Preface to First Edition - lib

234 ANALYSING LONGITUDINAL DATA IIform introduced in Chapter 12. For the respiratory data in Table 13.1 wecould then apply logistic regression and for epilepsy in Table 13.2, Poissonregression. It can be shown that this approach will give consistent estimates ofthe regression coefficients, i.e., with large samples these point estimates shouldbe close to the true population values. But the assumption of the independenceof the repeated measurements will lead to estimated standard errors that aretoo small for the between-subjects covariates (at least when the correlationbetween the repeated measurements are positive) as a result of assuming thatthere are more independent data points than are justified.We might begin by asking if there is something relatively simple that canbe done to ‘fix-up’ these standard errors so that we can still apply the Rglm function to get reasonably satisfactory results on longitudinal data witha non-normal response? Two approaches which can often help to get moresuitable estimates of the required standard errors are bootstrapping and useof the robust/sandwich, Huber-White variance estimator.The idea underlying the bootstrap (see Chapter 8 and Chapter 9), a techniquedescribed in detail in Efron and Tibshirani (1993), is to resample fromthe observed data with replacement to achieve a sample of the same size eachtime, and to use the variation in the estimated parameters across the set ofbootstrap samples in order to get a value for the sampling variability of theestimate (see Chapter 8 also). With correlated data, the bootstrap sampleneeds to be drawn with replacement from the set of independent subjects, sothat intra-subject correlation is preserved in the bootstrap samples. We shallnot consider this approach any further here.The sandwich or robust estimate of variance (see Everitt and Pickles, 2000,for complete details including an explicit definition), involves, unlike the bootstrapwhich is computationally intensive, a closed-form calculation, based onan asymptotic (large-sample) approximation; it is known to provide good resultsin many situations. We shall illustrate its use in later examples.But perhaps more satisfactory would be an approach that fully utilises informationon the data’s structure, including dependencies over time. In thelinear mixed models for Gaussian responses described in Chapter 12, estimationof the regression parameters linking explanatory variables to the responsevariable and their standard errors needed to take account of the correlationalstructure of the data, but their interpretation could be undertaken independentof this structure. When modelling non-normal responses this independenceof estimation and interpretation no longer holds. Different assumptionsabout how the correlations are generated can lead to regression coefficientswith different interpretations. The essential difference is between marginalmodels and conditional models.13.2.1 Marginal ModelsLongitudinal data can be considered as a series of cross-sections, and marginalmodels for such data use the generalised linear model (see Chapter 7) to fit© 2010 by Taylor and Francis Group, LLC