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mixed2.doc - FROM SAS DOCUMENTATION OF PROC MIXED Mixed...

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FROM SAS DOCUMENTATION OF PROC MIXED Mixed Models Theory This section provides an overview of a likelihood-based approach to general linear mixed models. This approach simplifies and unifies many common statistical analyses, including those involving repeated measures, random effects, and random coefficients. The basic assumption is that the data are linearly related to unobserved multivariate normal random variables. Extensions to nonlinear and nonnormal situations are possible but are not discussed here. Additional theory and examples are provided in Littell et al. (1996), Verbeke and Molenberghs (1997 2000), and Brown and Prescott (1999). Formulation of the Mixed Model The previous general linear model is certainly a useful one (Searle 1971), and it is the one fitted by the GLM procedure. However, many times the distributional assumption about is too restrictive. The mixed model extends the general linear model by allowing a more flexible specification of the covariance matrix of . In other words, it allows for both correlation and heterogeneous variances, although you still assume normality. The mixed model is written as where everything is the same as in the general linear model except for the addition of the known design matrix, Z , and the vector of unknown random-effects parameters, . The matrix Z can contain either continuous or dummy variables, just like X . The name mixed model comes from the fact that the model contains both fixed-effects parameters, , and random-effects parameters, . Refer to Henderson (1990) and Searle, Casella, and McCulloch (1992) for historical developments of the mixed model. A key assumption in the foregoing analysis is that and are normally distributed with The variance of y is, therefore, V = Z G Z ' + R . You can model V by setting up the random-effects design matrix Z and by specifying covariance structures for G and R . Note that this is a general specification of the mixed model, in contrast to many texts and articles that discuss only simple random effects. Simple random effects are a special case of the general specification with Z containing dummy variables, G
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containing variance components in a diagonal structure, and , where I n denotes the n × n identity matrix. The general linear model is a further special case with Z = 0 and . The following two examples illustrate the most common formulations of the general linear mixed model. Example: Growth Curve with Compound Symmetry Suppose that you have three growth curve measurements for s individuals and that you want to fit an overall linear trend in time. Your X matrix is as follows: The first column (coded entirely with 1s) fits an intercept, and the second column (coded with times of 1,2,3) fits a slope. Here, n = 3 s and p = 2. Suppose further that you want to introduce a common correlation among the observations from a single individual, with correlation being the same for all individuals.
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