Multiple Imputation

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

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

Missing Data mechanism

Ignorable nonresponse– MCAR– Parameter for missing process different

from data parameters Example for discussion

– Growth curve models for largemouth bass

Computer Example

5 Teachers, 3 methods, Y=relative improvement

Method Teacher 1

Teacher 2

Teacher 3

Teacher 4

Teacher 5

A 10, 7 6 11 6 6

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.

Multiple Imputation simulation

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

XYXXYXYYXXXYXY

XXXY

YXYY

X

Y

XMVNXY

MVNX

Y

11 ,~|then

,,~ If

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

ˆ i and s ˆ i, i 1,...M

Multiple Imputation Estimation

Our estimate and its standard error can be computed as:

1

ˆ and

1

,1

,ˆ1

2

1

2ˆ

2

1

MBs

MW

BM

MWs

M

MiM

M

iM

MM

M

iiM

i

M

Multiple Imputation Estimation

Combining estimates in SAS is non-standard.

Our example with LSMeans is atypical, and more straightforward than most.