Imputation and variable selection in linear regression models with missing covariates

Across multiply imputed data sets, variable selection methods such as stepwise regression and other criterion-based strategies that include or exclude particular variables typically result in models with different selected predictors, thus presenting a problem for combining the results from separate complete-data analyses. Here, drawing on a Bayesian framework, we propose two alternative strategies to address the problem of choosing among linear regression models when there are missing covariates. One approach, which we call "impute, then select" (ITS) involves initially performing multiple imputation and then applying Bayesian variable selection to the multiply imputed data sets. A second strategy is to conduct Bayesian variable selection and missing data imputation simultaneously within one Gibbs sampling process, which we call "simultaneously impute and select" (SIAS). The methods are implemented and evaluated using the Bayesian procedure known as stochastic search variable selection for multivariate normal data sets, but both strategies offer general frameworks within which different Bayesian variable selection algorithms could be used for other types of data sets. A study of mental health services utilization among children in foster care programs is used to illustrate the techniques. Simulation studies show that both ITS and SIAS outperform complete-case analysis with stepwise variable selection and that SIAS slightly outperforms ITS.