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Approximate Inferences for Nonlinear Mixed Effects Models with Scale Mixtures of Skew-Normal Distributions

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Abstract

Nonlinear mixed effects models have received a great deal of attention in the statistical literature in recent years because of their flexibility in handling longitudinal studies, including human immunodeficiency virus viral dynamics, pharmacokinetic analyses, and studies of growth and decay. A standard assumption in nonlinear mixed effects models for continuous responses is that the random effects and the within-subject errors are normally distributed, making the model sensitive to outliers. We present a novel class of asymmetric nonlinear mixed effects models that provides efficient parameters estimation in the analysis of longitudinal data. We assume that, marginally, the random effects follow a multivariate scale mixtures of skew-normal distribution and that the random errors follow a symmetric scale mixtures of normal distribution, providing an appealing robust alternative to the usual normal distribution. We propose an approximate method for maximum likelihood estimation based on an EM-type algorithm that produces approximate maximum likelihood estimates and significantly reduces the numerical difficulties associated with the exact maximum likelihood estimation. Techniques for prediction of future responses under this class of distributions are also briefly discussed. The methodology is illustrated through an application to Theophylline kinetics data and through some simulating studies.

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Acknowledgements

We are grateful to the anonymous referee, the associate editor and the editor for very useful comments and suggestions, which greatly improved this paper. Fernanda L. Schumacher acknowledges the partial support of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico—Brasil (CNPq).

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Authors and Affiliations

  1. Department of Statistics, Campinas State University, Campinas, Brazil

    Fernanda L. Schumacher

  2. Department of Statistics, University of Connecticut, Storrs, CT, 06269, USA

    Dipak K. Dey & Victor H. Lachos

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  1. Fernanda L. Schumacher
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  2. Dipak K. Dey
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  3. Victor H. Lachos
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Correspondence to Dipak K. Dey.

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This article is part of the topical collection “Celebrating the Centenary of Professor C. R. Rao” guest edited by, Ravi Khattree, Sreenivasa Rao Jammalamadaka, and M. B. Rao.

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Schumacher, F.L., Dey, D.K. & Lachos, V.H. Approximate Inferences for Nonlinear Mixed Effects Models with Scale Mixtures of Skew-Normal Distributions. J Stat Theory Pract 15, 60 (2021). https://doi.org/10.1007/s42519-021-00172-5

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