The MI Procedure

Multiple Imputation with Pattern-Mixture Models

For upper Y equals left-parenthesis upper Y Subscript o b s Baseline comma upper Y Subscript m i s Baseline right-parenthesis, the joint distribution of Y and bold upper R can be expressed as

normal p normal r left-parenthesis upper Y Subscript o b s Baseline comma upper Y Subscript m i s Baseline comma bold upper R right-parenthesis equals normal p normal r left-parenthesis upper Y Subscript m i s Baseline vertical-bar upper Y Subscript o b s Baseline comma bold upper R right-parenthesis normal p normal r left-parenthesis upper Y Subscript o b s Baseline comma bold upper R right-parenthesis

Under the MAR assumption,

normal p normal r left-parenthesis bold upper R vertical-bar upper Y Subscript o b s Baseline comma upper Y Subscript m i s Baseline right-parenthesis equals normal p normal r left-parenthesis bold upper R vertical-bar upper Y Subscript o b s Baseline right-parenthesis

and it can be shown that

normal p normal r left-parenthesis upper Y Subscript m i s Baseline vertical-bar upper Y Subscript o b s Baseline comma bold upper R right-parenthesis equals normal p normal r left-parenthesis upper Y Subscript m i s Baseline vertical-bar upper Y Subscript o b s Baseline right-parenthesis

That is,

normal p normal r left-parenthesis upper Y Subscript m i s Baseline vertical-bar upper Y Subscript o b s Baseline comma bold upper R equals 0 right-parenthesis equals normal p normal r left-parenthesis upper Y Subscript m i s Baseline vertical-bar upper Y Subscript o b s Baseline comma bold upper R equals 1 right-parenthesis

Thus the posterior distribution normal p normal r left-parenthesis upper Y Subscript m i s Baseline vertical-bar upper Y Subscript o b s Baseline comma bold upper R equals 1 right-parenthesis can be used to create imputations for missing data.

Under the MNAR assumption, each pattern that has missing upper Y Subscript m i s values might have a different distribution than the corresponding pattern that has observed upper Y Subscript m i s values. For example, in a clinical trial, suppose the data set contains an indicator variable Trt, with a value of 1 for patients in the treatment group and a value of 0 for patients in the placebo control group, a variable upper Y 0 for the baseline efficacy score, and a variable Y for the efficacy score at a follow-up visit. Assume that Trt and upper Y 0 are fully observed and Y is not fully observed. The indicator variable bold upper R is 0 or 1, depending on whether Y is missing or observed.

Then, under the MAR assumption,

normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 0 comma upper Y 0 comma bold upper R equals 0 right-parenthesis equals normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 0 comma upper Y 0 comma bold upper R equals 1 right-parenthesis

and

normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 1 comma upper Y 0 comma bold upper R equals 0 right-parenthesis equals normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 1 comma upper Y 0 comma bold upper R equals 1 right-parenthesis

Under the MNAR assumption,

normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 0 comma upper Y 0 comma bold upper R equals 0 right-parenthesis not-equals normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 0 comma upper Y 0 comma bold upper R equals 1 right-parenthesis

or

normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 1 comma upper Y 0 comma bold upper R equals 0 right-parenthesis not-equals normal p normal r left-parenthesis upper Y vertical-bar sans-serif upper T sans-serif r sans-serif t equals 1 comma upper Y 0 comma bold upper R equals 1 right-parenthesis

Thus, under MNAR, missing Y values in the treatment group can be imputed from a posterior distribution generated from observations in the control group, and the imputed values can be adjusted to reflect the systematic difference between the distributions for missing and observed Y values.

Multiple imputation inference, under either the MAR or MNAR assumption, involves three distinct phases:

  1. The missing data are filled in m times to generate m complete data sets.

  2. The m complete data sets are analyzed by using other SAS procedures.

  3. The results from the m complete data sets are combined for the inference.

For sensitivity analysis, you must specify the MNAR statement together with a MONOTONE statement or an FCS statement. When you specify a MONOTONE statement, the variables that have missing values are imputed sequentially in each imputation. When you specify an FCS statement, each imputation is carried out in two phases: the preliminary filled-in phase, followed by the imputation phase. The variables that have missing values are imputed sequentially for a number of burn-in iterations before the imputation.

Under the MNAR assumption, the following steps are used to impute missing values for each imputed variable in each imputation (when you specify a MONOTONE statement) or in each iteration (when you specify an FCS statement):

  1. For each imputed variable, a conditional model, such as a regression model for continuous variables, is fitted using either all applicable observations or a specified subset of observations.

  2. A new model is simulated from the posterior predictive distribution of the fitted model.

  3. Missing values of the variable are imputed based on the new model, and the imputed values for a specified subset of observations can be adjusted using specified shift and scale parameters.

The next two sections provide details for specifying subsets of observations for imputation models and for adjusting imputed values.

Last updated: December 09, 2022