following form:
(3) In this equation, and remain and matrices of known covariates, respectively. , however, is now a matrix of regression coefficients that are common to all respondents and is a matrix of coefficients specific to individual (Schafer & Yucel, 2002). is now also an matrix of which the rows are independently distributed as , where is an unstructured covariance matrix with dimensions (Shah et al., 1997). We can vectorize the model in (3) by stacking the columns of the matrices, thus obtaining (4) Here, which is an dimensional vector and in the same way which has the same dimension as . and are block diagonal matrices in which the blocks consist of the and matrices of explanatory variables for each of the dependent variables (Morrell et al., 2012). has dimension and is an matrix, which is usually a subset of as in the univariate model (Shah et al., 1997). Here and denote the total number of fixed effects for all response variables and the total number of individual random effects, respectively (Morrell et al., 2012). is a vector of fixed-effects regression parameters and is a vector of individual random effects. is assumed to be where has dimensions (Shah et al., 1997). The error terms are with having dimensions (Morrell et al., 2012). Conditional on the random effects we define responses to be independent over time as before, such that , where denotes the Kronecker product and is an identity matrix (Morrell et al., 2012). We assume different individuals to be independent so that for we have and . The fixed effects are also independent from the random effects such that (Shah et al., 1997). The multivariate responses of an individual are tied together through the covariance matrix of the random effects and of the error terms. We define to be an unstructured covariance matrix which allows for covariance between the random effects within a particular response variable and it also allows for covariance among the random effects of different response variables (Fieuws
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& Verbeke, 2004; Morrell et al., 2012). The error terms in the covariance matrix are correlated by allowing offdiagonal elements to be different from zero. The different responses of an individual are therefore joined by allowing both the random effects and the measurement error to be correlated. Parameter Estimation in the Multivariate LMM The parameters in the multivariate linear mixed model are estimated by means of the pairwise fitting approach that was introduced by Fieuws and Verbeke (2006). They developed this approach as a solution to dealing with models that have a large number of outcome variables. Previous methods were not able to overcome computational restrictions that are related to fitting such high dimensional models. This dimensionality problem can be solved by a two-step approach using pairwise estimation. For each response variable a univariate model is defined but instead of maximizing the likelihood of the multivariate model, it uses pairwise bivariate models which are fitted separately (Fieuws & Verbeke, 2006). If we define to be the vector containing all fixed effects and covariance parameters in the multivariate model, then represents the log-likelihood contribution of individual to the multivariate mixed model (Fieuws & Verbeke, 2006). In the first step, we fit bivariate models, namely all joint models for all possible pairs of outcomes: (Fieuws et al., 2007). Hence, the log-likelihoods that will be estimated have the following form (Fieuws & Verbeke, 2006) (5) and . is the vector containing all parameters from the bivariate joint mixed model that correspond to the pair . Hereafter, we will refer to the pair by , where is the total number of pairs. Equation (5) then becomes (Fieuws & Verbeke, 2006) (6) We stack the pair-specific parameter vectors into the vector . Clearly, an estimate of is obtained by separately maximizing each of the bivariate log-likelihoods (Fieuws et al., 2007). Fieuws and Verbeke (2006) stress the distinction between and . Whereas some parameters
MET | Volume 20 | Ă&#x;ETA Special | 2013