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After vacation, I was able to revisit this issue trigged by David Kondrat's question of what to do with just 8 level-2 units. As you may recall, Dan McNeish stepped in with a reference to a pre-publication paper by Bethany Bell et al., and others mentioned relevant sections of Snijders and Bosker and of Ullah and Giles. Bell et al report good performance for multilevel modeling with REML and Kenward-Rogers options for level-sample sizes as small as 10.
Section 12.2 of Snijders and Bosker took me back to work by a different Bell, Robert M. Bell (with Dan McCaffrey) in Survey Methodology. They showed that the sandwich estimator works for level-2 n>1 provided that the relevant portion of the design matrix is constant across clusters. I am going to oversimplify their work and say that if the independent variable of central interest has itself an ICC of zero, then there is no need to worry about using the sandwich estimator that is common in survey-sensitive regression software to adjust for the effects of clustering on the variances of the estimated parameters in a single-level model. On the other hand, if there is substantial ICC in the independent variable of interest, then there might be severe problems in sandwich variance estimators.
I am particularly interested in multi-site individually randomized trials. Here the independent variable of interest is randomly assigned treatment status. So Bell and McCaffrey's work would seem to suggest that as long as the randomization fraction is constant across sites, there are no problems in the sandwich variance estimator. I verified this with a simulation study with as few as 3 sites and 100 level-1 units per site. I used a constant randomization fraction of 0.67 within each site. I also had a level-1 covariate that explained a widely varying portion of the variance of the dependent variable across sites. Even with variable treatment effects across sites, the SAS procedure Surveyreg performed nearly perfectly. The same cannot be said of the SAS procedure MIXED with REML and Kenward-Rogers options, random slopes, and random intercepts. It worked just as well as Surveyreg for 8 sites, but started to be slightly liberal with 5 sites, and was considerably liberal with 3 sites.
Dave Judkins
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