### Monotone and FCS Logistic Regression Methods

The logistic regression method is another imputation method available for classification variables. In the logistic regression method, a logistic regression model is fitted for a classification variable with a set of covariates constructed from the effects. For a binary classification variable, based on the fitted regression model, a new logistic regression model is simulated from the posterior predictive distribution of the parameters and is used to impute the missing values for each variable (Rubin, 1987, pp. 167–170).

For a binary variable with responses 1 and 2, a logistic regression model is fitted using observations with observed values for the imputed variable and its covariates , , …, :

where are covariates for ,   ,   and

The fitted model includes the regression parameter estimates and the associated covariance matrix .

The following steps are used to generate imputed values for a binary variable with responses 1 and 2:

1. New parameters are drawn from the posterior predictive distribution of the parameters.

where is the upper triangular matrix in the Cholesky decomposition, , and is a vector of independent random normal variates.

2. For an observation with missing and covariates , compute the expected probability that :

where .

3. Draw a random uniform variate, u, between 0 and 1. If the value of u is less than , impute ; otherwise impute .

The preceding logistic regression method can be extended to include the ordinal classification variables with more than two levels of responses. The options ORDER= and DESCENDING can be used to specify the sort order for the levels of the imputed variables.