Monotone and FCS Discriminant Function Methods |
The discriminant function method is the default imputation method in the MONOTONE and FCS statements for classification variables.
For a nominal classification variable with responses 1, ..., and a set of effects from its preceding variables, if the covariates , , ..., associated with these effects within each group are approximately multivariate normal and the within-group covariance matrices are approximately equal, the discriminant function method (Brand 1999, pp. 95–96) can be used to impute missing values for the variable .
Denote the group-specific means for covariates , , ..., by
then the pooled covariance matrix is computed as
where is the within-group covariance matrix, is the group-specific sample size, and is the total sample size.
In each imputation, new parameters of the group-specific means (), pooled covariance matrix (), and prior probabilities of group membership () can be drawn from their corresponding posterior distributions (Schafer 1997, p. 356).
For each imputation, the MI procedure uses either the fixed observed pooled covariance matrix (PCOV=FIXED) or a drawn pooled covariance matrix (PCOV=POSTERIOR) from its posterior distribution with a noninformative prior. That is,
where is an inverted Wishart distribution.
The group-specific means are then drawn from their posterior distributions with a noninformative prior
See the section Bayesian Estimation of the Mean Vector and Covariance Matrix for a complete description of the inverted Wishart distribution and posterior distributions that use a noninformative prior.
The prior probabilities are computed through the drawing of new group sample sizes. When the total sample size is considered fixed, the group sample sizes have a multinomial distribution. New multinomial parameters (group sample sizes) can be drawn from their posterior distribution by using a Dirichlet prior with parameters .
After the new sample sizes are drawn from the posterior distribution of , the prior probabilities are computed proportionally to the drawn sample sizes.
See Schafer (1997, pp. 247–255) for a complete description of the Dirichlet prior.
The discriminant function method uses the following steps in each imputation to impute values for a nominal classification variable with responses:
Draw a pooled covariance matrix from its posterior distribution if the PCOV=POSTERIOR option is used.
For each group, draw group means from the observed group mean and either the observed pooled covariance matrix (PCOV=FIXED) or the drawn pooled covariance matrix (PCOV=POSTERIOR).
For each group, compute or draw , prior probabilities of group membership, based on the PRIOR= option:
PRIOR=EQUAL, , prior probabilities of group membership are all equal.
PRIOR=PROPORTIONAL, , prior probabilities are proportional to their group sample sizes.
PRIOR=JEFFREYS=, a noninformative Dirichlet prior with is used.
PRIOR=RIDGE=, a ridge prior is used with for and for .
With the group means , the pooled covariance matrix , and the prior probabilities of group membership , the discriminant function method derives linear discriminant function and computes the posterior probabilities of an observation belonging to each group
where is the generalized squared distance from to group .
Draw a random uniform variate , between 0 and 1, for each observation with missing group value. With the posterior probabilities, , the discriminant function method imputes if the value of is less than , if the value is greater than or equal to but less than , and so on.