Example: multiple imputation for a series of polls#

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

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Important

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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.

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