Auxiliary variables in multiple imputation in regression with missing X: a warning against including too many in small sample research

BACKGROUND: Multiple imputation is becoming increasingly popular. Theoretical considerations as well as simulation studies have shown that the inclusion of auxiliary variables is generally of benefit. METHODS: A simulation study of a linear regression with a response Y and two predictors X1 and X2 was performed on data with n = 50, 100 and 200 using complete cases or multiple imputation with 0, 10, 20, 40 and 80 auxiliary variables. Mechanisms of missingness were either 100% MCAR or 50% MAR + 50% MCAR. Auxiliary variables had low (r=.10) vs. moderate correlations (r=.50) with X's and Y. RESULTS: The inclusion of auxiliary variables can improve a multiple imputation model. However, inclusion of too many variables leads to downward bias of regression coefficients and decreases precision. When the correlations are low, inclusion of auxiliary variables is not useful. CONCLUSION: More research on auxiliary variables in multiple imputation should be performed. A preliminary rule of thumb could be that the ratio of variables to cases with complete data should not go below 1 : 3.