fully exponential laplace approximation em algorithm for nonlinear mixed effects models

204 Pages·2017·2.64 MB·English

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Dissertations and Theses in Statistics Statistics, Department of
12-2009
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Meijian Zhou
University of Nebraska-Lincoln, [e-mail]
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Zhou, Meijian, "FULLY EXPONENTIAL LAPLACE APPROXIMATION EM ALGORITHM FOR NONLINEAR
MIXED EFFECTS MODELS" (2009). Dissertations and Theses in Statistics. 4.
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FULLY EXPONENTIAL LAPLACE APPROXIMATION EM ALGORITHM
FOR NONLINEAR MIXED EFFECTS MODELS

by

Meijian Zhou

A DISSERTATION

Presented to the Faculty of

The Graduate College at the University of Nebraska

In Partial Fulfillment of Requirements

For the Degree of Doctor of Philosophy

Major: Statistics

Under the Supervision of Professor Anne M. Parkhurst

Lincoln, Nebraska

December, 2009
FULLY EXPONENTIAL LAPLACE APPROXIMATION EM ALGORITHM
FOR NONLINEAR MIXED EFFECTS MODELS

Meijian Zhou, Ph.D.
University of Nebraska, 2009

Advisor: Anne M. Parkhurst

Nonlinear mixed effects models provide a flexible and powerful platform for the analysis
of clustered data that arise in numerous fields, such as pharmacology, biology, agriculture,
forestry, and economics. This dissertation focuses on fitting parametric nonlinear mixed
effects models with single- and multi-level random effects. A new, efficient, and
accurate method that gives an error of order O(1/n2) , fully exponential Laplace
approximation EM algorithm (FELA-EM), for obtaining restricted maximum likelihood
(REML) estimates in nonlinear mixed effects models is developed. Sample codes for
implementing FELA-EM algorithm in R are given. Simulation studies have been
conducted to evaluate the accuracy of the new approach and compare it with the Laplace
approximation as well as four different linearization methods for fitting nonlinear mixed
effects models with single-level and two-crossed-level random effects. Of all
approximations considered in the thesis, FELA-EM algorithm is the only one that gives
unbiased or close-to-unbiased (%Bias < 1%) estimates for both the fixed effects and
variance-covariance parameters. Finally, FELA-EM algorithm is applied to a real dataset
to model feeding pigs’ body temperature and a unified strategy for building crossed and
nested nonlinear mixed effects models with treatments and covariates is provided.
iii
ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Anne M. Parkhurst, for offering me a graduate
research assistantship [1,2,3] and for all her inspiring guidance, full support and patience
throughout this research project.

I would like to thank my supervisory committee, Dr. Kent M. Eskridge and Dr. Shunpu
Zhang for giving helpful advice and perceptive comments on my dissertation, and Dr.
Roger A. Eigenberg for sharing his data and helping me understand his research and
experiments on heat dissipation in swine.

I would like to thank B. C. Pollard and Dr. Robert J. Collier from University of Arizona
and Dr. John A. Nienaber and Dr. G. Leroy Hahn from U.S. Meat Animal Research
Center for sharing their data to allow me to pursue my interest in applications of
nonlinear mixed effects models.

I would also like to thank Dr. David A. Fournier for helping me develop codes for
crossed and nested nonlinear mixed models in ADMB-RE and Dr. Oliver Schabenberger
for providing valuable advice on fitting nonlinear mixed models with both crossed and
nested random effects in the SAS macro %NLINMIX.

Special thanks to my wife Hui Shen, my daughters, Amy Zhou and Claire Zhou, my
parents, Mr. Yiben Zhou and Mrs. Yuehua Zhu, my sisters, Guangqin Zhou and Xueqin
Zhou and their families, for their love and support during the five years of study.

Reference
[1] U.S. Department of Agriculture, Agricultural Research Division
[2] Institute of Agriculture and Natural Resources, University of Nebraska-Lincoln
[3] Multi-State Research Project W173: Stress Factors of Farm Animals and Their
Effects on Performance
iv
TABLE OF CONTENTS

Acknowledgments............................................................................................................. iii
Table of Contents ............................................................................................................... iv
List of Figures ................................................................................................................. viii
List of Tables ..................................................................................................................... ix

1 Introduction ................................................................................................................. 1
1.1 Motivation ............................................................................................................ 1
1.2 Summary of the remaining chapters .................................................................... 4
1.3 References ............................................................................................................ 5

2 Literature Review ....................................................................................................... 8
2.1 Introduction .......................................................................................................... 8
2.2 Linear mixed effects models .............................................................................. 10
2.3 Nonlinear regression models.............................................................................. 11
2.4 Nonlinear mixed effects models ........................................................................ 12
2.5 Four categories of nonlinear mixed effects models ........................................... 13
2.5.1 Parametric nonlinear mixed effects models ........................................... 13
2.5.2 Nonparametric nonlinear mixed effects models .................................... 14
2.5.3 Semi-parametric nonlinear mixed effects models .................................. 15
2.5.4 Bayesian approach to nonlinear mixed effects models .......................... 15
v
2.6 Estimation methods of parametric nonlinear mixed effects models .................. 16
2.6.1 Linearization methods ............................................................................ 17
2.6.2 Integral approximation methods ............................................................ 19
2.6.3 EM algorithms ....................................................................................... 24
2.7 Software review ................................................................................................. 27
2.8 References .......................................................................................................... 32

3 REML Estimation in Nonlinear Mixed Effects Models via the Fully Exponential
Laplace Approximation EM Algorithm .................................................................. 41
3.0 Abstract .............................................................................................................. 41
3.1 Introduction ........................................................................................................ 42
3.2 Model and likelihood ......................................................................................... 45
3.3 FELA-EM algorithm for REML estimates of variance-covariance
components ........................................................................................................ 47
3.3.1 E-step ..................................................................................................... 48
3.3.2 M-step .................................................................................................... 50
3.3.3 Fully exponential Laplace approximation ............................................. 51
3.3.4 Calculating the information matrix ........................................................ 56
3.3.5 Estimating the fixed and random effects ............................................... 57
3.4 Comparing the approximations .......................................................................... 58
3.4.1 Logistic model ....................................................................................... 59
3.4.2 First-order compartment model ............................................................. 71
3.5 Discussion .......................................................................................................... 81
vi
3.6 Conclusions ........................................................................................................ 83
3.7 Summary ............................................................................................................ 85
3.8 References .......................................................................................................... 86

4 Extension of the Fully Exponential Laplace Approximation EM Algorithm for
Nonlinear Mixed Models with two Levels of Crossed Random Effects ............... 90
4.0 Abstract .............................................................................................................. 90
4.1 Introduction ........................................................................................................ 92
4.2 Model and likelihood ......................................................................................... 95
4.3 Laplace approximation to the likelihood ........................................................... 97
4.4 FELA-EM algorithm ........................................................................................ 100
4.4.1 E-step ................................................................................................... 101
4.4.2 M-step .................................................................................................. 104
4.4.3 Fully exponential Laplace approximation ........................................... 106
4.4.4 Calculating the information matrix ...................................................... 111
4.4.5 Estimating the fixed and random effects ............................................. 111
4.5 Comparing the approximations ........................................................................ 112
4.5.1 Logistic model ..................................................................................... 112
4.5.2 First-order compartment model ........................................................... 122
4.6 Discussion ........................................................................................................ 129
4.7 Conclusions ...................................................................................................... 131
4.8 Summary .......................................................................................................... 132
4.9 References ........................................................................................................ 133
vii
5 Multilevel Nonlinear Mixed Effects Models with both Crossed and Nested
Random Effects Applied in a Replicated Latin Square Design for Modeling
Temperature of Feeding Pigs ................................................................................. 137
5.0 Abstract ............................................................................................................ 137
5.1 Introduction ...................................................................................................... 138
5.2 Materials and methods ..................................................................................... 141
5.2.1 Data ...................................................................................................... 141
5.2.2 Statistical model ................................................................................... 144
5.2.3 Crossed and nested random effects ...................................................... 146
5.2.4 Model building ..................................................................................... 146
5.2.5 Review of FELA-EM algorithm .......................................................... 149
5.3 Results and discussion ..................................................................................... 155
5.3.1 Specification of random effects ........................................................... 155
5.3.2 Specification of within-event error correlation structure ..................... 157
5.3.3 Model diagnostics ................................................................................ 158
5.3.4 Comparison of the three thermal environmental treatments
and test of the feed intake and meal duration effects ........................... 162
5.4 Conclusions ...................................................................................................... 165
5.5 Summary .......................................................................................................... 166
5.6 References ........................................................................................................ 167

Appendix: R program for fitting the logistic model formulated by the equation
(4.4.1) using FELA-EM algorithm ........................................................................ 171
viii
LIST OF FIGURES

3.1 Example of simulated logistic curves for two settings of variance and
covariance parameters .............................................................................................. 60
3.2 Example of simulated first-order compartment curves for two settings
of variance and covariance parameters .................................................................... 72

5.1 Example of changes in tympanic temperature (ºC) and feed intake (kg)
of pigs over Julian calendar time for pig 27 (a member of the heavy
group) during first experimental period in the second run under
treatment 2 (28ºC and high air speed) .................................................................... 143
5.2 Autocorrelation function corresponding to the within-event errors of
Model 7 .................................................................................................................. 157
5.3 Autocorrelation function corresponding to the within-event errors of
Model 9 .................................................................................................................. 157
5.4 Scatter plot of standardized residuals versus fitted values for Model 9 ................ 160
5.5 Normal plot of standardized residuals for Model 9 ............................................... 160
5.6 Normal plot of the estimated random effects for Model 9 ..................................... 160
5.7 Observed (○) and predicted (―) tympanic temperatures (C) over
time (min) for eighteen feeding events .................................................................. 161
ix
LIST OF TABLES

3.1 Simulation results for the fixed effects in the logistic model for
small D and σ2 ...................................................................................................... 64
3.2 Simulation results for the fixed effects in the logistic model for
large D and σ2 ....................................................................................................... 65
3.3 Simulation results for the variance-covariance components in the
logistic model for small D and σ2 ......................................................................... 69
3.4 Simulation results for the variance-covariance components in the
logistic model for large D and σ2 ......................................................................... 70
3.5 Simulation results for the fixed effects in the first-order compartment
model for small D and σ2 ..................................................................................... 75
3.6 Simulation results for the fixed effects in the first-order compartment
model for large D and σ2 ...................................................................................... 76
3.7 Simulation results for the variance-covariance components in the
first-order compartment model for small D and σ2 .............................................. 79
3.8 Simulation results for the variance-covariance components in the
first-order compartment model for large D and σ2 ............................................... 80

4.1 Simulation results for the fixed effects in the logistic model ................................ 117
4.2 Simulation results for the variance-covariance components in the
logistic model ......................................................................................................... 120