A 1990 Slovenian public opinion survey asked 2,074 adults three questions related to the upcoming independence plebiscite:

  1. Are you in favor of independence?
  2. Are you in favor of secession?
  3. Will you attend the plebiscite?

Plebiscite rules defined nonattendance as a “no” vote; only those who both attend and vote “yes” count as supporting independence. The survey includes “don’t know” (DK) responses, which are treated as missing because each individual will ultimately choose “yes” or “no.”

Let $α$ denote the proportion of the population planning to attend and vote “yes,” the primary estimand.

Crude estimates

Ignoring DK responses yields
$ \hat{α} = 1439 / 1549 = 0.93 $,
suggesting overwhelming support. However, only $1439/2074 = 0.69$ of all respondents clearly planned to attend and vote “yes.”
Assuming all DK responses correspond to “no” produces a very different estimate, indicating the need for a principled missing-data treatment.

Likelihood and prior

The complete data form a multinomial sample of size 2074 over 8 cells (a $2 \times 2 \times 2$ table). Let $θ = (θ_{ijk})$ with
$i, j, k ∈ {0,1}$ representing “no/yes” for questions 1, 2, and 3.

The target estimand is:α=θ101+θ111.α = θ_{101} + θ_{111}.

DK responses are missing values in this multinomial structure.

A Dirichlet prior with all parameters equal to 0.1 provides a proper, diffuse prior. One observed cell count equals 0, so a prior with parameters equal to 0 would be improper unless a structural-zero assumption is made, which is not appropriate here.

Model for the missing data

Let $n = (n_{ijk})$ denote complete data counts and $m = (m_{ijk})$ the fully observed counts. Missing responses fall into 18 partially observed patterns.

For pattern $p$:

  • $r_p$ = number of respondents in this pattern
  • $S_p$ = subset of cells the individual could belong to

Under missing at random (MAR), the probability that an observation in pattern $p$ belongs to cell $(i,j,k)$ isπijk(p)=θijk Iijk∈Sp∑i′j′k′θi′j′k′Ii′j′k′∈Sp.π_{ijk}^{(p)} = \frac{θ_{ijk} \, I_{ijk \in S_p}}{\sum_{i’j’k’} θ_{i’j’k’} I_{i’j’k’ \in S_p}} .

EM algorithm for posterior mode of θ

The EM algorithm computes the posterior mode of $θ$.

E-step

Compute expected complete-data counts:nijkold=mijk+∑prp πijk(p).n_{ijk}^{\text{old}} = m_{ijk} + \sum_{p} r_p \, π_{ijk}^{(p)} .

M-step

Update:θijknew=nijkold+0.1n+0.8.θ_{ijk}^{\text{new}} = \frac{n_{ijk}^{\text{old}} + 0.1}{n + 0.8}.

Convergence occurs when parameter changes fall below a tolerance.

The posterior mode of $α$ is 0.882.

Posterior variance via SEM

A normal approximation requires the posterior variance matrix.
SEM computes this using:

  • the EM updates
  • the complete-data variance for multinomial outcomes

Logit transforms of $θ$ components improve normal approximation accuracy.
The 95% interval for $α$ is $[0.857,, 0.903]$, adjusted for the clustered sampling design.

Multiple imputation via data augmentation

A full Bayesian approach avoids reliance on asymptotic normality.

Data augmentation iterates:

  1. Imputation step:
    Allocate partially observed cases via multinomial draws using $π_{ijk}^{(p)}$.
  2. Posterior draw:
    Given completed data, draw θ∼Dirichlet(nijk+0.1).θ \sim \text{Dirichlet}(n_{ijk} + 0.1).

A total of 1000 posterior draws were used after convergence diagnostics ($\hat{R} < 1.1$).

Posterior inference for α

The posterior median is:α=0.883.α = 0.883.

After variance inflation for survey design, the 95% interval is:[0.859,  0.904].[0.859,\; 0.904].

This aligns with the actual plebiscite, where 88.5% attended and voted “yes.” Crude estimates—either ignoring DK or treating them as “no”—were misleading.