Missing values with counted data

Missing discrete data can be handled by extending methods used for multinomial and loglinear count models.

Multinomial samples

Assume the complete data come from a multinomial sample of size $n$ with $J$ cells, probabilities
$\theta = (\pi_1, \pi_2, \pi_3, \ldots, \pi_J)$,
and cell counts $(n_1, \ldots, n_J)$.

A Dirichlet prior is a natural conjugate prior for $\theta$.

The observed data consist of:

• $m$ fully classified observations with counts $(m_1, \ldots, m_J)$
• $n – m$ partially classified observations whose exact cell membership is unknown but restricted to a subset of possible cells

Example: In a $2 \times 2 \times 2$ table ($J = 8$), if two dimensions are observed but one is missing, an observation belongs to one of two possible cells.

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Partially classified observations

For the $n – m$ partially classified units, group observations by the subset of cells they could belong to.

• Let $K$ be the number of types of partially observed patterns.
• Let $S_k$ be the subset of cells possible for pattern $k$.
• Let $r_k$ be the number of observations with pattern $k$.

Imputation step (data augmentation)

For each partially classified observation in group $k$, assign it to cell $j \in S_k$ with probability:$$\frac{\pi_j\, I_{j \in S_k}}{\sum_{l=1}^{J} \pi_l\, I_{l \in S_k}}$$

where $I_{j \in S_k} = 1$ when cell $j$ is allowed for pattern $k$, and $0$ otherwise.

This produces a complete dataset by stochastically allocating missing classifications.

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Posterior updating step

Given a Dirichlet prior for $\theta$, the complete-data posterior is also Dirichlet with updated parameters equal to:$$m_j + \text{(imputed count for cell } j)$$

Nonconjugate priors can be used, but then posterior simulation is more complex.

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EM algorithm interpretation

A deterministic variant of data augmentation forms an EM algorithm:

E-step

Compute expected counts for each cell:

$\text{E}[\text{\# in cell } j] = m_j + r_k \cdot \frac{\pi_j\, I_{j \in S_k}}{\sum_{l=1}^J \pi_l\, I_{l \in S_k}}$

M-step

Update cell probabilities by normalizing the expected counts.

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Monotone or nearly monotone missing-data patterns

Computation becomes simpler when the partially classified data follow a monotone pattern:

• The likelihood factors into successive conditional multinomial components.
• If the prior factors accordingly (e.g., product of Dirichlets), posterior draws for each block can be obtained directly without simulation.
• For nearly monotone patterns, iterate:
  1. Impute values needed to restore monotonicity
  2. Draw parameters from factored posterior distributions

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Complications with modeled cell probabilities

When modeling $\theta$ structurally (e.g., loglinear relationships among cells), additional computation is required because cell probabilities depend on parameters through a nonlinear model. Posterior simulation must reflect this dependency.

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Example (Slovenia poll)

In the 1990 Slovenian preplebiscite survey, counts form a $3 \times 3 \times 3$ table. “Don’t know” responses are treated as missing. The objective is to estimate the proportion of the electorate whose underlying true responses are “Yes” for both independence and attendance.