762notes.pdfCHAPTER 14 ST 762, M. DAVIDIAN14 Estimating equation methods for population-average-models14.1 IntroductionIn this chapter, we focus on methods for estimation of the parameters in a population-averaged marginalmodel of the general form discussed in Chapter 13. In particular, we assume that the pairs (Yi, xi),i = 1, . . . , m, are independent, where each Yiis (ni× 1), and we consider the general mean-covariancematrix modelE(Yi|xi) = fi(xi, β), var(Yi|xi) = Vi(β, ξ, xi) (ni× ni), (14.1)where fi(xi, β) is the (ni× 1) vector with jth element f (xij, β). The covariance model Vi(β, ξ , xi) istaken to have the formvar(Yi|xi) = Vi(β, ξ , xi) = T1/2i(β, θ, xi)Γi(α, xi)T1/2i(β, θ, xi), (14.2)where Ti(β, θ, xi) is the diagonal matrix whose diagonal elements are the models for var(Yij|xi), e.g.,involving a variance functionvar(Yij|xi) = σ2g2(β, θ, xij)depending on possibly unknown variance parameters θ (as in the previous chapter, we will generallyabsorb σ2into θ for brevity and just refer to θ as all the variance parameters). The (ni× ni) matrixΓi(α, xi) is a correlation matrix that generally depends on xionly through the within-individual timesor other conditions tijat which observations in Yiare taken. Here, α is a vector of unknown correlationparameters.The vector of variance and correlation parameters ξ = (θT, αT)Tmay be entirely unknown, or it maybe that only α is unknown in models where the form of the variance function is entirely specified.As discussed in Chapter 13, the correlation model Γi(α, xi) is likely not to be correct. Rather, it isspecified as a “working” model that hopefully captures some of the main features of the overall patternof correlation. This is acknowledged when inference is carried out under model (14.1) with assumedcovariance structure (14.2); we will discuss this explicitly in Section 14.5.Model (14.1) may be viewed as a multivariate analog to the univariate mean-variance models discussedin Chapters 2–12. Thus, it should come as no surprise that inferential strategies for (14.1) exploit someof the same ideas as in the univariate case. In particular, estimation of β and ξ is typically carriedout by solution of linear or quadratic estimating equations th at are similar in spirit to those used forunivariate response. Of necessity, these equations are more complicated in the multivariate setting, aswe will see in Sections 14.2, 14.3, and 14.4, although the basic principles are the same.PAGE 371CHAPTER 14 ST 762, M. DAVIDIANThe terminology generalized estimating equations (GEEs), first coined by Liang and Zeger (1986), hascome to refer broadly to the body of techniques for inference for β and ξ based on solution of appropriateestimating equations.• As suggested by the title of Liang and Zeger (1986), “Longitudinal data analysis using generalizedlinear models,” this paper cast the idea of posing estimating equations for multivariate responsein the context where the the response is collected longitudinally for each experimental unit.• The development was also restricted to responses such as binary data, counts, and so on, thus onmean models f and models for var(Yij|xij) that are of the generalized linear model type. However,this restriction is unnecessary; solving estimating equations for any model of the form (14.1) isfeasible more generally.This restriction does explain why, in much of the literature, there are n o unknown varianceparameters θ, and interest focuses exclusively on estimating correlation parameters α. For ex-ample, for binary response, the m odel for var(Yij|xij) might be taken to be the usual modelf(xij, β){1 − f(xij, β)} and is assu med to be correctly specified. A “working” correlation modelmight be postulated depending on unknown α, so that only α remains to be estimated.• In our development here, we will allow the possibility th at the model Vi(β, ξ, xi) may involveboth unknown variance parameters θ and unknown correlation parameters α. We will notesimplifications that would occur in the case where the variance function does not depend on anyunknown parameters θ.There are numerous references that cover th e types of estimating equations we are about to discuss.Some of the key references are Prentice (1988), Zhao and Prentice (1990), Prentice and Zhao (1991),and Liang, Zeger, and Qaqish (1992). See also Section 7.5 and Chapter 8 of Diggle, Liang, and Zeger(1995), Chapter 9 of Vonesh and Chinchilli (1997), and Chap ter 3 of Fitzmaurice et al. (2009).14.2 Linear estimating equations for βJust as with univariate response, it is natural to start by considering the normal likelihood as a basisto derive an estimation method for β in (14.1). Thus, analogous to the case of “known weights” in theunivariate case, assume that the m atrices var(Yi|xi) = Vi, say, are known. Under these conditions,assuming that the Yi|xiare normally distributed, the normal loglikelihood has the formlog L = −(1/2)mXi=1hlog |Vi| + {Yi− fi(xi, β)}TV−1i{Yi− fi(xi, β)}i. (14.3)PAGE 372CHAPTER 14 ST 762, M. DAVIDIANWritingXi(β) =fTβ(xi1, β)...fTβ(xini, β)(ni× p),taking derivatives of (14.3) with respect to the (p × 1) vector β, and setting equal to zero yieldsmXi=1XTi(β)V−1i{Yi− fi(xi, β)} = 0, (14.4)which follow s by using the following standard m atrix differentiation results.• For quadratic form q = xTAx and A symmetric, ∂q/∂x = 2Ax. Note that this is a vector.• The chain rule gives ∂q/∂β = (∂x/∂β)(∂q/∂x). Note that if both x and β are vectors, then∂x/∂β is a matrix. (See Section 2.4 for a refresher.)Section 1.4 of Vonesh and Chinchilli (1997) is an excellent source for many useful and sometimesdifficult-to-find results on matrices and matrix differentiation.The equation (14.4) has the form of a multivariate analog to the usual, univariate WLS equation,where, here, the response and mean are now vectors, and the “weights” V−1iand “gradient” Xi(β) arematrices.In fact, we may write (14.4) in a way that makes it clear that there is really no fundamental differencein the general forms in the univariate and mu ltivariate cases. Let Y = (YT1, . . . , YTm)Tbe th e vector oflength N =Pmi=1ni(total number of observations). Definef(β) = {fT1(x1, β), . . . , fTm(x1, β)}T(N × 1),V = block diag(V1, . . . , Vm) (N × N ),andX(β) =X1(β)...Xm(β)(N × p).Note that