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The MI Procedure |
Discriminant Function Method for Monotone Missing Data |
The discriminant function method is the default imputation method for classification variables in a data set with a monotone missing pattern.
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
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then the pooled covariance matrix is computed as
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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,
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where is an inverted Wishart distribution.
The group-specific means are then drawn from their posterior distributions with a noninformative prior
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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
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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.
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