34 J OURNAL OF FOREST PRODUCTS & INDUSTRIES , 2013, 2(1), 34-42 ISSN:2325 – 4513(PRINT) ISSN 2325 - 453X (ONLINE )

Modelling Height-Diameter Relationships of Selected Economically Important Natural Forests Species EL MAMOUN H. OSMAN (1), EL ZEIN A. IDRIS (2) and EL MUGIRA M. IBRAHIM (3) (1) College of Natural Resources and Environmental Studies-Department of Forestry, University of Bahri. Khartoum- Sudan E mail: elmamounosman@yahoo.co.uk Telephone: +249911347139 (2) College of Natural Resources and Environmental Studies-Department of Forestry, University of Bahri. Khartoum- Sudan E mail: elzeinadam@yahoo.com Telephone: +249911270310 2) College of Natural Resources and Environmental Studies-Department of Forestry, University of Bahri. Khartoum- Sudan E mail: elmugheira1984@yahoo.com Telephone: +249911986534

(Received November 24, 2012; Accepted December 22, 2012) Abstract— This paper developed and evaluated the performance of twenty two functional models that predict total tree height from diameter at breast height. The models were applied to five selected economically important natural tree species common to central Sudan, namely, Acacia senegal, Acacia seyal, Anogeissus leocarpus, Balanites aegyptiaca, and Combretum hartmannianum. A total of 5774 tree heights and corresponding diameters at breast heights were extracted from a working plan inventory data of a natural reserved forest in the Blue Nile state. The model goodness of fit were evaluated in terms of adjusted coefficient of determination (Ra2), root mean squared error (RMSE), Akaike’s information criterion (AIC), homogeneity of the residuals and significance of the regression parameters. The results of the study indicated that the height-diameter relationship can best be described by non-linear models. Out of the evaluated models, at least 15 models were found to give reasonable results for each species with Ra2range of 0.73 – 0.89 and RMSE of 0.5 – 0.9 m. The number of individual model parameters was found to have little or no significant influence on the ranking of the model performance except for the Acacias. The best five models for each species and their ranking in terms of goodness of fit differ from species to another with each species having different combination of models. Comparison of the range of data for the five studied species suggested that representation of various diameter-height classes had significant influence on the accuracy of prediction outside the range of the fitted data. This implies that application of the selected models is only useful at local stand level or at best in similar biological and stand structure conditions.

Index Terms— Natural reserved forest, Height–diameter relationship, modelling, non-linear regression, model evaluation.

I. INTRODUCTION

T

he relationship between diameter at breast height (dbh) (1.3 m) and total height is a structural characteristic of a tree that describes key elements of stem form, and thus the volume of the harvestable stem. The diameter to height (d-h) relationship also affects product quality, as it influences wood structure such as lignin and cellulose content, and thus important properties of the stem (e.g., stiffness) [1]. The dbh and total height are the commonly measured variables in forest inventories as they are frequently required for both routine forest management activities and for research purposes. As such, accurate data on these two variables is quite important especially for reliable future management plans. Unlike dbh, total height is less frequently used for construction or application of forest models because measurement of dbh is more cost effective, easy and accurate than total height [2], [3]. Routine forest management usually requires updating forest growth parameters at regular intervals to refresh the future plans and decisions as affected by market and environmental variables. Investment in forest inventory operations usually weighed against the tangible benefits gained from the information obtained i.e. the direct economic return from the concerned forest. Natural forests in Sudan are generally known to be under the public ownership supervised by the local government authorities. The control over such types of forests and protective measures offered to them are generally poor due to the absence of up-to-date growth information. Any effort to ease and reduce the cost of forest inventories in such cases would be highly appreciated and could save the existence of a number of very valuable indigenous tree species

35 J OURNAL OF FOREST PRODUCTS & INDUSTRIES , 2013, 2(1), 34-42 ISSN:2325 â€“ 4513(PRINT) ISSN 2325 - 453X (ONLINE )

in these forests. The main objective of this study is therefore to select the most suitable model that could efficiently estimate tree heights without incurring unaffordable costs and time. Height-diameter relationships are used to estimate the heights of trees measured for their diameter at breast height. Such relationship describes the correlation between height and diameter of the trees in a stand on a given date and can be represented by a linear or non-linear mathematical model. In forest inventory designs diameter at breast height is measured for all trees within sample plots, while height is measured for only some selected trees, normally the dominant ones in terms of their dbh. However, for height-diameter models, more care is needed and a representative sample of accurately measured total-height is used as the response variable and dbh as the predictor variable. A major difficulty in modelling the height-diameter relationship is the large number of variables influencing it and thus hindering the construction of generic models based on empirical methods such as linear and nonlinear regression [4]. Height-diameter equations can either be used for local application or they can have a more generalised use. Equations dependant on only tree diameters, without considering other stand variables, are only applicable locally to the stand where the height-diameter data were gathered. A large number of local tree height-diameter equations have been developed and/or validated and reported in the literature (e.g. [5], [6], [7], [3]). The generalized height-diameter equations, on the other hand, are normally functions of tree diameter in addition to other stand variables, and as such can be applied in similar conditions at the regional level [8]. Numerous examples of generalized height-diameter equations were tested. In addition to dbh as independent variable, reference [7] used Crown diameter, [9] used competition factor (mean neighbouring tree

Tree species Acacia senegal (AS) Acacia seyal (AY) Anogeissus leocarpus (AL) Balanites aegyptiaca (BE) Combretum hartmannianum (CH)

distance, frequency of the neighbouring tree and position of the crown), [10] used stand age, site index and altitude, [8] used dominant height, dominant under bark diameter at breast height, [11] used density and developmental effects, [12], and [13] used trees/ha and basal area, [4] used tree age and genetic material, and [14] used stocking. Statistical modelling has to balance simplicity (i.e., fewer parameters in a model, lower variability in the predicted response, but with more modelling bias) against complexity (i.e., more parameters in a model, higher variability in the predicted response, but with smaller modelling bias). According to reference [15] Statistical model selection criteria have to seek a proper balance between overfitting (i.e., a model with too many parameters, more than actually needed) and underfitting (i.e., a model with too few parameters, not capturing the right signal).

II. MATERIALS AND METHODS The data for this study was extracted from an inventory carried out in El Nour Natural Forest Reserve for the purpose of updating its management plan. The forest is located in the Blue Nile State, Sudan, between longitudes 11o 48/ 19// N and 11o 53/ 30// N and latitudes 34o 28/ 47// E and 34o 32/ 35// E. The forest hosts more than 55 tree species dominated by Acacia seyal, Sterculia setigera, Balanites aegyptiaca, Anogeissus leocarpus, Combretum hartmannianum and Terminalia brownii.

The selection of the species for the study was mainly based on their economic importance (Table 1) to the local people around the forest [16]. The data for the selected species consists of dbh >7cm and total tree heights of dominant trees (Table 2).

Table1 local uses of the selected tree species Local uses fuelwood, dune control, gums, charcoal, honey, timber, fodder, fruit, medicine, hedging, agroforestry fuelwood, charcoal, dune control, forage, timber, gum, smoke (insects, local sauna), medicine tannin, hedging live and dead) Timber and poles, fuelwood and charcoal, soap manufacture folk medicine Timber, furniture, tools handles, firewood and charcoal, medicine, shade, fodder, , fruits, hedging, soap manufacture Fire wood, charcoal, fence posts and framework of thatched houses. Some portions of the tree are used as perfume.

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At the first stage, non-linear relationship between height and diameter was confirmed with a scattered plot diagram of height against dbh using polynomial equation of the second order. Twenty two different non-linear models described by [2], [10] and [3] (Table 3) were used to fit the height-diameter

Species AS AY AL BE CH

relationship. All these models ensure that the total height equals breast height (1.3 m) when dbh = 0. The fitting of the candidate models were carried out using “DataFit-9” statistical package of the Oakadale Engineering.

Table 2 Species diameter-height characteristics Diameter at Breast Height (cm) Total Height (m) n Max Min Range Mean Stdev n Max Min Range 172 42 8 34 20.98 8.9 172 13 5 8 2746 45 7 38 20.04 5.95 2746 12.5 5 7.5 563 83 13.3 69.7 44.81 12.78 579 18 7 11 985 60.5 10 50.5 30.62 11.13 985 15 5 10 9.45 1292 65 9.25 55.75 32.26 1292 15.4 5.83 9.57

The evaluation of the models was based on Akaike’s information criterion (AIC), root mean squared error (RMSE), adjusted coefficient of determination (Ra2), significance of parameters estimate (P<0.05) and on numerical and graphical analyses of the residuals. Model resulting in the largest Ra2, least RMSE, and smallest values of AIC and average bias was selected as the best model for the selected tree species. The formula for AIC is (-2) times the maximized log likelihood function plus 2 times the number of free parameters, with the former term describing lack of fit and the latter penalizing the number of free parameters in the model [15] and [17]. According to reference [18] AIC can also be calculated as indicated in equation 1, while RMSE is calculated as indicated in equation 2 [19]. For the model performance, a candidate showing increased values of Ra2 and

Mean 8.4 8.29 13.08 10.21 11.54

Stdev 2.16 1.32 1.88 2.15 1.61

decreased values of RMSE and AIC would be regarded as the best model. When comparing regression models that use the same dependent variable and the same estimation period, the RMSE goes down as Ra2 goes up. Hence, the model with the highest Ra2 will have the lowest RMSE, and the Ra2 could be used as a guide [20]. AIC = n*ln (RSS/n) + 2*K

(1)

RMSE = (RSS/n-P)

(2)

Where: n = Number of data points (observations) RSS = Residual sums of squares K= Number of parameters in the model P = Number of regression coefficients

Table 3 Regression models Model Formula Model Formula H = 1.3+a*x^b H=1.3+a* x/(b+x) M1 M12 H = 1.3+a*exp(b/x) H=1.3+x^2/(a+b*x)^2 M2 M13 H = 1.3+exp(a+(b*(x^c))) H=1.3+(a+b/x)^-2.5 M3 M14 H = 1.3+x^2/(a+b*x+c*(x^2)) H=1.3+(a+b/x)^-8 M4 M15 H = 1.3+a*x^(b+(c*x)) H=1.3+a* (1+(1/x))^-b M5 M16 H = 1.3+a*(x^2)/(x+b)^2 H=1.3+exp(a+b/x) M6 M17 H = 1.3+x^2/(a+b*x)^2 H=1.3+a*(ln(1+x))^b M7 M18 H = 1.3+(a+b/x)^-5 H=1.3+a*(1-exp(b*x^c)) M8 M19 H = 1.3+a*(1-exp(b*x))^c H=1.3+a/(1+b^-1* x^-c) M9 M20 H= 1.3+(exp(a+(b/(x+1)))) H=1.3+exp(a+b* (x^-c)) M10 M21 H= 1.3+(a*exp(-b*(x^-c))) H=1.3+x^a/(b+(c*(x^a))) M11 M22 H = total height; dbh = diameter at breast height; and a, b, c = regression coefficients; exp = exponent

37 J OURNAL OF FOREST PRODUCTS & INDUSTRIES , 2013, 2(1), 34-42 ISSN:2325 â€“ 4513(PRINT) ISSN 2325 - 453X (ONLINE ) prediction, or produced a height value greater than 1.3 m at zero dbh. The successful models showed different rankings of III. RESULTS AND DISCUSSION Tables 4 - 8 list, in descending order, the fit statistics of the performance with different type of tree species. Similarly, five best models for the five selected species, while Figures 1- there was great variation between the values of the regression 7 display the corresponding height curves predicted by the parameters of the same model across the species. In terms of same models and residual plots of some selected models. In prediction error, the entire range of the successful models general terms, at least fifteen out of the twenty two fitted yielded very low standard error values of one meter or less. models were found to give satisfactory results with R2 range of Such results confirm the positive relationship between the dbh 0.73 - 0.87. The rest of the models were either failed to give and the total tree height and the possibility of predicting tree any results, or produced some negative values of height height from the dbh ([2], [7], [3]). Table 4 Regression parameter estimates and fit statistics for Acacia senegal Model (n = 172) Regression coefficients Code S. Error R^2 Ra^2 RMSE AIC Parameter Value S. Error t-ratio Prob. (t) 0.9075 0.0549 16.5227 0.0000 a 0.7256 0.8887 0.8881 0.7256 -106 E1 0.6823 0.0188 36.2392 0.0000 b 99.2204 127.02 0.7811 0.4358 a 5.7900 0.5887 9.8346 0.0000 E11 0.7535 0.8807 0.8793 0.7535 -94 b 0.2627 0.1303 2.0168 0.0453 c 0.6008 0.0456 13.1688 0.0000 a E18 0.7520 0.8805 0.8798 0.7520 -94 2.2193 0.0648 34.2635 0.0000 b 25.1667 1.6152 15.5809 0.0000 a E12 0.7525 0.8803 0.8796 0.7525 -94 50.8887 4.8353 10.5245 0.0000 b 2.6875 0.0923 29.1054 0.0000 a 0.8144 0.8599 0.8590 0.8144 -67 E7 0.2373 0.0040 59.4007 0.0000 b Table 5 Regression parameter estimates and fit statistics for Acacia seyal Model (n =2746) Regression coefficients Code S. Error R^2 Ra^2 RMSE AIC Parameter Value S. Error t-ratio Prob. (t) 1.3276 0.0231 57.5425 0.0000 a 0.6055 0.7891 0.7890 0.6055 -2751 E1 0.5576 0.0057 98.3546 0.0000 b 61.922 26.214 2.3621 0.0182 a 4.6073 0.2364 19.4885 0.0000 E11 0.6181 0.7804 0.7802 0.6181 -2640 b 0.2517 0.0485 5.1870 0.0000 c 1.0118 0.0212 47.6859 0.0000 a E18 0.6187 0.7798 0.7797 0.6187 -2633 1.7503 0.0185 94.3791 0.0000 b 16.106 0.2203 73.1153 0.0000 a E12 0.6297 0.7719 0.7718 0.6297 -2536 25.202 0.6305 39.9738 0.0000 b 1.9827 0.0239 83.0376 0.0000 a 0.6535 0.7543 0.7542 0.6535 -2332 E7 0.2742 0.0012 233.663 0.0000 b Table 6 Regression parameter estimates and fit statistics for Anogeissus leocarpus Model (n =579) Regression coefficients Code S. Error R^2 Ra^2 RMSE AIC Parameter Value S. Error t-ratio Prob. (t) 103.75 73.722 1.4074 0.1599 a 5.0623 0.3221 15.7151 0.0000 b E11 0.9582 0.8068 0.8062 0.9582 -47 0.2237 0.0708 3.1592 0.0017 c 0.9581 0.0560 17.0954 0.0000 a E18 0.9575 0.8068 0.8064 0.9575 -46 1.8809 0.0434 43.3705 0.0000 b 1.7764 0.0773 22.9946 0.0000 a E1 0.9637 0.8042 0.8039 0.9637 -39 0.4996 0.0113 44.1765 0.0000 b 22.840 0.5309 43.0255 0.0000 a E12 0.9691 0.8021 0.8017 0.9691 -32 40.387 1.9897 20.2978 0.0000 b 2.8357 0.0779 36.3904 0.0000 a E7 1.0017 0.7885 0.7881 1.0017 6 0.2245 0.0018 127.010 0.0000 b

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Table 7 Regression parameter estimates and fit statistics for Balanites aegyptiaca Model (n =985) Regression coefficients Code S. Error R^2 Ra^2 RMSE AIC Parameter Value S. Error t-ratio Prob. (t) 3.1513 2.2703 1.3880 0.1654 a 1.6886 0.1648 10.2473 0.0000 b E4 0.9628 0.7990 0.7986 0.9628 -72 0.0497 0.0027 18.1362 0.0000 c 21.674 0.4977 43.5476 0.0000 a E12 0.9633 0.7986 0.7984 0.9633 -70 41.469 1.7039 24.3379 0.0000 b 2.8350 0.0517 54.8488 0.0000 a E7 0.9645 0.7981 0.7979 0.9645 -67 0.2353 0.0016 143.931 0.0000 b 2.8350 0.0517 54.8488 0.0000 a E13 0.9645 0.7981 0.7979 0.9645 -67 0.2353 0.0016 143.931 0.0000 b 18.063 0.2510 71.9653 0.0000 a E6 0.9645 0.7981 0.7979 0.9645 -67 12.049 0.2999 40.1701 0.0000 b

Code E11 E14 E8 E10 E15

Table 8 Regression parameter estimates and fit statistics for Combretum hartmannianum Model (n =1292) Regression coefficients S. Error R^2 Ra^2 RMSE AIC Parameter Value S. Error t-ratio Prob. (t) 18.678 0.7226 25.847 0.0000 a 9.9508 1.1286 8.8172 0.0000 b 0.5807 0.8700 0.8698 0.5807 -1401 0.8216 0.0521 15.7705 0.0000 c 0.3159 0.0009 355.910 0.0000 a 0.5810 0.8697 0.8696 0.5810 -1399 2.3910 0.0284 84.2128 0.0000 b 0.5657 0.0007 798.179 0.0000 a 0.5810 0.8697 0.8696 0.5812 -1399 1.8912 0.0223 84.7208 0.0000 b 2.8377 0.0058 487.391 0.0000 a 0.5815 0.8695 0.8694 0.5815 -1397 -16.015 0.1870 -85.643 0.0000 b 0.7014 0.0005 1328.52 0.0000 a 0.5815 0.8695 0.8694 0.5815 -1397 1.4028 0.0165 84.8551 0.0000 b

Fig. 1: Acacia senegal predicted heights beyond the observed data

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Fig. 2: Acacia seyal predicted heights beyond the observed data

Fig. 3: Anogeissus leocarpus predicted heights beyond the observed data

Fig.4: Balanites aegyptiaca predicted heights beyond the observed data

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Fig. 5: Comb. hartmannianum predicted heights beyond the observed data

Fig 6: Residual plot of some selected models

Fig 7: Residual plot of some selected models

41 J OURNAL OF FOREST PRODUCTS & INDUSTRIES , 2013, 2(1), 34-42 ISSN:2325 – 4513(PRINT) ISSN 2325 - 453X (ONLINE ) In terms of individual models performance, Model 11 (M11) was found to be either the best or the second best height predictor for all species except one. M1 was the best predictor for two of the five species, while M11 was the best predictor for another two species and M4 for the remaining species. M1, M11 and M18 were found to be among the best three models for three of the species but in different ranking order, while M12 and M7 were found to maintain the 4th and 5th ranking position respectively for the same three species. In general, only M1, M4, M11, M12, M14 and M18 were able to maintain the best or the second best performance level for all the five tree species. Species wise E4 was found to be the best predictor model for A. senegal in terms of the evaluation criteria RMSE, Ra2 and AIC. However, this model was eliminated as it was found to underestimate height of small diameter classes and overestimate those of the upper diameter classes, especially outside the observed data. Similarly E21 was also eliminated as best predictor for A. seyal as it was found to overestimate heights over all diameter classes and end up with height value of greater than 1.3 m at dbh value of zero.

was found to yield insignificant effect on the quality of their prediction. The results of the study indicated that a successful predictor model for a given species might not be so for other species growing in the same forest under similar conditions. The study also suggested that prediction outside the range of fitting data might give misleading or unsatisfactory results. Another fact revealed by this study was that multiple criteria for examining the quality of fit would be much better to cast any doubts concerning the quality of the tested models. In General, M1, M4, M11 and M18 are recommended for the investigated species and their prediction quality could further be improved by the inclusion of other stand and growth variables. REFERENCES [1] Kroon, J., Andersson, B. and Mullin, T. J. (2008). Genetic variation in the diameter–height relationship in Scots pine (Pinus sylvestris). Canadian Journal of Forest Research, 2008, 38(6): 1493-1503, 10.1139/X07-233. [2]

It is clear from the results that within the best five models for all the species only two models (M4 and M11) have three regression coefficients (a, b, c), while the rest have only two (a and b). All the parameters for the two-parameter models were statistically significant (p<0.0001) compared to those of M4 and M11. However, in all the cases where M4 or M11 occupied a better ranking position than the two-parameter models, there was no significant difference in their values of RMSE, Ra2 or AIC compared to those of the models below them. In addition, some of the regression parameters of these two models (M4, M11) were statistically insignificant with large values of standard errors (42% – 128%). Such results will give preference to the two-parameter models to avoid the unnecessary overfitting. According to [19] preference should go for the model with fewer parameters, other things being approximately equal.

Moore, J. A., Zhang, L. and Stuck, D. (1996). Height-Diameter Equations for Ten Tree Species in the Inland Northwest WJAF 11(4).

[3] Sharma, R.P. (2009).Modelling height-diameter relationship for Chir pine trees. Banko Janakari, Vol. 19, No. 2. [4] Guimarães, M. A. M., Calegário,N., Carvalho, L. M. T. and Trugilho, P. F. (2009). Height-diameter models in forestry with inclusion of covariates. Cerne, Lavras, v. 15, n. 3, p. 313-321. [5] Nowak, D. J. (1990). Height-diameter relations of Maple street trees. Journal of Arboriculture 16(9): 231-235. [6] Walter Zucchini, W., Schmidt, M. and Klaus, K. V. (2001). A Model for the Diameter-Height Distribution in an Uneven-Aged Beech Forest and a Method to Assess the Fit of Such Models. Silva Fennica 35(2), 169183. [7] Avsar M. D. (2004). The relationship between Diameter at breast height, tree height and crown diameter in Calabrain Pinus (Pinus brutia Ten) of Baskonus Mountain, Kahramanmaras, Turkey. Journal of Biological Science 4(4): 437-440, 2004.

Figures 1 – 5 display graphical representation of the predicted height curves given by top five successful models for each tree species. Examination of these height curves reveals that all the successful models performed quite well within the range of the observed data. Prediction outside this range produced different responses by different models and species especially at the extreme ends of the curves. However, the best sets of models were the predictors of B. aegyptiaca and C. hartmannianum where almost all the models overlap as one curve. Such trend could be attributed to the quality of the sample data or to some unexplained effects that need further investigation to improve the performance of prediction for the other species.

[8] González, S. M., Cañellas, I Montero, G (2007). Generalized heightdiameter and crown diameter prediction models for cork oak forests in Spain. Investigación Agraria: Sistemas y Recursos Forestales, 16(1), 76-88. ISSN: 1131-7965

IV. CONCLUSIONS

[12] Sharma, M. and Parton, J. (2007). Height–diameter equations for boreal tree species in Ontario using a mixed-effects modelling approach. Forest Ecology and Management 249 (2007) 187–198. [13] Saunders, M. R. and Wagner, R. G. (2008). Height-diameter models with random coefficients and site variables for tree species of Central Maine. Annals of Forest Science - ANN FOR SCI , vol. 65, no. 2, pp. 203-203, 2008.

The height-diameter model developed in this study gave reasonably precise estimates of tree heights and could be used to predict the height of the species under consideration. Inclusion of additional prediction parameter to some models

[9]

Dauda. T. O., Ojo, L. O. and Nokoe, S. K (2004).Unexplained relationships of height-diameter of three species in a tropical forest. Global Nest: the Int. J. Vol 6, No 3, pp 196-204. [10] Zhao, W., Mason, E.G., and Brown, J. (2006) Modelling heightdiameter relationships of Pinus radiata plantations in Canterbury, New Zealand, New Zealand Journal of Forestry 51 (1): 23-27. [11] Newton, P. F. and Amponsah, I. G. (2007). Comparative evaluation of five height–diameter models developed for black spruce and jack pine stand-types in terms of goodness-of-fit, lack-of-fit and predictive ability. Forest Ecology and Management, 247(1-3), 149-166.

42 J OURNAL OF FOREST PRODUCTS & INDUSTRIES , 2013, 2(1), 34-42 ISSN:2325 â€“ 4513(PRINT) ISSN 2325 - 453X (ONLINE ) [14] Vanclay, J. K. (2009). Tree diameter, height and stocking in even-aged forests. Ann. For. Sci. 66 (2009) 702. [15] Yang, H. and Bozdogan, H. (2011). Model Selection with Information Complexity in Multiple Linear Regression Modeling. Multiple Linear Regression Viewpoints, 2011, Vol. 37(2). [16]

Osman E. h. and El Zain A. Idris (2012). Species Dynamics and Potential Disturbances in El Nour Natural Forest Reserve, Sudan. journal of Forest products and industries, 2012, 1(2), 10-20, ISSN:2166. [17] Burnham, K. P., and Anderson, D.R. (2004), "Multimodel inference: understanding AIC and BIC in Model Selection", Sociological Methods and Research, 33: 261-304.

[18] Motulsky, H. J. and Christopoulos, A. (2003). fitting models to biological data using linear and non-linear regression. a practical guide to curve fitting. GraphPad software Inc., San Diago, CA 2 nd ed. [19] Beal, D. J. (2005). SAS Code to Select the Best Multiple Linear Regression Model for Multivariate Data Using Information Criteria. Paper SA01_05. [20] Nau, R. F (2005). How to compare models. http://people.duke.edu/~rnau/compare.htm

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