Unbalanced cluster sizes and rates of convergence in mixed-effects models for clustered data

Convergence problems often arise when complex linear mixed-effects models are fitted. Previous simulation studies (see, e.g. [Buyse M, Molenberghs G, Burzykowski T, Renard D, Geys H. The validation of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics. 2000;1:49-67, Renard D, Geys H, Molenberghs G, Burzykowski T, Buyse M. Validation of surrogate endpoints in multiple randomized clinical trials with discrete outcomes. Biom J. 2002;44:921-935]) have shown that model convergence rates were higher (i) when the number of available clusters in the data increased, and (ii) when the size of the between-cluster variability increased (relative to the size of the residual variability). The aim of the present simulation study is to further extend these findings by examining the effect of an additional factor that is hypothesized to affect model convergence, i.e. imbalance in cluster size. The results showed that divergence rates were substantially higher for data sets with unbalanced cluster sizes - in particular when the model at hand had a complex hierarchical structure. Furthermore, the use of multiple imputation to restore balance' in unbalanced data sets reduces model convergence problems.