Value of sample size for computation of the Bayesian information criterion (BIC) in multilevel modeling

The Bayesian information criterion (BIC) can be useful for model selection within multilevel-modeling studies. However, the formula for the BIC requires a value for sample size, which is unclear in multilevel models, since sample size is observed for at least two levels. In the present study, we used simulated data to evaluate the rate of false positives and the power when the level 1 sample size, the effective sample size, and the level 2 sample size were used as the sample size value, under various levels of sample size and intraclass correlation coefficient values. The results indicated that the appropriate value for sample size depends on the model and test being conducted. On the basis of the scenarios investigated, we recommend using a BIC that has different penalty terms for each level of the model, based on the number of fixed effects at each level and the level-based sample sizes.