Misspecification of the covariance structure in generalized linear mixed models

When fitting marginal models to correlated outcomes, the so-called sandwich variance is commonly used. However, this is not the case when fitting mixed models. Using two data sets, we illustrate the problems that can be encountered. We show that the differences or the ratios between the naive and sandwich standard deviations of the fixed effects estimators provide convenient means of assessing the fit of the model, as both are consistent when the covariance structure is correctly specified, but only the latter is when that structure is misspecified. When the number of statistical units is not too small, the sandwich formula correctly estimates the variance of the fixed effects estimator even if the random effects are misspecified, and it can be used in a diagnostic tool for assessing the misspecification of the random effects. A simple comparison with the naive variance is sufficient and we propose considering a ratio of the naive and sandwich standard deviation out of the [3/4; 4/3] interval as signaling a risk of erroneous inference due to a model misspecification. We strongly advocate broader use of the sandwich variance for statistical inference about the fixed effects in mixed models.