Preface to First Edition - lib

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236 ANALYSING LONGITUDINAL DATA IIbe used in the so-called generalised estimating equations to obtain estimatesof the regression parameters. The GEE algorithm proceeds by iterating between(1) estimation of the regression parameters using the correlation anddispersion parameters from the previous iteration and (2) estimation of thecorrelation and dispersion parameters using the regression parameters fromthe previous iteration.The estimated regression coefficients are ‘robust’ in the sense that they areconsistent from misspecified correlation structures assuming that the meanstructure is correctly specified. Note however that the GEE estimates of marginaleffects are not robust against misspecified regression structures, such asomitted covariates.The use of GEE estimation on a longitudinal data set in which some subjectsdrop out assumes that they drop out completely at random (see Chapter 12).13.2.2 Conditional ModelsThe random effects approach described in the previous chapter can be extendedto non-normal responses although the resulting models can be difficultto estimate because the likelihood involves integrals over the random effectsdistribution that generally do not have closed forms. A consequence is that itis often possible to fit only relatively simple models. In these models estimatedregression coefficients have to be interpreted, conditional on the random effects.The regression parameters in the model are said to be subject-specificand such effects will differ from the marginal or population averaged effects estimatedusing GEE, except when using an identity link function and a normalerror distribution.Consider a set of longitudinal data in which Y ij is the value of a binaryresponse for individual i at say time t j . The logistic regression model (seeChapter 7) for the response is now written aslogit (P(y ij = 1|u i )) = β 0 + β 1 t j + u i (13.1)where u i is a random effect assumed to be normally distributed with zeromean and variance σ 2 u. This is a simple example of a generalised linear mixedmodel because it is a generalised linear model with both a fixed effect, β 1 , anda random effect, u i .Here the regression parameter β 1 again represents the change in the log oddsper unit change in time, but this is now conditional on the random effect. Wecan illustrate this difference graphically by simulating the model (13.1); theresult is shown in Figure 13.1. Here the thin grey curves represent subjectspecificrelationships between the probability that the response equals one anda covariate t for model (13.1). The horizontal shifts are due to different valuesof the random intercept. The thick black curve represents the population averagedrelationship, formed by averaging the thin curves for each value of t. Itis, in effect, the thick curve that would be estimated in a marginal model (see© 2010 by Taylor and Francis Group, LLC
METHODS FOR NON-NORMAL DISTRIBUTIONS 237P(y = 1)0.0 0.2 0.4 0.6 0.8 1.0−0.4 −0.2 0.0 0.2 0.4TimeFigure 13.1Simulation of a positive response in a random intercept logistic regressionmodel for 20 subjects. The thick line is the average over all20 subjects.previous sub-section). The population averaged regression parameters tend tobe attenuated (closest to zero) relative to the subject-specific regression parameters.A marginal regression model does not address questions concerningheterogeneity between individuals.Estimating the parameters in a logistic random effects model is undertakenby maximum likelihood. Details are given in Skrondal and Rabe-Hesketh(2004). If the model is correctly specified, maximum likelihood estimates areconsistent when subjects in the study drop out at random (see Chapter 12).© 2010 by Taylor and Francis Group, LLC
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A Handbook ofStatisticalAnalysesUsi
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DedicationTo our wives, Mary-Elizab
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Preface to First EditionThis book i
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List of Figures1.1 Histograms of th
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6.8 Normal probability plot of resi
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12.2 R output of the linear mixed-e
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18.7 Within-cluster sum of squares
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4.6 Lanza data. Misoprostol randomi
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15.2 BCG data. Meta-analysis on BCG
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6 Simple and Multiple Linear Regres
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CHAPTER 1An Introduction to R1.1 Wh
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INSTALLING R 3One can change the ap
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DATA OBJECTS IN R 5http://CRAN.R-pr
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DATA OBJECTS IN R 7R> help("Forbes2
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DATA IMPORT AND EXPORT 9As a simple
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BASIC DATA MANIPULATION 11The funct
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BASIC DATA MANIPULATION 13name sale
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COMPUTING WITH DATA 15Max. : 20.960
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COMPUTING WITH DATA 17Error in quan
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COMPUTING WITH DATA 19R> layout(mat
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© 2010 by Taylor and Francis Group
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SUMMARY 23examples of these functio
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26 DATA ANALYSIS USING GRAPHICAL DI
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Table 2.2:CHFLS data. Chinese Healt
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Table 2.5:USstates data. Socio-demo
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CHAPTER 3Simple Inference: Guessing
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INTRODUCTION 47table. Here there ar
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STATISTICAL TESTS 49Table 3.4:pisto
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STATISTICAL TESTS 51assumed to have
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ANALYSIS USING R 53procedure is McN
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ANALYSIS USING R 551 R> layout(matr
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ANALYSIS USING R 57R> wilcox.test(I
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ANALYSIS USING R 59R> t.test(moorin
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ANALYSIS USING R 61R> cor.test(~ mo
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SUMMARY 63R> mcnemar.test(rearrests
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CHAPTER 4Conditional Inference: Gue
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INTRODUCTION 67Table 4.4:Lanza data
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CONDITIONAL TEST PROCEDURES 69and x
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ANALYSIS USING R 71R> hist(meandiff
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ANALYSIS USING R 73R> wilcox_test(y
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ANALYSIS USING R 75For the first st
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SUMMARY 77R> mh_test(anomalies)Asym
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CHAPTER 5Analysis of Variance: Weig
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INTRODUCTION 81The data in Table 5.
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ANALYSIS USING R 83set out in the f
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ANALYSIS USING R 85being far more t
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ANALYSIS USING R 87R> plot.design(f
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ANALYSIS USING R 89always be the ca
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ANALYSIS USING R 91The cbind statem
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ANALYSIS USING R 93Df Roy approx F
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SUMMARY 95Table 5.4:schooldays data
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CHAPTER 6Simple and Multiple Linear
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SIMPLE LINEAR REGRESSION 99hours be
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MULTIPLE LINEAR REGRESSION 101The e
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ANALYSIS USING R 1036.3.1 Regressio
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ANALYSIS USING R 107R> data("clouds
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ANALYSIS USING R 109R> summary(clou
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ANALYSIS USING R 111R> psymb plot(
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SUMMARY 113R> plot(clouds_fitted, c
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SUMMARY 115R> plot(clouds_lm)Cook's
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CHAPTER 7Logistic Regression and Ge
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INTRODUCTION 119Table 7.2:womensrol
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LOGISTIC REGRESSION AND GENERALISED
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ANALYSIS USING R 123R> data("plasma
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ANALYSIS USING R 125R> summary(plas
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ANALYSIS USING R 1277.3.2 Women’s
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ANALYSIS USING R 1357.3.4 Driving a
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SUMMARY 137Table 7.5:bladdercancer
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CHAPTER 8Density Estimation: Erupti
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DENSITY ESTIMATION 141The Hertzspru
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DENSITY ESTIMATION 143rectangular:
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DENSITY ESTIMATION 1451 R> plot(xgr
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ANALYSIS USING R 1478.3 Analysis Us
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2.2ANALYSIS USING R 149R> library("
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ANALYSIS USING R 1510.360891 54.612
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ANALYSIS USING R 153The results are
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SUMMARY 155R> layout(matrix(1:2, nc
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SUMMARY 157Table 8.4: birthdeathrat
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SUMMARY 159Table 8.5:schizophrenia
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162 RECURSIVE PARTITIONINGTable 9.1
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164 RECURSIVE PARTITIONINGBoth sets
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166 RECURSIVE PARTITIONINGR> librar
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168 RECURSIVE PARTITIONINGR> DEXfat
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170 RECURSIVE PARTITIONINGOne way o
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172 RECURSIVE PARTITIONINGR> librar
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174 RECURSIVE PARTITIONINGR> plot(g
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CHAPTER 10Scatterplot Smoothers and
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INTRODUCTION 179Table 10.2:USairpol
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SMOOTHERS AND GENERALISED ADDITIVE
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Page 267 and 268: SUMMARY 251Table 13.3:schizophrenia
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SUMMARY 295R> biplot(heptathlon_pca
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SUMMARY 297rainJuly: rainfall in Ju
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Table 17.1:watervoles data. Water v
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302 MULTIDIMENSIONAL SCALINGn in nu
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304 MULTIDIMENSIONAL SCALINGwill be
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306 MULTIDIMENSIONAL SCALINGR> x y
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308 MULTIDIMENSIONAL SCALINGR> libr
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310 MULTIDIMENSIONAL SCALINGR> plot
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Table 17.3:eurodist data (package d
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CHAPTER 18Cluster Analysis: Classif
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INTRODUCTION 317mass, mass), the pe
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CLUSTER ANALYSIS 319x 20 2 4 6 8 10
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CLUSTER ANALYSIS 32141031678925Figu
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CLUSTER ANALYSIS 3231. Find some in
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ANALYSIS USING R 32518.3 Analysis U
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BibliographyAdler, D. and Murdoch,
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BIBLIOGRAPHY 337Chambers, J. M. and
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BIBLIOGRAPHY 339Everitt, B. S. and
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BIBLIOGRAPHY 341Heitjan, D. F. (199
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BIBLIOGRAPHY 343Leisch, F. and Ross
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BIBLIOGRAPHY 345Proudfoot, J., Gold
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BIBLIOGRAPHY 347Stevens, J. (2001),
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