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Chapter 3.2 144 multiple markers was considered by Morrell et al. 7 and Marshall et al. 8 who based their discrimination rule on a multivariate (non)linear mixed model (M(N)LMM). Finally, Fieuws et al. 9 describe a discrimination procedure based on a multivariate generalized linear mixed model (MGLMM) which allowed them to include discrete markers as well. All of the above approaches assume a normal distribution for the random effects in the underlying mixed model. Nevertheless, it is known that it is difficult to check this assumption which cannot be evaluated using commonly used empirical Bayes estimates of individual random effects due to their shrinkage (Verbeke and Lesaffre10 ). Consequently, Verbeke and Molenberghs11 , Chapter 7 conclude that non-normality of the random effects can only be detected by comparing the results obtained under the normality assumption with results obtained from fitting a mixed model with relaxed distributional assumptions for the random effects. Moreover, according to Komárek et al. 12 , the most promising approach for discrimination based on longitudinal profiles is based on predictors of individual random effects and the distribution of these random effects. It is therefore natural that the correct specification of the random effects distribution plays an important role. For these reasons, we are targeting a method in which the normality assumption of random effects is relaxed. A suitable semi-parametric model for an unknown distribution is a normal mixture. Verbeke and Lesaffre10 used the homoscedastic version (variances of the mixture components are equal) as a model for random effects in LMM for a single marker. This approach relaxes the strong parametric assumption of the normal random effects distribution and also allows to cluster the longitudinal profiles in the absence of a training data set. The first objective of this paper is to generalize the model of Verbeke and Lesaffre10 to (a) allow for multiple longitudinal markers in a computationally tractable manner; (b) consider more general heteroscedastic normal mixtures (variances of the mixture components are unequal) in the random effect distribution. The second objective of this paper is to apply the developed model to the training (Dutch PBC) data set and to discriminate future patients using their multivariate longitudinal profiles. The paper proceeds as follows. The first Section describes the multivariate linear mixed model with a normal mixture in the random effects distribution. This approach will be used to model in each prognostic group the longitudinal evolution of the markers and their dependence on possible covariates. The estimation procedure for the proposed model is based on the Markov chain Monte Carlo methodology and is given in Section ’Estimation’. Section ’Discrimination procedure’ explains how the fitted mixed models can be used to discriminate future patients into prognostic
Discriminant analysis using a MLMM with a normal mixture 145 groups. The methodology is illustrated on the Dutch PBC Study data in Section ’Application to PBC data’. We have also extended the R (R Development Core Team13 ) package mixAK (Komárek14 ) to apply the proposed methods in practice. The use of the package is briefly explained in the Appendix. A multivariate linear mixed model with normal mixture in the random effects distribution In the multivariate linear mixed model (MLMM, Morrell et al. 7 ), a standard linear mixed model is first specified for the r-th marker. That is, Y i,r = Xi,rαr + Zi,rbi,r + εi,r (i =1,...,N, r =1,...,R), (1) where Xi,r is a ni,r×pr covariate matrix for fixed effects and Zi,r is a ni,r×qr covariate matrix for random effects in a model for marker r. Further, αr =(αr,1,...,αr,pr ) ′ isavectoroffixedeffectsformarkerr, andbi,r =(bi,r,1,...,bi,r,qr ) ′ is a vector of random effects for marker r specific for the i-th subject. For computational convenience of the approach outlined in Section ’Estimation’, a hierarchically centered parametrization of the LMM will be used here, i.e. E(bi,r)=βr =(βr,1,...,βr,qr ) ′ with matrices (Xi,r, Zi,r) of full column rank. That is, the vector (α ′ r , β ′ r ) ′ is a vector of fixed effects in a classical sense. In the remainder of the paper we let α =(α ′ 1 ,...,α′ R )′ and β =(β ′ 1,...,β ′ R) ′ be the vectors of fixed effects and means of random effects for all considered markers, respectively. Further, let p = �R r=1 pr be the length of the vector α and q = �R r=1 qr be the length of the vector β also equal to the total dimension of random effects. Finally, εi,r =(εi,r,1,...,εi,r,ni,r )′ is the vector of random errors for the measurements of the r-thmarkeronthe i-th subject. The errors are assumed to be mutually independent and normally distributed. However, we allow the residual variances corresponding to different markers to differ. Hence, for εi =(ε ′ i,1 ,...,ε′ i,R )′ we assume εi ∼N(0, Σi), where Σi is a block-diagonal matrix with each diagonal block being equal to σ 2 r Ini,r , where σ2 r is the residual variance of the r-th marker. Note that the MLMM (1) can now be written as a standard LMM as where Xi is a ni ×p block-diagonal matrix with matrices Xi,1,...,Xi,R on the diagonal and similarly Zi is a ni × q block-diagonal matrix with matrices Zi,1,...,Zi,R on the diagonal. Y i = Xiα + Zibi + εi (i =1,...,N), (2)
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Statistical Models of Treatment Eff
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ISBN: 978-90-8559-069-9 © Bettina
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PROMOTIECOMMISSIE Promotor: Prof.dr
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CONTENTS Chapter 1. Introduction 11
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Cha pter 1 Introduction
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Chapter 1 14 Epidemiology Worldwide
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Chapter 1 16 Prediction models with
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Chapter 1 18 standard method fails
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Chapter 1 20 The extended non-linea
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Chapter 1 22 References 1. Blumberg
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Cha pter 2 Prediction of response t
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Chapter 2.1 28 ABSTRACT The aim of
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Chapter 2.1 30 been hepatitis B sur
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Chapter 2.1 32 HBV DNA decline The
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Chapter 2.1 34 Figure 2 continued.
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Chapter 2.1 36 Figure 4. Estimated
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Chapter 2.1 38 DNA was observed in
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Chapter 2.1 40 15. Kao JH. Role of
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Chapter 2.2 44 ABSTRACT Background:
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Chapter 2.2 46 low baseline HBV DNA
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Chapter 2.2 48 patient subgroups wa
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Chapter 2.2 50 p=0.002). Previous I
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Chapter 2.2 52 Application of the m
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Chapter 2.2 54 PEG-IFN, the proport
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Chapter 2.2 56 References 1. Lavanc
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Chapter 2.2 58 26. Bonino F, Lau GK
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Chap ter 2.3 Prediction of the resp
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INTRODUCTION Dynamic prediction of
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Statistical analysis Dynamic predic
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Dynamic prediction of response to P
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References Dynamic prediction of re
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Cha pter 2.4 Dynamic prediction of
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INTRODUCTION Dynamic prediction of
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For the indirect approach we rewrit
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logit pi,j = logit Pr(Ri =1|visit =
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Dynamic prediction of response usin
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Page 108 and 109: Chapter 2.5 106 ABSTRACT Peginterfe
Page 110 and 111: Chapter 2.5 108 and placebo daily.
Page 112 and 113: Chapter 2.5 110 for patients with a
Page 114 and 115: Chapter 2.5 112 Mean HBV DNA declin
Page 116 and 117: Chapter 2.5 114 peginterferon and n
Page 118 and 119: Chapter 2.5 116 hepatocellular carc
Page 120 and 121: Chapter 2.5 118 13. Werle-Lapostoll
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Page 127 and 128: Chap ter 3.1 Long-term clinical out
Page 129 and 130: INTRODUCTION Glycyrrhizin for IFN-n
Page 131 and 132: Statistics Glycyrrhizin for IFN-non
Page 133 and 134: Glycyrrhizin for IFN-non responders
Page 135 and 136: Table 2. Time dependent Cox regress
Page 143 and 144: Cha pter 3.2 Discriminant analysis
Page 145: Introduction Discriminant analysis
Page 149 and 150: Estimation Discriminant analysis us
Page 151 and 152: Discriminant analysis using a MLMM
Page 153 and 154: where wi,k(y i, θ) = (k =1,...,K).
Page 171 and 172: Appendix Discriminant analysis usin
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Page 178 and 179: Chapter 4.1 176 ABSTRACT Nucleoside
Page 180 and 181: Chapter 4.1 178 at day 1 at t =0 an
Page 182 and 183: Chapter 4.1 180 Table 1. Baseline c
Page 184 and 185: Chapter 4.1 182 Figure 1.
Page 186 and 187: Chapter 4.1 184 Figure 1. Viral dec
Page 188 and 189: Chapter 4.1 186 suppression in seru
Page 190 and 191: Chapter 4.1 188 13. Seifer M, Hamat
Page 193 and 194: Cha pter 4.2 Viral dynamics during
Page 195 and 196: INTRODUCTION Viral dynamics during
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Viral dynamics during tenofovir the
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log HBV DNA (copies/mL) 10.0 9.0 8.
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Chap ter 4.3 Modelling of early vir
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INTRODUCTION HBV viral kinetics dur
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HBV viral kinetics during PEG-IFN 2
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SUMMARY AND CONCLUSION Summary and
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Summary and conclusion 233 The mode
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Summary and conclusion 235 The infe
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Summary and conclusion 237 Early st
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SAMENVATTING EN CONCLUSIE Samenvatt
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Samenvatting en conclusie 243 Er we
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Samenvatting en conclusie 245 Data
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Samenvatting en conclusie 247 Behan
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Dan kwoord
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Dankwoord 254 Lieve Marion en Margr
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BIBLI OGRAPHY Bibliography 257 1. K
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Bibliography 259 Interferon-ribavir
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Bibliography 261 Long term clinical
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Bibliography 263 Flares in chronic
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Bibliography 265 Genetic characteri
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Bibliography 267 106. Reijnders JG,
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