# The MI Procedure

### Summary of Issues in Multiple Imputation

Subsections:

This section summarizes issues that are encountered in applications of the MI procedure.

#### The MAR Assumption

Multiple imputation usually assumes that the data are missing at random (MAR). But the assumption cannot be verified, because the missing values are not observed. Although the MAR assumption becomes more plausible as more variables are included in the imputation model (Schafer 1997, pp. 27–28; Van Buuren, Boshuizen, and Knook 1999, p. 687), it is important to examine the sensitivity of inferences to departures from the MAR assumption.

#### Number of Imputations

Based on the theory of multiple imputation, only a small number of imputations are needed for a data set with little missing information (Rubin, 1987, p. 114). The number of imputations can be informally verified by replicating sets of m imputations and checking whether the estimates are stable (Horton and Lipsitz, 2001, p. 246).

#### Imputation Model

Generally you should include as many variables as you can in the imputation model (Rubin, 1996), At the same time, however, it is important to keep the number of variables in control, as discussed by Barnard and Meng (1999, pp. 19–20). For the imputation of a particular variable, the model should include variables in the complete-data model, variables that are correlated with the imputed variable, and variables that are associated with the missingness of the imputed variable Schafer 1997, p. 143; Van Buuren, Boshuizen, and Knook 1999, p. 687).

#### Multivariate Normality Assumption

Although the regression and MCMC methods assume multivariate normality, inferences based on multiple imputation can be robust to departures from the multivariate normality if the amount of missing information is not large (Schafer, 1997, pp. 147–148).

You can use variable transformations to make the normality assumption more tenable. Variables are transformed before the imputation process and then back-transformed to create imputed values.

#### Monotone Regression Method

With the multivariate normality assumption, either the regression method or the predictive mean matching method can be used to impute continuous variables in data sets with monotone missing patterns.

The predictive mean matching method ensures that imputed values are plausible and might be more appropriate than the regression method if the normality assumption is violated (Horton and Lipsitz, 2001, p. 246).

#### Monotone Propensity Score Method

The propensity score method can also be used to impute continuous variables in data sets with monotone missing patterns.

The propensity score method does not use correlations among variables and is not appropriate for analyses involving relationship among variables, such as a regression analysis (Schafer, 1999, p. 11). It can also produce badly biased estimates of regression coefficients when data on predictor variables are missing (Allison, 2000).

#### MCMC Monotone-Data Imputation

The MCMC method is used to impute continuous variables in data sets with arbitrary missing patterns, assuming a multivariate normal distribution for the data. It can also be used to impute just enough missing values to make the imputed data sets have a monotone missing pattern. Then, a more flexible monotone imputation method can be used for the remaining missing values.

#### Checking Convergence in MCMC

In an MCMC method, parameters are drawn after the MCMC is run long enough to converge to its stationary distribution. In practice, however, it is not simple to verify the convergence of the process, especially for a large number of parameters.

You can check for convergence by examining the observed-data likelihood ratio statistic and worst linear function of the parameters in each iteration. You can also check for convergence by examining a plot of autocorrelation function, as well as a trace plot of parameters (Schafer, 1997, p. 120).

#### EM Estimates

The EM algorithm can be used to compute the MLE of the mean vector and covariance matrix of the data with missing values, assuming a multivariate normal distribution for the data. However, the covariance matrix associated with the estimate of the mean vector cannot be derived from the EM algorithm.

In the MI procedure, you can use the EM algorithm to compute the posterior mode, which provides a good starting value for the MCMC method (Schafer, 1997, p. 169).

#### Sensitivity Analysis

Multiple imputation inference often assumes that the data are missing at random (MAR). But the MAR assumption cannot be verified, because the missing values are not observed. For a study that assumes MAR, the sensitivity of inferences to departures from the MAR assumption should be examined.

In the MI procedure, you can use the MNAR statement to imputes missing values for scenarios under the MNAR assumption. You can then generate inferences and examine the results. If the results under MNAR differ from the results under MAR, then the conclusion under MAR is in question.