Preface to First Edition - lib

METHODS FOR NON-NORMAL DISTRIBUTIONS 235each cross-section. In this approach the relationship of the marginal meanand the explanatory variables is modelled separately from the within-subjectcorrelation. The marginal regression coefficients have the same interpretationas coefficients from a cross-sectional analysis, and marginal models are naturalanalogues for correlated data of generalised linear models for independentdata. Fitting marginal models to non-normal longitudinal data involves the useof a procedure known as generalised estimating equations (GEE), introducedby Liang and Zeger (1986). This approach may be viewed as a multivariateextension of the generalised linear model and the quasi-likelihood method (seeChapter 7). But the problem with applying a direct analogue of the generalisedlinear model to longitudinal data with non-normal responses is that thereis usually no suitable likelihood function with the required combination ofthe appropriate link function, error distribution and correlation structure. Toovercome this problem Liang and Zeger (1986) introduced a general methodfor incorporating within-subject correlation in GLMs, which is essentially anextension of the quasi-likelihood approach mentioned briefly in Chapter 7. Asin conventional generalised linear models, the variances of the responses giventhe covariates are assumed to be of the form Var(response) = φV(µ) wherethe variance function V (µ) is determined by the choice of distribution family(see Chapter 7). Since overdispersion is common in longitudinal data, thedispersion parameter φ is typically estimated even if the distribution requiresφ = 1. The feature of these generalised estimation equations that differs fromthe usual generalised linear model is that different responses on the sameindividual are allowed to be correlated given the covariates. These correlationsare assumed to have a relatively simple structure defined by a small numberof parameters. The following correlation structures are commonly used (Y ijrepresents the value of the jth repeated measurement of the response variableon subject i).An identity matrix leading to the independence working model in whichthe generalised estimating equation reduces to the univariate estimatingequation given in Chapter 7, obtained by assuming that the repeated measurementsare independent.An exchangeable correlation matrix with a single parameter similar tothat described in Chapter 12. Here the correlation between each pair ofrepeated measurements is assumed to be the same, i.e., corr(Y ij , Y ik ) = ρ.An AR-1 autoregressive correlation matrix, also with a single parameter,but in which corr(Y ij , Y ik ) = ρ |k−j| , j ≠ k. This can allow the correlationsof measurements taken farther apart to be less than those takencloser to one another.An unstructured correlation matrix with K(K −1)/2 parameters whereK is the number of repeated measurements andcorr(Y ij , Y jk ) = ρ jkFor given values of the regression parameters β 1 , ...β q , the ρ-parametersof the working correlation matrix can be estimated along with the dispersionparameter φ (see Zeger and Liang, 1986, for details). These estimates can then© 2010 by Taylor and Francis Group, LLC