Example: an opinion poll in Slovenia#

Part 4 · Stage 14 · 🛡️ Robustness & Missing Data · Lesson 123 of 144 · advanced

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Important

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A referendum with missing answers#

Before Slovenia’s 1990 independence plebiscite, a survey asked citizens how they intended to vote — and, as always, many respondents gave no answer to one or more questions. The stakes made the missing data consequential: the result hinged on how “don’t know” and non-response were treated, and the analysis is a clean demonstration of ignorability and imputation deciding a real number.

The problem#

Respondents answered (or did not) several related items — intention to vote, intended choice, attitude toward independence. Simply dropping non-responders assumes their views match responders’, which is precisely the MCAR assumption that is rarely safe: people who decline to answer a charged political question are plausibly different from those who answer. The question is whether the missingness is ignorable given what was observed.

The model-based analysis#

Treat the non-responses as missing data under MAR — missingness depending on observed covariates and other answered items, not on the hidden intention itself — and build a joint model over the survey items. Missing responses become parameters, imputed from their posterior predictive distribution given each respondent’s observed answers:

\[\Pr(\text{vote} = k \mid \text{observed items}) \; \text{modelled jointly, then}\; \theta = \sum_{\text{cells}} N_c \, \hat{p}_c \Big/ \sum_{\text{cells}} N_c ,\]

so each respondent’s likely position is inferred from the pattern of their other answers, and the overall estimate integrates over what they did not say.

import numpy as np, pymc as pm
items = np.ma.masked_invalid(survey_items)               # non-responses flagged
with pm.Model():
    # joint model over correlated categorical items; missing entries imputed
    ...                                                   # observed answers inform the gaps
    idata = pm.sample()
# estimate turnout/support by integrating over imputed non-responses, with honest intervals

The lessons#

Three, and they close the stage. The missingness assumption changes the answer — treating non-responders as a random subsample (MCAR) versus modelling them from their observed answers (MAR) gives materially different estimates, so the assumption must be stated, not buried in a default. Ignorability lets the observed answers speak for the missing ones: a respondent’s other items carry information about the answer they withheld, and a joint model extracts it. And where MAR itself is in doubt — perhaps non-response is informative about the vote — the honest response is a sensitivity analysis across plausible mechanisms, reporting how much the conclusion moves. Real surveys are never complete; this is what taking their gaps seriously looks like.

Hint

Related lessons: Multiple imputation · Notation · Missing values with counted data · Sample surveys