Preface to First Edition - lib

218 ANALYSING LONGITUDINAL DATA Iother and of the time t j . The model in (12.1) is known as a random interceptmodel, the u i being the random intercepts. The repeated measurements for anindividual vary about that individual’s own regression line which can differ inintercept but not in slope from the regression lines of other individuals. Therandom effects model possible heterogeneity in the intercepts of the individualswhereas time has a fixed effect, β 1 .The random intercept model implies that the total variance of each repeatedmeasurement is Var(y ij ) = Var(u i +ε ij ) = σ 2 u +σ 2 . Due to this decompositionof the total residual variance into a between-subject component, σ 2 u, and awithin-subject component, σ 2 , the model is sometimes referred to as a variancecomponent model.The covariance between the total residuals at two time points j and k in thesame individual is Cov(u i +ε ij , u i +ε ik ) = σ 2 u. Note that these covariances areinduced by the shared random intercept; for individuals with u i > 0, the totalresiduals will tend to be greater than the mean, for individuals with u i < 0they will tend to be less than the mean. It follows from the two relations abovethat the residual correlations are given byCor(u i + ε ij , u i + ε ik ) =σ2 uσ 2 u + σ 2 .This is an intra-class correlation interpreted as the proportion of the totalresidual variance that is due to residual variability between subjects. A randomintercept model constrains the variance of each repeated measure to be thesame and the covariance between any pair of measurements to be equal. This isusually called the compound symmetry structure. These constraints are oftennot realistic for repeated measures data. For example, for longitudinal data it ismore common for measures taken closer to each other in time to be more highlycorrelated than those taken further apart. In addition the variances of the laterrepeated measures are often greater than those taken earlier. Consequentlyfor many such data sets the random intercept model will not do justice tothe observed pattern of covariances between the repeated measures. A modelthat allows a more realistic structure for the covariances is one that allowsheterogeneity in both slopes and intercepts, the random slope and interceptmodel.In this model there are two types of random effects, the first modellingheterogeneity in intercepts, u i , and the second modelling heterogeneity inslopes, v i . Explicitly the model isy ij = β 0 + β 1 t j + u i + v i t j + ε ij (12.2)where the parameters are not, of course, the same as in (12.1). The two randomeffects are assumed to have a bivariate normal distribution with zero meansfor both variables and variances σ 2 u and σ 2 v with covariance σ uv . With thismodel the total residual is u i + u i t j + ε ij with varianceVar(u i + v i t j + ε ij ) = σ 2 u + 2σ uv t j + σ 2 vt 2 j + σ 2© 2010 by Taylor and Francis Group, LLC