.. _bda-example-an-opinion-poll-in-slovenia:

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Example: an opinion poll in Slovenia
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**Part 4 · Stage 14 · 🛡️ Robustness & Missing Data**  ·  Lesson 123 of 144  ·  *advanced*

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.. important::

   **✨ AI-generated content.** This page was written with the assistance of an
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   practitioner, treat it as a starting point rather than an authoritative
   reference: read it critically and independently verify anything you act on
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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:

.. math::

   \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.

.. code-block:: python

   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:** :doc:`Multiple imputation <119-multiple-imputation>`  ·  :doc:`Notation <118-notation>`  ·  :doc:`Missing values with counted data <122-missing-values-with-counted-data>`  ·  :doc:`Sample surveys <052-sample-surveys>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/09/example-an-opinion-poll-in-slovenia/ <https://insightful-data-lab.com/2025/12/09/example-an-opinion-poll-in-slovenia/>`__ (insightful-data-lab.com).

.. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: advanced