1. Multiple Imputation

2. Multiple Imputation Missing data method developed by Donald Rubin
Simulate multiple samples of “complete” data, and compute estimates and standard errors from the complete data.
Rubin distinguished multiple imputation from
Different models
Same model
We will focus on same-model multiple imputation

3. Missing Data mechanism
Missing data mechanisms
MCAR (Missing completely at random)—missing data are a random subsample of complete data
MAR (Missing at random)—missing data mechanism may depend on independent variables, but not the response

4. Missing Data mechanism
Ignorable nonresponse
MCAR
Parameter for missing process different from data parameters
Example for discussion
Growth curve models for largemouth bass

5. Computer Example 5 Teachers, 3 methods, Y=relative improvement Method
10, 7 6 11 B 4 .
8.5
4,5 3 C 9 13 16 8

6. Multiple Imputation simulation
Repeated draws i=1,…,M from the posterior predictive distribution of the missing data.
The complete data sets have the same set of fully observed responses.
In practice, there are numerous ways to generate complete data.
Introductory methods rely on monotone missingness, and classic results for conditional distributions of jointly multivariate normal random variables.

7. Multiple Imputation simulation
In a multivariate normal setting (some values of Y missing), we generate our draws from Y|X:

8. Multiple Imputation Estimation
Combining results from imputation for parameters of interest is surprisingly straightforward. E.g., let q represent the PMM’s for Method. We can compute

9. Multiple Imputation Estimation
Our estimate and its standard error can be computed as:

10. Multiple Imputation Estimation
Combining estimates in SAS is non-standard.
Our example with LSMeans is atypical, and more straightforward than most.