Variable selection when confronted with missing data

Variable selection is a common problem in linear regression. Stepwise methods, such as forward selection, are popular and are easily available in most statistical packages. The models selected by these methods have a number of drawbacks: they are often unstable, with changes in the set of variable selected due to small changes in the data, and they provide upwardly biased regression coefficient estimates. Recently proposed methods, such as the lasso, provide accurate predictions via a parsimonious, interpretable model. Missing data values are also a common problem, especially in longitudinal studies. One approach to account for missing data is multiple imputation. The simulation studies were conducted comparing the lasso to standard variable selection methods under different missing data conditions, including the percentage of missing values and the missing data mechanism. Under missing at random mechanisms, missing data were created at the 25 and 50 percent levels with two types of regression parameters, one containing large effects and one containing several small, but nonzero, effects. Five correlation structures were used in generating the data: independent, autoregressive with correlation 0.25 and 0.50, and equicorrelated again with correlation 0.25 and 0.50. Three different missing data mechanisms were used to create the missing data: linear, convex and sinister. Least angle regression performed well under all conditions when the true regression parameter vector contained large effects, with its dominance increasing as the correlation between the predictor variables increased. This is consistent with complete data simulations studies suggesting the lasso performed poorly in situations where the true beta vector contained small, nonzero effects. When the true beta vector contained small, nonzero effects, the performance of the variable selection methods considered was situation dependent. Ordinary least squares had superior performance in terms confidence interval coverage under the independent correlation structure and with correlated data when the true regression parameter vector consists of small, nonzero effects. A variety of methods performed well when the regression parameter vector consisted of large effects and the predictor variables were correlated depending on the missing data situation.