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
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