Normalized Information Criteria and Model Selection in the Presence of Missing Data

Abstract

Information criteria such as the Akaike information criterion (AIC) and Bayesian information criterion (BIC) are commonly used for model selection. However, the current theory does not support unconventional data, so naive use of these criteria is not suitable for data with missing values. Imputation, at the core of most alternative methods, is both distorted as well as computationally demanding. We propose a new approach that enables the use of classic well-known information criteria for model selection when there are missing data. We adapt the current theory of information criteria through normalization, accounting for the different sample sizes used for each candidate model (focusing on AIC and BIC). Interestingly, when the sample sizes are different, our theoretical analysis finds that AICj/nj is the proper correction for AICj that we need to optimize (where nj is the sample size available to the jth model) while −(BICj−BICi)/(nj−ni) is the correction of BIC. Furthermore, we find that the computational complexity of normalized information criteria methods is exponentially better than that of imputation methods. In a series of simulation studies, we find that normalized-AIC and normalized-BIC outperform previous methods (i.e., normalized-AIC is more efficient, and normalized BIC includes only important variables, although it tends to exclude some of them in cases of large correlation). We propose three additional methods aimed at increasing the statistical efficiency of normalized-AIC: post-selection imputation, Akaike sub-model averaging, and minimum-variance averaging. The latter succeeds in increasing efficiency further.