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6 Linear Mixed Effects Models CHAPTER 6 LONGITUDINAL DATA ANALYSIS 6 Linear Mixed Effects Models 6.1 Introduction In the last chapter, we discussed a general class of linear models for continuous response arising from a population-averaged point of view. Here, population mean response is represented directly by a linear model that incorporates among- and within-individual covariate information. In keeping with the population-averaged perspective, the overall aggregate covariance matrix of a response vector is also modeled directly. These models are appropriate when the questions of scientific interest are questions about features of population mean response profiles. As we observed, selecting among candidate covariance models to represent the overall covariance structure is an inherent challenge. The aggregate pattern of variance and correlation may be suf- ficiently complex that, for example, standard correlation models like those reviewed in Section 2.5 cannot faithfully represent it. Moreover, when the number of observations per individual ni differs across individual and/or the observations are at different time points for different individuals, simple exploratory approaches like those in Section 2.6 are not possible, and some correlation models may not be feasible. Also, care must be taken in implementation. Of course, as discussed in Section 5.6, the reasons for imbalance must be carefully considered from the point of view of missing data mechanisms. In this chapter, we instead take a subject-specific perspective , which leads to the so-called lin- ear mixed effects model , the most popular framework for longitudinal data analysis in practice. Here, individual inherent response trajectories are represented by a linear model incorporating covariates, and, as in Chapter 2, within- and among-individual sources of correlation are explicitly acknowledged and modeled separately. Following the conceptual point of view in Chapter 2, it is natural to acknowledge individual response profiles in this way, and many scientific questions can be interpreted as pertaining to the “typical” features of individual trajectories; e.g., the “typical slope.” As discussed in Section 2.4, because of the use of linear models , this approach implies a linear model for overall population mean response and induces a model for the overall aggregate covariance matrix , so that a linear population-averaged model is a byproduct. Thus, as we noted there, the linear mixed effects model is a relevant framework for addressing questions of either a subject-specific or population-averaged nature. 169 CHAPTER 6 LONGITUDINAL DATA ANALYSIS Moreover, as we observe shortly, the induced covariance structure ameliorates the problems asso- ciated with direct specification of the overall pattern and implementation with unbalanced data dis- cussed in Section 5.2 when a population-averaged model is adopted directly and offers the analyst great flexibility for modeling variance and correlation structure. It follows that the same methods, namely, maximum likelihood under the assumption of normality and REML , can be used to fit a linear mixed effects model, and the same large sample theory results deduced in Section 5.5 hold and are used for the basis approximate inference. Likewise, the same concerns discussed in Section 5.6 regarding missing data continue to apply. Unlike the population-averaged approach in Chapter 5, however, because the subject-specific per- spective here represents explicitly individual behavior , it is possible to characterize features of in- dividual behavior and to develop an alternative approach to implementation via maximum likelihood, which we discuss later in this chapter. 6.2 Model specification BASIC MODEL: For convenience, we restate that the observed data are (Y i , zi , ai ) = (Y i , xi ), , i = 1, ... , m, T independent across i, where Y i = (Yi1, ... , Yini ) , with Yij recorded at time tij , j = 1, ... , ni (possibly different times for different individuals); z = (zT , ... , zT )T comprising within-individual covariate i i1 ini T T T information ui and the tij ; ai is a vector of among-individual covariates; and xi = (zi , ai ) . We introduce the basic form of the linear mixed effects model and then present examples that demonstrate how it provides a general framework in which various subject-specific models can be placed. The model is Y i = X i β + Z i bi + ei , i = 1, ... , m. (6.1) • In (6.1), X i (ni ×p) and Z i (ni ×q) are design matrices for individual i that depend on individual i’s covariates xi and time; we present examples of how X i and Z i arise from a subject-specific perspective momentarily. • The vector β (p × 1) in (6.1) is referred to as the fixed effects parameter. 170 CHAPTER 6 LONGITUDINAL DATA ANALYSIS • bi is a (q × 1) vector of random effects characterizing among-individual behavior; i.e., where individual i “sits” in the population. The standard and most basic assumption is that the bi are independent of the covariates xi and satisfy, for (q × q) covariance matrix D, E(bi jxi ) = E(bi ) = 0, var(bi jxi ) = var(bi ) = D, (6.2) bi ∼ N (0, D). (6.3) As we demonstrate, D characterizes variance and correlation due to among-individual sources. The specifications (6.2) and (6.3) can be relaxed to allow the distribution to differ depending on the values of among-individual covariates ai , as we discuss shortly, so that E(bi jxi ) = 0, var(bi jxi ) = var(bi jai ) = D(ai ), bi jxi ∼ N f0, D(ai )g. (6.4) T • The within-individual deviation ei = (ei1, ... , eini ) represents the aggregate effects of the within-individual realization and measurement error processes operating at the level of the individual. The standard and most basic assumption is that the ei are independent of the random effects bi and the covariates xi and satisfy E(ei jxi , bi ) = E(ei ) = 0, var(ei jxi , bi ) = var(ei ) = Ri (γ). (6.5) for some (ni × ni ) covariance matrix Ri (γ) depending on parameters γ. The most common assumption, often adopted by default without adequate thought, is that 2 2 Ri (γ) = σ Ini , γ = σ , for all i = 1, ... , m; (6.6) we discuss considerations for specification of Ri (γ) shortly. Ordinarily, it is further assumed that ei ∼ N f0, Ri (γ)g. (6.7) (6.5) and (6.7) can be relaxed to allow dependence of ei on xi and bi . We consider dependence of ei on ai here and defer discussion of more general specifications to Chapter 9. INTERPRETATION: From the perspective of the conceptual model (2.9) in Section 2.3, Y i = µi + Bi + ei = µi + Bi + ePi + eMi , (6.8) inspection of (6.1) shows that we can identify µi = X i β as the (ni × 1) overall population mean re- sponse vector, Bi = Z i bi as the (ni ×1) vector of deviations from the population mean characterizing where individual i “sits” in the population and thus among-individual variation, and ei as the (ni × 1) vector of within-individual deviations due to the realization process and measurement error. 171 CHAPTER 6 LONGITUDINAL DATA ANALYSIS Thus, in the linear mixed effects model (6.1), X i β + Z i bi , characterizes the individual-specific trajectory for individual i. As we demonstrate in examples shortly, this general form offers great latitude for representing individual profiles. IMPLIED POPULATION-AVERAGED MODEL: It follows from (6.1) and (6.2) – (6.7) that, conditional on bi and xi , Y i is ni -variate normal with mean vector X i β + Z i bi and covariance matrix Ri (γ); i.e., Y i jxi , bi ∼ N fX i β + Z i bi , Ri (γ)g. Thus, this conditional distribution characterizes how response observations for individual i vary and covary about the inherent trajectory X i β+Z i bi for i due to the realization process and measurement error. Letting p(y i jxi , bi ; β, γ) denote the corresponding normal density, and, from (6.3), letting p(bi ; D) be the q-variate normal density corresponding to (6.3), the density of Y i given xi is then given by Z p(y i jxi ; β, γ, D) = p(y i jxi , bi ; β, γ) p(bi ; D) dbi , (6.9) which is easily shown (try it) to be the density of a ni -variate normal with mean vector X i β and covariance matrix T T T T V i (γ, D, xi ) = V i (ξ, xi ) = Z i DZ i + Ri (γ), ξ = fγ , vech(D) g , (6.10) where vech(D) is the vector of distinct elements of D (see Appendix A). Summarizing, the linear mixed effects model framework above implies that E(Y i jxi ) = X i β, var(Y i jxi ) = V i = V i (ξ, xi ), Y i jxi ∼ N fX i β, V (ξ, xi )g, i = 1, ... , m, (6.11) where V i (ξ, xi ) is defined in (6.10). • As in Chapter 5, we will sometimes write V i and Ri for brevity , suppressing dependence on parameters for brevity. • As (6.11) shows, consistent with the discussion above and that in Section 2.4, the subject- specific linear mixed effects model implies a population-averaged model with overall popu- lation mean of the same form as in (5.4) and overall aggregate covariance matrix of the particular form (6.10). 172 CHAPTER 6 LONGITUDINAL DATA ANALYSIS • The specific form of the overall covariance matrix (6.10) is induced by specific choices of Ri (γ), reflecting the belief about the nature of the within-individual realization and measurement error processes , and of the covariance matrix D of the random effects, which characterizes among-individual variability in individual trajectories X i β + Z i bi . • This development generalizes in the obvious way when the covariance matrix var(bi jxi ) = var(bi jai ) = D(ai ) depends on among-individual covariates ai .
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