The effective sample size in Bayesian information criterion for level-specific fixed and random-effect selection in a two-level nested model

Popular statistical software provides the Bayesian information criterion (BIC) for multi-level models or linear mixed models. However, it has been observed that the combination of statistical literature and software documentation has led to discrepancies in the formulas of the BIC and uncertainties as to the proper use of the BIC in selecting a multi-level model with respect to level-specific fixed and random effects. These discrepancies and uncertainties result from different specifications of sample size in the BIC's penalty term for multi-level models. In this study, we derive the BIC's penalty term for level-specific fixed- and random-effect selection in a two-level nested design. In this new version of BIC, called ${\mathrm{BIC}}_{E1}$, this penalty term is decomposed into two parts if the random-effect variance–covariance matrix has full rank: (a) a term with the log of average sample size per cluster and (b) the total number of parameters times the log of the total number of clusters. Furthermore, we derive the new version of BIC, called ${\mathrm{BIC}}_{E2}$, in the presence of redundant random effects. We show that the derived formulae, ${\mathrm{BIC}}_{E1}$ and ${\mathrm{BIC}}_{E2}$, adhere to empirical values via numerical demonstration and that ${\mathrm{BIC}}_E$ ($E$ indicating either $E1$ or $E2$) is the best global selection criterion, as it performs at least as well as BIC with the total sample size and BIC with the number of clusters across various multi-level conditions through a simulation study. In addition, the use of ${\mathrm{BIC}}_{E1}$ is illustrated with a textbook example dataset.