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Example 30
Bayesian imputation is like stochastic regression imputation except that it takes
into account the fact that the parameter values are only estimated and not known.
Multiple Imputation
In multiple imputation (Schafer, 1997), a nondeterministic imputation method (either
stochastic regression imputation or Bayesian imputation) is used to create multiple
completed datasets. While the observed values never change, the imputed values vary
from one completed dataset to the next. Once the completed datasets have been
created, each completed dataset is analyzed alone. For example, if there are m
completed datasets, then there will be m separate sets of results, each containing
estimates of various quantities along with estimated standard errors. Because the m
completed datasets are different from each other, the m sets of results will also differ
from one to the next.
After each of the m completed datasets has been analyzed alone, the data analyst has
m sets of estimates and standard errors that must be combined into a single set of results.
Well-known formulas attributed to Rubin (1987) are available for combining the results
from multiple completed datasets. Those formulas will be used in Example 31.
Model-Based Imputation
In this example, imputation is performed using a factor analysis model. Model-based
imputation has two advantages. First, you can impute values for any latent variables in
the model. Second, if the model is correct and has positive degrees of freedom, the
implied covariance matrix and implied means will be estimated more accurately than
with a saturated model. (Imputation is based on the implied covariance matrix and
means.) However, a saturated model like the model in Example 1 can be used for
imputation when no other model is appropriate.
Performing Multiple Data Imputation Using Amos Graphics
For this example, we will perform Bayesian multiple imputation using the
confirmatory factor analysis model from Example 17. The dataset is the incomplete
Holzinger and Swineford (1939) dataset in the file grant_x.sav. The imputation of
missing values is only the first step in obtaining useful results from multiple
imputation. Eventually, all three of the following steps need to be carried out.