The MI Procedure

Monotone and FCS Discriminant Function Methods

Subsections:

The discriminant function method is the default imputation method in the MONOTONE and FCS statements for classification variables.

For a nominal classification variable $Y_{j}$ with responses 1, …, g and a set of effects from its preceding variables, if the covariates $X_{1}$, $X_{2}$, …, $X_{k}$ 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 $Y_{j}$.

Denote the group-specific means for covariates $X_{1}$, $X_{2}$, …, $X_{k}$ by

\[ \overline{\mb{X}}_{t} = ( \overline{X}_{t1}, \overline{X}_{t2}, \ldots , \overline{X}_{tk} ), \, t= 1, 2, \ldots , g \]

then the pooled covariance matrix is computed as

\[ \mb{S} = \frac{1}{n-g} \sum _{t=1}^{g} (n_{t}-1) \mb{S}_{t} \]

where $\mb{S}_{t}$ is the within-group covariance matrix, $n_{t}$ is the group-specific sample size, and $n= \sum _{t=1}^{g} n_{t}$ is the total sample size.

In each imputation, new parameters of the group-specific means ($\mb{m}_{*t}$), pooled covariance matrix ($\mb{S}_{*}$), and prior probabilities of group membership ($q_{*t}$) can be drawn from their corresponding posterior distributions (Schafer 1997, p. 356).

Pooled Covariance Matrix and Group-Specific Means

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,

\begin{eqnarray*} \bSigma | \Strong{X} \quad \sim & & W^{-1} \left( \, n-g, \, (n-g)\Strong{S} \right) \end{eqnarray*}

where $W^{-1}$ is an inverted Wishart distribution.

The group-specific means are then drawn from their posterior distributions with a noninformative prior

\begin{eqnarray*} \bmu _{t} | ( \bSigma , \overline{\Strong{X}}_{t}) \quad \sim & & N \left( \, \overline{\Strong{X}}_{t}, \, \, \frac{1}{\, n_{t} \, } \bSigma \right) \end{eqnarray*}

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.

Prior Probabilities of Group Membership

The prior probabilities are computed through the drawing of new group sample sizes. When the total sample size n is considered fixed, the group sample sizes $(n_{1}, n_{2}, \ldots , n_{g})$ have a multinomial distribution. New multinomial parameters (group sample sizes) can be drawn from their posterior distribution by using a Dirichlet prior with parameters $({\alpha }_{1}, {\alpha }_{2}, \ldots , {\alpha }_{g})$.

After the new sample sizes are drawn from the posterior distribution of $(n_{1}, n_{2}, \ldots , n_{g})$, the prior probabilities $q_{*t}$ are computed proportionally to the drawn sample sizes.

See Schafer (1997, pp. 247–255) for a complete description of the Dirichlet prior.

Imputation Steps

The discriminant function method uses the following steps in each imputation to impute values for a nominal classification variable $Y_{j}$ with g responses:

  1. Draw a pooled covariance matrix $\mb{S}_{*}$ from its posterior distribution if the PCOV=POSTERIOR option is used.

  2. For each group, draw group means $\mb{m}_{*t}$ from the observed group mean $\overline{\mb{X}}_{t}$ and either the observed pooled covariance matrix (PCOV=FIXED) or the drawn pooled covariance matrix $\mb{S}_{*}$ (PCOV=POSTERIOR).

  3. For each group, compute or draw $q_{*t}$, prior probabilities of group membership, based on the PRIOR= option:

    • PRIOR=EQUAL, $q_{*t}=1/g$, prior probabilities of group membership are all equal.

    • PRIOR=PROPORTIONAL, $q_{*t}=n_{t}/n$, prior probabilities are proportional to their group sample sizes.

    • PRIOR=JEFFREYS=$\Argument{c}$, a noninformative Dirichlet prior with ${\alpha }_{t}=c$ is used.

    • PRIOR=RIDGE=$\Argument{d}$, a ridge prior is used with ${\alpha }_{t} = d * n_{t}/n$ for $d \geq 1$ and ${\alpha }_{t} = d * n_{t}$ for $d < 1$.

  4. With the group means $\mb{m}_{*t}$, the pooled covariance matrix $\mb{S}_{*}$, and the prior probabilities of group membership $q_{*t}$, the discriminant function method derives linear discriminant function and computes the posterior probabilities of an observation belonging to each group

    \[ p_{t}(\mb{x}) = \frac{ \mr{exp}(-0.5 D_{t}^{2}(\mb{x}) )}{ \sum _{u=1}^{g} \mr{exp}(-0.5 D_{u}^{2}(\mb{x}) )} \]

    where $D_{t}^{2}(\mb{x}) = {(\mb{x}-\mb{m}_{*t})}^{\prime } \mb{S}_{*}^{-1} (\mb{x}-\mb{m}_{*t}) - 2 \, \mr{log}(q_{*t})$ is the generalized squared distance from $\mb{x}$ to group t.

  5. Draw a random uniform variate u, between 0 and 1, for each observation with missing group value. With the posterior probabilities, $p_{1}(\mb{x}) + p_{2}(\mb{x}) + \ldots , + p_{g}(\mb{x}) = 1$, the discriminant function method imputes $Y_{j}= 1$ if the value of u is less than $p_{1}(\mb{x})$, $Y_{j}= 2$ if the value is greater than or equal to $p_{1}(\mb{x})$ but less than $p_{1}(\mb{x})+p_{2}(\mb{x})$, and so on.