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The Bayesian information criterion (BIC), defined as the observed data loglikelihood minus a penalty term based on the sample size $N$, is a popularmodel selection criterion for factor analysis with complete data. Thisdefinition has also been suggested for incomplete data. However, the penaltyterm based on the `complete' sample size $N$ is the same no matter whether in acomplete or incomplete data case. For incomplete data, there are often only$N_i<N$ observations for variable $i$, which means that using the `complete'sample size $N$ implausibly ignores the amounts of missing information inherentin incomplete data. Given this observation, a novel criterion calledhierarchical BIC (HBIC) for factor analysis with incomplete data is proposed.The novelty is that it only uses the actual amounts of observed information,namely $N_i$'s, in the penalty term. Theoretically, it is shown that HBIC is alarge sample approximation of variational Bayesian (VB) lower bound, and BIC isa further approximation of HBIC, which means that HBIC shares the theoreticalconsistency of BIC. Experiments on synthetic and real data sets are conductedto access the finite sample performance of HBIC, BIC, and related criteria withvarious missing rates. The results show that HBIC and BIC perform similarlywhen the missing rate is small, but HBIC is more accurate when the missing rateis not small.

Choosing the number of factors in factor analysis with incomplete data via a hierarchical Bayesian information criterion