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Chapter 3.2 154 Focus of marginal, conditional and random effects prediction Spiegelhalter et al. 25 define the ‘focus’ of the Bayesian model which is given by considered factorization of the marginal distribution p(y new) of new data. We shall show that the marginal, conditional and random effects predictions correspond to different model focuses in the spirit of Spiegelhalter et al. 25 . In our context, there are two obvious possibilities for factorization of p(y new). Let Θ g be the parameter space for θ g . With factorization � p(y new) = p � � y � g new θ � p(θ g ) dθ g , (31) Θ g the model is focused on Θ g , i.e., on the mean evolution of the markers over time. Further, let θ g 1 = � αg′ ,σ g2 g 2 1 ,...,σR � ′ g and θ2 = � w g′ , μ g′ g ′ 1 ,...,μK , vec(D g 1 ),...,vec(Dg K )� ′ g be the parts of θ corresponding to (1) the fixed effects and variances of random errors and (2) the distribution of the random effects. Let Θ g 1 and Θg 2 be the corresponding parameter spaces. Finally, let Ψ = Rq be the parameter space for the vector bnew of random effects. The marginal distribution p(y new) can alternatively be factorized as � p(y new) = p � � y � new bnew, θ g� 1 p(bnew) p(θ g 1 ) dbnew dθ g 1 , (32) where Ψ×Θ g 1 � p(bnew) = p � � � bnew � g g θ2 p(θ2 ) dθg2 , (33) Θ g 2 and the model is focused on Ψ×Θ g 1 , i.e., on the patient specific evolution of markers over time. That is, comparing expressions (28) and (31) we conclude that the marginal prediction is based on the likelihood of the Bayesian model focused on the mean evolution of the markers over time. Further, expressions (29) and (33) reveal that the conditional prediction is based on the likelihood of the Bayesian model focused on the patient specific evolution of the markers over time where, however, the nuisance parameter (the vector of random effects) is replaced by a plug-in estimate. Hence, the conditional prediction ignores variability in the estimation of the individual random effects. It is seen from expressions (30) and (33) that the random effects prediction in fact uses the likelihood of the Bayesian model for random effects focused on the parameters of the distribution of random effects, i.e., focused on the patient specific evolution of the markers over time. Finally, note that the marginal and conditional predictions work on the level of the observables and have to take into account also the error (within subjects) variability. On the other hand, the random effects pre-
Discriminant analysis using a MLMM with a normal mixture 155 diction uses only the latent characteristics of the subjects with the between subjects variability. Application to PBC data As an illustration, we have applied our methodology to the Dutch PBC study data described in the introduction. Of the 375 patients, 178 patients are known to be alive at T = 10 years without needing a liver transplantation. They will be further referred to as prognostic group 0. A total of 41 patients died from a liver related cause or needed a liver transplantation during the first 10 years and will be further referred as prognostic group 1. The remaining 156 patients can be divided in three categories for whom the prognostic group is unknown because loss of follow-up (14 patients), or is unknown because the follow-up time was less than 10 years (112 patients). The third category consists of 30 patients who died in the first T = 10 years from another than a liver related cause. These 156 patients could be considered as the third prognostic group in our methodology. However, the results of our discrimination procedure can be better exemplified with G = 2 groups by means of sensitivity, specificity and receiver operating curves (ROC). Since the main objective of this section is to illustrate and explore the performance of our methodology, we will therefore consider only the two prognostic groups mentioned above. Hence our training data consist of 178 + 41 patients. The markers used to predict the prognostic group include values of: bilirubin, albumin, alkaline phosphatase (AP). Hence, R = 3 in model (1). To better satisfy mixed model (1), the natural logarithm of AP was used as the marker instead of the original AP value. Figure 1 shows the observed longitudinal profiles of the markers, separately for Group 0 and Group 1. We can observe that the bilirubin levels in Group 0 are rather low and stable over time whereas they start to increase dramatically from a specific time point for many patients in Group 1. Albumin levels are in general higher in Group 0 than in Group 1 while the reverse is true for AP. For practically all patients in Group 1, log(AP) is almost never negative whereas in Group 0 negative values of log(AP) are quite frequent. Although not required with our approach, the mixed models (1) for the PBC data will have the same structure in both prognostic groups. Namely, for each of three markers (r =1, 2, 3) and both groups (g =0, 1), the X g i,r matrix contains two columns: age of the patient at start of treatment (median 54.7) and a dose of ursodeoxycholic acid (UDCA) in mg/day (median 750). Consequently, the vector αg of fixed effects has a length of 6 for g = 0, 1. Matrices Z correspond to
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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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Table 1. Results of different stopp
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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 and 146: Introduction Discriminant analysis
Page 147 and 148: Discriminant analysis using a MLMM
Page 149 and 150: Estimation Discriminant analysis us
Page 153 and 154: where wi,k(y i, θ) = (k =1,...,K).
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Page 171 and 172: Appendix Discriminant analysis usin
Page 175: Cha pter 4 Statistical models of ea
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
Page 197 and 198: Viral dynamics during tenofovir the
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Viral dynamics during tenofovir the
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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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