negative) while the additive genetic variance is increasing, it appears that genotype

× age interaction was becoming more and more important with increasing age. For log IGFBP-3, the relatively large standard errors of the respective parameter estimates would

127

Log IGF-I 6 5

Vg

4 3 2 1

null

0 15

35

55 Age (years)

75

95

Ve

Log IGF-I 1E-19 9E-20 8E-20 7E-20 6E-20 5E-20 4E-20 3E-20 2E-20 1E-20 1E-56 15

35

55

75

95

Age (years) Figure 38. Genotype Ă— age interaction for log IGF-I in SAFHS Mexican Americans. Top panel: Additive genetic variance function and its null. Bottom panel: Environmental variance function (displaced far away from its null).

128

Log IGF-I 1

ρG

0.8 0.6 0.4 0.2 0 0

2

4

6

8

10

Age Differences (years)

Log IGF-I 1

ρG

0.8 0.6 0.4 0.2 0 0

20

40

60

80

Age Differences (years) Figure 39. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans: The same genetic correlation function is displayed on 2 scales. The line y = 1 is the null function.

129

seem to invalidate the apparent evidence of genotype × age interaction (Table 8; Fig. 40). Lastly, as regards log Ratio3, it appears that there is evidence of genotype × age interaction and heteroscedasticity in the environmental variance (Table 9). However, the pattern is almost the complete reverse of that exhibited by log IGF-I. That is, the additive genetic variance is significantly decreasing while the environmental variance is significantly increasing (Fig. 41). It appears therefore that environmental effects were becoming more and more important in the determination of Ratio3 levels while additive genetic effects were becoming less and less important.

Log IGFBP-3 3.5 Vp 3

Vg Ve

Variances

2.5

Null-p

2 1.5 1 0.5 0 15

35

55

75

95

Age (years) Figure 40. Questionable genotype × age interaction for log IGFBP-3 in SAFHS Mexican Americans: Phenotypic, additive genetic and environmental variance functions. The null function for the phenotypic variance function is indicated as Null-p. For clarity, the null functions for the additive genetic and environmental variance functions are omitted.

130

Vg

Log Ratio3 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 15

35

55

75

95

Age (years)

Ve

Log Ratio3 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

Ve Null-e

15

35

55

75

95

Age (years) Figure 41. Apparent genotype Ă— age interaction due to significant heteroscedasticity in the environmental variance function for log Ratio3 in the environmental variance in SAFHS Mexican Americans. Top panel: Additive genetic variance function. The curve is displaced far away from its null function (not shown). Bottom panel: Environmental variance function and its null function, Null-e.

131

Power Analyses and Results of the Genotype × Age Interaction Model By a series of involved derivations, it can be shown that not only is the likelihood ratio statistic Λ asymptotically distributed as a noncentral χ′2 (ν, ζ ) , but also Λ itself gives the noncentrality parameter (Anderson, 1984: 75-77; Stuart and Ord, 1991: §§ 23.423.8; Williams and Blangero, 1999a&b; Blangero et al., 2001): ′ ζ = θˆ A − θˆ N FI θˆ A − θˆ N = Λ .

(

) (

)

Eq. 161

Since Λ = W = R s asymptotically, W (or R s ) is a suitable surrogate test statistic, the power of which can be computed according to Equation 153. For preliminary power analyses, W (but neither Λ nor R s ) is all the more suitable because it does not require the simulation of phenotypes. It just so happens that the computation of W does not require phenotype values, whereas the computation of both Λ and R s ultimately require a phenotype vector. This latter point concerning the input requirements of these statistics can be confirmed by inspecting the components that go into their computation. The likelihood ratio statistic Λ (Eq. 143) requires the ln-likelihood function (Eq. 120), and, on examining Equation 120, the ln-likelihood function requires Δ = y − Xβ , where y is the phenotype vector. Similarly, Rao’s score statistic R s (Eq. 159) requires the score vector (and the sampling covariance matrix), and the elements of the score vector (Eqs. 129-131) all require Δ . As for the Wald-type statistic W , inspection of Equation 157 (and Eq. 158 for the scalar parameter case) shows that W only requires a parameter vector, which can be specified, and FI . It will be seen in Appendix C that none of the elements in FI require Δ . In deriving the elements in FI in Appendix C, each element consisted in part of the form E[ΔΔ ′] = Σ , which, when pre-multiplied by its inverse,

132

âˆ’1

effectively canceled out of the equation; i.e., ÎŁ ÎŁ = I (where I is the identity matrix of matrix algebra). In particular, W requires computing FI under a given set of parameter values, which can be specified by the investigator for a plausible range to study the asymptotic statistical behavior of the model. Proceeding with W then, one can write a program in GaussTM to compute FI , W , and power under a range of parameter values for a fixed pedigree structure and age distribution. At this point, the author must express his immense debt of gratitude to Dr. Thomas D. Dyer (staff scientist at the SFBR, Department of Genetics) not only for writing such a program in GaussTM but also for carrying out preliminary simulations to validate the program and for teaching him some of the basics in GaussTM. This section of the dissertation simply would not have been possible were it not for the expert help of Dr. Dyer. Strictly speaking, the program is a computational program that computes FI , W , and power given certain input specifications, but it is not a simulation program. As choices for pedigree structure and age distribution, the author requested of Dr. Dyer that these be directly determined by the pedigree structure and age distribution for the analyses of the log IGF-I data so that inferences concerning the empirical analyses (in the first component) could be made. The sample size used for the current computations is N = 690 individuals. The pedigree structure was described in the first methods chapter of this dissertation. The age distribution of all individuals in the SAFHS with IGF-I data is compared to the age distribution used for the present computations (Fig. 42). As can be seen, the latter is a fairly representative sub-sample of the total given that a number of

133

Age Distribution for Log IGF-I 70 60 Frequency

50 40 30 20 10 0 15.48

40.67709677

65.87419355

Age

Simulated Age Distribution: N = 690 70 60 Frequency

50 40 30 20 10 0 15.75

45.1

74.45

Ages Figure 42. Top panel: age distribution in the sub-sample of the SAFHS with log IGF-I data. Bottom panel: age distribution used in the program used to compute power.

134

individuals would have been excluded from the empirical analyses for lack of covariate data or if their data were deemed to be outliers at values greater than ± 4 SD from the mean log IGF-I level. The ages of individuals in the sample were used to determine the sample mean age and the elements of the “age” matrices, A , B , C , and D (see Appendix C for the definition of C and D ). The sample mean age is 38.24 years of age. Three analyses were carried out, one each for the effect on the power function of variable values for γ g , λ , and γ e (Table 10). The variable parameters were varied so as to achieve a p-value = 1 and power = 1. This ensures that the power function is filled out. The other parameters (other than the parameter under analysis) were assigned their corresponding MLE parameter values for log IGF-I. The results of these analyses are reported in Figures 43-45.

Table 10. Power Analyses: Parameter Sets Parameter Values Parameter

γ g analysis

λ analysis

γ e analysis

αg

-0.16339

-0.16339

-0.16339

γg

variable

0.01857

0.01857

λ

0.33675

variable

0.33675

αe

-24.15996

-24.15996

-24.15996

γe

-1.10072

-1.10072

variable

135

Wald Statistic

50 40 30 MLE >

20 10

3.841... 0 0.848

0.858

0.868

parameter values of Vg

1 MLE p = 0.9976

power

0.8 0.6 0.4 0.2

p = 0.05 0 0.848

0.858

0.868

parameter values of Vg

Figure 43. Top panel: values of W for parameter values of Îł g expressed as the additive genetic variance, denoted by Vg. Bottom panel: power curve for parameter values of Îł g expressed as the additive genetic variance, Vg.

136

16

Wald Statistic

14 12 MLE

10 8 6 4

2.705...

2 0 0

0.5

1

parameter values of ρG

1 MLE p = 0.9334

power

0.8 0.6 0.4 0.2

p = 0.1

0 0

0.5

1

parameter values of ρG Figure 44. Top panel: values of W for parameter values of λ expressed as the genetic correlation, denoted by ρ G . Bottom panel: power curve for parameter values of λ expressed as the genetic correlation, ρ G .

137

Wald Statistic

50 40 30 20 10

MLE <

3.841...

0 0.0085 0.0135 0.0185 0.0235 0.0285 9 parameter values of Ve x 10

1

power

0.8 0.6 0.4 0.2

MLE p = 0.3477 p = 0.05

0 0.0085 0.0135 0.0185 0.0235 0.0285 9 parameter values of Ve x 10

Figure 45. Top panel: values of W for parameter values of γ e expressed as the environmental variance × 109, denoted by Ve × 109. Bottom panel: power curve for parameter values of γ e expressed as Ve × 109.

138

Before interpreting the results, there are several details concerning their make-up that should be addressed. Several interpretational guideposts that are provided in the various graphs are in need of explanation. The significance levels, α = 0.05 in the case of γ g and γ e and α = 0.1 in the case of λ , are indicated on the graphs of the power curve. The significance level for λ is adjusted upward because the parameter gives rise ⎛1 2 1 2⎞ to the mixture ⎜ χ 0 + χ1 ⎟ . To see this, fix α = 0.05 . Then, for parameter λ , 2 ⎠ ⎝2

significance obtains for: 0.05 ≤

1 2

( )

p χ0 + 2

0.05 ≤ 0 +

1 2

1 2

( )

p χ1 ⇒ 2

( )

p χ1 ⇒ 2

Eq. 162

( )

0.1 ≤ p χ1 , 2

( )

where p χ ν is the p-value obtained by evaluating W as a χ ν random variable on ν 2

2

degrees of freedom. The χ1 values corresponding to α = 0.05 and α = 0.1 are 2

2

2

2

χ1 ≅ 3.84146 and χ1 ≅ 2.70554 , respectively. The appropriate χ1 values are indicated

on the graphs of W under the three analyses. For the γ g and γ e analyses, the line at W = 50 , corresponding to a power that is effectively 1 for unbounded parameters, is

indicated on their graphs. Recalling Equation 153, let P(Λ ) = P(W ) , where P(W ) is the power of W . For the γ g and γ e analyses, P (W ) was computed as follows: P (W ) = 1 − ∫

χ α′ 2 (ν,0 ) 0

dχ ′2 (ν,ζ = W ) = 1 − ∫

3.84146 0

dχ ′2 (ν,ζ = W ) ; ∀ ν = 1 .

Eq. 163

139

For the λ analysis, P (W ) was computed as follows: P (W ) = 1 − ∫

χ α′ 2 (ν,0 ) 0

dχ ′2 (ν,ζ = W ) = 1 − ∫

2.70554 0

dχ ′2 (ν,ζ = W ) ; ∀ ν = 1 .

Eq. 164

Notice that the integrals differ at the upper limit of integration, where the upper limits are 2

2

given by χ1 ≅ 3.84146 and χ1 ≅ 2.70554 , respectively, which, as just noted above, correspond to α = 0.05 and α = 0.1 , respectively. It will have been noticed that the γ g , λ , and γ e parameters were respectively expressed in terms of the additive genetic variance (Vg), genetic correlation ( ρ G ), and environmental variance (Ve) functions. The information needed to do this is to be found in Table 10, and the equations describing how exactly this is to be done were reviewed earlier. For the variance functions, an age term of 1 was used. For the genetic correlation function, the age term was also set at 1. The calculated Ve term was rescaled by multiplying by 109 because of its extremely small values. The likelihood ratio statistic Λ —and W or R s asymptotically in most cases—is known to have at least two optimum properties (for ample discussion, see Das Gupta et al., 1964; Anderson, 1984: ch. 8; Freund, 1992: ch. 12; Kuriki, 1993; Shao, 1999: ch. 8). The first of these is known as unbiasedness of the test statistic, where a test statistic is said to be unbiased if it achieves its minimum at the null hypothesis. The second of these is that the corresponding power function of the test statistic is monotonic, which is said to obtain when the power increases with increasing distance between the null and alternative hypotheses. As can be seen, W is unbiased and its corresponding power function is monotonic in respect to γ g , λ , and γ e (Figs. 43-45). For the γ g analysis, W achieves

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its minimum at the null value, which for γ g = 0 is: exp(α g ) = exp(−0.16339) ≈ 0.85 (Fig. 43: Top panel). Moreover, its corresponding power function increases with increasing distance between this null value and other point-wise alternatives (Fig. 43: Bottom panel). For the γ e analysis, W achieves its minimum at the null value, which for γ e = 0 is: exp(α e ) × 10 = exp(−24.15996) × 10 ≈ 0.032 (Fig. 45: Top panel). Further, its 9

9

corresponding power function increases for point-wise alternatives that increasingly depart from the null value (Fig. 45: Bottom panel). For the λ analysis, recall that the null is λ = 0 or ρ G = 1 . Clearly, W achieves it minimum at the null value of ρ G = 1 (Fig. 44: Top panel) and its corresponding power function increases for increasing departures from the null (Fig. 44: Bottom panel). The specific implications of the results presented here are now taken up. By now, it is perhaps sufficiently clear that the power of a test statistic, W in the present case, depends on the hypothesis being tested, and hence on the parameter being analyzed and its estimated effect size (besides other factors such as the significance level given by α , study design and sample size). As defined earlier, power is the probability of rejecting the null hypothesis when it is false. A complementary view holds that power is the probability that a phenomenon exists for a given estimated effect size, where the phenomenon is defined in contradistinction to what the null hypothesis is formulated to negate (Cohen, 1977). Here, the phenomenon is G × E interaction, which obtains when either or both of the null hypotheses of γ g = 0 or λ = 0 are rejected. In particular, if the null hypothesis is γ g = 0 , then, strictly speaking, the phenomenon is heteroscedasticity in the additive genetic variance; its negation is homoscedasticity in the additive genetic

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variance. Similarly, if the null hypothesis is λ = 0 , then the phenomenon is nonstationarity in the genetic correlation; its negation is correlation stationarity at ρ G = 1 . Similar to the convention of a significance level of α = 0.05 , the convention for deciding that a given set-up (to include significance level α , study design, sample size, and the estimated parameter effect size) has adequate power seems to be a power of 0.80 (Berry et al., 1998; for statistical genetic, variance components models, cf. Williams and Blangero, 1999a; Blangero et al., 2001, where the cited authors studied the combinations of study design, sample size, and parameter effect sizes needed to achieve a power of 0.80). The powers with respect to the MLEs of γ g , λ , and γ e are reported in the bottom panels of Figures 43, 44, and 45, respectively. For both γ g and λ , the power to detect their particular MLEs was greater than 0.90, well above the 0.80 convention. Given that both of their corresponding null hypotheses were rejected (Table 7), for a 1 d.f. and 2 d.f. test, respectively, these high probabilities of observing the MLEs strengthen the conclusion that genotype × age interaction was discovered at least for log IGF-I. In contrast, the γ e analysis reveals that the set-up did not have sufficient power to detect the phenomenon that it underlies, namely heteroscedasticity in the environmental variance. While the null hypothesis of H 0 : γ e = 0 was apparently rejected (Table 7), the finding of a low power is consistent with the fact that the parameter estimate is less than 2.35 times its standard error. Precisely because the environmental variance was extremely small to begin with due to the effect size of α e and that it was declining to smaller values still due to the negative γ e , both the power to detect γ e and the ability to measure γ e with

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precision would have been low. Ironically, it is therefore logical to believe that there was significant heteroscedasticity in the environmental variance in log IGF-I. Taken together, these results indicate that genotype Ă— age interaction was becoming an increasingly important component of phenotype determination, specifically in relation to log IGF-I levels. How can this latest conclusion be made to agree with the conclusion regarding the declining influence of the environmental variance, if the age continuum is regarded as a continuous environment? The answer to this apparent conundrum lies in the fact that what is called the environmental variance is really just the residual variance, after accounting for other variance components, which in the present case are those components representative of polygenic and interaction effects. The inference here is that genotype Ă— age interaction, at least for the system under study, was absorbing the variance that normally would have gone into the environmental (i.e., residual) variance. This is yet another reason to believe that there was significant heteroscedasticity in the environmental variance. Thus, all of the power results exhibit an encouraging level of internal consistency.

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Chapter 9 Discussion The discussion is divided into three sections. The first outlines the findings of this study and these will be discussed in relation to the literature. The next two sections will focus on the biomedical and evolutionary ramifications of these findings. Statistical Genetic Findings It will be useful to first state clearly what the statistical genetic findings are: 1) All four of the traits analyzed are significantly heritable. 2) The additive genetic variance function for log IGF-I was significantly increasing with age. The genetic correlation function for log IGF-I significantly departed from ρ G = 1 . Taken separately or together, these findings indicate that the determination of IGF-I levels is affected by genotype × age interaction. 3) There was more than adequate power to detect an increasing additive genetic variance function and a changing genetic correlation function whereas there was not adequate power to detect a decreasing environmental variance function. The power results strengthen the conclusion that genotype × age interaction is important for IGF-I. 4) IGFBP-1 showed no evidence of genotype × age interaction. 5) IGFBP-3 showed some evidence of genotype × age interaction by the likelihood ratio test, but this inference was not supported by the conservative tests. 6) Ratio3 initially showed evidence of genotype × age interaction, but on further analysis it appeared that the signal was due to heteroscedasticity in the environmental variance. Thus far, there are four examples of the G × E interaction model developed by Blangero (1993). These will be discussed in the order they were published. The first

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example comes from Blangero (1993). In a study on captive bred baboons at the SFBR, Blangero (1993) found significant G × E interaction for serum levels of apolipoprotein B (apo B). For this case, the continuous environment was the temperature at which apo B was measured. This is relevant to CVD because apo B is a major component of LDL, which is a major CVD risk factor. Blangero (1993) interpreted this result in relation to seasonal variation in lipoprotein levels. He argued that ambient temperatures, through their effects on enzymatic activity, might bring about variation in lipoprotein levels. In the second study employing the G × E interaction model and the first for genotype × age interaction, Jaquish et al. (1997) analyzed ultrasound fetal morphometric measurements of 438 male and 454 female baboon fetuses at the SFBR. They found significant genotype × age interaction for biparietal diameter and femur length (Fig. 46). This study demonstrated that genotype × age interaction is manifest during the critical intrauterine period of development. The third example is provided by the work of Duggirala et al. (2000), who analyzed genotype × age interaction in CVD risk factors in a Mennonite population. Duggirala et al. (2000) discovered significant genotype × age interaction for serum levels of high density lipoprotein-cholesterol (HDL-C) and creatinine, both of which are important quantitative correlates of CVD (Fig. 47). This study is the first to demonstrate genotype × age interaction effects in a human population using the model of Blangero (1993). The fourth example is provided by the work of Diego et al. (2003). For the Genetic Analysis Workshop 13, Diego et al. (2003) analyzed the Framingham Heart Study data and found significant genotype × age interaction for systolic blood pressure (SBP) and fasting glucose levels (Fig. 48) and significant quantitative trait locus (QTL) × age interaction for a QTL influencing SBP. Taken together with the present study, it is

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Biparietal Diameter 55 50 45

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Gestational Age (days) Figure 46. Additive genetic variance in fetal ultrasound morphometrics in the baboon, Papio hamadryas (spp.). Top panel: biparietal diameter. Parameter estimates are provided in Jaquish et al. (1997: 837, Table 1 therein). Bottom panel: femur length. Parameter estimates are provided in Jaquish et al. (1997: 843, Table 5 therein).

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Age (years) Figure 47. Additive genetic variances in phenotypes associated with atherosclerosis. Top panel: High density lipoprotein-cholesterol (HDL-C). Bottom panel: serum creatinine. Parameter estimates were obtained from Duggirala et al. (2000: 93-94, Tables 10 and 11 therein, respectively).

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Figure 48. Genetic parameters for atherosclerosis risk factors in the Framingham Heart Study. Top panel: variance functions for systolic blood pressure (SBP; solid line) and fasting glucose levels (FG; diamonds). Bottom panel: correlation functions for SBP and FG. Modified from Diego et al. (2003: 3, Figure 1 therein).

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justifiable to conclude that G × E interaction is an important component of phenotype determination and that the model of Blangero (1993) provides a feasible approach for the study of G × E interaction in anthropological and/or biomedical settings. Power analyses of G × E interaction models from a statistical genetic perspective are extremely rare in the literature. After an intensive search of this literature, there appear to be only two such reports, one by Fry (1992) and the other by Boomsma and Martin (2002). These will be discussed in relation to the present work. The λ analysis results compare well with those of Fry (1992). Fry (1992), using an analysis of variance (ANOVA) approach, showed that the power to detect G × E interaction via departures from ρ G = 1 declines as the true value approaches unity, which is classical monotonicity with respect to the null hypothesis of H 0 : ρ G = 1 (Fig. 49). In terms of achieving maximum power, Fry (1992) also showed that as the true value of the genetic correlation coefficient approaches unity the optimal design, while keeping the total sample number constant at 500 individuals, approaches designs that use a collection of large extended families from designs that use a lesser number of larger extended families. For instance, the three maximum power values in Figure 49 correspond respectively from left to right to designs that use a collection of about 10.5 families of about 48 individuals, about 9 families of about 56 individuals, and about 5 families of 100 individuals. This seems to imply that as the effect size in terms of departure from the null of ρ G = 1 becomes larger, requirements on the amount of genetically related individuals relative to the total sample become more permissive. Conversely, as the null hypothesis is approached, rejection of the null increasingly requires a greater amount of genetic information for the same sample size, as would be provided by larger and larger

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1

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0.8 ~ 0.56

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Figure 49. Power to detect G × E interaction by ANOVA. The power values are the maximum power values across a range of study designs (N = 500 in all cases). Data and results are from Fry (1992: his Figure 1, p. 545).

extended families. The power analyses of the present study were carried out for a sample size of 690 individuals who constitute a random subset of a larger sample of 1,047 individuals from 48 families. Moreover, judging from Figure 39 of this dissertation, the effect size was rather large. Boomsma and Martin (2002) carried out power analyses of G × E interaction using the approach known as genetic covariance structure modeling (GCSM), which falls under the more general approach known as structural equation modeling. GCSM is equivalent to the variance components approach in that the total phenotypic variance is decomposed in exactly the same way. The main difference is that GCSM tends to employ samples of monozygotic (MZ) or dizygotic (DZ) twins or some mixture of MZ

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and DZ twins. In fact, Boomsma and Martin (2002) carried out simulation studies on two designs utilizing different mixtures of MZ and DZ twins. In one design, which they denoted as N1, they simulated a sample comprised of 50% MZ and 50% DZ twin pairs. In the other design, which they denoted as N2, they simulated a sample comprised of 40% MZ and 60% DZ twin pairs. As their measure of G × E interaction, Boomsma and 2

Martin (2002) used the change in heritability, here denoted by Δh , on going from one environment to another (cf. the discrete case of two environments in the second section of Chapter 5). The results of Boomsma and Martin (2002) need to be treated with some caution because, as shown earlier, G × E interaction depends on two components, heteroscedasticity in the additive genetic variance and departure from a genetic 2

correlation of ρ G = 1 . Therefore, Δh will be comprised of some unknown mix of the 2

two components. Their simulation strategy was to vary sample size and Δh for fixed power values of 0.50, 0.80, and 0.90. Their results are reported in Figure 50. To interpret their results, recall the well-known relationship between power and sample size: As the sample size increases, the power increases. Figure 50 shows that as Δh

2

increases, a smaller sample size is needed to achieve the same level of power. This relationship implies that the power inherent in the set-up is increasing with increasing 2

Δh , which is an implication of a monotone power function. Because of the fact that 2

Δh does not allow for heteroscedasticity in the additive genetic variance and nonstationarity in the genetic correlation, it is difficult to compare the results of Boomsma and Martin (2002) with the results of the present study. Overall, however, the results of the present study are consistent with those of Boomsma and Martin (2002) in that it was shown that there is sufficient power to detect both components of G × E 151

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Figure 50. Power analysis of G Ă— E interaction in samples of twin pairs. Open symbols represent the N1 design of 50% MZ and 50% DZ twin pairs and shaded symbols represent the N2 design of 40% MZ and 60% DZ twin pairs. Data and results are from Boomsma and Martin (2002: their Table XIII.3, p. 185).

interaction. Other points can be gleaned from Figure 50. Figure 50 shows that the better study design in all cases seems to be the N1 design, which has proportionately more MZ twins. This indicates that more information in terms of genetic relatedness translates to higher power. The same observation also implies that the inclusion of longitudinal measurements in the modeling framework of the present study will increase power; that

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is, to the extent any given individual may be conceptualized as a “twin” of him- or herself at any two longitudinal points. While the results of the present study are in good agreement with those reported by Fry (1992) and by Boomsma and Martin (2002), it is desirable to carry out in the near future more intensive simulation investigations that would vary sample sizes, study designs, parameter effect sizes and levels of power. To the best of the author’s knowledge, Fry (1992) and Boomsma and Martin (2002) are the only reports in the literature of power analyses of G × E interaction based on the same (or similar) underlying theory as the one espoused in this study, namely the “Falconer-Robertson” formulation. Further, Fry (1992) only studied the power to detect departures from ρ G = 1 (à la Falconer), which, as explained earlier, is only partly responsible for G × E interaction, whereas Boomsma and Martin (2002) did not distinguish between the two components of G × E interaction. Therefore, the present study is the first to carry out power analyses of a model—formulated in the spirit of the Falconer-Robertson formulation—that incorporates G × E interaction due to heteroscedasticity in the additive genetic variance and to departure of the genetic correlation from ρ G = 1 . Biomedical Ramifications

Recall the 3-phase model for the behavior of the IGF-I axis throughout ontogeny: 1) the autocrine/paracrine mode predominates during late fetal development; 2) the endocrine mode becomes increasingly important postnatally for somatic growth and is maximally important in this regard over the course of the pubertal growth spurt; and 3) the endocrine mode undergoes a transition from being a regulator of somatic growth to being a regulator of metabolism and somatic maintenance over the course of adulthood.

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The present analyses address the third phase of the above model of the behavior of the IGF-I axis throughout ontogeny. Relation to Metabolism in Adulthood and the Metabolic Syndrome There are at least two physiological hypotheses that can explain the discovery of genotype × age interaction for IGF-I. Both of these relate in complex ways to the physiology of insulin and leptin, which are secreted by β-cells of the pancreas and adipocytes of adipose tissue, respectively. A brief account of the relevant processes is given just following. It is known that the condition of obesity is associated with the up-regulation of pancreatic secretion of insulin (Polonsky et al., 1988; Polonsky, 2000) and adipose-tissue secretion of leptin (Ahima and Flier, 2000; Baile et al., 2000; Harris, 2000). Insulin promotes leptin secretion indirectly by promoting adipogenesis and fat mass accumulation and directly by stimulating adipose-tissue secretion of leptin (Fried et al., 2000; Harris, 2000; Kieffer and Habener, 2000). This contrasts with the reverse effect that leptin has on insulin secretion, where leptin indirectly down-regulates insulin secretion by exerting effects at the hypothalamus and negatively regulates β-cell insulin secretion (Harris, 2000; Kieffer and Habener, 2000; Havel, 2004). The complementary signaling systems of insulin and leptin in peripheral tissues taken together with their complementary and overlapping actions in the hypothalamus (Porte et al., 1998, 2002; Niswender and Schwartz, 2003; Benoit et al., 2004) has been called the “adipoinsular” axis (Kieffer and Habener, 2000). It was hypothesized by several investigative groups that prolonged conditions of obesity are associated with a state of leptin resistance due to the elevated secretion of this

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hormone, which in turn contributes to defects in the leptin-specific blood-brain-barrier (BBB) transport system, and desensitization in the signal transduction networks that are targets of leptin action (Maffei et al., 1995; Caro et al., 1996; Considine and Carro, 1996; Considine et al., 1996; Hassink et al., 1996; Schwartz et al., 1996). However, the issue of whether leptin resistance is pathogenic with respect to obesity or merely pathognomonic of same still remains to be conclusively resolved. Subsequent investigations along this line have generally supported the hypothesis of obesity-associated leptin resistance by defects in the leptin-specific BBB transport system to the hypothalamus and in the components of the leptin-specific signal transduction network (reviewed in Friedman and Halaas, 1998; Jéquier and Tappy, 1999; Friedman, 2002; Cummings and Schwartz, 2003; Sahu, 2004). The latest extension of the hypothesis of obesity-associated leptin resistance suggests that the sustained high levels of circulating leptin typical of prolonged obesity may render β-cells unresponsive to the leptin signal which would in turn result sequentially in dysregulated β-cell insulin secretion, hyperinsulinemia, β-cell exhaustion and/or insulin resistance and the attendant sequelae of dysfunctional glucose homeostasis including the eventual progression to frank, full-blown T2D (Seufert et al., 1999a&b; Kieffer and Habener, 2000; Seufert, 2004). This may be referred to as the hypothesis of dysregulation of the adipoinsular axis (Kieffer and Habener, 2000; Seufert, 2004). Work on a rat model by Vickers et al. (2001) suggested that dysregulation of the adipoinsular axis may originate in the fetus. Given that the studies giving rise to the dysregulated adipoinsular hypothesis were carried out on a mouse model (Seufert et al., 1999a) and a cell culture system of human pancreatic β-cells (Seufert et al., 1999b), it is notable that Söderberg et al. (2002), in a study of adult men and women, have confirmed the

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prediction of a diminishing infuence of leptin on pro-insulin concentrations with increasing obesity. It is also notable that a recent review of the pathogenesis of T2D has suggested that β-cell hypersecretion of insulin now appears to be the fundamental defect that initiates the metabolic derangements culminating in T2D (Cusi and DeFronzo, 2001). On average, the SAFHS Mexican Americans are clinically obese with a combined-sex mean BMI slightly above 30 (Comuzzie et al., 1996). By criteria for defining hyperinsulinemia, the Mexican Americans of the San Antonio Heart Study (SAHS)—the epidemiological precursor of the SAFHS—are known to be relatively hyperinsulinemic (Han et al., 2002). It is therefore highly likely that the SAFHS Mexican Americans are also relatively hyperinsulinemic. One hypothesis that can explain the IGF-I patterns is that the relatively obese and hyperinsulinemic condition of the SAFHS Mexican Americans results in the up-regulation of IGF-I, which is consistent with the knowledge that insulin is a potent stimulator of liver secretion of IGF-I (Jones and Clemmons, 1995). Another hypothesis is that because IGF-I exhibits much overlap with insulin structure and function (Froesch and Zapf, 1985), the conditions of β-cell exhaustion due to chronic hyperinsulinemia and of insulin resistance may result in the mobilization of additional compensatory mechanisms such as the potentially up-regulated IGF-I axis (the author would like to thank Dr. Anthony G. Comuzzie, who is a Scientist at the SFBR Department of Genetics, for suggesting this hypothesis to him). Indeed, the knowledge that the IGF-I axis (to include the actions of the IGFBPs and IGF-I receptor) may compensate for insulin resistance provides the physiological basis for its potential clinical uses in the control of insulin resistance and T2D (Froesch et al., 1994, 1996a&b; Hussain and Froesch, 1995; Hussain et al., 1995, 1996; Froesch, 1997; Simpson et al.,

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1998; Holt et al., 2003). Both of these hypotheses are consistent with the findings that the additive genetic variance function in log IGF-I significantly increases with age, the genetic correlation function significantly departs from ρ G = 1 , and the mean circulating level of IGF-I decreases with increasing age in the SAFHS Mexican Americans, as is depicted in Figure 51. The explanation of Figure 51 needs to be prefaced by some caveats regarding the relation between measures of obesity and insulin secretion. While

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Figure 51. Schematic diagram of changes in rank and scale along n segments of a continuous environment. I. Here, A and a represent a parent population and a fraction of the same population, respectively, measured at different points along a continuous environment. The parent population measure decreases throughout while the population fraction measure decreases at a slower rate and then increases. Due to pathophysiological events—occurring at 4 in the figure—the population fraction measure increases while the parent population measure is still decreasing.

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BMI and insulin secretion are no doubt highly correlated, the relation is a nonlinear one (Kahn et al., 2001). Furthermore, the pattern of body fat distribution and not just BMI has significant influences on several metabolic profiles, including insulin sensitivity and secretion indices (Wajchenberg, 2000; Kahn et al., 2001). Given that measures of obesity, insulin secretion and leptin secretion are continuous traits that may exhibit complex nonlinear relations, it follows that the mechanisms proposed regarding downstream effects on IGF-I secretion would be differentially manifested in a manner roughly reflective of their joint distribution. Thus, a scenario of jointly heterogeneous obesity status and leptin and insulin secretion patterns would be expected to generate an increasing additive genetic variance function and a significantly changing genetic correlation function in IGF-I in the face of declining mean circulating levels. In Figure 51, heterogeneity is simplistically depicted by supposing that a parent population and a fraction of the same population exhibit different behaviors in the mean and variance in some generic measure. Now why should Ratio3 exhibit a distinctly different behavior from IGF-I? As discussed in the first methods chapter, Ratio3 is an index of free IGF-I, and hence a coarse marker of the bioavailability of IGF-I at the level of the individual. At finer levels such as the organ- and/or tissue-levels, however, IGF-I bioavailability is largely determined by the local milieu of hormones (to include insulin, leptin, estrogen, etc.; see below), cytokines, generically named â€œfactorsâ€?, as well as by the suite of supporting proteins specific to the IGF-I axis, which include the receptor, binding, phosphorylating, and proteolytic proteins, and the acid labile sub-unit (ALS). Moreover, there is no reason to expect a priori that the same local milieu of determinants will be present at say the

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growth plate in longitudinal bones and at the complex interface of the blood with the smooth muscle and epithelial cells of the vascular system, although we know that both â€œcompartmentsâ€? involve an extraordinarily complex mix of the aforementioned components (for the local milieu at the growth plate in relation to IGF-I, see Lindahl et al., 1996; Rosen and Donahue, 1998; Robson et al., 2002; van der Eerden et al., 2003; for the local milieu in the vascular system in relation to IGF-I, see Bar et al., 1988; Raines and Ross, 1995, 1996; Sowers, 1997; Delafontaine et al., 2004). Indeed, intuition suggests the contrary. That is, we would more likely expect that the average mix of components of the local milieu would be reflective of the biological functions that need to be carried out, and to the extent that these biological functions differ at the tissueand/or organ-levels, the local milieu would also differ. Thus, the statistical genetic findings that IGF-I exhibits significant genotype Ă— age interaction and that Ratio3 exhibits significantly increasing environmental variation are consistent with the fact that there is vastly more opportunity for the determinants of free IGF-I to exhibit individualspecific or, equivalently, environmental variation than for the determinants of IGF-I secretion, since the latter are a subset of the former and where the determinants of the latter predominantly act at a single place in the body, namely the liver. This argument is recast in terms of the underlying genetic architecture in the evolutionary section. Ontogeny, Aging, and Neuroendocrine Cascades It is tempting to speculate that periods of intense hormonal activity, as in the normal conditions of puberty (see below) or the pathological consequences of a chronic state of obesity, are sufficient to mobilize the IGF-I axis GEN such that we observe signals in the variance and genetic correlation functions. In the case of puberty, it is

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known that the IGF-I axis is in fact maximally up-regulated during this period (Giustina and Veldhuis, 1998; MĂźller et al., 1999; Rogol et al., 2002; Grumbach and Styne, 2003). In the case of chronic obesity, it is highly likely that the IGF-I axis is mobilized in response to dysregulation of the adipoinsular axis, although by precisely what mechanisms we currently do not know. In both cases, hormones with wide ranging physiological effects and/or tissue targets are mobilized at relatively high concentrations. The up-regulation of the IGF-I axis is a normal physiological process in the case of puberty and a potential, endogenous mechanism to restore metabolic homeostasis in the case of chronic obesity. The results of the present study and the above speculations may be related to the hypothesis elaborated by Finch (1975, 1977, 1976, 1979, 1987, 1988, 1990, 1993; Finch and Landfield, 1985) that the pathologies of senescence are mediated by neuroendocrine cascades that are late-life occurrences of physiological control systems responsible for homeostasis throughout ontogeny. Finchâ€™s neuroendocrine hypothesis implicitly assumes that homeostatic systems decline with age and so on this assumption the neuroendocrine cascades may be seen as inducers of pathology or progressively inefficacious mechanisms for restoring homeostasis. Finchâ€™s neuroendocrine cascade hypothesis may be understood as a more recent and refined version of the long established concept of systemic homeostatic decline with advancing age. The 3-phase model in general and the second phase thereof in particular is consistent with the widely-held belief that the dramatic increase in GH and endocrine IGF-I secretion during puberty is causally related to the adolescent growth spurt in humans (Martha and Reiter, 1991; Clark and Rogol, 1996; Bogin, 1998; Hibi and

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Tanaka, 1998; Mauras, 2001; Rogol et al., 2002; Grumbach and Styne, 2003; Styne, 2003; van der Eerden et al., 2003) and nonhuman primates (Copeland et al., 1981, 1982, 1985; Liu et al., 1991; Styne, 1991; Crawford and Handelsman, 1996; Crawford et al., 1997; Suzuki et al., 2003). It will be instructive to briefly review the main hormonal determinants of pubertal growth. Pubertal growth is a consequence of the concerted actions of the gonadotropin/sex steroid hormone and GH/IGF-I axes. At the onset of puberty, elevated pituitary secretion of gonadotropin brings about elevated gonadal secretion of the sex hormones, which are estrogen in females and androgens in males (Terasaw and Fernandez, 2001; Grumbach, 2002), and the gene encoding the intracellular enzyme aromatase, which carries out biosynthesis of estrogen from steroid precursors, exhibits increased expression in the ovaries and testes (Grumbach and Auchus, 1999; Alonso and Rosenfield, 2002). The aromatase gene (CYP19) is expressed in non-gonadal tissues as well, most notably adipose and bone tissues (Simpson, 2000; Simpson et al., 2002). Therefore, the total estrogen in circulation derives from endocrine and autocrine/intracrine sources, which is somewhat similar to the case for IGF-I. In both sexes, the rise in estrogen synthesis shortly after the onset of puberty eventually promotes increased secretion of GH and then this of course brings about increased circulation levels of IGF-I (Martha and Reiter, 1991; Clark and Rogol, 1996; Caufriez, 1997; Grumbach and Auchus, 1999; Mauras, 2001; Rogol et al., 2002; Grumbach and Styne, 2003; Styne, 2003; Veldhuis, 2003). But the role of estrogen is not limited to the elevation of GH and IGF-I levels. In fact, skeletal growth is a function of the additive actions of estrogen on the one hand and GH and IGF-I on the other and of their synergistic interactions (Martha and Reiter, 1991; Clark and Rogol, 1996; Caufriez, 1997;

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Grumbach and Auchus, 1999; Grumbach, 2000; Soyka et al., 2000; Mauras, 2001; Riggs et al., 2002; Rogol et al., 2002; Grumbach and Styne, 2003; Styne, 2003; Veldhuis, 2003). Estrogen and IGF-I also synergistically interact at the level of the central nervous system (CNS) in the regulation of reproductive physiology (Melcangi et al., 2002) and their interactions may even confer neuroprotective effects on brain tissue (CardonaGómez et al., 2001, 2003). The latter point is supported by independent lines of research on the neuroprotective effects of IGF-I (D’Ercole et al., 1996, 2002; Trejo et al., 2004) and estrogen (Garcia-Segura et al., 2003; Norbury et al., 2003; Maggi et al., 2004). The effects of the physiological upheaval during puberty on the IGF-I axis GEN are theoretically depicted in Figure 52. The similarities between the hormonal regulation of puberty and adult metabolic pathophysiology run deeper than their common ground in the IGF-I axis. Although the gonadotropin/sex steroid hormone and GH/IGF-I axes are most important for pubertal growth regulation, leptin and insulin are also active and of some importance. Similarly, estrogen is not restricted to pubertal and reproductive endocrinology, as it is thought to be involved in numerous aspects of the metabolic syndrome. To establish these points, the roles of leptin and insulin in puberty and, conversely, the role of estrogen in the metabolic syndrome will be reviewed. Leptin is thought to be a permissive factor that contributes to the suite of complex signals in the CNS initiating the onset of puberty (Grumbach, 2002; Margetic et al., 2002; Grumbach and Styne, 2003; Shalitin and Phillip, 2003; Styne, 2003; Veldhuis, 2003). Ong et al. (1999) suggested that leptin’s role in regard to weight regulation and

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n-1

n

Figure 52. Schematic diagram of changes in rank and scale along n segments of a continuous environment. II. Here, A and B are genotypes measured at different points along a continuous environment. The change in environment from 1 to 2 elicits a change in both rank and scale and is reflected by genetic correlation less than 1 and changing additive genetic variance (first decreasing then increasing). Subsequently, similar incremental changes in environment elicit only changes in scale and are reflected by a decreasing additive genetic variance.

maintenance contributes to the physiological mechanism underlying the positive association between adiposity levels and the onset of puberty (Frisch, 1985, 1987). Further, Maor et al. (2002) recently reported that leptin has skeletal growth factor properties in a mouse model of endochondral ossification. They hypothesized that leptin may help to explain the accelerated growth in obese adolescents relative to nonobese adolescents. Given that the study of leptin biology has only recently been emphasized in

163

biomedical research, it appears likely that leptin will be found to play more roles in pubertal growth regulation. Studies have shown that normal puberty (i.e., defined by the absence of endocrine disorders) is associated with increased insulin resistance and compensatory hyperinsulinemia in non-Hispanic Whites and Hispanic children (Bloch et al., 1987; Caprio et al., 1989, 1993, 1994a&b; Amiel et al., 1991; Savage et al., 1992; Cook et al., 1993; Caprio and Tamborlane, 1994; Potau et al., 1997; Moran et al., 1999; Goran and Gower, 2001). There are important population differences, however. Arslanian and colleagues have shown that elevated compensatory Î˛-cell insulin secretion in African American adolescents does not occur despite an increase in insulin resistance relative to their prepubertal counterparts, which is in contradistinction to what occurs in White American adolescents (Arslanian and Suprasongsin, 1996; Arslanian and Danadian, 1998; Arslanian, 1998, 2002; Saad et al., 2002). However, several studies have found that African American adolescents seem to compensate for insulin resistance by reduction in the rate of hepatic insulin extraction, which renders Î˛-cell compensation unnecessary (Jiang et al., 1996; Goran et al., 2002; Gower et al., 2002; cf. Mittelman et al., 2000 for the same mechanism in dogs). This finding is likely to be robust because studies comparing adults of African ancestry to White American, Mexican American and White European adults have reported reduced hepatic insulin extraction rates in the foremost group (Cruickshank et al., 1991; Osei and Schuster, 1994; Osei et al., 1997; Harris et al., 2002). Therefore, hyperinsulinemia may arise by either of two mechanisms, by Î˛-cell compensation (Kahn, 1996) or reduction in the rate of hepatic insulin extraction (Goran et al., 2002; Gower et al., 2002).

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The effects of pubertal insulin resistance (PIR) (sensu Goran et al., 2003) are largely restricted to carbohydrate metabolism and seem not to be manifested in protein or lipid metabolism (Amiel et al., 1991; Caprio et al., 1993, 1994a&b; Caprio and Tamborlane, 1994; Arslanian and Kalhan, 1994). However, as regards protein metabolism in particular, Arslanian and Kalhan (1996) found that PIR and the resultant hyperinsulinemia seemed to suppress proteolysis. It was hypothesized that the hyperinsulinemia following PIR would therefore help to promote protein anabolism during pubertal growth (Amiel et al., 1991; Caprio et al., 1993, 1994a; Caprio and Tamborlane, 1994). This hypothesis is consistent with the knowledge that insulin, GH and IGF-I exert coordinated, anabolic actions on muscle tissues (Fryburg and Barrett, 1995; Liu and Barrett, 2002). Moreover, one of insulinâ€™s more important roles in protein anabolism is to inhibit protein degradation (Fryburg and Barrett, 1995; Wolfe and Volpi, 2001), which is consistent with the finding of Arslanian and Kalhan (1996) mentioned above. Caprio (1999a&b) further suggested that hyperinsulinemia suppresses circulating levels of IGFBP-1 and this would in turn increase circulating levels of free IGF-I. But recall that insulin up-regulates liver secretion of IGF-I (Jones and Clemmons, 1995) and so this too may play a role. There are now numerous studies implicating estrogen as a major player in the pathophysiology of the metabolic syndrome and these fall roughly into three classes: 1) studies on postmenopausal women, 2) studies on the effect of estrogen or aromatase deficiency in men and 3) animal models of estrogen or aromatase deficiency. These studies have demonstrated that estrogen deficiency is associated with insulin resistance and impaired glucose tolerance in adults and that estrogen treatment, usually involving

165

estrogen replacement therapy (ERT), tends to ameliorate dysfunction in carbohydrate metabolism (see reviews by Sharp and Diamond, 1993; Gaspard et al., 1995; FaustiniFustini et al., 1999; Meinhardt and Mullis, 2002; Rochira et al., 2002; Murata et al., 2002). Estrogen also has demonstrable cardiovascular protective effects, such as associated reductions in lipid levels and suppression of the vascular response to chronic inflammatory stress (Gaspard et al., 1995; Farhat et al., 1996; Nathan and Chaudhuri, 1997; Mendelsohn and Karas, 2001; Mendelsohn, 2002; Baker et al., 2003). It should be noted, however, that there is still considerable controversy surrounding estrogen’s role in cardiovascular protection (Barrett-Connor and Grady, 1998; Mendelsohn and Karas, 2001; Mikkola and Clarkson, 2002; Pradhan and Sumpio, 2004). Taken all together, the two main features of Finch’s neuroendocrine cascade hypothesis seem to be upheld. The pathologies of senescence, such as those associated with the metabolic syndrome, do in fact involve neuroendocrine cascades. Further, these neuroendocrine cascades are late-life occurrences of homeostatic mechanisms that operate in coordinated fashion during developmentally critical periods in ontogeny well before the onset of senescence. Evolutionary Ramifications: Relation to the Evolution of Senescence

As indicated in the background chapter on senescence and the IGF-I axis, the evolution of senescence can be explained by the disposable soma (DS) theory. However, the DS theory—in its present formulation at least—can say nothing about the statistical genetic expectations regarding senescence. On the other hand, there are two other theories of the evolution of senescence that are formulated in terms of population genetics and that make statistical genetic predictions (these should be viewed as being

166

complementary to the DS theory; cf. Kirkwood and Rose, 1991). These are the mutation accumulation (MA) and antagonistic pleiotropy (AP) theories of the evolution of senescence (Medawar, 1952; Williams, 1957; Rose, 1991; Charlesworth, 1994a&b). In its modern form, the MA model posits that senescence evolves as the result of agestructured mutation-selection balance, where the mutation rate across age classes is assumed uniform and the sensitivity of fitness or, equivalently, the selection intensity can be shown to decline with increasing age (Charlesworth, 1994a&b, 2001). The AP model posits that senescence evolves as the result of an age related tradeoff in the beneficial and detrimental effects of genes (Williams, 1957; Rose, 1991; Charlesworth, 1994a&b). Under AP theory, genes that confer beneficial fitness effects early in the lifespan are maintained by selection, but later in the lifespan, when selection intensity declines significantly, it may happen that the very same genes confer detrimental effects. This model is conceptually similar to the so-called â€œhitchhikingâ€? population genetic models that explain the higher-than-expected frequencies of neutral alleles as the result of the linkage of a neutral locus to a locus that is selectively advantageous. In the case of the AP model, alleles that are harmful late in the lifespan (but effectively neutral with respect to fitness effects in the evolutionary sense) can evolve to higher-than-expected frequencies if they are selectively advantageous early in the lifespan. Charlesworth and Hughes (1996) showed that both the MA and AP theories predict that the additive genetic variance in life history traits increases with increasing age. This prediction has been upheld in a number of studies (e.g., Charlesworth and Hughes, 1996; Snoke and Promislow, 2003; see also the most recent review by Hughes and Reynolds, 2005), but there is an important, observed deviation from expectations

167

discussed below. It should be noted that there are other testable predictions that are derivable from MA and AP theory (see reviews by Zwaan, 1999; Kirkwood and Austad, 2000; Partridge and Gems, 2002; Hughes and Reynolds, 2005). However, only the prediction of an increase in the additive genetic variance of life history traits will be addressed in relation to the present study. While the IGF-I axis is not a classical life history trait, it is ostensibly one of the more important “endophenotypes” of such for it significantly affects growth rate, size at maturity, fecundity and mortality, all of which are classical life history traits. Therefore, the IGF-I axis should be taken as a “microcosm” for testing the MA and AP theories. Another reason for using the IGF-I axis as a microcosm of life history traits is that for traits such as fecundity and mortality there is always additional statistical error arising from the fact that such traits have to be estimated rather than directly measured (Shaw et al., 1999). Therefore, a focus on endophenotypes of life history traits is likely to be less hindered by the introduction of additional error. The overall results of the present study are not consistent with the MA and AP models in that their joint prediction of increasing additive genetic variance is not born out. Only IGF-I exhibits significantly increasing additive genetic variance, but IGFBP-1 and perhaps IGFBP-3 show a stable additive genetic variance with increasing age and Ratio3 exhibits a significantly declining additive genetic variance with increasing age. This overall disagreement of results with theory is similar to what was reported for the additive genetic variance in log or ln mortality in Drosophila melanogaster by the laboratory and colleagues of J. W. Curtsinger (Curtsinger et al., 1995; Promislow et al., 1996; Shaw et al., 1999). Their group reported that the additive genetic variance first

168

increased and then started to decline at the most advanced age groups. In a reanalysis of the data analyzed by Charlesworth and Hughes (1996) and by Promislow et al. (1996), Shaw et al. (1999) found that, whereas both studies exhibit the pattern first discovered by Curtsinger et al. (1995), only the study by Promislow et al. (1996) had enough power to detect the declining additive genetic variance at the oldest age groups (Fig. 53). As discussed in the previous section, IGF-I secretion is largely a consequence of multiple determinants acting at a single site, namely the liver, whereas the levels of free IGF-I throughout the body will always include the determinants of IGF-I secretion and, in addition to these, all of the tissue- and/or organ-specific determinants. Moreover, liver IGF-I secretion is a classical endocrine response in that several to many signals converge at a site to elicit a common response, which is usually the increased or decreased expression of certain genes. In the case of liver IGF-I secretion, many signals, which are mainly hormonal and/or nutritional or the two acting together (Clemmons and Underwood, 1991; Corpas et al., 1993; Thissen et al., 1994; Jones and Clemmons, 1995; Ketelslegers et al., 1995; Giustina and Veldhuis, 1998; M端ller et al., 1999), converge at the liver to effectuate increased liver expression of the gene encoding IGF-I and the secretion of these gene products. Recalling Figure 11 in the background chapter on senescence and the IGF-I axis, the increasing additive genetic variance in IGF-I and the changing genetic correlation coefficient may have been reflective of the process of increasing mobilization of the genetic elements of the IGF-I axis GEN. In contrast, the compartment-wise determinants of free IGF-I will have exhibited environmental variation across individuals. This idea explains why Ratio3 should exhibit significantly increasing environmental variance in the face of the IGF-I pattern.

169

Figure 53. Additive genetic variance in ln mortality in Drosophila melanogaster. Top panel: The figure here is modified from a reanalysis by Shaw et al. (1999: 559, Figure 1 therein) of the data from Charlesworth and Hughes (1996). The general increase in additive genetic variance is significant whereas the decline exhibited at the ends of the trajectories are not significant due to lack of power (Shaw et al., 1999). Bottom panel: The figure here is modified from Shaw et al. (1999: 560, Figure 3 therein) as well. Additive genetic variance in ln mortality for females (top curve) and males (bottom curve). All parts of the trajectories are significant.

170

In response to the lack of agreement between the population genetic theories of senescence and data, a number of investigators have called for revisions in the way the evolution of senescence is conceptualized and modeled (Promislow et al., 1996; Pletcher et al., 1998; Promislow and Tatar, 1998; Mangel, 2001; Promislow and Pletcher, 2002). Indeed, Promislow and Pletcher (2002) argued that the over-reliance on classical models of senescence has been a hindrance to advances in understanding the evolution of senescence. In line with this appeal, the present study supports the following two suggestions: 1) Evolutionary models of senescence need to be conceptualized so that they are in closer agreement with the underlying physiological processes of senescence. This may be achieved by conceptualizing a model that unites the DS theory with the neuroendocrine cascades theory. In this regard, an attractive approach that links life history evolution with physiological processes is provided by the reliability models reviewed earlier. 2) The statistical genetic approach advocated herein allows one to draw inferences and biological interpretations that would be useful in the conceptualization of such a united model.

171

Chapter 10 Conclusions This last chapter is divided into three sections, caveats, prospectus and conclusions. The first section emphasizes the limitations of this study. The second section presents an extension of the genotype × age interaction model to accommodate the theories of oxidative stress and mitochondrial dysfunction in senescence. The last section summarizes the conclusions of this dissertation research. Caveats As with all studies, there are limitations that need to be recognized. A main limitation of the present study is related to the way in which the “environment” is accounted for. Strictly speaking, variance components models do not account for the environment but rather relegate all factors that cannot be accounted for in genetic terms to the environment. This at once confounds numerous aspects of the environment that are ostensibly important. Moreover, the random environmental term may even include non-additive genetic factors, such as dominance and/or epistasis. Therefore, this study must be regarded as being rather preliminary. Indeed, there is much more that needs to be done in terms of adding to the genotype × age interaction model. Prospectus The genotype × age interaction model is easily extended to the existing framework for a statistical genetic analysis of mitochondrial effects. Maternal effects sensu stricto as opposed to maternally-inherited cytoplasmic factors (i.e., mitochondria in animals and mitochondria and chloroplasts in plants) can be easily distinguished within the framework of the multivariate mixed linear model (Beavis et al., 1987; Schork and

172

Guo, 1993; Zhu and Weir, 1994, 1997; Czerwinski et al., 2001; Kent et al., in press; Lease et al., in press). It should be recalled that mitochondrial effects exhibit strong age dependencies (Shoffner and Wallace, 1992; Wallace, 1992a&b, 1995, 1999; Ozawa, 1995, 1997, 1998, 1999; Melov et al., 1999; Kokoszka et al., 2001; Shoffner, 2001; Wallace et al., 2001). In particular, reactive oxygen species (ROS) that are generated largely as a result of oxidative phosphorylation (OXPHOS), which takes place in mitochondria, contribute in a cumulative manner to the total cellular and intracellular damage incurred over the life span (Wallace, 1992a&b, 1995, 1999; Ames et al., 1993; Shigenaga et al., 1994; Ozawa, 1995, 1997, 1998, 1999; Sohal and Weindruch, 1996; Nagley and Wei, 1998; Wei, 1998; Wei et al., 1998; Ashok and Ali, 1999; Cortopassi and Wong, 1999; Esposito et al., 1999; Finkel and Holbrook, 2000; Van Remmen and Richardson, 2001; Shoffner, 2001; Wallace et al., 2001; Sastre et al., 2003). Perhaps the most important consequence of an age-related ROS load is the high rate of mutation in the mitochondrial DNA (mtDNA), which in turn is strongly associated with an age-related decline in OXPHOS capacity (Wallace, 1992a&b, 1995, 1999; Ames et al., 1993; Shigenaga et al., 1994; Lee et al., 1997; Lenaz, 1998; Nagley and Wei, 1998; Wei, 1998; Wei et al., 1998; DiMauro and Schon, 2001, 2003; Kokoszka et al., 2001; Shoffner, 2001; Lenaz et al., 2002; Pak et al., 2003). It has been pointed out that the age-related processes of increasing ROS and somatic mutation loads and of decreasing OXPHOS capacity are inherently stochastic within individuals (Wallace, 1999; Stadtman, 2002). These considerations lead to the prediction that, across individuals at the population level, the variance in mitochondrial effects is itself age-dependent. Therefore,

173

on strong biochemical and physiological grounds, the variance in mitochondrial effects is expected a priori to be heteroscedastic across the age continuum. The high mtDNA mutation rate gives rise to a more subtle age dependency, which has important effects on the correlation structure (of mitochondrial effects) inherent in a given population of relatives. This other type of age dependency is due to the phenomenon known as â€œreplicative segregationâ€?, which refers to the mutation-driven departure from homoplasmy (mitochondrial genome comprised of wild-type mtDNA) towards increasing heteroplasmy (mitochondrial genome comprised of mutant mtDNA) (Shoffner and Wallace, 1992; Wallace, 1992a&b, 1995, 1999; Lightowlers et al., 1997; Ozawa, 1997; DiMauro and Schon, 2001, 2003; Shoffner, 2001; Wallace et al., 2001). This process has been modeled as a genetic drift process and the model behavior seems to be consistent with data (Chinnery and Samuels, 1999; Chinnery et al., 2002; Elson et al., 2001). Because of this genetic drift type process, the correlation structure in mitochondrial effects is expected a priori to decay with increasing age differences between any two individuals belonging to the same maternal lineage. In terms of assumptions, the preliminary mitochondrial model assumes homoscedasticity in the mitochondrial variance and a stationary correlation structure at complete, positive correlation for all individuals belonging to the same maternal lineage. Both of these assumptions can be relaxed by modeling the mitochondrial variance and correlation in mitochondrial effects as functions of age and age differences, respectively, in the same manner as under the genotype Ă— age interaction model. It will be interesting to fully develop and analyze these models in relation to the IGF-I axis. The IGF-I axis seems to be a universal regulator of senescence, as the axis

174

and its homologs have been studied in relation to senescence in yeast, nematodes, fruit flies, and mammals (Ghigo et al., 1996, 2000; Arvat et al., 1999, 2000; Guarente and Kenyon, 2000; Kenyon, 2000; Finch and Ruvkun, 2001; Gems and Partridge, 2001; Longo and Finch, 2002, 2003; Barbieri et al., 2003; Tatar et al., 2003; Browner et al., 2004). The IGF-I axis has been integrated with oxidative stress and mitochondrial dysfunction in relation to senescence. The most supported physiological model along the lines of combining oxidative stress, mitochondrial dysfunction, and neuroendocrine factors holds that the decline in IGF-I axis activity over the life span in turn decreases metabolic activity and hence oxidative stress and mitochondrial dysfunction (Carter et al., 2002a&b, Bartke et al., 2003; Brown-Borg, 2003; Brown-Borg and Harman, 2003; Hursting et al., 2003; Holzenberger, 2004). Work on several murine models has demonstrated that the decline in IGF-I axis activity in conjunction with caloric restriction is a significant determinant of life span extension (Shimokawa et al., 2002, 2003; Tirosh et al., 2003, 2004; Al-Regaiey et al., 2005; Miskin et al., 2005). In one of these murine models, interaction of IGF-I signalling pathways with mitochondrial function was thought to be important (Tirosh et al., 2003, 2004; Miskin et al., 2005). Conclusions In multicellular organisms, the IGF-I axis is central to processes that are fundamental to life, including development, growth, somatic maintenance and metabolism. The importance of the IGF-I axis holds for most of the duration of ontogeny, although its precise roles may vary dramatically over the lifespan. It was hypothesized that this dynamic endocrine system is reflected by a GEN and, hence, it would be an ideal system to study genotype Ă— age interaction in humans. Convincing

175

evidence of genotype × age interaction was presented. Specifically, it was found that in Mexican Americans in the San Antonio Family Heart Study, the additive genetic variance and genetic correlation functions change significantly with age for IGF-I. These findings were discussed in terms of their implications for the pathophysiology of the metabolic syndrome, the neuroendocrine cascade hypothesis of senescence, and evolutionary theories of senescence. The idea that the IGF-I axis is mobilized as an integral component of neuroendocrine cascades, that are age-specific in the case of puberty and age-associated in the case of obesity, is consistent with the treatment of the age continuum as a continuous index of the range of environments experienced by organisms (Hegele, 1992; Zerba and Sing, 1992; Zerba et al., 1996, 2000). The results of the present study justify the belief that the genotype × age interaction model can detect cases where the expression of genotype is highly dependent on environment, which, for the IGF-I axis, includes the prevailing hormonal milieu. The IGF-I axis is only one kind of complex trait. Similarly, age is only one kind of continuous environment that is likely to have important influences in the determination of complex traits. The genotype × age interaction model is a specific version of a general G × E interaction model that can be applied to virtually any other complex trait and any other continuous environment. Further, this model has been extended to the level of quantitative trait loci (QTLs) and can be used for QTL × environment interaction analyses (Almasy et al., 2001; Diego et al., 2003). Further still, these models fall under the general class of variance components models (Blangero et al., 2000, 2001), which also includes multivariate models that can assess pleiotropy at both the polygenic and linkage levels (Comuzzie et al., 1996; Almasy et al., 1997; Williams et al., 1999).

176

Indeed, there is some evidence of pleiotropy with respect to metabolic syndrome traits and the IGF-I axis (Comuzzie et al., 1996). There is an emerging picture of how complex physiological networks are modulated by dynamic modulation in their critical regulatory factors, such as the IGF-I axis (Finch and Ruvkun, 2001; Gems and Partridge, 2001; Longo and Finch, 2002). Therefore, it is justifiable to conclude that analyses using these models hold much promise for understanding the biology of dynamic, complex traits in general and of senescence in particular.

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Appendix A: A Geometric Proof of the G × E Interaction Theorem Mathematical concepts emanating from one branch of mathematics—if they are ultimately derived from some deep underlying set of truths—may often be translated into the language of another branch. Thus, theorems and their respective proofs may often be (and have often been) delivered in the different languages of mathematics. It turns out that this is the case in regard to the theorem on G × E interaction and its proof, for now a geometric representation of the theorem and a proof of its validity can be constructed. The following constitutes more than an independent proof of the theorem, however, for while offering an independent proof, it contributes a novel perspective on G × E interaction that may one day prove useful. The restatement and proof require certain definitions from vector space geometry. In particular, the proof will be confined to the vector space in the plane, ℜ 2 , or 2-space, but the underlying theory is easily generalized to ℜ n for n arbitrarily large because all that is required is that two vectors lie in the same plane. Hausner (1965) provides a useful reference for the vector space approach to geometry. Extensions of this perspective to multivariate statistics are presented in Dempster (1969) and Wickens (1995). Further, the insightful articles by Herr (1980) and Bryant (1984) inspired the current approach. Some axioms and definitions from Euclidean geometry are also needed but these can be mentioned as the exposition unfolds. The following can be found in linear algebra and calculus texts that cover vector ⎡u ⎤ ⎡v ⎤ space. Let there be two vectors u = ⎢ 1 ⎥ and v = ⎢ 1 ⎥ . The Euclidean norm (also ⎣u 2 ⎦ ⎣v 2 ⎦ known as the length, magnitude, or absolute value) of u , denoted by u , is given as:

178

1

1

⎛ ⎞2 u = (u′u ) = ⎜ ∑ u i2 ⎟ = u 21+ u 22 , ⎝ i =1 ⎠ 2

n

Eq. A1

and similarly for v . For now, let the inner product or dot product of u and v , denoted by u ⋅ v , be defined as: ⎡u ⎤⎡v ⎤ u ⋅ v = ⎢ 1 ⎥ ⎢ 1 ⎥ = (u1v1 + u 2 v 2 ) , ⎣u 2 ⎦ ⎣ v 2 ⎦

Eq. A2

and so it follows that: ⎡u ⎤⎡u ⎤ u ⋅ u = ⎢ 1 ⎥ ⎢ 1 ⎥ = (u12 + u 22 ) = ⎣u 2 ⎦ ⎣u 2 ⎦

[ (u

2 1

+ u 22 )

]

2

= u

2

.

Eq. A3

Both operations give rise to scalars. Suppose that u and v are centered at the origin in

ℜ 2 as depicted in Figure A1. Addition and subtraction hold as in the general case for vectors but note now the geometric meaning of vector subtraction, u − v , in Figure A1.

u − v

v

θ

u

∠ u,v = θ Figure A1. Schematic Representation of Vector Space in ℜ 2 . Ideally, the vector, u − v , should not be offset but its direction is more clearly seen this way. That is, the three vectors form a triangle. The angle between u and v , from u to v is θ .

179

An important law holds here, namely the Law of Cosines, which is given as: u−v

2

= u

2

2

+ v

−2 u

v cos θ .

Eq. A4

At this point, we may begin to see where all of this is leading for the Law of Cosines evokes a familiar form. Now, by the property of inner products implied by Equation A3, the vector given by u − v has the following inner product: u−v

= (u − v ) ⋅ (u − v ) .

2

Eq. A5

Further, inner products are distributive, associative, and commutative as in scalar algebra, and their like products are additive. Thus, Equation A5 may be rewritten to yield:

(u − v )⋅ (u − v ) = u ⋅ (u − v ) − v ⋅ (u − v ) = (u ⋅ u ) − (u ⋅ v ) − (v ⋅ u ) + (v ⋅ v ) = (u ⋅ u ) + (v ⋅ v ) − 2(u ⋅ v ) .

Eq. A6

Invoking the property implied by Equation A3 once more gives:

(u ⋅ u ) + ( v ⋅ v ) − 2(u ⋅ v ) = u

2

+ v

2

− 2u ⋅ v .

Eq. A7

Equating the right hand side of Equation A7 with the right hand side of Equation A4 in this order and discarding like terms gives a new definition for the inner product: u⋅v = u

v cos θ .

Eq. A8

On rearranging Equation A8, we have that:

cos θ =

u⋅v u

,

Eq. A9

v

which leads to the theorem that two vectors are orthogonal (perpendicular) if and only if: u⋅v = 0 ,

Eq. A10

which holds when u and v are at a right angle to each other. Angle θ is restricted to the range 0 ≤ θ ≤ π . While the cosine function is periodic on 2 π , it declines monotonically

180

from θ = 0 to θ = π , taking values from +1 to –1 so that we have for the range of cos θ , under the restriction, the closed interval [− 1,+1] , or − 1 ≤ cos θ ≤ +1 . The range of cos θ in vector space follows directly from the Cauchy-Schwarz Inequality (Halmos, 1958: 125-126; Horn and Johnson, 1985: 15), which holds that: u⋅v ≤ u

v ⇒

u⋅v u

v

≤1

Eq. A11

u⋅v u⋅v ⇒ ≤ 1 ⇒ −1 ≤ ≤ +1 . u v u v

The Cauchy-Schwarz Inequality plays fundamental roles in vector space geometry (Hausner, 1965), matrix analysis (Horn and Johnson, 1985), and in probability theory (Parzen, 1960). In terms of random variables Y and Z, the probabilistic version of the Cauchy-Schwarz Inequality is given as (Parzen, 1960: 363): σY,Z ≤ σ Y σ Z ⇒ ⇒

σ Y,Z σYσZ

σ Y,Z σYσZ

≤ 1 ⇒ −1 ≤

≤1

σ Y,Z σYσZ

Eq. A12 ≤ +1 ,

which may be immediately recognized as the correlation coefficient (Eq. 94 in the text). We are almost in a position to restate the theorem. It is a common practice in multivariate statistics to express random variables or statistical parameters as vectors endowed with the properties of such in vector space (Dempster, 1969; Herr, 1980; Bryant, 1984; Wickens, 1995). Following this tradition and on comparing Equations A11 and A12 element by element, let now Euclidean norms u and v be understood as metrics in vector space of σ g1 and σ g 2 , respectively. As immediate consequences, we find that the squared Euclidean norms u

2

, v

2

, and u − v

2

become metrics of

181

σ g21 , σ g2 2 , and σ g2Δ , respectively. As the analog of ρ G in vector space, we have cos θ ,

where θ is restricted to the closed interval [0, π ] . In the language of vector space, G × E interaction holds for u − v

2

≠ 0 . The theorem on G × E interaction may now be

restated. There is no G × E interaction, i.e., u − v

2

= 0 , if and only if u

2

= v

2

and cos θ = 1 . Similar to the algebraic proof in the text, the trivial cases corresponding to

u

2

= v

2

= 0 will not be considered below. But again, there is little loss in

generality with these concessions. It can now be seen that the fundamental equation for G × E interaction arises from the Law of Cosines, which may now be rewritten to yield:

u−v

2

⎧ u 2 + v 2 −2 u ⎪ =⎨ ⎪ 2 ⎩2 u (1 − cos θ )

v cos θ ; ∀ u

2

;∀ u

2

≠ v

2

= v

2

.

Eq. A13 .

Further still, by specifying for cos θ the values 1, 0, and –1 (corresponding to 0o , 90o ,

and 180o , respectively), we recover the six cases under the algebraic approach in Chapter 6 in the text, but this time under the assumptions that u Assuming u

u−v

2

= u

2

2

≠ v

+ v

2

2

2

≠ v

−2 u

2

and u

2

= v

2

.

, Equation A13 gives:

−2 u

⎧ u ⎪ ⎪ ⎪ v cos θ = ⎨ u ⎪ ⎪ ⎪ u ⎩

2

+ v

2

2

+ v

2

2

+ v

2

v ; ∀ cos θ = 1. ; ∀ cos θ = 0.

+2 u

v ; ∀ cos θ = −1.

Eq. A14

182

Assuming u

2

= v

u−v

2

2

, Equation A13 gives: ⎧0 ⎪ ⎪ ⎪ 2 = 2 u (1 − cos θ ) = ⎨2 u ⎪ ⎪ ⎪4 u ⎩

; ∀ cos θ = 1 . 2

; ∀ cos θ = 0 .

2

; ∀ cos θ = −1 .

Eq. A15

These cases will be taken in turn as before, but this time with a focus on the geometry of the situation. In fact, the proof requires six figures to treat each case (Figs. A2-A7). These are treated from top to bottom for Equation A14 and then for Equation A15. The geometric systems depicted in Figure A2 on the left hand side are known as degenerate triangles (Pedoe, 1970). The axioms and derived theorems of Euclidean geometry are satisfied, the main theorem for the present case being: ∠ u, v + ∠ u, (u − v ) + ∠ v, (u − v ) = 180o ,

Eq. A16

where the theorem is tailored to the present circumstances. For Figure A2, the angles are: ∠ u, v = 0o , ∠ u, u − v = 180o , and ∠ u, u − v = 0o . Clearly, these sum to 180o . The

important point here is that the vector, (u − v ) , is nonzero and so u − v

2

is nonzero by

definition (Equations A1 and A3). For the second case, Figure A3 amounts to an illustration of the fact that the Pythagorean theorem, namely: u−v

2

= u

2

+ v

2

,

Eq. A17

is merely a special case of the Law of Cosines. It is the Law of Cosines for θ = 90o ; that is, for right triangles. As regards the theorem of G × E interaction, u − v

2

is clearly

nonzero for the squared magnitudes of u and v are nonzero.

183

2

2

u ≠ v

•

•

∠ u, v

θ = 0 o ; cos θ = 1

•

∠ u,u − v

=

∠ u, v

•

v u

∠ u,u − v

•

u−v

•

u−v

=

∠ u,u − v

v

∠ u,u − v

u

Figure A2. Geometry of Heteroscedasticity I: The Degenerate Triangle in Vector Space. The geometric systems on the left hand side arise from vector subtraction under the stated conditions.

θ = 90 o ; cos θ = 0

u−v

v

u−v v u

u 2

u < v

2

2

u > v

2

Figure A3. Geometry of Heteroscedasticity II: The Law of Pythagoras in Vector Space.

184

One of the more surprising results herein arises from the geometric system presented in Figure A4 (cf. the corresponding homoscedastic system). It turns out that all three vectors constitute the side opposite θ = 180o , which is the vertex of u and v (the other two sides are given by zero vectors; see below). This result represents the maximum squared magnitude that the vector (u − v ) can attain. Therefore, u − v

2

≠ 0.

The geometric system depicted in Figure A5 is the crucial case under the theorem of G × E interaction, for here arises the geometric lower limit on u − v

2

. The system

is not a triangle in vector space. In fact, the system degenerates even further to a 0 angle and a point represented by the zero vector in vector space. That is, (u − v ) does exist but it is the zero vector. For when v = u , the vector (u − v ) is:

2

u ≠ v

2

θ = 180o ; cos θ = −1 u−v

•

∠ v,u − v

• ∠ u, v

•

=

∠ u,u − v v

u

Figure A4. Geometry of Heteroscedasticity III: The Degenerate Triangle in Vector Space. The vectors u and v are separated by a line to indicate their relative magnitudes.

185

⎡ u ⎤ ⎡ u ⎤ ⎡0 ⎤ u − u = ⎢ 1 ⎥ − ⎢ 1 ⎥ = ⎢ ⎥ = 0 2×1 . ⎣ u 2 ⎦ ⎣ u 2 ⎦ ⎣0 ⎦

Eq. A18

Further, the zero vector has a defined magnitude, given by: 0 2×1 = 0 2 + 0 2 = 0 = 0 .

Eq. A19

Moreover, the zero vector extends in all directions in vector space (Hausner, 1960). It is a proper vector. As regards the theorem of G × E interaction, u − v

2

u = v

2

2

= 0 in this case.

θ = 0 o ; cos θ = 1

u−v

• =

•

v

u Figure A5. Geometry of Homoscedasticity I: The Zero Vector in Vector Space.

The next two cases depicted in Figures A6 and A7 need little comment since similar arguments to the ones given for their corresponding cases under heteroscedasticity apply just as well under homoscedasticity. By those arguments, the cases depicted in Figures A6 and A7 give rise to u − v

2

≠ 0.

186

2

u = v

2

θ = 90 o ; cos θ = 0

u−v v

u

Figure A6. Geometry of Homoscedasticity II: The Law of Pythagoras in Vector Space.

2

u = v

2

θ = 180o ; cos θ = −1

u−v

•

•

• = v

u

Figure A7. Geometry of Homoscedasticity III: The Degenerate Triangle in Vector Space.

187

Appendix B: Relation to the Gaussian and Ornstein-Uhlenbeck Stochastic Processes The derivation of genotype × age interaction model has its counterpart in the theory of Gaussian stationary stochastic processes. It will be instructive to briefly discuss such processes in relation to the genotype × age interaction model to draw out common themes. Gaussian stationary stochastic processes can be shown to be covariance stationary (sensu Parzen, 1962) in translation along some environmental continuum of interest. Moreover, according to Karlin and Taylor (1975: 446), “For covariance stationary processes, the crux of the matter . . . is whether or not the covariance function converges to zero as the time difference [age difference in the genotype × age interaction model] . . . becomes large, and if it does so vanish, the rate at which this convergence takes place has relevance.” By a limit equation presented below, the overall covariance function vanishes exponentially for large age differences under the genotype × age interaction model. For phenotypes, the Gaussian nature of the stochastic process comes from the assumption that a given phenotype at any point along the age continuum follows a Gaussian or multivariate normal distribution. The process of phenotype determination along the age continuum may therefore be conceptualized as a Gaussian covariance stationary stochastic process. In fact, several investigators have developed the stochastic process approach as a model of phenotype determination independently of the concept of G × E interaction (Kirkpatrick and Heckman, 1989; Kirkpatrick and Lofsvold, 1989; Kirkpatrick et al., 1990, 1994; Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002).

188

The Gaussian covariance stationary stochastic process model is a surprisingly straightforward extension of Equations 83-84. Let y( t ) denote a phenotype function in time, t , where t ∈ T ⊂ ℜ . Analogous to the classical case, the linear model for y( t ) is: y ( t ) = μ ( t ) + g ( t ) + e( t ) ,

Eq. B1

where μ ( t ) is the mean function, g( t ) and e( t ) are independent Gaussian processes, and the following expectations hold: E[y( t )] = μ( t ) , and E[g( t )] = E[e( t )] = 0 . By the assumptions of independent Gaussian processes and of additivity in the random effects 2

functions, the phenotypic covariance function, denoted by σ y ( t ) , can be decomposed as follows (Kirkpatrick and Heckman, 1989; Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002): 2

2

2

σ y(t ) = σ g (t ) + σ e(t ) , 2

Eq. B2

2

where σ g ( t ) and σ e ( t ) are the genetic and environmental covariance functions, respectively. From general treatments of stochastic processes (Parzen, 1962; Karlin and Taylor, 1975), covariance stationarity requires that: σ 2υ( 0 ) = ... = σ 2υ(s −1) = σ 2υ(s ) = σ 2υ( t ) ; ∀ υ = g, e ; s, t ∈ T ⊂ ℜ ,

Eq. B3

which means that the variance is stationary in translation along the time axis. It is also required that the correlation function is a function in absolute time differences (Parzen, 1962; Karlin and Taylor, 1975). Doob (1942) had pointed this out for the OrnsteinUhlenbeck stochastic process, which is a special case of Gaussian stochastic processes and is the inspiration of Equation 108 (see below). For the process of phenotype determination along the time (or age) continuum, we can restrict the correlation requirement to the genetic correlation function, denoted by ρg ( t ) ( s − t ) , by the assumption

189

that the genetic and environmental effects are independent Gaussian processes. Therefore, the phenotypic covariance function may be written as: σ y ( t ) = ρ g ( t ) ( s − t )σ g ( t ) + σ e ( t ) . 2

2

2

Eq. B4

Significantly, the variance stationarity requirement can be relaxed by modeling variance heterogeneity with any suitable parametric model as long as it maintains positivity (Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002). Thus, the parameterizations in Equations 107 and 108 for the variance and correlation functions, respectively, are both acceptable even under the theory of Gaussian stationary stochastic processes. It should be emphasized that the idea of environmental insensitivity or invariance under translation is taken as the null process in both the stochastic process and G × E interaction concepts. That is, no G × E interaction or insensitivity to environmental change is equivalent to stationarity or invariance under translation and both concepts are defined with respect to a specific environmental continuum. The Ornstein-Uhlenbeck (O-U) stochastic process is clearly fundamental to the current formulation of the G × E interaction model for this is where the genetic correlation function comes from. The question of the origin of the correlation function of Equation 112 will now be addressed. The ensuing is in no way an exhaustive account or a rigorous derivation of the O-U stochastic process. Lange (1986) provides a rigorous development of the theory in relation to statistical genetics. General treatments of the OU stochastic process in the light of modern stochastic calculus can be found in Karatzas and Shreve (1991), Durrett (1996), and Krylov (2002). The standard derivation of the O-U stochastic process starts with Langevin’s Equation (cited in Chandrasekhar, 1943; Karatz and Shreve, 1991; see also Doob, 1942;

190

Nelson, 2001). Incidentally, Langevin’s Equation is thought to mark the origin of the theory of stochastic differential equations (Karatz and Shreve, 1991; Nelson, 2001). We will start with the original treatment by Uhlenbeck and Ornstein (1930) and take it to the point where the formal integration machinery developed by Doob (1942) and others applies. For clarity of exposition, we will pretend for the moment that the context is known and that all terms have been defined. After the crucial manipulations have been carried out, we will then describe the context and terms. Also, the Leibniz and prime notations will be used simultaneously, as this will be advantageous. The following is known as Langevin’s Equation: du (t ) dt

+ λ u (t ) = A (t ) .

Eq. B5

Langevin’s Equation is a first-order, linear differential equation. As such, it will fall to the integration factor method. That is, we seek an integration factor, h (t ) say. Recall the Product Rule of differential calculus for the product of two functions, u (t ) and h (t ) :

[u (t ) ⋅ h (t )] ′ = u′(t ) ⋅ h (t ) + u (t ) ⋅ h′(t ) .

Eq. B6

Multiplying h (t ) across Equation B5, temporarily suppressing the function notation, and switching to the Leibniz notation, we have: u ′h + uλh = Ah .

Eq. B7

Now we have two equations and one unknown so that we may solve for h (t ) . To do so, we equate the right hand side of Equation B6 to the left hand side of Equation B7. Then, on dropping redundant terms, we will have: h ′ = λh ⇒

h′ h

=λ,

Eq. B8

191

which is a first-order, separable differential equation and is immediately integrable. Recalling that we have functions in t, integration with respect to t gives: h′

∫ h = ∫ λdt ⇒ ln h = λt ⇒ h = e

λt

.

Eq. B9

Having solved for h (t ) , we rewrite Equation B7 accordingly to give:

u′e λt + uλe λt = Aeλt .

Eq. B10

But the left hand side of Equation B10 is merely the derivative of ⎡⎢u ⋅ e λt ⎤⎥ , by the ⎦ ⎣ Product Rule. Therefore, Equation B10 may be rewritten as:

[u ⋅ e ] ′ = Ae λt

λt

,

Eq. B11

which is also immediately integrable. Integrating across the interval from t = 0 to t , and no longer suppressing the function notation in t , we find: ′ ∫ [u(s )⋅ e ] = ∫ A(s )⋅ e t

t

λs

0

0

λs

ds ⇒ Eq. B12

t

u (t ) ⋅ e λt − u 0 = ∫ A(s ) ⋅ e λs ds , 0

which gives the formal solution as: t

u (t ) = u 0 ⋅ e −λt + e −λt ∫ A(s ) ⋅ e λs ds . 0

Eq. B13

We are now at the point where the details can be filled in. Langevin’s Equation is an ingenious variant of Newton’s Second Law (after dividing by the mass): F = ma , where F is force, m is mass, and a is acceleration. Rearranging gives:

( ) = −λu(t )+ A(t ).

du t dt

192

()

()

Now, u t gives the velocity of a particle, the term −λu t gives a deterministic frictional

()

()

()

effect on u t and the term A t gives the residual effects on u t , which are assumed to be stochastic (Doob, 1942, 1953; Nelson, 2001). Hence, Langevin’s Equation was a

()

rather bold hypothesis for it claimed that the rate of change in u t ,

( ) , is given by a

du t dt

linear combination of deterministic and stochastic effects. Doob (1942) noted that Langevin’s Equation caused much controversy. Indeed, the equation must have railed against the dogma of deterministic theories in theoretical physics (on the reign of deterministic philosophy in theoretical physics up until the rise of quantum mechanics in the late 1920s see Popper, 1977: ch. 6). Now, the solution given by Equation B13 is a formal solution. However, the integral involving the stochastic term did not at the time of Uhlenbeck and Ornstein (1930) admit a straightforward solution. The reason for that state of affairs is obvious in retrospect. The formal apparatus of measure theory for probabilistic phenomena had not yet been laid down. Indeed, the probability calculus would not receive its fundamental enunciations until the 1930s (see Doob, 1941, 1953: Supplement and Appendix, 1996). Moreover, not until after this period do we observe the formalization of the theory of stochastic integrals by the pioneering work of Doob (1942, 1953) and K. Itô (Itô’s early works in the 1940s were published in Japanese journals but they are considered fundamental in stochastic calculus; see Karatzas and Shreve, 1991; Durrett, 1996; Brzeźniak and Zastawniak, 1999). The formal apparatus of the stochastic calculus is quite intricate and lies beyond the scope of this section (for general treatments, see Karatzas and Shreve, 1991; Durrett,

193

1996; Brzeźniak and Zastawniak, 1999). At this point, we wish only to report the fundamental results of the stochastic calculus applied to Equation B13. It is sufficient to point out that it has been proven that the O-U stochastic process is multivariate Gaussian across the time continuum (using the methods of statistical physics, see Uhlenbeck and Ornstein, 1930; Chandrasekar, 1943; using the methods of stochastic calculus, see Doob, 1942; Durrett, 1996). Further, Doob (1942) showed that for a standardized O-U stochastic process, we have for the mean: E[u (t )] = μ(t ) = μ ,

Eq. B14

and, for the covariance function: E{[u (t ) − μ ] [u (s ) − μ ]} = σ 2 exp(− λ t − s ) ; ∀ s, t ∈ ℜ ,

Eq. B15

where the exponential term is the correlation function. Limits at infinity and zero, for fixed λ ≠ 0 , provide boundaries on the correlation function as follows: lim exp[− λ t − s ] = 0 ;

Eq. B16

lim exp[− λ t − s ] = 1 .

Eq. B17

t -s → ∞

t -s → 0

Equations B15-B17 have a simple interpretation: 1) The covariance function may ultimately be expressed as a function in increments in time (or whatever the continuum may be) and, by the limits imposed, the covariance function is 2) sufficiently stationary for small increments or approximates stationarity exponentially fast or 3) approaches 0 exponentially fast for large increments. So finally, where does the correlation function come from? Apparently, the correlation function comes from the integration factor method for the formal solution of

194

Langevin’s Equation. Further, on rearranging Equation B13, we see that the solution is obtained by computing the stochastic integral, as Chandrasekhar (1943) noted: t

u (t ) − u 0 ⋅ e −λt = e −λt ∫ A(s ) ⋅ e λs ds . 0

By certain assumptions under the theory of stochastic calculus, the right hand side can be rewritten as (for a similar form of this particular stochastic integral under rigorous definitions, see Karatzas and Shreve, 1991: 358; Krylov, 2002: 106): t

u (t ) − u 0 ⋅ e −λt = ∫ A(s ) ⋅ e −λt e λs ds 0

t

= ∫ A(s ) ⋅ e −λt e (−λ )(−s )ds 0

Eq. B18

t

= ∫ A(s ) ⋅ e −λ (t −s )ds . 0

One last point should be made before leaving this appendix. It should be reiterated that the O-U stochastic process is a special case of Gaussian covariance stationary stochastic processes. As such, it is immediately applicable to Gaussian phenomena manifest along a continuum as a model of their probabilistic behavior. In nature, environments will more often than not exhibit continuous rather than discrete variation. This is particularly true of the age continuum. Hence, any phenotype manifest along the age continuum, or any other continuous environment of interest, can be modeled using the approach discussed herein.

195

Appendix C: Derivation of the Elements in the Expected Fisher Information Matrix To compute Σ θˆ for the genotype × age interaction model, we need to first write the ln-likelihood function for the model. Taking the case of a single pedigree, the lnlikelihood function of the genotype × age interaction model is given as: ln L(β, α g , γ g , λ, α e , γ e y , X ) = −

1

[N ln(2π) + ln Σ + Δ′Σ Δ] . −1

2

Eq. C1

Let the parameter vector under the genotype × age interaction model be denoted by: ′ θ = [β, α g , γ g , λ, α e , γ e ] , where the carats have been dropped for easier notation. The partial derivatives of ln L(θ ) with respect to effects in β will not have changed under the genotype × age interaction model. Note that on taking the first partial derivative of ln L(θ ) with respect to any parameter θ in θ , the right hand side will always involve the

derivative of a constant, which is always 0, thus leaving only two terms to differentiate: −1 −1 ∂ ln Σ ∂Δ ′Σ Δ ⎤ 1 ⎡ ∂N ln(2π) ∂ ln Σ ∂Δ′Σ Δ ⎤ 1⎡ ∂ ln L(θ ) =− ⎢ + + + ⎥ = − ⎢0 + ⎥ 2 ⎣⎢ 2 ⎣⎢ ∂θ ∂θ ∂θ ∂θ ⎦⎥ ∂θ ∂θ ⎦⎥

Eq. C2 −1 1 ⎡ ∂ ln Σ ∂Δ′Σ Δ ⎤ =− ⎢ + ⎥ . 2 ⎣⎢ ∂θ ∂θ ⎦⎥

Recall that the genetic covariance function in Equation 112 is really just one function:

{exp[α

g

(

+ γ g p x − age

)] } {exp[α 1 2

g

(

+ γ g q z − age

)] } {exp(− λ p 1 2

x

− qz

⎡αg γg ⎤ ⎡αg γg ⎤ = exp ⎢ + + p x − age ⎥ exp ⎢ q z − age ⎥ exp(− λ p x − q z 2 2 ⎣ 2 ⎦ ⎣ 2 ⎦

(

)

γg ⎡ = exp ⎢α g + p x − age + q z − age 2 ⎣

[(

) (

(

)]⎤⎥ exp(− λ p ⎦

)

x

− qz

)}

)

) 196

γg ⎡ ⎤ p x + q z − 2age ⎥ exp(− λ p x − q z = exp ⎢α g + 2 ⎣ ⎦

(

)

)

γg ⎡ ⎤ p x + q z − 2age − λ p x − q z ⎥ . = exp ⎢α g + 2 ⎣ ⎦

(

)

Further, for x = z , the covariance function just gives the variance function:

γg γg ⎤ ⎤ ⎡ ⎡ exp ⎢α g + p x + p x − 2age − λ p x − p x ⎥ = exp ⎢α g + 2p x − 2age − λ 0 ⎥ 2 2 ⎦ ⎦ ⎣ ⎣

(

)

(

(

)

)

⎡ ⎤ γ g 2 p x − age = exp ⎢α g + − 0⎥ = exp α g + γ g p x − age . ⎢⎣ ⎥⎦ 2

[

It will simplify matters to put: c =

)]

(

(p + q i

j

− 2age

2

) and d = p − q i

j

. Let there be matrices

of ages, C n×n = {cij } , and of age differences, D n×n = {d ij }, i = 1, . . . , n ; j = 1, . . . , n , where:

(

) (

) (

⎧ 2p i − 2age 2 p i − age = = p i − age ; ∀ i = j ⎪ ⎪ 2 2 ⎪ c ij = ⎨ ⎪ ⎪ p i + q j − 2age ;∀ i ≠ j, ⎪⎩ 2

(

)

)

Eq. C3

and ⎧ pi − pi = 0 = 0 ; ∀ i = j ⎪⎪ d ij = ⎨ ⎪ ; ∀ i ≠ j. ⎪⎩ p i − q j

Eq. C4

In finding the partial derivatives, the Product Rule will be invoked at various stages. In general, the Product Rule applies to products of functions. For matrices, the Product Rule applies for a product of matrices of variables, since matrices can be viewed as

197

matrix-valued functions. This fact holds for standard or Hadamard matrix multiplication, even under trace operations because the trace operation is linear. Having made these remarks, it should be pointed out that Φ , C , and D are matrices of constants and so the Product Rule does not apply to products involving them. This can be seen by taking their derivatives, which will be 0 n×n (compare the case for the derivatives of scalar constants). Lastly, a fact that will prove useful in evaluating the second partial derivatives in the ensuing is that a second partial derivative is merely the partial derivative of a first partial derivative. To compute the partial derivative of ln L(θ ) with respect to α g , we write:

[

]

[

]

−1 ∂lnL(θ ) 1 ∂[ln Σ ] 1 ∂ Δ′Σ −1Δ 1 ∂ N ln(2π) + ln Σ + Δ′Σ Δ , =− =− − 2 ∂α g ∂α g ∂α g 2 ∂α g 2

Eq. C5

where the right hand side makes explicit the fact that we can differentiate the remainder term-by-term, which follows from the linearity of differential operators. Now,

−

∂Σ ⎞⎟ 1 ∂[ln Σ ] 1 ⎛ = − Tr ⎜ Σ −1 , ∂α g ⎟⎠ 2 ∂α g 2 ⎜⎝

Eq. C6

and, on recalling that Σ = 2Φ o A + B , we have: ∂Σ ∂[2Φ o A + B] ∂ (2Φ o A ) ∂B ∂A ∂A . = = + = 2Φ o + 0 = 2Φ o ∂α g ∂α g ∂α g ∂α g ∂α g ∂α g

Eq. C7

The partial derivative of a matrix with respect to a scalar parameter is the matrix of the partial derivatives of its elements with respect to the parameter, whereby differentiation is carried out element-by-element (Cullen, 1990: 265; Horn and Johnson, 1991: 490; Lange, 1997: 125). All of the elements in A are given by the covariance function: exp ⎡α g + γ g c ij − λd ij ⎤ , ⎢⎣ ⎥⎦

198

and so their partials with respect to α g are given by: ∂ exp ⎡α g + γ g cij − λd ij ⎤ ⎢⎣ ⎥⎦ = exp ⎡α g + γ g c ij − λd ij ⎤ . ⎢⎣ ⎥⎦ ∂α g

Eq. C8

In terms of a matrix A 2×2 say, we have: ⎤ ∂ exp ⎡α g + γ g c12 − λd12 ⎤ ⎥ ⎢⎣ ⎥⎦ ⎥ ∂α g ⎥ ⎥ ⎥ ⎡ ⎤ ∂ exp α g + γ g c 22 − λd 22 ⎥ ⎢⎣ ⎥⎦ ⎥ ⎥ ∂α g ⎥⎦

⎡ ⎡ ⎤ ⎢ ∂ exp ⎢α g + γ g c11 − λd11 ⎥ ⎣ ⎦ ⎢ ∂α g ⎢ ∂A ⎢ = ∂α g ⎢ ⎢ ∂ exp ⎡α + γ c − λd ⎤ g 21 21 ⎥⎦ ⎢⎣ g ⎢ ⎢ ∂α g ⎢⎣

Eq. C9 ⎡ ⎡ ⎤ ⎡ ⎤⎤ ⎢ exp ⎢α g + γ g c11 − λd11 ⎥ exp ⎢α g + γ g c12 − λd12 ⎥ ⎥ ⎣ ⎦ ⎣ ⎦⎥ ⎢ =⎢ ⎥ = A. ⎢ ⎡ ⎤ ⎡ ⎤⎥ ⎢exp ⎢⎣α g + γ g c 21 − λd 21 ⎥⎦ exp ⎢⎣α g + γ g c 22 − λd 22 ⎥⎦ ⎥ ⎦ ⎣ Thus, in this case and this case only, it happens that: 2Φ o

∂A = 2Φ o A , ∂α g

Eq. C10

and so

−

1 ∂[ln Σ ] 1 −1 −1 = − Tr Σ (2Φ o A ) = −Tr Σ (Φ o A ) . 2 ∂α g 2

[

]

[

]

Eq. C11

Now to the remainding term. We have: −

[

]

−1 −1 ⎤ 1 ∂ Δ′Σ Δ 1 ⎡ ∂Σ Δ⎥ , = − ⎢Δ ′ 2 ∂α g 2 ⎢⎣ ∂α g ⎥⎦

and, by the above evaluation of

Eq. C12

∂Σ , we find that: ∂α g

199

−1

∂Σ −1 ∂Σ −1 −1 −1 Σ = − Σ (2Φ o A )Σ , = −Σ ∂α g ∂α g

Eq. C13

thereby giving: −

[

−1

]

[ {

}]

1 ∂ Δ′Σ Δ 1 −1 −1 −1 −1 = − Δ ′ − Σ (2Φ o A )Σ Δ = Δ′Σ (Φ o A )Σ Δ . 2 ∂α g 2

Eq. C14

Combining results, we find that: ∂ ln L(θ ) −1 −1 −1 = −Tr Σ (Φ o A ) + Δ′Σ (Φ o A )Σ Δ . ∂α g

[

]

Eq. C15

For the other two variance components terms, we will need: ∂ exp ⎡α g + γ g cij − λd ij ⎤ ⎢⎣ ⎥⎦ = cij exp ⎡α g + γ g cij − λd ij ⎤ , ⎢⎣ ⎥⎦ ∂γ g

Eq. C16

∂ exp ⎡α g + γ g c ij − λd ij ⎤ ⎢⎣ ⎥⎦ = −d ij exp ⎡α g + γ g cij − λd ij ⎤ . ⎢⎣ ⎥⎦ ∂λ

Eq. C17

and

∂ exp ⎡α g + γ g cij − λd ij ⎤ ⎢⎣ ⎥⎦ are Notice that the differences in these cases in comparison to ∂α g given by c ij and − d ij , respectively. Indeed, we find for A 2×2 : ⎡ ∂ exp[α g + γ g c11 − λd 11 ] ∂ exp[α g + γ g c12 − λd 12 ] ⎤ ⎥ ⎢ ∂γ g ∂γ g ⎥ ⎢ ∂A ⎥ =⎢ ∂γ g ⎢ ∂ exp[α g + γ g c 21 − λd 21 ] ∂ exp[α g + γ g c 22 − λd 22 ]⎥ ⎥ ⎢ ⎥⎦ ⎢⎣ ∂γ g ∂γ g ⎡ c11 exp[α g + γ g c11 − λd 11 ] c12 exp[α g + γ g c12 − λd 12 ]⎤ ⎥ = Co A, =⎢ ⎥ ⎢ ⎢⎣c 21 exp[α g + γ g c 21 − λd 21 ] c 22 exp[α g + γ g c 22 − λd 22 ]⎥⎦

Eq. C18

200

and similarly, we find: ⎡ ∂ exp[α g + γ g c11 − λd 11 ] ∂ exp[α g + γ g c12 − λd 12 ] ⎤ ⎥ ⎢ ∂λ ∂λ ∂A ⎢ ⎥ =⎢ ⎥ ∂λ ⎢ ∂ exp[α g + γ g c 21 − λd 21 ] ∂ exp[α g + γ g c 22 − λd 22 ]⎥ ⎥ ⎢ ⎦ ⎣ ∂λ ∂λ ⎡ − d 11 exp[α g + γ g c11 − λd 11 ] − d 12 exp[α g + γ g c12 − λd 12 ]⎤ ⎥ =⎢ ⎢ ⎥ ⎣⎢− d 21 exp[α g + γ g c 21 − λd 21 ] − d 22 exp[α g + γ g c 22 − λd 22 ]⎦⎥

Eq. C19

⎡ d 11 exp[α g + γ g c11 − λd 11 ] d 12 exp[α g + γ g c12 − λd 12 ]⎤ ⎥ = −D o A . = (− 1) ⎢ ⎢ ⎥ ⎣⎢d 21 exp[α g + γ g c 21 − λd 21 ] d 22 exp[α g + γ g c 22 − λd 22 ]⎦⎥

That is, the same differences hold for differentiation of the matrix A . Therefore, the elements of the score vector involving the two other variance components parameters are: ∂ ln L(θ ) −1 −1 −1 = Δ′Σ (Φ o C o A )Σ Δ − Tr Σ (Φ o C o A ) , ∂γ g

Eq. C20

∂ ln L(θ ) −1 −1 −1 = Tr Σ (Φ o D o A ) − Δ′Σ (Φ o D o A )Σ Δ . ∂λ

Eq. C21

[

]

and

[

]

The partials with respect to the environmental parameters are similarly computed. For the first of these, we have:

[

]

∂ ln L(θ ) 1 ∂[ln Σ ] 1 ∂ Δ′Σ −1Δ =− − . ∂α e 2 ∂α e 2 ∂α e

Eq. C22

Starting with the first term, we find that: −

∂Σ ⎞ 1 ∂[ln Σ ] 1 ⎛ ⎟, = − Tr ⎜⎜ Σ −1 ∂α e ⎟⎠ 2 ∂α e 2 ⎝

Eq. C23

and that: 201

∂B ∂B ∂Σ ∂ 2Φ o A ∂B . = =0+ + = ∂α e ∂α e ∂α e ∂α e ∂α e

Eq. C24

For say B 2×2 , we have: ⎡ ⎢ ∂ exp ⎡⎢α e + γ e c11 ⎤⎥ ⎦ ⎣ ⎢ ∂B ⎢ ∂α e =⎢ ∂α e ⎢ 0 ⎢ ⎢⎣

⎤ ⎥ ⎥ 0 ⎥ ⎥ ∂ exp ⎡α e + γ e c 22 ⎤ ⎥ ⎢⎣ ⎥⎦ ⎥ ∂α e ⎥⎦

Eq. C25 ⎡ ⎤ 0 ⎢exp ⎡⎢α e + γ e c11 ⎤⎥ ⎥ ⎦ ⎢ ⎣ ⎥ =⎢ ⎥ =B, ⎢ ⎥ 0 exp ⎡α e + γ e c 22 ⎤ ⎥ ⎢ ⎥⎦ ⎢⎣ ⎣ ⎦

and so the first term reduces to: −

1 ∂[ln Σ ] 1 = − Tr (Σ −1B ) . 2 ∂α e 2

Eq. C26

The second term is given by: −

[

]

−1 −1 ⎤ 1 ∂ Δ′Σ Δ 1 ⎡ ∂Σ Δ⎥ , = − ⎢Δ ′ 2 ∂α e 2 ⎢⎣ ∂α e ⎥⎦

and, by the above evaluation of

Eq. C27

∂Σ , we also have: ∂α e

−1

∂Σ −1 ∂Σ −1 −1 −1 Σ = − Σ BΣ , = −Σ ∂α e ∂α e

Eq. C28

which gives: −

[

]

1 ∂ Δ′Σ −1Δ 1 1 = − Δ′(− Σ −1BΣ −1 )Δ = Δ′Σ −1BΣ −1Δ . 2 ∂α e 2 2

[

]

Eq. C29

202

Combining results yields: ∂ ln L(θ ) 1 = Δ′Σ −1BΣ −1Δ − Tr Σ −1B . ∂α e 2

[

It will come as no surprise to find that ⎡ ∂ exp[α e + γ e c11 ] ∂B ⎢ ∂γ e =⎢ ∂γ e ⎢ 0 ⎢⎣

(

)]

Eq. C30

∂B ∂A is similar to in final form. To wit: ∂γ e ∂γ g

⎤ ⎥ ⎥ ∂ exp[α e + γ e c 22 ]⎥ ⎥⎦ ∂γ e 0

Eq. C31 0 ⎤ ⎡c11 exp[α e + γ e c11 ] ⎥ = CoB. ⎢ = ⎥ ⎢ 0 c 22 exp[α e + γ e c 22 ]⎦ ⎣

Therefore, we have for the last element of the score vector: ∂ ln L(θ ) 1 −1 −1 −1 = Δ ′Σ (C o B )Σ Δ − Tr Σ (C o B ) . ∂γ e 2

[

{

}]

Eq. C32

The elements in FI can now be derived. The following fact will be crucial in all the derivations (Searle, 1982: 27; McCulloch and Searle, 2001: 309): Tr (z ) = z ; ∀ z ∈ ℜ ,

Eq. C33

which states that a scalar is equal to its own trace. Now consider the generic quadratic form Δ′ZΔ , where Δ = y − Xβ and Z is some matrix of dimensions n × n . As pointed out earlier, Δ′ZΔ is a scalar quadratic function. So the above fact can be used in the following theorem (Magnus and Neudecker, 1999: 247; cf. Lange, 1997: 127; McCulloch and Searle, 2001: 309):

Δ′ZΔ = Tr (Δ′ZΔ ) = Tr (ZΔΔ′) ; ∀ Z n×n ∈ ℜ

n×n

.

Eq. C34

203

It will be useful to note some facts regarding the expectation operator (Magnus and Neudecker, 1999: 244; cf. Lange, 1997: 127; McCulloch and Searle, 2001: 309):

E[z ] = z ; ∀ z ∈ ℜ ;

E[z ] = z ; ∀ z ∈ ℜ ;

E[Z] = Z ; ∀ Z n×n ∈ ℜ

n

n ×n

,

Eq. C35

which respectively state that the expectation of a scalar is the scalar itself, the expectation of vector of constants is the vector itself, and the expectation of a matrix of constants is the matrix itself. For say a matrix of constants and a vector-valued function, we have: E[Zz ] = Z(E[z ]) ; ∀ z ∈ ℜ ; Z n×n ∈ ℜ n

n ×n

,

Eq. C36

and E[Tr (ZΔΔ′)] = Tr[E(ZΔΔ ′)] = Tr [Z(E{ΔΔ′})] ; ∀ Δ ∈ ℜ ; Z n×n ∈ ℜ n

n×n

.

Eq. C37

Further, we also have:

′ ′ E[ΔΔ′] = E ⎡(y − Xβ )(y − Xβ ) ⎤ = E ⎡(y − E[y ])(y − E[y ]) ⎤ = Σ , ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦

Eq. C38

which is the standard definition of Σ (Magnus and Neudecker, 1999: 246). For the diagonal element in FI for parameter α g , we have: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂α g ∂α g ⎟ ⎜ ∂α g ⎠ ⎝ ⎝

[ {

⎡ ∂ ln L(θ ) ⎤ ⎥ ⎢ ⎣⎢ ∂α g ⎦⎥

⎞ ⎟ ⎟ ⎠

}

⎛ ∂ − Tr Σ −1 (Φ o A ) + Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ − ⎜ ∂α g ⎝

[ {

}

⎛ ∂ Tr Σ −1 (Φ o A ) − Δ ′Σ −1 (Φ o A )Σ −1Δ = E⎜ ⎜ ∂α g ⎝

{

} ⎤⎥ ⎞⎟ + E⎛⎜ ∂[− Δ′Σ

⎛ ⎡ ∂ Σ −1 (Φ o A ) = E⎜ Tr ⎢ ⎜ ⎢ ∂α g ⎝ ⎣

⎥⎦ ⎟⎠

⎜ ⎝

−1

] ⎞⎟ ⎟ ⎠ Eq. C39

] ⎞⎟ ⎟ ⎠

(Φ o A )Σ −1Δ] ⎞⎟

∂α g

⎟ ⎠

.

204

It will be convenient to evaluate the resultant terms separately. Taking the first term:

{

}⎞⎟⎤⎥ = Tr ⎡⎢ ∂{Σ

⎡ ⎛ ∂ Σ −1 (Φ o A ) E ⎢Tr ⎜ ∂α g ⎢⎣ ⎜⎝

⎟⎥ ⎠⎦

−1

⎢⎣

(Φ o A )}⎤ ⎥ ⎥⎦

∂α g

⎡ −1 ⎧⎪ ∂ (Φ o A ) ⎫⎪ ⎛ ∂Σ −1 ⎞ ⎤ ⎟(Φ o A )⎥ = Tr ⎢ Σ ⎨ ⎬ + ⎜⎜ ⎢⎣ ⎪⎩ ∂α g ⎪⎭ ⎝ ∂α g ⎟⎠ ⎥⎦ ⎡ −1 ⎧⎪ ⎤ ∂A ⎫⎪ −1 ∂Σ −1 Σ (Φ o A )⎥ = Tr ⎢ Σ ⎨Φ o ⎬−Σ ∂α g ⎪⎭ ∂α g ⎢⎣ ⎪⎩ ⎥⎦

[

−1

[

−1

= Tr Σ (Φ o A ) − Σ

−1

Eq. C40

(2Φ o A )Σ −1 (Φ o A )]

]

{

}

= Tr Σ (Φ o A ) − 2Tr ⎡ Σ (Φ o A ) ⎢⎣ −1

2

⎤. ⎥⎦

For the second term, we find:

[

⎛ ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ E⎜ ⎜ ∂α g ⎝

[

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎟ ⎠

⎜ ⎝

]

−1

(Φ o A )Σ −1 ] ∂α g

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ = E ⎢− Tr ⎜ Δ′ Δ ⎟ ⎥ = −Tr ⎜ E [ΔΔ′] ⎟ ⎜ ⎟⎥ ⎜ ⎟ ∂α g ∂α g ⎢⎣ ⎝ ⎠⎦ ⎝ ⎠

{

}

⎛ = −Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o A ) −1 ∂Σ ⎤ −1 Σ + Σ (Φ o A ) ⎢ ⎥Σ ∂α g ⎦⎥ ∂α g ⎣⎢

⎛ = −Tr ⎜ ⎜ ⎝

⎡ −1 ∂ (Φ o A ) ∂Σ −1 ∂Σ ⎤ −1 (Φ o A ) − Σ −1 (Φ o A )Σ −1 + ⎢Σ ⎥Σ Σ ∂α g ⎥⎦ ∂α g ∂α g ⎢⎣

⎛ = −Tr ⎜ ⎜ ⎝

⎡ −1 ⎛ ∂A ⎢ Σ ⎜⎜ Φ o ∂α g ⎢⎣ ⎝

⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

⎤ ⎞ Σ ⎟ − Σ −1 ∂ Σ −1 (Φ o A ) − Σ −1 (Φ o A )Σ −1 (2Φ o A )⎥ I ⎟ ∂α g ⎥⎦ ⎠

⎞ ⎟ ⎟ ⎠

Eq. C41

205

([

−1

(Φ o A ) − Σ −1 (2Φ o A )Σ −1 (Φ o A ) − 2Σ −1 (Φ o A )Σ −1 (Φ o A )] )

([

−1

(Φ o A ) − 2Σ −1 (Φ o A )Σ −1 (Φ o A ) − 2Σ −1 (Φ o A )Σ −1 (Φ o A )] )

= −Tr Σ = −Tr Σ

[

= −Tr Σ

−1

(Φ o A )]+ 4Tr ⎡⎢{Σ −1 (Φ o A )}

2

⎣

⎤ . ⎥⎦

Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ 2 ⎟ = Tr Σ −1 (Φ o A ) − 2Tr ⎡ Σ −1 (Φ o A ) ⎤ E⎜ − ⎢⎣ ⎥⎦ ⎜ ∂α g ∂α g ⎟ ⎝ ⎠ 2 −1 −1 − Tr Σ (Φ o A ) + 4Tr ⎡ Σ (Φ o A ) ⎤ ⎢⎣ ⎥⎦

[

]

{

[

]

{

= 2Tr ⎡ Σ ⎢⎣

−1

(Φ o A )}

}

{

2

}

Eq. C42

⎤ . ⎥⎦

For the diagonal element in FI for parameter γ g , we find: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂γ g ∂γ g ⎟ ⎜ ∂γ g ⎝ ⎠ ⎝

⎡ ∂ ln L(θ ) ⎤ ⎢ ⎥ ⎢⎣ ∂γ g ⎥⎦

[ {

⎞ ⎟ ⎟ ⎠

}

⎛ ∂ − Tr Σ −1 (Φ o C o A ) + Δ′Σ −1 (Φ o C o A )Σ −1Δ = E⎜ − ⎜ ∂γ g ⎝

[ {

] ⎞⎟

}] {

⎟ ⎠

} ⎞⎟

⎛ ∂ Tr Σ −1 (Φ o C o A ) ∂ − Δ′Σ −1 (Φ o C o A )Σ −1Δ + = E⎜ ⎜ ∂γ g ∂γ g ⎝

{

} ⎤⎥ ⎞⎟ + E⎛⎜ ∂{− Δ′Σ

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) = E⎜ Tr ⎢ ⎜ ⎢ ∂γ g ⎝ ⎣

⎥⎦ ⎟⎠

Eq. C43

−1

⎜ ⎝

⎟ ⎠

(Φ o C o A )Σ −1Δ} ⎞⎟ ⎟ ⎠

∂γ g

.

We find for the first term:

{

} ⎞⎟ ⎤⎥ = Tr⎛⎜ ∂{Σ

⎡ ⎛ ∂ Σ −1 (Φ o C o A ) E ⎢Tr ⎜ ∂γ g ⎢⎣ ⎜⎝

⎟⎥ ⎠⎦

⎜ ⎝

−1

(Φ o C o A )} ⎞⎟ ∂γ g

⎟ ⎠

Eq. C44

206

⎡ −1 ∂ (Φ o C o A ) ∂Σ −1 ⎤ (Φ o C o A )⎥ = Tr ⎢ Σ + ∂γ g ∂γ g ⎣⎢ ⎦⎥ ⎡ −1 ⎛ ⎤ ∂A ⎞⎟ −1 ∂Σ −1 −Σ Σ (Φ o C o A )⎥ = Tr ⎢ Σ ⎜ Φ o C o ⎜ ∂γ g ∂γ g ⎟⎠ ⎣⎢ ⎝ ⎦⎥

[

−1

(Φ o C o C o A ) − Σ −1 (2Φ o C o A )Σ −1 (Φ o C o A )]

[

−1

(Φ o C o C o A )] − 2Tr ⎡⎢{Σ −1 (Φ o C o A )}

= Tr Σ = Tr Σ

⎣

2

⎤ . ⎥⎦

For the second term, we have:

[

⎛ ∂ − Δ′Σ −1 (Φ o C o A )Σ −1Δ E⎜ ⎜ ∂γ g ⎝

[

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎜ ⎝

⎟ ⎠

]

−1

(Φ o C o A )Σ −1 ] ∂γ g

⎞ Δ⎟ ⎟ ⎠

[

]

⎡ ⎞ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎟ ⎜ E [ΔΔ′] ⎟ Δ ⎥ = −Tr ⎜ = E ⎢− Tr Δ′ ⎟⎥ ⎟ ⎜ ⎜ ∂γ g ∂γ g ⎢⎣ ⎠⎦ ⎠ ⎝ ⎝

{

}

⎛ = −Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o C o A ) −1 ∂Σ ⎤ −1 ( ) + o o Σ Σ Φ C A ⎢ ⎥Σ ∂γ g ∂γ g ⎥⎦ ⎢⎣

⎛ ⎜ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎤ ⎡ −1 ∂ (Φ o C o A ) ∂Σ −1 (Φ o C o A )K⎥ + ⎢Σ ∂γ g ∂γ g ⎥ Σ −1 Σ ⎢ ⎥ ⎢ −1 −1 ∂Σ ⎥ ⎢ K − Σ (Φ o C o A )Σ ∂γ g ⎥⎦ ⎢⎣

⎛ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎡ −1 ⎛ ⎤ ∂A ⎞⎟ −1 ∂Σ −1 −Σ Σ (Φ o C o A )K⎥ ⎢ Σ ⎜⎜ Φ o C o ∂γ g ∂γ g ⎟⎠ ⎢ ⎝ ⎥I ⎢ ⎥ −1 −1 K − Σ (Φ o C o A )Σ (2Φ o C o A )⎥⎦ ⎢⎣

⎛ = −Tr ⎜ ⎜ ⎝

⎡ Σ −1 (Φ o C o C o A ) − Σ −1 (2Φ o C o A )Σ −1 (Φ o C o A )K ⎤ ⎢ ⎥ −1 −1 K − 2Σ (Φ o C o A )Σ (Φ o C o A )⎥⎦ ⎢⎣

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

Eq. C45

⎞ ⎟ ⎟ ⎠ 207

{

}

2 −1 −1 = −Tr ⎛⎜ ⎡ Σ (Φ o C o C o A ) − 4 Σ (Φ o C o A ) ⎤ ⎞⎟ ⎥⎦ ⎠ ⎝ ⎢⎣

[

= −Tr Σ

−1

(Φ o C o C o A )]+ 4Tr ⎡⎢{Σ −1 (Φ o C o A )} ⎣

2

⎤ . ⎥⎦

Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ 2 ⎟ = Tr Σ −1 (Φ o C o C o A ) − 2Tr ⎡ Σ −1 (Φ o C o A ) ⎤ E⎜ − ⎢⎣ ⎥⎦ ⎜ ∂γ g ∂γ g ⎟ ⎝ ⎠ 2 −1 −1 − Tr Σ (Φ o C o C o A ) + 4Tr ⎡ Σ (Φ o C o A ) ⎤ ⎢⎣ ⎥⎦

[

]

{

[

]

{

= 2Tr ⎡ Σ ⎢⎣

−1

(Φ o C o A )}

2

}

{

}

Eq. C46

⎤ . ⎥⎦

The diagonal element in FI for parameter λ is similarly computed as follows: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎛⎜ − ∂ ⎡ ∂ ln L(θ ) ⎤ E⎜ − ⎜ ∂λ ⎢ ∂λ ⎥ ⎜ ∂λ∂λ ⎟⎠ ⎣ ⎦ ⎝ ⎝

[

⎞ ⎟⎟ ⎠

}] ⎞⎟

{

⎛ ∂ − Δ′Σ −1 (Φ o D o A )Σ −1Δ + Tr Σ −1 (Φ o D o A ) = E⎜ − ⎜ ∂λ ⎝

[ {

}] [

⎛ ∂ Tr − Σ −1 (Φ o D o A ) ∂ Δ′Σ −1 (Φ o D o A )Σ −1Δ + = E⎜ ⎜ ∂λ ∂ λ ⎝

{

} ⎤⎥ ⎞⎟ + E⎛⎜ ∂{Δ′Σ

⎛ ⎡ ∂ Σ −1 (Φ o D o A ) = E⎜ − Tr ⎢ ⎜ ∂λ ⎢⎣ ⎝

⎥⎦ ⎟⎠

⎜ ⎝

−1

Eq. C47

⎟ ⎠

] ⎞⎟ ⎟ ⎠

(Φ o D o A )Σ −1Δ} ⎞⎟ ⎟ ⎠

∂λ

.

We find for the first term:

{

} ⎞⎟ ⎤⎥ = −Tr⎛⎜ ∂{Σ

⎡ ⎛ ∂ Σ −1 (Φ o D o A ) E ⎢− Tr ⎜ ⎜ ∂λ ⎝ ⎣⎢

⎟⎥ ⎠⎦

⎜ ⎝

−1

(Φ o D o A )} ⎞⎟ ∂λ

⎟ ⎠

Eq. C48

208

⎡ −1 ∂ (Φ o D o A ) ∂Σ −1 ⎤ (Φ o D o A )⎥ Tr = − ⎢Σ + ∂λ ∂λ ⎣⎢ ⎦⎥ ⎡ −1 ⎧ ⎤ ∂A ⎫ −1 ∂Σ −1 Σ (Φ o D o A )⎥ = −Tr ⎢ Σ ⎨Φ o D o ⎬−Σ ∂λ ∂λ ⎭ ⎣ ⎩ ⎦

[

= −Tr Σ

−1

(− Φ o D o D o A ) − Σ −1 (− 2Φ o D o A )Σ −1 (Φ o D o A )]

[

]

{

}

2 −1 −1 = Tr Σ (Φ o D o D o A ) − 2Tr ⎡ Σ (Φ o D o A ) ⎤ . ⎢⎣ ⎥⎦

For the second term, we have:

[

⎛ ∂ Δ′Σ −1 (Φ o D o A )Σ −1Δ E⎜ ⎜ ∂λ ⎝

[

] ⎞⎟ = E⎛⎜ Δ′ ∂[Σ ⎟ ⎠

⎜ ⎝

]

−1

(Φ o D o A )Σ −1 ] ∂λ

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ ∂ Σ −1 (Φ o D o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o D o A )Σ −1 ⎞ ⎜ ⎟ ′ Δ ⎥ = Tr ⎜ E [ΔΔ ′] ⎟ = E ⎢Tr Δ ⎜ ⎟⎥ ⎜ ⎟ ∂λ ∂λ ⎠⎦ ⎝ ⎠ ⎣⎢ ⎝

[

]

⎛ = Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o D o A ) −1 ∂Σ ⎤ −1 Σ + Σ (Φ o D o A ) ⎢ ⎥Σ ∂λ ⎥⎦ ∂λ ⎢⎣

⎛ ⎜ ⎜ = Tr ⎜ ⎜⎜ ⎝

⎤ ⎡ −1 ∂ (Φ o D o A ) ∂Σ −1 (Φ o D o A )K⎥ −1 + ⎢Σ ∂λ ∂λ ⎥Σ Σ ⎢ ⎢ −1 −1 ∂Σ ⎥ K − Σ (Φ o D o A )Σ ⎥ ⎢ ∂λ ⎦ ⎣

⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠

⎛ ⎜ = Tr ⎜ ⎜⎜ ⎝

⎡ −1 ⎛ ⎤ ∂A ⎞ −1 ∂Σ −1 Σ (Φ o D o A )K⎥ ⎟−Σ ⎢Σ ⎜ Φ o D o ∂λ ⎠ ∂λ ⎢ ⎝ ⎥I − 1 − 1 ⎢ K − Σ (Φ o D o A )Σ (− 2Φ o D o A ) ⎥⎦ ⎣

⎞ ⎟ ⎟ ⎟⎟ ⎠

⎛ = Tr ⎜ ⎜ ⎝

⎡ Σ −1 (− Φ o D o D o A ) − Σ −1 (− 2Φ o D o A )Σ −1 (Φ o D o A )K ⎤ ⎢ ⎥ −1 −1 K + 2 Σ (Φ o D o A )Σ (Φ o D o A )⎥⎦ ⎢⎣

⎞ ⎟ ⎟ ⎠

Eq. C49

⎞ ⎟ ⎟ ⎠

209

{

}

2 −1 −1 = Tr ⎛⎜ ⎡− Σ (Φ o D o D o A ) + 4 Σ (Φ o D o A ) ⎤ ⎞⎟ ⎥⎦ ⎠ ⎝ ⎢⎣

[

]

{

}

2 −1 −1 = −Tr Σ (Φ o D o D o A ) + 4Tr ⎡ Σ (Φ o D o A ) ⎤ . ⎢⎣ ⎥⎦

Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ 2 ⎟ = Tr Σ −1 (Φ o D o D o A ) − 2Tr ⎡ Σ −1 (Φ o D o A ) ⎤ E⎜ − ⎢⎣ ⎥⎦ ⎜ ∂λ∂λ ⎟⎠ ⎝ 2 −1 −1 − Tr Σ (Φ o D o D o A ) + 4Tr ⎡ Σ (Φ o D o A ) ⎤ ⎢⎣ ⎥⎦

[

]

{

[

]

{

= 2Tr ⎡ Σ ⎢⎣

−1

(Φ o D o A )}

2

}

{

}

Eq. C50

⎤ . ⎥⎦

We now look to the environmental component of the model. For the diagonal element in FI for parameter α e , we find: ⎛ ∂ 2 ln L(θ ) ⎞ ⎛ ⎤ ⎡ ⎟ = E⎜ − ∂ ⎢ ∂ ln L(θ ) ⎥ E⎜ − ⎜ ∂α ⎜ ∂α e ∂α e ⎟ ∂α e ⎦ e ⎣ ⎝ ⎠ ⎝

[ (

)]

[

−1 −1 ⎛ 1 ∂ Tr Σ −1B 1 ∂ − Δ′Σ BΣ Δ + = E⎜ ⎜2 2 ∂α e ∂α e ⎝

[ (

)

⎛ 1 ∂ − Tr Σ −1B + Δ′Σ −1BΣ −1Δ ⎞ ⎟ = E⎜ − ⎟ ⎜ 2 ∂α e ⎠ ⎝

] ⎞⎟

] ⎞⎟ = E⎛⎜ 1 Tr ⎡⎢ ∂{Σ B}⎤⎥ ⎞⎟ + E⎛⎜ 1 ∂[− Δ′Σ ⎜2 ⎝

⎢⎣ ∂α e ⎥⎦ ⎟⎠

]

−1 BΣ Δ ⎞⎟ . ⎟ ∂α e ⎠

−1

⎟ ⎠

⎟ ⎠

⎜2 ⎝

−1

Eq. C51 For the first term, we find:

[

⎛ 1 ∂ Σ −1B E⎜ Tr ⎜2 ∂α e ⎝

=

] ⎞⎟ = 1 Tr ⎡⎢ ∂(Σ B)⎤⎥ = 1 Tr ⎡⎢Σ −1

⎟ ⎠

2

⎣⎢ ∂α e ⎦⎥

2

[

⎣⎢

−1

∂B ∂α e

+

⎤ B⎥ ∂α e ⎦⎥

∂Σ

]

−1

[

]

⎡ −1 2 1 1 1 −1 ∂Σ −1 ⎤ −1 −1 −1 −1 −1 Tr ⎢ Σ B − Σ Σ B ⎥ = Tr Σ B − Σ BΣ B = Tr Σ B − Tr ⎡ Σ B ⎤ . ⎥⎦ 2 2 ⎢⎣ 2 ⎢⎣ ∂α e ⎥⎦ 2

1

(

)

Eq. C52

210

For the second term, we have:

[

⎛ 1 ∂ − Δ ′Σ −1BΣ −1Δ E⎜ ⎜2 ∂α e ⎝

[

] ⎞⎟ = E⎛⎜ − 1 Δ′ ∂[Σ ⎟ ⎠

⎜ 2 ⎝

−1

BΣ ∂α e

]

−1

]Δ ⎞⎟ = E ⎡⎢− 1 Tr⎛⎜ Δ′ ∂[Σ ⎟ ⎠

[

⎣⎢ 2

⎜ ⎝

]

−1 −1 −1 ⎛ ⎡ ∂ Σ −1B −1 ⎤ ⎞ ∂ Σ 1 1 ⎛⎜ ∂ Σ BΣ −1 ⎜ = − Tr Σ +Σ B E[ΔΔ′] ⎟ = − Tr ⎢ ⎥Σ ⎟ ⎜ ⎢ ∂α e ∂ α ∂α e 2 2 ⎜⎝ ⎥⎦ e ⎠ ⎝⎣

−1 1 ⎛⎜ ⎡ −1 ∂B ∂Σ −1 −1 ∂Σ ⎤ −1 + = − Tr ⎢ Σ B − Σ BΣ ⎥Σ Σ ∂α e ⎦⎥ ∂α e ∂α e 2 ⎜⎝ ⎣⎢

1 ⎛ ⎡ −1 −1 ∂Σ −1 −1 −1 ⎤ = − Tr ⎜⎜ ⎢ Σ B − Σ Σ B − Σ BΣ B ⎥ I ∂α e 2 ⎝⎣ ⎦

[

{ }

]

−1

BΣ ∂α e

−1

]Δ ⎞⎟ ⎤⎥ ⎟⎥ ⎠⎦

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

(

)

2 2 1 1 −1 −1 −1 −1 = − Tr ⎛⎜ ⎡ Σ B − 2 Σ B ⎤ ⎞⎟ = − Tr Σ B + Tr ⎡ Σ B ⎤ . ⎥⎦ ⎠ ⎢⎣ ⎥⎦ 2 2 ⎝ ⎢⎣

Eq. C53 Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ 1 2 ⎟ = Tr Σ −1B − 1 Tr ⎡ Σ −1B ⎤ E⎜ − ⎢ ⎥⎦ ⎜ ∂α ∂α ⎟ 2 2 ⎣ e e ⎠ ⎝ 2 1 −1 −1 − Tr Σ B + Tr ⎡ Σ B ⎤ ⎢⎣ ⎥⎦ 2

[

]

(

[ ]

=

1

(

)

(

)

Eq. C54

)

2 −1 Tr ⎡ Σ B ⎤ . ⎥⎦ 2 ⎢⎣

For the diagonal element in FI for parameter γ e , we have: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎡ ⎤ ⎟ = E⎜ − ∂ ⎢ ∂ ln L(θ ) ⎥ E⎜ − ⎜ ∂γ ∂γ ⎟ ⎜ ∂γ ⎢ ∂γ e e ⎠ e ⎣ e ⎦⎥ ⎝ ⎝

⎞ ⎟ ⎟ ⎠

Eq. C55

211

[ {

}

⎛ 1 ∂ − Tr Σ −1 (C o B ) + Δ′Σ −1 (C o B )Σ −1Δ = E⎜ − ⎜ 2 ∂γ e ⎝

[ {

}]

[

] ⎞⎟ ⎟ ⎠

⎛ 1 ∂ Tr Σ −1 (C o B ) 1 ∂ − Δ′Σ −1 (C o B )Σ −1Δ = E⎜ + ⎜2 2 ∂ γ ∂γ e e ⎝

[

] ⎞⎟ + E⎛⎜ 1 ∂[− Δ′Σ

⎛ 1 ∂ Σ −1 (C o B ) = E⎜ Tr ⎜2 ∂γ e ⎝

⎜2 ⎝

⎟ ⎠

−1

] ⎞⎟ ⎟ ⎠

(C o B )Σ −1Δ] ⎞⎟ ⎟ ⎠

∂γ e

.

For the first term, we have:

[

] ⎞⎟ = 1 Tr ∂[Σ

⎛ 1 ∂ Σ −1 (C o B ) E⎜ Tr ⎜2 ∂γ e ⎝

⎟ ⎠

2

−1

(C o B )]

∂γ e

=

−1 ⎤ 1 ⎡ −1 ∂ (C o B ) ∂Σ (C o B )⎥ Tr ⎢ Σ + ∂γ e 2 ⎢⎣ ∂γ e ⎥⎦

=

⎤ ∂Σ −1 1 ⎡ −1 ⎛ ∂B ⎞ ⎟⎟ − Σ −1 Tr ⎢ Σ ⎜⎜ C o Σ (C o B )⎥ ∂γ e 2 ⎣ ⎝ ∂γ e ⎠ ⎦

=

1 −1 −1 −1 Tr Σ (C o C o B ) − Σ (C o B )Σ (C o B ) 2

=

2 1 1 −1 −1 Tr Σ (C o C o B ) − Tr ⎡ Σ (C o B ) ⎤ . ⎥⎦ 2 2 ⎢⎣

Eq. C56

[

]

[

]

{

}

For the second term, we find:

[

⎛ 1 ∂ − Δ′Σ −1 (C o B )Σ −1Δ E⎜ ⎜2 ∂γ e ⎝

[

] ⎞⎟ = E⎛⎜ − 1 Δ′ ∂[Σ ⎟ ⎠

⎜ 2 ⎝

]

−1

(C o B )Σ −1 ] ∂γ e

[

⎞ Δ⎟ ⎟ ⎠

]

Eq. C57

⎡ 1 ⎛ ∂ Σ −1 (C o B )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (C o B )Σ −1 ⎞ 1 = E ⎢− Tr ⎜ Δ′ Δ ⎟ ⎥ = − Tr ⎜ E[ΔΔ′] ⎟ ⎟⎥ ⎟ ∂γ e ∂γ e 2 ⎜⎝ ⎢⎣ 2 ⎜⎝ ⎠⎦ ⎠

212

[

]

−1 ⎛ ⎡ ∂ Σ −1 (C o B ) −1 ∂Σ ⎤ −1 ⎜ Σ + Σ (C o B ) = − Tr ⎢ ⎥Σ 2 ⎜⎝ ⎣⎢ ∂γ e ⎦⎥ ∂γ e

1

⎞ ⎟ ⎟ ⎠

−1 1 ⎛ ⎡ −1 ∂ (C o B ) ∂Σ −1 −1 ∂Σ ⎤ −1 ( C o B ) − Σ (C o B )Σ + = − Tr ⎜ ⎢ Σ ⎥Σ Σ ⎜ ∂γ e ⎥⎦ 2 ⎝ ⎢⎣ ∂γ e ∂γ e

⎞ ⎟ ⎟ ⎠

⎤ 1 ⎛ ⎡ −1 −1 ∂Σ −1 −1 −1 Σ (C o B ) − Σ (C o B )Σ (C o B )⎥ I = − Tr ⎜ ⎢ Σ (C o C o B ) − Σ 2 ⎜⎝ ⎣⎢ ∂γ e ⎦⎥

{

}

{

}

⎞ ⎟ ⎟ ⎠

2 1 −1 −1 = − Tr ⎛⎜ ⎡ Σ (C o C o B ) − 2 Σ (C o B ) ⎤ ⎞⎟ ⎥⎦ ⎠ 2 ⎝ ⎢⎣

[

]

2 1 −1 −1 = − Tr Σ (C o C o B ) + Tr ⎡ Σ (C o B ) ⎤ . ⎢⎣ ⎥⎦ 2

On summing terms, we find: ⎛ ∂ 2 ln L(θ ) ⎞ 1 2 ⎟ = Tr Σ −1 (C o C o B ) − 1 Tr ⎡ Σ −1 (C o B ) ⎤ E⎜ − ⎥⎦ ⎜ ∂γ e ∂γ e ⎟ 2 2 ⎢⎣ ⎝ ⎠ 2 1 −1 −1 − Tr Σ (C o C o B ) + Tr ⎡ Σ (C o B ) ⎤ ⎢⎣ ⎥⎦ 2

[

]

[

=

{

{

]

{

}

}

Eq. C58

}

2 1 ⎡ −1 Tr Σ (C o B ) ⎤ . ⎥⎦ 2 ⎢⎣

By the theorem on the identity of mixed partial derivatives regardless of differentiation order (Eq. 124), there are ten, unique, mixed partial derivatives involving the variance components parameters and, together with the above results, these will give the sampling variances and covariances in Ω 5×5 in Σ θˆ after inversion of FI . By symmetry, there will be four, three, two, and one unique, mixed partial derivatives with respect to the first partial derivatives evaluated with respect to say α g , γ g , λ , and α e , respectively. Incidentally, any permutation of four of the five variables of differentiation

213

could have been taken. This order is consistent, however, with the order adhered to thus far. All together, we will have 15 unique, second partial derivatives. ⎛ ∂ 2 ln L(θ ) ⎞ ⎟, Given the above order, the first of the mixed elements in FI is E⎜ − ⎜ ∂γ g ∂α g ⎟ ⎝ ⎠ which is computed as follows: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂γ g ∂α g ⎟ ⎜ ∂γ g ⎝ ⎠ ⎝

[ {

⎡ ∂ ln L(θ ) ⎤ ⎢ ⎥ ⎢⎣ ∂α g ⎥⎦

⎞ ⎟ ⎟ ⎠

}

⎛ ∂ − Tr Σ −1 (Φ o A ) + Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ − ⎜ ∂γ g ⎝

[ {

}

{

}

⎛ ∂ Tr Σ −1 (Φ o A ) − Δ ′Σ −1 (Φ o A )Σ −1Δ = E⎜ ⎜ ∂γ g ⎝ ⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ γ g ⎣ ⎦⎥ ⎝

] ⎞⎟ ⎟ ⎠ Eq. C59

] ⎞⎟ ⎟ ⎠

[

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ′Σ (Φ o A )Σ Δ ⎜ ⎟ ∂γ g ⎝ ⎠

] ⎞⎟ . ⎟ ⎠

Taking the first term, we have:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂γ g ⎥⎦ ⎝ ⎣

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o A ) ⎥ ⎟ ∂γ g ⎢⎣ ⎥⎦ ⎠

Eq. C60 ⎡ −1 ∂ (Φ o A ) ∂Σ −1 ⎤ (Φ o A )⎥ = Tr ⎢ Σ + ∂γ g ∂γ g ⎢⎣ ⎥⎦ ⎤ ⎡ −1 ⎛ ∂A ⎞⎟ −1 ∂Σ −1 Σ (Φ o A )⎥ −Σ = Tr ⎢ Σ ⎜ Φ o ∂γ g ∂γ g ⎟⎠ ⎥⎦ ⎢⎣ ⎜⎝

[

−1

(Φ o C o A ) − Σ −1 (2Φ o C o A )Σ −1 (Φ o A )]

[

−1

(Φ o C o A )] − 2Tr [Σ −1 (Φ o C o A )Σ −1 (Φ o A )] .

= Tr Σ = Tr Σ

214

The second term is evaluated as follows:

[

⎛ ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ E⎜ ⎜ ∂γ g ⎝

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎟ ⎠

−1

⎜ ⎝

[

]

(Φ o A )Σ −1 ] ∂γ g

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ ⎜ ⎟ ′ E[ΔΔ ′]⎟ Δ ⎥ = −Tr ⎜ = E ⎢− Tr Δ ⎜ ⎟⎥ ⎜ ⎟ ∂γ g ∂γ g ⎝ ⎠⎦ ⎝ ⎠ ⎣⎢ ⎛ = −Tr ⎜ ⎜ ⎝

{

}

−1 ⎡ ∂ Σ −1 (Φ o A ) −1 ∂Σ ⎤ ⎞⎟ −1 Σ + Σ (Φ o A ) ⎢ ⎥Σ ∂γ g ⎥⎦ ⎟⎠ ∂γ g ⎢⎣

⎛ ⎡ −1 ∂ (Φ o A ) ∂Σ −1 ⎞ −1 −1 ∂Σ ⎤ −1 ( Φ o A ) − Σ (Φ o A )Σ Σ Σ⎟ + = −Tr ⎜ ⎢ Σ ⎥ ⎜⎢ ⎟ ∂γ g ⎦⎥ ∂γ g ∂γ g ⎝⎣ ⎠ ⎛ = −Tr ⎜ ⎜ ⎝

⎤ ⎞ ⎡ −1 ⎛ ∂A ⎞⎟ −1 ∂Σ −1 −1 −1 ⎜ ( ) ( ) ( ) Σ Φ Σ Σ Φ A Σ Φ A Σ 2 Φ C A o o o o o − − ⎥ I⎟ ⎢ ⎜ ⎟ ∂γ g ∂γ g ⎠ ⎥⎦ ⎟⎠ ⎢⎣ ⎝

([

= −Tr Σ

[

= −Tr Σ

−1

−1

(Φ o C o A ) − Σ −1 (2Φ o C o A )Σ −1 (Φ o A ) − 2Σ −1 (Φ o A )Σ −1 (Φ o C o A )] )

(Φ o C o A )]+ 2Tr [Σ −1 (Φ o C o A )Σ −1 (Φ o A )]+ 2Tr [Σ −1 (Φ o A )Σ −1 (Φ o C o A )] . Eq. C61

On summing terms, we have: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1 (Φ o C o A ) − 2Tr Σ −1 (Φ o C o A )Σ −1 (Φ o A ) E⎜ − ⎜ ∂γ g ∂α g ⎟ ⎠ ⎝

[ − Tr [Σ

] [ (Φ o C o A )] + 2Tr [Σ (Φ o C o A )Σ K + 2Tr [Σ (Φ o A )Σ (Φ o C o A )] [

−1

= 2Tr Σ

−1

−1

−1

−1

−1

] (Φ o A )]K Eq. C62

(Φ o A )Σ −1 (Φ o C o A )] .

215

⎛ ∂ 2 ln L(θ ) ⎞ ⎟, In the stated order, the second of the mixed elements in FI is E⎜ − ⎜ ∂λ∂α g ⎟ ⎝ ⎠ which is computed as follows: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎡ ⎤ ⎟ = E⎜ − ∂ ⎢ ∂ ln L(θ ) ⎥ E⎜ − ⎜ ∂λ∂α g ⎟ ⎜ ∂λ ⎢ ∂α g ⎥ ⎣ ⎦ ⎝ ⎠ ⎝

[ {

⎞ ⎟ ⎟ ⎠

}

⎛ ∂ − Tr Σ −1 (Φ o A ) + Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ − ⎜ ∂λ ⎝

[ {

}] [

] ⎞⎟

⎛ ∂ Tr Σ −1 (Φ o A ) ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ + ⎜ ∂λ ∂λ ⎝

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ λ ⎥⎦ ⎝ ⎣

[

⎟ ⎠ Eq. C63

] ⎞⎟ ⎟ ⎠

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ′Σ (Φ o A )Σ Δ ⎜ ⎟ ∂λ ⎝ ⎠

] ⎞⎟ . ⎟ ⎠

The first term is found to be:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂λ ⎥⎦ ⎝ ⎣

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o A ) ⎥ ⎟ ∂λ ⎢⎣ ⎥⎦ ⎠

⎡ −1 ∂ (Φ o A ) ∂Σ −1 ⎤ (Φ o A )⎥ = Tr ⎢ Σ + ∂λ ∂λ ⎣⎢ ⎦⎥

Eq. C64

⎤ ⎡ −1 ⎛ ∂A ⎞ −1 ∂Σ −1 Σ (Φ o A )⎥ = Tr ⎢ Σ ⎜ Φ o ⎟−Σ ∂λ ⎠ ∂λ ⎦ ⎣ ⎝

[

= Tr Σ

[

−1

= −Tr Σ

(− Φ o D o A ) − Σ −1 (− 2Φ o D o A )Σ −1 (Φ o A )]

−1

(Φ o D o A )]+ 2Tr [Σ −1 (Φ o D o A )Σ −1 (Φ o A )] .

The second term is evaluated as follows:

216

[

⎛ ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ E⎜ ⎜ ∂λ ⎝

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎜ ⎝

⎟ ⎠

[

−1

]

(Φ o A )Σ −1 ] ∂λ

⎞ Δ⎟ ⎟ ⎠

[

]

⎡ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ ⎜ ⎟ = E ⎢− Tr Δ′ Δ ⎥ = −Tr ⎜ E[ΔΔ′]⎟ ⎜ ⎟⎥ ⎜ ⎟ ∂λ ∂λ ⎢⎣ ⎝ ⎠⎦ ⎝ ⎠ ⎛ = −Tr ⎜ ⎜ ⎝

{

}

−1 ⎡ ∂ Σ −1 (Φ o A ) −1 ∂Σ −1 ( ) Σ Σ Φ A o + ⎢ ∂λ ∂λ ⎢⎣

⎤ ⎞ ⎥ Σ⎟ ⎥⎦ ⎟⎠

⎛ ⎡ −1 ∂ (Φ o A ) ∂Σ −1 ⎞ −1 −1 ∂Σ ⎤ −1 ( Φ o A ) − Σ (Φ o A )Σ + = −Tr ⎜ ⎢ Σ ⎥ Σ Σ⎟ ⎜⎢ ⎟ ∂λ ⎦⎥ ∂λ ∂λ ⎝⎣ ⎠ ⎛ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

−1 ⎡ −1 ⎛ ⎤ ⎞ ∂A ⎞ −1 ∂Σ −1 − Σ Φ Σ Σ (Φ o A )K⎥ ⎟ o ⎟ ⎢ ⎜ ∂λ ∂λ ⎠ ⎢ ⎝ ⎥ I⎟ −1 −1 ⎢ ⎥ ⎟⎟ ( ) ( ) − − Σ Φ A Σ Φ D A K o o o 2 ⎣ ⎦ ⎠

⎛ = −Tr ⎜ ⎜ ⎝

⎡ Σ −1 (− Φ o D o A ) − Σ −1 (− 2Φ o D o A )Σ −1 (Φ o A )K⎤ ⎢ ⎥ −1 −1 K + 2 Σ (Φ o A )Σ (Φ o D o A ) ⎥⎦ ⎢⎣

⎛ = −Tr ⎜ ⎜ ⎝

⎡− Σ −1 (Φ o D o A ) + 2 Σ −1 (Φ o D o A )Σ −1 (Φ o A )K⎤ ⎢ ⎥ −1 −1 K + 2 Σ (Φ o A )Σ (Φ o D o A ) ⎦⎥ ⎣⎢

[

⎞ ⎟ ⎟ ⎠

Eq. C65

⎞ ⎟ ⎟ ⎠

(Φ o D o A )] − 2Tr [Σ −1 (Φ o D o A )Σ −1 (Φ o A )]K −1 −1 K − 2Tr [Σ (Φ o A )Σ (Φ o D o A )] .

= Tr Σ

−1

On summing terms, we have:

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1 (Φ o D o A ) + 2Tr Σ −1 (Φ o D o A )Σ −1 (Φ o A ) E⎜ − ⎜ ∂λ∂α ⎟ g ⎠ ⎝

[

[

]

[

]

[

] ]

+ Tr Σ (Φ o D o A ) − 2Tr Σ (Φ o D o A )Σ (Φ o A ) K −1

[

−1

−1

]

K − 2Tr Σ (Φ o A )Σ (Φ o D o A ) −1

−1

Eq. C66

217

[

]

= −2Tr Σ (Φ o A )Σ (Φ o D o A ) . −1

−1

The third of the mixed elements in FI under the specified order is ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ . It is computed as: E⎜ − ⎜ ∂α e ∂α g ⎟ ⎝ ⎠ ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂α e ∂α g ⎟ ⎜ ∂α e ⎝ ⎠ ⎝

[ {

⎡ ∂ ln L(θ ) ⎤ ⎢ ⎥ ⎢⎣ ∂α g ⎥⎦

⎞ ⎟ ⎟ ⎠

}

⎛ ∂ − Tr Σ −1 (Φ o A ) + Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ − ⎜ ∂α e ⎝

[ {

}] [

] ⎞⎟

⎛ ∂ Tr Σ −1 (Φ o A ) ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ + ⎜ ∂α e ∂α e ⎝

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ α e ⎣ ⎦⎥ ⎝

[

⎟ ⎠ Eq. C67

] ⎞⎟ ⎟ ⎠

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ′Σ (Φ o A )Σ Δ ⎜ ⎟ ∂α e ⎝ ⎠

] ⎞⎟ . ⎟ ⎠

Starting with the first term, we find:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ α ⎥⎦ e ⎝ ⎣

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o A ) ⎥ ⎟ ∂α e ⎢⎣ ⎥⎦ ⎠

Eq. C68 ⎡ −1 ∂ (Φ o A ) ∂Σ −1 ⎤ (Φ o A )⎥ = Tr ⎢ Σ + ∂α e ∂α e ⎢⎣ ⎥⎦ ⎡ −1 ⎛ ⎤ ∂Σ −1 ∂A ⎞ ⎟⎟ − Σ −1 Σ (Φ o A )⎥ = Tr ⎢ Σ ⎜⎜ Φ o ∂α e ∂α e ⎠ ⎣ ⎝ ⎦

[

= Tr Σ

−1

(0) − Σ −1BΣ −1 (Φ o A )] = −Tr [Σ −1BΣ −1 (Φ o A )] .

The second term is evaluated as follows:

218

[

⎛ d − Δ′Σ −1 (Φ o A )Σ −1Δ E⎜ ⎜ ∂α e ⎝

] ⎞⎟ = E⎛⎜ − Δ′ d[Σ ⎜ ⎝

⎟ ⎠

[

−1

(Φ o A )Σ −1 ]

]

∂α e

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ d Σ −1 (Φ o A )Σ −1 ⎞ ⎤ ⎞ ⎛ d Σ −1 (Φ o A )Σ −1 ⎟ ⎜ ′ Δ ⎥ = −Tr ⎜ E[ΔΔ′]⎟ = E ⎢− Tr Δ ⎟⎥ ⎜ ⎟ ⎜ ∂α e ∂α e ⎠⎦ ⎝ ⎠ ⎝ ⎣⎢

{

}

−1 ⎛ ⎡ d Σ −1 (Φ o A ) −1 ∂Σ ⎤ −1 Σ + Σ (Φ o A ) = −Tr ⎜ ⎢ ⎥Σ ⎜⎢ ∂ α ∂ α ⎥⎦ e e ⎝⎣

⎞ ⎟ ⎟ ⎠

⎛ ⎡ −1 ∂ (Φ o A ) ∂Σ −1 −1 −1 ∂Σ ⎤ −1 ( Φ o A ) − Σ (Φ o A )Σ Σ Σ = −Tr ⎜ ⎢ Σ + ⎥ ⎜⎢ ∂α e ∂α e ∂α e ⎦⎥ ⎝⎣

⎞ ⎟ ⎟ ⎠

⎛ = −Tr ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎠

⎡ −1 ⎛ ∂Σ −1 ∂A ⎞ −1 −1 ⎤ ⎟⎟ − Σ −1 Σ (Φ o A ) − Σ (Φ o A )Σ B ⎥ I ⎢ Σ ⎜⎜ Φ o ∂α e ∂α e ⎠ ⎦ ⎣ ⎝

([

= −Tr Σ

[

−1

−1

(0) − Σ −1BΣ −1 (Φ o A ) − Σ −1 (Φ o A )Σ −1B] )

= Tr Σ BΣ

−1

(Φ o A )] + Tr [Σ −1 (Φ o A )Σ −1B] .

Eq. C69

Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1BΣ −1 (Φ o A ) E⎜ − ⎜ ∂α e ∂α g ⎟ ⎝ ⎠

[

]

[

−1

+ Tr Σ BΣ

[

= Tr Σ

−1

−1

(Φ o A )]+ Tr [Σ −1 (Φ o A )Σ −1B]

Eq. C70

(Φ o A )Σ −1B] .

The fourth of the mixed elements in FI under the specified order is ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ , which is found as follows: E⎜ − ⎜ ∂γ e ∂α g ⎟ ⎝ ⎠

219

⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂γ ⎜ ∂γ ∂α ⎟ e g ⎠ e ⎝ ⎝

[ {

⎡ ∂ ln L(θ ) ⎤ ⎢ ⎥ ⎢⎣ ∂α g ⎥⎦

⎞ ⎟ ⎟ ⎠

}

⎛ ∂ − Tr Σ −1 (Φ o A ) + Δ′Σ −1 (Φ o A )Σ −1Δ = E⎜ − ⎜ ∂γ e ⎝

[ {

] ⎞⎟

}] [

⎛ ∂ Tr Σ −1 (Φ o A ) ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ + = E⎜ ⎜ ∂ γ ∂γ e e ⎝

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ γ e ⎣ ⎦⎥ ⎝

[

⎟ ⎠ Eq. C71

] ⎞⎟ ⎟ ⎠

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ′Σ (Φ o A )Σ Δ ⎜ ⎟ ∂γ e ⎝ ⎠

] ⎞⎟ . ⎟ ⎠

The first term is given as:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ γ e ⎣ ⎦⎥ ⎝

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o A ) ⎥ ⎟ ∂γ e ⎣⎢ ⎦⎥ ⎠

⎡ −1 ∂ (Φ o A ) ∂Σ −1 ⎤ (Φ o A )⎥ = Tr ⎢ Σ + ∂γ e ∂γ e ⎢⎣ ⎥⎦ ⎡ −1 ⎛ ⎤ ∂Σ −1 ∂A ⎞ ⎟⎟ − Σ −1 Σ (Φ o A )⎥ = Tr ⎢ Σ ⎜⎜ Φ o ∂γ e ∂γ e ⎠ ⎣ ⎝ ⎦

[

= Tr Σ

[

−1

= −Tr Σ

Eq. C72

(0) − Σ −1 (C o B )Σ −1 (Φ o A )]

−1

(C o B )Σ −1 (Φ o A )] .

The next term is evaluated as follows:

[

⎛ ∂ − Δ′Σ −1 (Φ o A )Σ −1Δ E⎜ ⎜ ∂γ e ⎝

]⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎟ ⎠

⎜ ⎝

−1

(Φ o A )Σ −1 ] ∂γ e

⎞ Δ⎟ ⎟ ⎠

Eq. C73

220

[

]

[

]

⎡ ⎞ ⎛ ∂ Σ −1 (Φ o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o A )Σ −1 = E ⎢− Tr ⎜ Δ′ Δ ⎟ ⎥ = −Tr ⎜ E [ΔΔ′] ⎟ ⎜ ⎟ ⎜ ⎟⎥ ∂γ e ∂γ e ⎢⎣ ⎝ ⎠ ⎝ ⎠⎦

{

}

⎛ = −Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o A ) −1 ∂Σ ⎤ −1 Σ + Σ (Φ o A ) ⎢ ⎥Σ ∂γ e ∂γ e ⎦⎥ ⎣⎢

⎛ = −Tr ⎜ ⎜ ⎝

⎡ −1 ∂ (Φ o A ) ∂Σ −1 ∂Σ ⎤ (Φ o A ) − Σ −1 (Φ o A )Σ −1 ⎥ Σ −1Σ + ⎢Σ ∂γ e ∂γ e ∂γ e ⎥⎦ ⎢⎣

⎛ = −Tr ⎜ ⎜ ⎝

⎡ −1 ⎛ ⎤ ∂Σ −1 ∂A ⎞ −1 −1 ⎟ − Σ −1 ( ) − Σ (Φ o A )Σ (C o B )⎥ I o Σ Φ A ⎢ Σ ⎜⎜ Φ o ∂γ e ∂γ e ⎟⎠ ⎢⎣ ⎝ ⎥⎦

([

= −Tr Σ

[

−1

⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

(0) − Σ −1 (C o B )Σ −1 (Φ o A ) − Σ −1 (Φ o A )Σ −1 (C o B )] )

] [

]

= Tr Σ (C o B )Σ (Φ o A ) + Tr Σ (Φ o A )Σ (C o B ) . −1

−1

−1

−1

Summing terms gives:

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1 (C o B )Σ −1 (Φ o A ) E⎜ − ⎜ ∂γ e ∂α g ⎟ ⎝ ⎠

[

]

[

+ Tr Σ

[

= Tr Σ

−1

−1

(C o B )Σ −1 (Φ o A )]+ Tr [Σ −1 (Φ o A )Σ −1 (C o B )]

Eq. C74

(Φ o A )Σ −1 (C o B )] .

The next set of mixed elements in FI have their first partial derivative evaluated with respect to γ g . The first of these involves the two genetic slope parameters, γ g and λ , for the additive genetic variance and genetic correlation functions, respectively: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎡ ⎤ ⎟ = E⎜ − ∂ ⎢ ∂ ln L(θ ) ⎥ E⎜ − ⎜ ∂λ ⎢ ∂γ ⎜ ∂λ∂γ g ⎟⎠ ⎥⎦ g ⎝ ⎣ ⎝

⎞ ⎟ ⎟ ⎠

Eq. C75

221

{ [

]

} ⎞⎟

⎛ ∂ − Tr Σ −1 (Φ o C o A ) + Δ′Σ −1 (Φ o C o A )Σ −1Δ = E⎜ − ⎜ ∂λ ⎝

{ [

]} {

⎟ ⎠

} ⎞⎟

−1 −1 ⎛ ∂ Tr Σ −1 (Φ o C o A ) ∂ − Δ′Σ (Φ o C o A )Σ Δ = E⎜ + ⎜ ∂λ ∂λ ⎝

{

}

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ λ ⎣ ⎦⎥ ⎝

{

⎟ ⎠

} ⎞⎟ .

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ′Σ (Φ o C o A )Σ Δ ⎜ ⎟ ∂λ ⎝ ⎠

⎟ ⎠

Evaluating the first term gives:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ λ ⎥⎦ ⎝ ⎣

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o C o A ) ⎥ ⎟ ∂λ ⎢⎣ ⎥⎦ ⎠

⎡ −1 ∂ (Φ o C o A ) ∂Σ −1 ⎤ (Φ o C o A )⎥ = Tr ⎢ Σ + ∂λ ∂λ ⎢⎣ ⎥⎦ ⎡ −1 ⎛ ⎤ ∂A ⎞ −1 ∂Σ −1 = Tr ⎢ Σ ⎜ Φ o C o Σ (Φ o C o A )⎥ ⎟−Σ ∂λ ⎠ ∂λ ⎣ ⎝ ⎦

[

= Tr Σ

−1

[

= Tr − Σ

[

= −Tr Σ

Eq. C76

(− Φ o C o D o A ) − Σ −1 (− 2Φ o D o A )Σ −1 (Φ o C o A )] −1

−1

(Φ o C o D o A ) + 2Σ −1 (Φ o D o A )Σ −1 (Φ o C o A )]

(Φ o C o D o A )] + 2Tr [Σ −1 (Φ o D o A )Σ −1 (Φ o C o A )] .

The next term is found as follows:

[

⎛ ∂ − Δ′Σ −1 (Φ o C o A )Σ −1Δ E⎜ ⎜ ∂λ ⎝

[

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎜ ⎝

⎟ ⎠

]

−1

(Φ o C o A )Σ −1 ] ∂λ

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ = E ⎢− Tr ⎜ Δ′ Δ ⎟ ⎥ = −Tr ⎜ E[ΔΔ′]⎟ ⎜ ⎟⎥ ⎜ ⎟ ∂λ ∂λ ⎢⎣ ⎝ ⎠⎦ ⎝ ⎠

Eq. C77

222

{

}

⎛ = −Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o C o A ) −1 ∂Σ ⎤ ⎞⎟ −1 Σ + Σ (Φ o C o A ) ⎢ ⎥Σ ∂λ ∂λ ⎦⎥ ⎟⎠ ⎣⎢

⎛ ⎜ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎤ ⎞ ⎡⎧⎪ −1 ∂ (Φ o C o A ) ∂Σ −1 ⎫ (Φ o C o A )⎪⎬Σ −1 K ⎥ ⎟⎟ + ⎢⎨Σ ∂λ ∂λ ⎪⎭ ⎥Σ ⎢⎪⎩ ⎥ ⎟ ⎢ ⎟ Σ ∂ −1 −1 −1 ⎢ Σ ⎥ ⎟ K − Σ (Φ o C o A )Σ ⎥⎦ ⎠ ⎢⎣ ∂λ

⎛ ⎜ = −Tr ⎜ ⎜⎜ ⎝

⎞ ⎡ −1 ⎛ ⎤ ∂A ⎞ −1 ∂Σ −1 Σ (Φ o C o A )K⎥ −1 ⎟ ⎟−Σ ⎢Σ ⎜ Φ o C o ∂λ ∂λ ⎠ ⎢ ⎝ ⎥ Σ Σ⎟ ⎟⎟ − 1 − 1 − 1 ⎢ K − Σ (Φ o C o A )Σ (− 2Φ o D o A )Σ ⎥ ⎣ ⎦ ⎠

⎛ = −Tr ⎜ ⎜ ⎝

⎡ Σ −1 (− Φ o C o D o A ) − Σ −1 (− 2Φ o D o A )Σ −1 (Φ o C o A )K⎤ ⎞ ⎢ ⎥ I⎟ −1 −1 K + 2Σ (Φ o C o A )Σ (Φ o D o A )⎥⎦ ⎟⎠ ⎢⎣

⎛ = −Tr ⎜ ⎜ ⎝

⎡− Σ −1 (Φ o C o D o A ) + 2Σ −1 (Φ o D o A )Σ −1 (Φ o C o A )K ⎤ ⎢ ⎥ −1 −1 K + 2Σ (Φ o C o A )Σ (Φ o D o A )⎦⎥ ⎣⎢

[

]

[

⎞ ⎟ ⎟ ⎠

]

= Tr Σ (Φ o C o D o A ) − 2Tr Σ (Φ o D o A )Σ (Φ o C o A ) K −1

[

−1

−1

]

K − 2Tr Σ (Φ o C o A )Σ (Φ o D o A ) . −1

−1

Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1 (Φ o C o D o A ) + 2Tr Σ −1 (Φ o D o A )Σ −1 (Φ o C o A ) E⎜ − ⎜ ∂λ∂γ g ⎟⎠ ⎝

[

]

[ + Tr [Σ (Φ o C o D o A )] − 2Tr [Σ (Φ o D o A )Σ K − 2Tr [Σ (Φ o C o A )Σ (Φ o D o A )] −1

−1

−1

[

= −2Tr Σ

−1

−1

−1

] (Φ o C o A )] K

(Φ o C o A )Σ −1 (Φ o D o A )] . Eq. C78

The next mixed element in FI that has its first derivative evaluated with respect

223

to γ g is evaluated as follows: ⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂α e ∂γ g ⎟ ⎜ ∂α e ⎝ ⎠ ⎝

{ [

⎡ ∂ ln L(θ ) ⎤ ⎥ ⎢ ⎣⎢ ∂γ g ⎦⎥

⎞ ⎟ ⎟ ⎠

]

} ⎞⎟

⎛ ∂ − Tr Σ −1 (Φ o C o A ) + Δ′Σ −1 (Φ o C o A )Σ −1Δ = E⎜ − ⎜ ∂α e ⎝

{ [

]} {

⎛ ∂ Tr Σ (Φ o C o A ) ∂ − Δ′Σ = E⎜ + ⎜ ∂α e ⎝ −1

{

}

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂α e ⎥⎦ ⎝ ⎣

−1

(Φ o C o A )Σ

−1

∂α e

{

⎟ ⎠ Eq. C79

}

Δ ⎞⎟ ⎟ ⎠

} ⎞⎟ .

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ ′Σ (Φ o C o A )Σ Δ ⎜ ⎟ ∂α e ⎝ ⎠

⎟ ⎠

The first term is found as follows:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂α e ⎦⎥ ⎝ ⎣

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o C o A ) ⎥ ⎟ ∂α e ⎣⎢ ⎦⎥ ⎠

Eq. C80 ⎡ −1 ∂ (Φ o C o A ) ∂Σ ⎤ (Φ o C o A )⎥ = Tr ⎢ Σ + ∂α e ∂α e ⎢⎣ ⎥⎦ −1

⎡ −1 ⎛ ⎤ ∂Σ −1 ∂A ⎞ ⎟⎟ − Σ −1 Σ (Φ o C o A )⎥ = Tr ⎢Σ ⎜⎜ Φ o C o ∂α e ∂α e ⎠ ⎣ ⎝ ⎦

[

= Tr Σ

[

−1

(0) − Σ −1BΣ −1 (Φ o C o A )]

−1

= −Tr Σ BΣ

−1

(Φ o C o A)] .

The second term is evaluated as follows:

[

⎛ ∂ − Δ′Σ −1 (Φ o C o A )Σ −1Δ E⎜ ⎜ ∂α e ⎝

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎟ ⎠

⎜ ⎝

−1

(Φ o C o A )Σ −1 ] ∂α e

⎞ Δ⎟ ⎟ ⎠

Eq. C81

224

[

]

[

]

⎡ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ ⎤ ⎜ ⎟ Δ ⎥ = −Tr ⎜ E[ΔΔ′]⎟ = E ⎢− Tr Δ′ ⎜ ⎟ ⎜ ⎟⎥ ∂α e ∂α e ⎝ ⎠ ⎝ ⎠⎦ ⎣⎢

{

}

⎛ = −Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o C o A ) −1 ∂Σ ⎤ ⎞⎟ −1 ( ) o o Σ Σ Φ C A + ⎢ ⎥Σ ∂α e ⎥⎦ ⎟⎠ ∂α e ⎢⎣

⎛ ⎜ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎞ ⎡ −1 ∂ (Φ o C o A ) ∂Σ −1 ⎤ ⎟ (Φ o C o A )K⎥ + ⎢Σ ∂ α ∂ α ⎢ ⎥ Σ −1Σ ⎟ e e ⎟ ⎢ ⎥ −1 −1 ∂Σ ⎟ ⎢ ⎥ K − Σ (Φ o C o A )Σ ⎟ ∂α e ⎦⎥ ⎣⎢ ⎠

⎛ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎡ −1 ⎛ ⎤ ⎞ ∂A ⎞ ∂Σ −1 ⎟ − Σ −1 Σ (Φ o C o A )K⎥ ⎟ ⎢ Σ ⎜⎜ Φ o C o ∂α e ⎟⎠ ∂α e ⎢ ⎝ ⎥ I⎟ ⎢ ⎥ ⎟⎟ −1 −1 K − Σ (Φ o C o A )Σ B ⎦⎥ ⎠ ⎣⎢

([

= −Tr Σ

[

−1

(0) − Σ −1BΣ −1 (Φ o C o A ) − Σ −1 (Φ o C o A )Σ −1B] )

] [

]

= Tr Σ BΣ (Φ o C o A ) + Tr Σ (Φ o C o A )Σ B . −1

−1

−1

−1

Summing terms gives: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1BΣ −1 (Φ o C o A ) E⎜ − ⎜ ∂α e ∂γ g ⎟ ⎝ ⎠

[

]

[

−1

+ Tr Σ BΣ

[

= Tr Σ

−1

−1

(Φ o C o A )]+ Tr [Σ −1 (Φ o C o A )Σ −1B]

Eq. C82

(Φ o C o A )Σ −1B] .

The last mixed element in FI with γ g as its first partial derivative is:

⎛ ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂γ ⎜ ∂γ ∂γ ⎟ e g ⎠ e ⎝ ⎝

⎡ ∂ ln L(θ ) ⎤ ⎥ ⎢ ⎢⎣ ∂γ g ⎥⎦

⎞ ⎟ ⎟ ⎠

Eq. C83

225

{ [

]

} ⎞⎟

⎛ ∂ − Tr Σ −1 (Φ o C o A ) + Δ′Σ −1 (Φ o C o A )Σ −1Δ = E⎜ − ⎜ ∂γ e ⎝

{ [

]} {

⎟ ⎠

} ⎞⎟

−1 −1 ⎛ ∂ Tr Σ −1 (Φ o C o A ) ∂ − Δ′Σ (Φ o C o A )Σ Δ = E⎜ + ⎜ ∂γ e ∂γ e ⎝

{

}

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) ⎤ = E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ γ e ⎣ ⎦⎥ ⎝

⎟ ⎠

{

} ⎞⎟ .

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ − Δ′Σ (Φ o C o A )Σ Δ ⎜ ⎟ ∂γ e ⎝ ⎠

⎟ ⎠

The first term gives:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o C o A ) ⎤ E⎜ Tr ⎢ ⎥ ⎜ ⎢ ∂ γ ⎥⎦ e ⎝ ⎣

{

}

−1 ⎞ ⎡ ⎤ ⎟ = Tr ⎢ ∂ Σ (Φ o C o A ) ⎥ ⎟ ∂γ e ⎢⎣ ⎥⎦ ⎠

⎡ −1 ∂ (Φ o C o A ) ∂Σ −1 ⎤ (Φ o C o A )⎥ = Tr ⎢ Σ + ∂γ e ∂γ e ⎢⎣ ⎥⎦ Eq. C84 ⎡ −1 ⎛ ⎤ ∂A ⎞ ∂Σ −1 ⎟⎟ − Σ −1 = Tr ⎢ Σ ⎜⎜ Φ o C o Σ (Φ o C o A )⎥ ∂γ e ⎠ ∂γ e ⎣ ⎝ ⎦

[

= Tr Σ

−1

(0) − Σ −1 (C o B )Σ −1 (Φ o C o A )] = −Tr [Σ −1 (C o B )Σ −1 (Φ o C o A )] .

The second term is evaluated as follows:

[

⎛ ∂ − Δ′Σ −1 (Φ o C o A )Σ −1Δ E⎜ ⎜ ∂γ e ⎝

[

] ⎞⎟ = E⎛⎜ − Δ′ ∂[Σ ⎟ ⎠

⎜ ⎝

]

−1

(Φ o C o A )Σ −1 ] ∂γ e

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o C o A )Σ −1 ⎞ ⎜ ⎟ = E ⎢− Tr Δ′ Δ ⎥ = −Tr ⎜ E[ΔΔ′]⎟ ⎜ ⎟⎥ ⎜ ⎟ ∂γ e ∂γ e ⎝ ⎠⎦ ⎝ ⎠ ⎣⎢ ⎛ = −Tr ⎜ ⎜ ⎝

{

Eq. C85

}

−1 ⎡ ∂ Σ −1 (Φ o C o A ) −1 ∂Σ ⎤ ⎞⎟ −1 ( ) + Σ Σ Φ o C o A ⎢ ⎥Σ ∂γ e ∂γ e ⎥⎦ ⎟⎠ ⎢⎣

226

⎛ ⎜ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎞ ⎤ ⎡ −1 ∂ (Φ o C o A ) ∂Σ −1 ⎟ (Φ o C o A )K⎥ + ⎢Σ ∂ γ ∂ γ − 1 ⎥ Σ Σ⎟ ⎢ e e ⎟ ⎥ ⎢ −1 −1 ∂Σ ⎟ ⎥ ⎢ K − Σ (Φ o C o A )Σ ⎟ ⎢⎣ ∂γ e ⎥⎦ ⎠

⎛ ⎜ = −Tr ⎜ ⎜ ⎜ ⎝

⎡ −1 ⎛ ⎤ ⎞ ∂A ⎞ −1 ∂Σ −1 ⎜ ⎟ ( ) Σ Φ C − Σ Σ Φ C A o o o o K ⎢ ⎜ ⎥ ⎟ ⎟ ∂γ e ⎠ ∂γ e ⎢ ⎝ ⎥ I⎟ ⎢ ⎥ ⎟⎟ −1 −1 ( ) ( ) Σ Φ C A Σ C B − K o o o ⎣⎢ ⎦⎥ ⎠

([

= −Tr Σ

[

−1

(0) − Σ −1 (C o B )Σ −1 (Φ o C o A ) − Σ −1 (Φ o C o A )Σ −1 (C o B )] )

] [

]

= Tr Σ (C o B )Σ (Φ o C o A ) + Tr Σ (Φ o C o A )Σ (C o B ) . −1

−1

−1

−1

On summing terms, we find: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1 (C o B )Σ −1 (Φ o C o A ) E⎜ − ⎜ ∂γ e ∂γ g ⎟ ⎝ ⎠

[

]

[

+ Tr Σ

[

= Tr Σ

−1

−1

(C o B )Σ −1 (Φ o C o A )]+ Tr [Σ −1 (Φ o C o A )Σ −1 (C o B )]

(Φ o C o A )Σ −1 (C o B )] . Eq. C86

The next two mixed elements in FI have their first partial derivatives evaluated with respect to λ . The first of these is: ⎛ ∂ 2 ln L(θ ) ⎞ ⎛ ⎟ = E⎜ − ∂ ⎡ ∂ ln L(θ ) ⎤ E⎜ − ⎜ ∂α ⎢⎣ ∂λ ⎥⎦ ⎜ ∂α e ∂λ ⎟⎠ e ⎝ ⎝

{ [

]

⎞ ⎟ ⎟ ⎠

} ⎞⎟

⎛ ∂ Tr Σ −1 (Φ o D o A ) − Δ′Σ −1 (Φ o D o A )Σ −1Δ = E⎜ − ⎜ ∂α e ⎝

{ [

]} {

Eq. C87

⎟ ⎠

} ⎞⎟

−1 −1 ⎛ ∂ − Tr Σ −1 (Φ o D o A ) ∂ Δ′Σ (Φ o D o A )Σ Δ ⎜ =E + ⎜ ∂α e ∂α e ⎝

⎟ ⎠

227

{

}

⎛ ⎡ ∂ Σ −1 (Φ o D o A ) ⎤ ⎜ = E − Tr ⎢ ⎥ ⎜ ∂ α ⎢ e ⎣ ⎦⎥ ⎝

{

} ⎞⎟ .

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ Δ′Σ (Φ o D o A )Σ Δ ⎜ ⎟ ∂α e ⎝ ⎠

⎟ ⎠

The first term is evaluated as follows:

{

}

⎛ ⎡ ∂ Σ −1 (Φ o D o A ) ⎤ ⎜ E − Tr ⎢ ⎥ ⎜ ∂ α ⎢ e ⎣ ⎦⎥ ⎝

{

}

−1 ⎞ ⎡ ⎤ ⎟ = −Tr ⎢ ∂ Σ (Φ o D o A ) ⎥ ⎟ ∂α e ⎣⎢ ⎦⎥ ⎠

⎡ −1 ∂ (Φ o D o A ) ∂Σ −1 ⎤ (Φ o D o A )⎥ = −Tr ⎢ Σ + ∂α e ∂α e ⎢⎣ ⎥⎦ ⎡ −1 ⎛ ⎤ ∂A ⎞ ∂Σ −1 ⎟⎟ − Σ −1 = −Tr ⎢ Σ ⎜⎜ Φ o D o Σ (Φ o D o A )⎥ ∂α e ⎠ ∂α e ⎣ ⎝ ⎦

[

= −Tr Σ

[

−1

−1

Eq. C88

(0) − Σ −1BΣ −1 (Φ o D o A )]

= Tr Σ BΣ

−1

(Φ o D o A )] .

The second term is found to be:

[

⎛ ∂ Δ′Σ −1 (Φ o D o A )Σ −1Δ E⎜ ⎜ ∂α e ⎝

[

] ⎞⎟ = E⎜⎛ Δ′ ∂[Σ ⎜ ⎝

⎟ ⎠

]

−1

(Φ o D o A )Σ −1 ] ∂α e

[

⎞ Δ⎟ ⎟ ⎠

]

⎡ ⎛ ∂ Σ −1 (Φ o D o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o D o A )Σ −1 ⎞ = E ⎢Tr ⎜ Δ′ Δ ⎟ ⎥ = Tr ⎜ E[ΔΔ′]⎟ ⎟⎥ ⎜ ⎟ ∂α e ∂α e ⎢⎣ ⎜⎝ ⎠⎦ ⎝ ⎠

{

}

⎛ = Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o D o A ) −1 ∂Σ ⎤ ⎞⎟ −1 Σ + Σ (Φ o D o A ) ⎢ ⎥Σ ∂α e ∂α e ⎦⎥ ⎟⎠ ⎣⎢

⎛ ⎜ ⎜ = Tr ⎜ ⎜ ⎜ ⎝

⎞ ⎡ −1 ∂ (Φ o D o A ) ∂Σ −1 ⎤ ⎟ ( ) + Σ Φ o D o A K ⎢ ⎥ ∂α e ∂α e ⎢ ⎥ Σ −1Σ ⎟ ⎟ ⎢ ⎥ −1 −1 ∂Σ ⎟ ⎢ ⎥ K − Σ (Φ o D o A )Σ ⎟ ⎢⎣ ∂α e ⎥⎦ ⎠

Eq. C89

228

⎛ ⎜ = Tr ⎜ ⎜ ⎜ ⎝

⎤ ⎞ ⎡ −1 ⎛ ∂A ⎞ ∂Σ −1 ⎟ − Σ −1 Σ (Φ o D o A )K⎥ ⎟ ⎢ Σ ⎜⎜ Φ o D o ∂α e ⎟⎠ ∂α e ⎥ I⎟ ⎢ ⎝ ⎥ ⎟⎟ ⎢ −1 −1 K − Σ (Φ o D o A )Σ B ⎥⎦ ⎠ ⎢⎣

([

−1

[

−1

= Tr Σ (0 ) − Σ BΣ (Φ o D o A ) − Σ (Φ o D o A )Σ B −1

−1

−1

] [

−1

])

]

= −Tr Σ BΣ (Φ o D o A ) − Tr Σ (Φ o D o A )Σ B . −1

−1

−1

On summing terms, we find: ⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1BΣ −1 (Φ o D o A ) E⎜ − ⎜ ∂α e ∂λ ⎟⎠ ⎝

[

]

[

−1

[

−1

− Tr Σ BΣ = −Tr Σ

−1

(Φ o D o A )]− Tr [Σ −1 (Φ o D o A )Σ −1B]

Eq. C90

(Φ o D o A )Σ −1B] .

The next mixed element in FI that has its first partial derivative evaluated with respect to λ is evaluated as follows: ⎛ ∂ 2 ln L(θ ) ⎞ ⎛ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂γ ⎜ ∂γ e ∂λ ⎟⎠ e ⎝ ⎝

{ [

⎡ ∂ ln L(θ ) ⎤ ⎢ ∂λ ⎥ ⎦ ⎣

]

⎞ ⎟⎟ ⎠

} ⎞⎟

⎛ ∂ Tr Σ −1 (Φ o D o A ) − Δ ′Σ −1 (Φ o D o A )Σ −1Δ = E⎜ − ⎜ ∂γ e ⎝

{ [

]} {

⎟ ⎠

} ⎞⎟

−1 −1 ⎛ ∂ − Tr Σ −1 (Φ o D o A ) ∂ Δ′Σ (Φ o D o A )Σ Δ + = E⎜ ⎜ ∂γ e ∂γ e ⎝

{

}

⎛ ⎡ ∂ Σ −1 (Φ o D o A ) ⎤ ⎜ = E − Tr ⎢ ⎥ ⎜ ∂ γ ⎢ e ⎦⎥ ⎣ ⎝

Eq. C91

{

⎟ ⎠

} ⎞⎟ .

−1 −1 ⎞ ⎛ ⎟ + E⎜ ∂ Δ′Σ (Φ o D o A )Σ Δ ⎜ ⎟ ∂γ e ⎝ ⎠

⎟ ⎠

The first term is evaluated as follows:

229

{

}

⎛ ⎡ ∂ Σ −1 (Φ o D o A ) ⎤ ⎜ E − Tr ⎢ ⎥ ⎜ ∂γ e ⎣⎢ ⎦⎥ ⎝

{

}

−1 ⎞ ⎡ ⎤ ⎟ = −Tr ⎢ ∂ Σ (Φ o D o A ) ⎥ ⎟ ∂γ e ⎣⎢ ⎦⎥ ⎠

⎡ −1 ∂(Φ o D o A ) ∂Σ −1 ⎤ (Φ o D o A )⎥ = −Tr ⎢ Σ + ∂γ e ∂γ e ⎣⎢ ⎦⎥ ⎡ −1 ⎛ ⎤ ∂Σ −1 ∂A ⎞ ⎟⎟ − Σ −1 Σ (Φ o D o A )⎥ = −Tr ⎢ Σ ⎜⎜ Φ o D o ∂γ e ∂γ e ⎠ ⎣ ⎝ ⎦

[

= −Tr Σ

[

= Tr Σ

−1

−1

Eq. C92

(0) − Σ −1 (C o B )Σ −1 (Φ o D o A )]

(C o B )Σ −1 (Φ o D o A )] .

The second term is found to be:

[

⎛ ∂ Δ′Σ −1 (Φ o D o A )Σ −1Δ E⎜ ⎜ ∂γ e ⎝

[

] ⎞⎟ = E⎛⎜ Δ′ ∂[Σ ⎟ ⎠

⎜ ⎝

]

−1

(Φ o D o A )Σ −1 ] ∂γ e

[

⎞ Δ⎟ ⎟ ⎠

]

Eq. C93

⎡ ⎛ ∂ Σ −1 (Φ o D o A )Σ −1 ⎞ ⎤ ⎛ ∂ Σ −1 (Φ o D o A )Σ −1 ⎞ ⎜ ⎟ ′ = E ⎢Tr Δ Δ ⎥ = Tr ⎜ E[ΔΔ ′]⎟ ⎜ ⎟⎥ ⎜ ⎟ ∂γ e ∂γ e ⎠⎦ ⎝ ⎠ ⎣⎢ ⎝

{

}

⎛ = Tr ⎜ ⎜ ⎝

−1 ⎡ ∂ Σ −1 (Φ o D o A ) −1 ∂Σ ⎤ ⎞⎟ −1 Σ + Σ (Φ o D o A ) ⎢ ⎥Σ ∂γ e ∂γ e ⎦⎥ ⎟⎠ ⎣⎢

⎛ ⎜ ⎜ = Tr ⎜ ⎜ ⎜ ⎝

⎞ ⎤ ⎡ −1 ∂ (Φ o D o A ) ∂Σ −1 ⎟ ( ) o o K Σ Φ D A + ⎥ ⎢ ∂γ e ∂γ e ⎥ Σ −1Σ ⎟ ⎢ ⎟ ⎥ ⎢ −1 −1 ∂Σ ⎟ ⎥ ⎢ K − Σ (Φ o D o A )Σ ⎟ ⎢⎣ ∂γ e ⎥⎦ ⎠

⎛ ⎜ = Tr ⎜ ⎜ ⎜ ⎝

⎡ −1 ⎛ ⎤ ⎞ ∂A ⎞ ∂Σ −1 ⎟ − Σ −1 ( ) o o K Σ Φ D A ⎢ Σ ⎜⎜ Φ o D o ⎥ ⎟ ⎟ ∂γ e ⎠ ∂γ e ⎢ ⎝ ⎥ I⎟ ⎢ ⎥ ⎟ −1 −1 K − Σ (Φ o D o A )Σ (C o B )⎥⎦ ⎟⎠ ⎢⎣

230

([

−1

[

−1

= Tr Σ

(0) − Σ −1 (C o B )Σ −1 (Φ o D o A ) − Σ −1 (Φ o D o A )Σ −1 (C o B )] )

]

[

]

= −Tr Σ (C o B )Σ (Φ o D o A ) − Tr Σ (Φ o D o A )Σ (C o B ) . −1

−1

−1

Summing term gives:

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1 (C o B )Σ −1 (Φ o D o A ) E⎜ − ⎜ ∂γ e ∂λ ⎟⎠ ⎝

[

]

[

(C o B )Σ −1 (Φ o D o A )]K −1 −1 K − Tr [Σ (Φ o D o A )Σ (C o B )] − Tr Σ

[

= −Tr Σ

−1

−1

Eq. C94

(Φ o D o A )Σ −1 (C o B )] .

The last mixed term in FI involves the environmental parameters. It is evaluated as follows: ⎛ ∂ 2 ln L(θ ) ⎞ ⎛ ⎟ = E⎜ − ∂ E⎜ − ⎜ ∂γ ⎜ ∂γ e ∂α e ⎟ e ⎝ ⎝ ⎠

{ [

⎡ ∂ ln L(θ ) ⎤ ⎥ ⎢ ⎣ ∂α e ⎦

]

]}

Eq. C95

} ⎞⎟

⎛ 1 ∂ − Tr Σ −1B + Δ′Σ −1BΣ −1Δ = E⎜ − ⎜ 2 ∂γ e ⎝

{ [

⎞ ⎟ ⎟ ⎠

{

⎟ ⎠

} ⎞⎟

−1 −1 ⎛ 1 ∂ Tr Σ −1B 1 ∂ − Δ ′Σ BΣ Δ = E⎜ + ⎜2 2 ∂γ e ∂γ e ⎝

{ }

{

⎟ ⎠

} ⎞⎟ .

⎛ 1 ⎡ ∂ Σ −1B ⎤ ⎞ ⎛ 1 ∂ − Δ′Σ −1BΣ −1Δ ⎜ ⎟ = E Tr ⎢ + E⎜ ⎜2 ⎜ 2 ⎢ ∂γ e ⎥⎥ ⎟ ∂γ e ⎣ ⎦⎠ ⎝ ⎝

⎟ ⎠

On evaluating the first term, we find:

(

)

(

)

⎛ 1 ⎡ ∂ Σ −1B ⎤ ⎞ 1 ⎡ ∂ Σ −1B ⎤ 1 ⎡ −1 ∂B ∂Σ −1 ⎤ E⎜ Tr ⎢ B⎥ + ⎥ = Tr ⎢ Σ ⎥ ⎟ = Tr ⎢ ⎜ 2 ⎢ ∂γ ⎥ ⎟ 2 ⎢ ∂γ ⎥ 2 ⎢ ∂γ e ∂γ e ⎦⎥ e e ⎣ ⎦ ⎣ ⎦⎠ ⎣ ⎝

Eq. C96

231

1 ⎡ −1 −1 ∂Σ −1 ⎤ = Tr ⎢Σ (C o B) − Σ Σ B⎥ 2 ⎣⎢ ∂γ e ⎦⎥

[

1 −1 −1 −1 = Tr Σ (C o B) − Σ (C o B)Σ B 2

[

]

]

[

]

1 1 −1 −1 −1 = Tr Σ (C o B) − Tr Σ (C o B)Σ B . 2 2

The second term is found to be:

[

⎛ 1 ∂ − Δ ′Σ −1BΣ −1Δ E⎜ ⎜2 ∂γ e ⎝

[

] ⎞⎟ = E⎛⎜ − 1 Δ′ ∂[Σ ⎟ ⎠

⎜ 2 ⎝

−1

BΣ ∂γ e

]

−1

]Δ ⎞⎟ ⎟ ⎠

[

]

−1 −1 ⎡ 1 ⎛ ∂ Σ −1BΣ −1 ⎞ ⎤ ⎞ 1 ⎛⎜ ∂ Σ BΣ ⎜ ⎟ Δ ⎥ = − Tr E[ΔΔ ′]⎟ = E ⎢− Tr Δ′ ⎟⎥ ⎟ 2 ⎜⎝ ∂γ e ∂γ e ⎢⎣ 2 ⎜⎝ ⎠⎦ ⎠

(

Eq. C97

)

−1 −1 1 ⎛ ⎡ ∂ Σ B −1 ∂Σ ⎤ −1 Σ +Σ B = − Tr ⎜ ⎢ ⎥Σ 2 ⎜⎝ ⎢⎣ ∂γ e ∂γ e ⎥⎦

⎞ ⎟ ⎟ ⎠

−1 1 ⎛⎜ ⎡ −1 ∂B ∂Σ −1 −1 ∂Σ ⎤ −1 B − Σ BΣ Σ Σ = − Tr ⎢ Σ + ⎥ 2 ⎜⎝ ⎣⎢ ∂γ e ∂γ e ∂γ e ⎦⎥

1 ⎛ = − Tr ⎜⎜ 2 ⎝

⎞ ⎟ ⎟ ⎠

⎡ −1 ⎤ −1 ∂Σ −1 −1 −1 Σ B − Σ BΣ (C o B )⎥ I ⎢ Σ (C o B ) − Σ ∂γ e ⎣ ⎦

([

⎞ ⎟ ⎟ ⎠

])

1 −1 −1 −1 −1 −1 = − Tr Σ (C o B ) − Σ (C o B )Σ B − Σ BΣ (C o B ) 2

[

]

[

]

[

]

1 1 1 −1 −1 −1 −1 −1 = − Tr Σ (C o B ) + Tr Σ (C o B )Σ B + Tr Σ BΣ (C o B ) . 2 2 2 On summing terms, we find:

232

⎛ ∂ 2 ln L(θ ) ⎞ 1 ⎟ = Tr Σ −1 (C o B ) − 1 Tr Σ −1 (C o B )Σ −1B E⎜ − ⎜ ∂γ e ∂α e ⎟ 2 2 ⎝ ⎠ 1 1 −1 −1 −1 − Tr Σ (C o B ) + Tr Σ (C o B )Σ B K 2 2 1 −1 −1 K + Tr Σ BΣ (C o B ) 2

[

]

[

[

]

[ ]

[

=

[

]

]

Eq. C98

]

1 −1 −1 Tr Σ BΣ (C o B ) . 2

It is perhaps desirable to summarize these results by presenting the elements of the score vector and all of the unique elements in FI in the order they were computed. For the sake of completeness, the partial derivatives with respect to the mean and covariate effects are also reported. For the elements in the score vector, we have: ∂ ln L(θ ) ⎛ ∂β = ⎜⎜ ∂β i ⎝ ∂β i

′ ⎞ ′ ⎟⎟ X′Σ −1Δ = (e i( n ) ) X′Σ −1Δ ; i = 0, 1, . . . , n ⎠

∂ ln L(θ ) −1 −1 −1 = −Tr Σ (Φ o A ) + Δ′Σ (Φ o A )Σ Δ ∂α g

[

]

∂ ln L(θ ) −1 −1 −1 = −Tr Σ (Φ o C o A ) + Δ′Σ (Φ o C o A )Σ Δ ∂γ g

[

]

∂ ln L(θ ) −1 −1 −1 = Tr Σ (Φ o D o A ) − Δ ′Σ (Φ o D o A )Σ Δ ∂λ

[

]

1 1 ∂ ln L(θ ) −1 −1 −1 = − Tr Σ B + Δ ′Σ BΣ Δ 2 2 ∂α e

(

)

∂ ln L(θ ) 1 1 −1 −1 −1 = − Tr Σ (C o B ) + Δ ′Σ (C o B )Σ Δ ∂γ e 2 2

{

}

For all of the unique elements in FI , we have:

233

′ ⎛ ∂ 2 ln L(θ ) ⎞ ⎛ ∂β ⎞ ⎟=⎜ ⎟ X′Σ −1X ∂β = e (jn ) ′ X′Σ −1Xei(n ) ∀ i, j, . . . , n E⎜ − ⎜ ∂β ∂β ⎟ ⎜ ∂β ⎟ ∂β i i j ⎠ ⎝ ⎝ j⎠

( )

⎛ ∂ 2 ln L(θ ) ⎞ 2 ⎟ = 2Tr ⎡ Σ −1 (Φ o A ) ⎤ E⎜ − ⎢⎣ ⎥⎦ ⎜ ∂α g ∂α g ⎟ ⎝ ⎠

{

}

⎛ ∂ 2 ln L(θ ) ⎞ 2 ⎟ = 2Tr ⎡ Σ −1 (Φ o C o A ) ⎤ E⎜ − ⎢⎣ ⎥⎦ ⎜ ∂γ g ∂γ g ⎟ ⎝ ⎠

{

}

⎛ ∂ 2 ln L(θ ) ⎞ 2 ⎟ = 2Tr ⎡ Σ −1 (Φ o D o A ) ⎤ E⎜ − ⎢⎣ ⎥⎦ ⎜ ∂λ∂λ ⎟⎠ ⎝

{

}

⎛ ∂ 2 ln L(θ ) ⎞ 1 ⎡ −1 2 ⎤ ⎟ = Tr Σ B E⎜ − ⎥⎦ ⎜ ∂α e ∂α e ⎟ 2 ⎣⎢ ⎝ ⎠

(

)

⎛ ∂ 2 ln L(θ ) ⎞ 1 ⎡ −1 2 ⎟ = Tr Σ (C o B ) ⎤ E⎜ − ⎥⎦ ⎜ ∂γ e ∂γ e ⎟ 2 ⎣⎢ ⎝ ⎠

{

}

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = 2Tr Σ −1 (Φ o A )Σ −1 (Φ o C o A ) E⎜ − ⎜ ∂γ g ∂α g ⎟ ⎝ ⎠

[

]

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −2Tr Σ −1 (Φ o A )Σ −1 (Φ o D o A ) E⎜ − ⎜ ∂λ∂α g ⎟ ⎠ ⎝

[

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1 (Φ o A )Σ −1B E⎜ − ⎜ ∂α e ∂α g ⎟ ⎝ ⎠

[

]

]

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1 (Φ o A )Σ −1 (C o B ) E⎜ − ⎜ ∂γ e ∂α g ⎟ ⎝ ⎠

[

]

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −2Tr Σ −1 (Φ o C o A )Σ −1 (Φ o D o A ) E⎜ − ⎜ ∂λ∂γ g ⎟⎠ ⎝

[

]

234

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1 (Φ o C o A )Σ −1B E⎜ − ⎜ ∂α e ∂γ g ⎟ ⎝ ⎠

[

]

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = Tr Σ −1 (Φ o C o A )Σ −1 (C o B ) E⎜ − ⎜ ∂γ e ∂γ g ⎟ ⎠ ⎝

[

]

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1 (Φ o D o A )Σ −1B E⎜ − ⎜ ∂α e ∂λ ⎟⎠ ⎝

[

]

⎛ ∂ 2 ln L(θ ) ⎞ ⎟ = −Tr Σ −1 (Φ o D o A )Σ −1 (C o B ) E⎜ − ⎜ ∂γ e ∂λ ⎟⎠ ⎝

[

]

⎛ ∂ 2 ln L(θ ) ⎞ 1 ⎟ = Tr Σ −1BΣ −1 (C o B ) E⎜ − ⎜ ∂γ e ∂α e ⎟ 2 ⎠ ⎝

[

]

The 15 unique elements corresponding to the variance components are arranged in the Fisher information matrix in the following page. This formulation assumes that (Williams and Blangero, 1999a): 1) Xβ = 0 ; and 2) Σ is determined completely by the variances in the genetic and environmental effects. Because FI is symmetrical, only the upper triangular part is reported.

235

⎡ ⎢ −1 ⎢2Tr Σ (Φo A) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ FI = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

[{

}] 2

[

]

2Tr Σ (Φo A)Σ (Φo Co A) −1

−1

[{

}]

2Tr Σ (Φo Co A) −1

2

[

]

[

− 2Tr Σ (Φo A)Σ (Φo Do A)

[

−1

−1

−1

[{

}]

2Tr Σ (Φo Do A) −1

]

] [

− 2Tr Σ (Φo Co A)Σ (Φo Do A) −1

⎤ ⎥ −1 −1 −1 −1 Tr Σ (Φo A)Σ B Tr Σ (Φo A)Σ (Co B) ⎥ ⎥ ⎥ −1 −1 −1 −1 Tr Σ (Φo Co A)Σ B Tr Σ (Φo Co A)Σ (Co B) ⎥ ⎥ ⎥ −1 −1 −1 −1 − Tr Σ (Φo Do A)Σ B − Tr Σ (Φo Do A)Σ (Co B) ⎥ ⎥ ⎥ ⎥ 2 1 1 −1 −1 −1 Tr Σ B Tr Σ BΣ (Co B) ⎥ 2 2 ⎥ ⎥ ⎥ 2 1 −1 Tr Σ (Co B) ⎥ 2 ⎥ ⎥⎦ .

2

[

]

[

[

]

[(

)]

]

]

[

]

[

]

[{

}]

Eq. C99 By Equation 139, we also have for the sampling covariance matrix of the parameter estimates: ⎡σ α2g ⎢ ⎢ FI−1 = Σ θˆ = ⎢ ⎢ ⎢ ⎢ ⎣

σα γ σ γ2

g g

g

σα λ σα α σγ λ σγ α σ λ2 σ λα σ α2 g

g

e

g

g

e

e

e

⎤ ⎥ g e ⎥ ⎥ e ⎥ ⎥ e e ⎥ e ⎦ .

σα γ σγ γ σ λγ σα γ σ γ2

g e

Eq. C100

The actual elements in the Fisher information and sampling covariance matrices can be solved for numerically using GaussTM.

Appendix D: Geometry of the Likelihood Function There is an elegant geometrical interpretation of Σ θˆ that follows from the results of vector calculus (Fig. D1). If FI is visualized as giving a hyperparaboloid tangent to the hypersurface at which S(θˆ ) = 0 or, approximately, the curvature where ln L(θˆ ) is a

maximum, then Σ θˆ , being the reciprocal of the curvature, gives the radius of curvature in the vicinity where ln L(θˆ ) is a maximum (Huzurbazar, 1949; Rao, 1960; Efron, 1975; Edwards, 1992). In other words, Σ θˆ measures the curvature under the maximum. As such, Σ θˆ also measures the precision of estimates in the parameter vector (Huzurbazar, 1949; Rao, 1960; Efron, 1975; Thompson, 1986; Edwards, 1992). To justify the geometric interpretation, we may take the case of a simple ln-likelihood function for θˆ scalar, i.e., ln L(θˆ ), which with advanced differential geometry approaches can be generalized to the multivariable case (see Huzurbazar, 1949; Rao, 1960; Efron, 1975; Kass, 1989). Let there be an osculating circle, defined as the circle that best fits under the maximum and is tangent to the point at the maximum. The osculating circle of radius r is given by a vector-valued function in θˆ : τ (θˆ ) = r (cos θˆ i + sin θˆ j) ,

Eq. D1

where i and j are vectors in the plane ℜ 2 . Equation D1 is a parametric equation in terms of x and y functions, namely: x (θˆ ) = r cos θˆ ,

Eq. D2

y(θˆ ) = r sin θˆ ,

Eq. D3

and

237

d ln L (θˆ ) = 0 d θˆ

2 r ; r = ± SE ln L (θˆ )

θˆ Figure D1. Geometry of the Ln-Likelihood Function. For a simple ln-likelihood function, the ideal maximum likelihood estimate is indicated by downward concavity and tight curvature in the vicinity of the maximum. In the figure, r denotes the radius of the osculating circle of diameter 2r.

respectively. The curvature of a vector-valued function, denoted by ψ , is given by: ψ=

x ′(θˆ )⋅ y′′(θˆ ) − y′(θˆ )⋅ x ′′(θˆ )

[{x′(θˆ )} + {y′(θˆ )} ] 2

2 32

,

Eq. D4

where the prime notation now indicates differentiation with respect to θˆ (instead of vector or matrix transpose). On differentiating accordingly and recalling that sin 2 θˆ + cos 2 θˆ = 1 is a Pythagorean identity, we have:

ψ=

(− r sin θˆ )(− r sin θˆ ) − (r cos θˆ )(− r cos θˆ ) [(− r sin θˆ ) + (r cos θˆ ) ] 2

2 32

=

r2 r

3

=

1 r

,

Eq. D5

238

which tells us that the curvature is equal to the reciprocal of the radius of curvature and vice versa. To complete the justification of the geometric interpretation, it may be argued that the Taylor expansion approximations (about the maximum likelihood estimate) of ln L(θˆ ) and τ (θˆ ) agree at least up to their quadratic terms (Efron, 1975). Kass (1989) reviews extensions of these concepts to more complicated likelihood functions using advanced differential geometry.

239

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My Dissertation

Published on Mar 8, 2010

Dissertation on the use of a genotype-by-age interaction model to the IGF-1 axis in relation to human senescence.

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