.. _bda-missing-values-with-counted-data:

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Missing values with counted data
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**Part 4 · Stage 14 · 🛡️ Robustness & Missing Data**  ·  Lesson 122 of 144  ·  *advanced*

:doc:`◀ Previous · Example: multiple imputation for a series of polls <121-example-multiple-imputation-for-a-series-of-polls>`   ·   :doc:`Next · Example: an opinion poll in Slovenia ▶ <123-example-an-opinion-poll-in-slovenia>`   ·   :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   review, it may still contain mistakes, omissions, or out-of-date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.

Imputation beyond the normal
------------------------------

The multivariate-normal imputation of the previous stage assumed continuous variables. **Count** data —
disease cases, survey tallies, event frequencies — need imputation on their own terms: a fractional or
negative imputed count is nonsense, so the imputation model must respect the discreteness.

Model the counts directly
---------------------------

The Bayesian principle is unchanged: treat missing counts as **unknown parameters** and give them a
count likelihood, so imputations are proper draws from a Poisson or binomial rather than rounded normals.

.. math::

   y_i^{\text{mis}} \sim \mathrm{Poisson}(\lambda_i), \qquad
   \log \lambda_i = X_i \beta,

with the same linear-predictor machinery as the GLM stage. A missing count is drawn from its posterior
predictive Poisson, guaranteeing a non-negative integer, and — if the counts are overdispersed — a
**negative binomial** imputation carries the extra variance, exactly as robustness demanded for observed
counts.

.. code-block:: python

   import pymc as pm
   with pm.Model():
       beta = pm.Normal("beta", 0, 1, shape=k)
       rate = pm.math.exp(X @ beta)
       # observed counts constrain beta; missing entries are drawn as integer parameters
       pm.Poisson("y", mu=rate, observed=y_counts_with_missing)   # masked array
       idata = pm.sample()

The offset subtlety
---------------------

Counts usually come with an **exposure** — population at risk, area, time — and a missing count often
sits beside a *known* exposure. The imputation must condition on it: impute the **rate** from the model,
then scale by the observed exposure to draw the count, so a small-population cell gets a correspondingly
small imputed count. Ignoring the offset would impute as if every cell had the same exposure, distorting
exactly the comparison the counts were meant to support.

Where it fits
---------------

The lesson generalises the point that **imputation inherits the likelihood**: use a normal model and you
impute normals; use a count model and you impute counts. Getting the imputation distribution right — its
support, its variance, its offset — is what keeps completed data coherent with the process that generated
them. It closes the mechanics of missing data; the next example puts the whole apparatus, ignorability
and imputation together, on a real survey.

.. hint::

   **Related lessons:** :doc:`Multiple imputation <119-multiple-imputation>`  ·  :doc:`Standard generalized linear model likelihoods <106-standard-generalized-linear-model-likelihoods>`  ·  :doc:`Overdispersed Poisson regression for police stops <109-overdispersed-poisson-regression-for-police-stops>`  ·  :doc:`Example: an opinion poll in Slovenia <123-example-an-opinion-poll-in-slovenia>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/09/missing-values-with-counted-data/ <https://insightful-data-lab.com/2025/12/09/missing-values-with-counted-data/>`__ (insightful-data-lab.com).

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