Departament d’Estadística Nonlinear Mixed-effects Models and Nonparametric Inference. A Method Based on Bootstrap for the Analysis of Non-normal Repeated Measures Data in Biostatistical Practice. Rachid El Halimi ii Programa de Doctorat Probabilitats i Estadística Bienni 1997-99 Memòria presentada per a optar al grau de Doctor en Matemàtiques Certifico que la present memòria ha estat realitzada per Rachid el Halimi, al Departament d’Estadística de la Universitat de Barcelona sota la meva direcció Prof. Jordi Ocaña Rebull Catedràtic del Departament d’Estadística Universitat de Barcelona Barcelona, a 1 de juny de 2005 iii iv Acknowledgements This work is the result of many years of effort and would not have been possible without the assistance of many people who gave their support in different ways. To them, I would like to convey my gratitude and sincere appreciation. I have been indebted with my supervisor, Dr. JORDI OCAÑA REBULL, whose guidance throughout the various revisions of this thesis, his previous instruction, suggestions, encouragement, constant availability, incredible patience and support in an unusual extent, as well as his academic experience, have been invaluable to me. He will always be remembered as the key factor that geared my career towards this path. It has been an honor and a pleasure working under his guidance. My sincere appreciation to Dra. MARIA DEL CARME RUIZ DE VILLA and Dr. CARLES M. CUADRAS AVELLANA from the Department of Statistics, University of Barcelona, for their recommendations and generous cooperation to obtain some real data sets. They contributed to greatly improve this work. I would like also to extend my sincere appreciation to all professors and students of the Department of Statistics, Faculty of Biology, University of Barcelona, for their support during the realization of this project. It is a pleasure to acknowledge their extraordinary talents. Special thanks to Dr. EDUARD ESCRICH, Department of Cellular Biology, Physiology and Immunology, Faculty of Medicine, Universitat Autónoma de Barcelona, Dra. GENOVEVA MONTSERRAT, Department of Productes Naturals, Faculty of Pharmacy, University of Barcelona, Dra. M.CRISTINA POBLET, Department of Economic Theory, University of Barcelona, and to MANUEL VARAS, PhD student and friend, from the Department of Ecology, Faculty of Biology, University of Barcelona. Their permission to use their data sets, and their reviews and helpful comments were decisive in the realization of this work. v My colleagues from the University of Pompeu Fabra deserve special thanks for giving me access to their computers. Thanks to their collaboration, many computer-intensive simulations presented in this thesis were possible. This thesis was possible thanks to the financial support of project FIS 00/1130 by the Instituto de Salud Carlos III and 2001/SGR00067 from the Generalitat de Catalunya,. vi CONTENTS vii viii CONTENTS Acknowledgements ---------------------------------------------------------------------------------------------------v 0. RESUMEN EN CASTELLANO ------------------------------------------------------------------------- xiii 0.1. Análisis de datos reales usando modelos mixtos------------------------------------------------ xiv 0.2. Métodos Establecidos --------------------------------------------------------------------------------xxiii 0.3. Robustez Del Enfoque Paramétrico --------------------------------------------------------------xxvi 0.4. Remuestreo Bootstrap sobre Modelos Mixtos no Lineales -------------------------------- xxxiv 1. INTRODUCTION AND MOTIVATION ----------------------------------------------------------------1 1.1. Introduction ------------------------------------------------------------------------------------------------1 1.2. Random two-stage model -------------------------------------------------------------------------------2 1.2.1. Nitrification data------------------------------------------------------------------------------------3 1.3. Linear mixed models -------------------------------------------------------------------------------------5 1.3.1. Global amonification data ------------------------------------------------------------------------5 1.3.2. Residual ammonification data -------------------------------------------------------------------7 1.4. Non-linear mixed effects models-----------------------------------------------------------------------8 1.4.1. Daily carbon data-----------------------------------------------------------------------------------8 1.4.2. Accumulated carbon data----------------------------------------------------------------------- 10 1.4.3. Breast cancer data-------------------------------------------------------------------------------- 10 1.5. Multilevel non-linear mixed models ---------------------------------------------------------------- 13 1.5.1. Fangar data: two levels-------------------------------------------------------------------------- 14 1.6. Discussion------------------------------------------------------------------------------------------------- 16 2. REVIEW OF STATISTICAL METHODS------------------------------------------------------------- 19 2.1. Introduction ---------------------------------------------------------------------------------------------- 19 2.2. Linear mixed model ------------------------------------------------------------------------------------ 19 2.2.1. Assumptions on random variation ------------------------------------------------------------ 20 2.2.2. Estimation of effects and covariance parameters------------------------------------------ 21 2.2.2.1. Maximum Likelihood----------------------------------------------------------------------- 21 ix 2.2.2.2. Best linear unbiased prediction---------------------------------------------------------- 23 2.2.2.3. Restricted maximum Likelihood--------------------------------------------------------- 26 2.3. Non-linear mixed effects------------------------------------------------------------------------------- 28 2.3.1. Estimation of effects and variance components-------------------------------------------- 31 2.3.1.1. Sheiner and Beal approximation -------------------------------------------------------- 32 2.3.1.2. Lindstrom and Bates approximation --------------------------------------------------- 35 2.3.1.3. STS- Standard two stage approximation---------------------------------------------- 37 2.3.1.4. GTS- Global two stage approximation ------------------------------------------------ 38 2.3.1.5. Laplace approximation -------------------------------------------------------------------- 40 2.4. Multilevel non-linear mixed models ---------------------------------------------------------------- 42 2.4.1. Likelihood function ------------------------------------------------------------------------------- 43 2.4.2. LB-Approximation-------------------------------------------------------------------------------- 45 2.4.3. Asymptotic properties of mixed models ----------------------------------------------------- 46 2.4.4. Methods for models comparison --------------------------------------------------------------- 46 2.5. Bibliographic Review----------------------------------------------------------------------------------- 49 3. SOME PARAMETRIC BIOSTATISTICAL ANALYSIS ------------------------------------------ 51 3.1. Introduction ---------------------------------------------------------------------------------------------- 51 3.2. Modelization under a parametric perspective ---------------------------------------------------- 51 3.2.1. Analysis of nitrification data ------------------------------------------------------------------- 51 3.2.1.1. Classical versus individual models ------------------------------------------------------ 51 3.2.1.2. A compromise between global and individual models: mixed models ---------- 55 3.2.2. Analysis of ammonification data -------------------------------------------------------------- 63 3.2.2.1. Ammonification global analysis---------------------------------------------------------- 63 3.2.2.2. Ammonification residual ------------------------------------------------------------------ 66 3.2.3. Analysis of daily carbon (CO2) respiration data------------------------------------------- 68 3.2.4. Analysis of accumulated carbon data -------------------------------------------------------- 73 3.2.5. Analysis of breast cancer data ----------------------------------------------------------------- 76 4. VALIDATION OF THE PARAMETRIC APPROACH ------------------------------------------- 91 x 4.1. Introduction ---------------------------------------------------------------------------------------------- 91 4.2. Simulation study on distributional assumptions ------------------------------------------------ 91 4.2.1. Breast cancer model simulation --------------------------------------------------------------- 92 4.2.1.1. Results for the estimations of the fixed effects parameters: ML- Method----- 94 4.2.1.2. Results for the estimations of the variance parameters: ML- method---------105 4.2.1.3. Results for the estimations fixed effects: REML- Method -----------------------117 4.2.1.4. Results for the estimations of the variance parameters: REML- Method ----122 4.2.1.5. Robustness of the LB method under misspecification of the structure of the covariance matrix of random effects and/or the correlation structure of the residuals --133 1.2.1.2.1. Results for fixed effects ---------------------------------------------------------------137 1.2.1.2.2. Results for variance components----------------------------------------------------143