N

If the number of level 2 units, J is large then the estimates generated through maximum likelihood are approximately equal to estimates generated through restricted maximum likelihood (REML). REML estimates of the covariance components are based on residuals which are computed after estimating the fixed effects (11) by WLS or by GLS and are estimates based on maximizing a marginal likelihood. REML estimates take into account the degrees of freedom used in estimating the fixed effects when estimating the covariance components. REML estimates of G and R are found by maximizing the following log-likelihood function (see also Littell et al.2 and Searle et al.10):

HLM/2L generates REML estimates by default and uses the EM algorithm to maximize (16). SAS Proc Mixed also produces REML estimates by default and uses a ridge-stabilized Newton-Raphson algorithm to maximize the likelihood.2 Maximum likelihood estimates can be requested in both HLM/2L and SAS Proc Mixed.

4.1.3. Estimating Random Effects (u)

Random effects are estimated using shrinkage estimators. SAS Proc Mixed generates estimates of random effects according to the following:

It is generally of interest to estimate individual (for example, physician-level) random coefficients (for example, and fifj). These can be obtained by substitution. For example, considering the model described in (2), = y00 + y0i + v0J, where v0J is from (17). A shrinkage estimate (also referred to as an empirical Bayes (EB) estimate in HLM/2L or a best linear unbiased prediction (BLUP) in SAS Proc Mixed2) for the ;'th random coefficient (for example, ¡loj of model 1 in Table I) is essentially an optimally weighted, linear combination of the estimated overall mean (for example, y00) and the jth level 2 mean (Yj). The degree of shrinkage depends on the magnitude of the variation in level 2 means (which is related to the number of level 1 units within the specific level 2 unit used to generate the estimate). Thus, when rij, the number of level 1 units within the jth level 2 unit, is small, the estimate of the j'th random coefficient is close to the overall mean y00, but, as an rij increases, the estimate of the jth random coefficient moves closer to the level 2 mean Yj. BLUPs are seen as estimates subject to regression toward the overall mean (y) based on the covariance components of model effects (for example, G, V). For more details, see also Littell et al.2 and Zeger et al.12 Empirical Bayes estimates are computationally efficient and results are asymptotically equivalent to Bayes solutions.13

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