Preface to First Edition - lib

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