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A Formal Basis for the Two-Stage Approach

As was explained in Section 4.3, an alternative strategy (not based on MI) that can be used to avoid the computational problems that arise in the estimation of the variance components of Model (15.1), consists of replacing the mixed-effects model representation by its fixed-effects (two-stage) counterpart (Tibaldi et al., 2003). The two-stage approach is based on the original paper by Laird and Ware (1982). Details can be found in Verbeke and Molenberghs (2000). Tibaldi et al. (2003) obtained very good results with the two-stage approach, confirmed in various chapters of this text. Of course, the question remains as to whether the method is formally valid, i.e., whether it leads to a consistent estimator.

A first observation is that the two-stage approach is consistent when the number of patients per trial approaches infinity, whereas the number of trials is either bounded, or increases sufficiently slowly. A first observation is that the two-stage approach is consistent when the number of patients per trial approaches infinity, whereas the number of trials is either bounded, or increases sufficiently slowly.

Second, as shown in Hermans et al. (2015), with a compound-symmetry

structure, or one related to it (i.e., where patients within a trial are exchangeable), weights that are proportional to the trial size can be used, while equal weights may not perform too badly, either. This is because there is a so- called finite information limit in compound-symmetry-type structures (Nas- siri, Molenberghs, and Verbeke, 2016).

Third, it is technically possible to derive a closed-form estimator for each of the trials separately, which implies that the weighted two-stage approach can be constructed with high computational efficiency, even in a big-data context.

 
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