Review of methods of estimating parameters in Nonlinear mixed-effects (nlme) models

Abstract

This study is a critical review of theoretical issues that underline the linear mixed effects (LME) and nonlinear mixed effects (NLME) models. These two areas are revisited under maximum likelihood and restricted maximum likelihood estimation frameworks. We also review methods of estimating parameters in both linear and nonlinear mixed effects models. In the case of LME, we consider different ways of developing the likelihood estimators, key among these methods are the "pseudo-data" approach, orthogonal triangular decomposition method and the use of penalized least squares problem. For NLME, we intended to investigate the computational efficiency and accuracy of com­ putational methods, like the b-splines, that could be used to approximate the log-likelihood function in non-linear mixed effects models. This was not achieved in this study but can be an interesting area for further research work. We critically review the four methods of estimating parameters by Pinheiro and Bates (1995) through proving a number of lemmas. Our proves led us to same stated results by different researchers in different papers. This is a key issue in the investigation of other expansion methods and comparing their computational efficiency and accuracy with these existing ones. We conclude by giving an insight into linear mixed effects models by analyzing a data set from livestock where we examine incorporation of random effects to study variations among rams (sires) and ewes (dams) and their influences on lamb weaning weight. Factors like year of birth of the lamb, sex of lamb, age at weaning, age of dam, ewe breed and ram breed are found to influence the weaning weight differently. With the random terms (ewes and rams) specified in the model the estimate of the residual among lamb variance is found to reduce due to taking into account the variations among rams and ewes within breeds. It was our intention to obtain heritability estimates which determine the proportion of the variation among offspring that have been handed down from parents out of these random estimates. Keywords: repeated-measures data, multilevel data, longitudinal data, LME, NLME, "pseudo­ data" and b-splines.