Multiple Imputation for Missing Data

Multiple imputation provides a useful strategy for dealing with data sets with missing values. Instead of filling in a single value for each missing value, Rubin’s (1987) multiple imputation procedure replaces each missing value with a set of plausible values that represent the uncertainty about the right value to impute. These multiply imputed data sets are then analyzed by using standard procedures for complete data and combining the results from these analyses. No matter which complete-data analysis is used, the process of combining results from different imputed data sets is essentially the same. This results in valid statistical inferences that properly reflect the uncertainty due to missing values. This paper reviews methods for analyzing missing data, including basic concepts and applications of multiple imputation techniques. The paper also presents new SAS R procedures for creating multiple imputations for incomplete multivariate data and for analyzing results from multiply imputed data sets. These procedures are still under development and will be available in experimental form in Release 8.1 of the SAS System. Introduction Most SAS statistical procedures exclude observations with any missing variable values from the analysis. These observations are called incomplete cases. While using only complete cases has its simplicity, you lose information in the incomplete cases. This approach also ignores the possible systematic difference between the complete cases and incomplete cases, and the resulting inference may not be applicable to the population of all cases, especially with a smaller number of complete cases. Some SAS procedures use all the available cases in an analysis, that is, cases with available information. For example, PROC CORR estimates a variable mean by using all cases with nonmissing values on this variable, ignoring the possible missing values in other variables. PROC CORR also estimates a correlation by using all cases with nonmissing values for this pair of variables. This may make better use of the available data, but the resulting correlation matrix may not be positive definite. Another strategy is simple imputation, in which you substitute a value for each missing value. Standard statistical procedures for complete data analysis can then be used with the filled-in data set. For example, each missing value can be imputed from the variable mean of the complete cases, or it can be imputed from the mean conditional on observed values of other variables. This approach treats missing values as if they were known in the complete-data analyses. Single imputation does not reflect the uncertainty about the predictions of the unknown missing values, and the resulting estimated variances of the parameter estimates will be biased toward zero. Instead of filling in a single value for each missing value, a multiple imputation procedure (Rubin 1987) replaces each missing value with a set of plausible values that represent the uncertainty about the right value to impute. The multiply imputed data sets are then analyzed by using standard procedures for complete data and combining the results from these analyses. No matter which complete-data analysis is used, the process of combining results from different data sets is essentially the same. Multiple imputation does not attempt to estimate each missing value through simulated values but rather to represent a random sample of the missing values. This process results in valid statistical inferences that properly reflect the uncertainty due to missing values; for example, valid confidence intervals for parameters. Multiple imputation inference involves three distinct phases: The missing data are filled in m times to generate m complete data sets. The m complete data sets are analyzed by using standard procedures. The results from the m complete data sets are combined for the inference. A new SAS/STAT R procedure, PROC MI, is a multiple imputation procedure that creates multiply imputed data sets for incomplete p-dimensional multivariate data. It uses methods that incorporate appropriate variability across the m imputations. Once the m complete data sets are analyzed by using standard procedures, another new procedure, PROC MIANALYZE, can be used to generate valid statistical inferences about these parameters by combining results from the m complete data sets.