.. _bda-example-multiple-imputation-for-a-series-of-polls:

========================================================================
Example: multiple imputation for a series of polls
========================================================================

**Part 4 · Stage 14 · 🛡️ Robustness & Missing Data**  ·  Lesson 121 of 144  ·  *advanced*

:doc:`◀ Previous · Missing data in the multivariate normal and t models <120-missing-data-in-the-multivariate-normal-and-t-models>`   ·   :doc:`Next · Missing values with counted data ▶ <122-missing-values-with-counted-data>`   ·   :doc:`↑ Section <index>`


.. important::

   **✨ AI-generated content.** This page was written with the assistance of an
   AI language model and is provided as a learning aid. Despite careful
   review, it may still contain mistakes, omissions, or out-of-date
   information. Whether you are new to the topic, a team lead, or a senior
   practitioner, treat it as a starting point rather than an authoritative
   reference: read it critically and independently verify anything you act on
   (code, commands, figures, and factual claims) against official
   documentation and primary sources before relying on it.

Missingness across many surveys
---------------------------------

Pre-election polling assembles many surveys, and no two ask exactly the same questions of exactly the
same people. Some respondents skip items; some questions appear in only a subset of polls; demographic
variables are recorded unevenly. Analysing the pooled series requires **filling these gaps coherently**,
and it is a natural showcase for multiple imputation at scale.

The structure
---------------

The data form a large, ragged table: respondents in rows, survey items in columns, with a scattered
pattern of missingness. Crucially the gaps are plausibly **MAR** — whether an item is missing depends on
*which survey* a respondent was in and on their recorded demographics, not on the hidden answer itself.
That is exactly the condition under which imputation is valid, and it holds here by design: the
missingness is a feature of survey construction, observed in the data.

The imputation model
----------------------

Fit a joint model rich enough to capture the relationships the analysis will use — demographic
predictors, survey indicators, and the correlations among responses — then draw completed datasets from
its posterior predictive distribution.

.. code-block:: python

   import numpy as np, pymc as pm
   Y = np.ma.masked_invalid(responses)                      # ragged missingness flagged
   with pm.Model():
       mu = pm.Normal("mu", 0, 5, shape=Y.shape[1])
       chol, _, _ = pm.LKJCholeskyCov("chol", n=Y.shape[1], eta=2,
                                      sd_dist=pm.HalfNormal.dist(1.0), compute_corr=True)
       pm.MvNormal("y", mu=mu, chol=chol, observed=Y)       # missing entries imputed
       idata = pm.sample()
   # each posterior draw carries a completed dataset; analyse across draws, not one

Why it matters here
---------------------

Three lessons the polling context makes vivid. **Completing the data unlocks the pooled analysis** — with
gaps filled coherently, every survey contributes to every question, rather than fragmenting into
question-by-question subsamples. The imputation **propagates uncertainty**: where a respondent's answer
was never observed, the spread across imputations widens the final interval honestly, so the pooled
estimate is not falsely precise. And it **respects survey structure** — conditioning on the survey
indicator means a poll that asked a question informs imputations for polls that did not, without
pretending they are identical. The same machinery underlies modern poll aggregation: incomplete,
heterogeneous surveys combined into one coherent picture, with missingness modelled rather than deleted.

.. hint::

   **Related lessons:** :doc:`Multiple imputation <119-multiple-imputation>`  ·  :doc:`Missing data in the multivariate normal and t models <120-missing-data-in-the-multivariate-normal-and-t-models>`  ·  :doc:`Missing values with counted data <122-missing-values-with-counted-data>`  ·  :doc:`State-level opinons from national polls <110-state-level-opinons-from-national-polls>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/07/example-multiple-imputation-for-a-series-of-polls/ <https://insightful-data-lab.com/2025/12/07/example-multiple-imputation-for-a-series-of-polls/>`__ (insightful-data-lab.com).

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