.. _bda-censoring-and-truncation:

========================================================================
Censoring and truncation
========================================================================

**Part 2 · Stage 7 · 🗳️ Data Collection & Decisions**  ·  Lesson 056 of 144  ·  *intermediate*

:doc:`◀ Previous · Observational studies <055-observational-studies>`   ·   :doc:`Next · Bayesian decision theory in different contexts ▶ <057-bayesian-decision-theory-in-different-contexts>`   ·   :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   review, it may still contain mistakes, omissions, or out-of-date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.

Two ways data go missing
--------------------------

Both words describe incomplete observation, and confusing them yields a wrong likelihood. The
distinction is what you know about the units you did not fully see.

* **Censoring**: the unit **is** in your dataset, but its value is known only to lie in a range. A
  patient still alive at the end of a trial has survival time :math:`> c`. A scale reading "over 100 kg".
* **Truncation**: the unit is **absent entirely**. Only patients who survived long enough to enrol
  appear; light sources fainter than the telescope's limit are never recorded. You do not know how many
  you missed.

The likelihoods differ
------------------------

For **censoring**, the censored observation contributes the probability of the event it represents —
the survival function, not the density:

.. math::

   p(y \mid \theta) = \prod_{i \, \text{obs}} f(y_i \mid \theta)
                      \prod_{i \, \text{cens}} \bigl[1 - F(c_i \mid \theta)\bigr] .

For **truncation**, every observed value must be **renormalised** by the probability of being observed
at all, because the sample space itself is restricted:

.. math::

   p(y \mid \theta) = \prod_{i} \frac{f(y_i \mid \theta)}{\Pr(\text{observed} \mid \theta)}
                    = \prod_{i} \frac{f(y_i \mid \theta)}{1 - F(c \mid \theta)} .

Ignore the denominator and you fit a model to a biased sample — the truncated-normal demonstration from
the data-collection lesson, where the sample mean converged confidently to the wrong number.

In code
---------

Both are one line in a modern PPL, which is precisely why the distinction must be made **before**
coding:

.. code-block:: python

   import pymc as pm
   with pm.Model():
       mu, sigma = pm.Normal("mu", 0, 10), pm.HalfNormal("sigma", 5)
       # censoring: value known only to exceed c (unit IS in the data)
       pm.Censored("y_cens", pm.Normal.dist(mu, sigma), lower=None, upper=c, observed=y)
       # truncation: values beyond c never enter the sample at all
       pm.Truncated("y_trunc", pm.Normal.dist(mu, sigma), lower=c, observed=y)

Censoring is ignorable, truncation less so
--------------------------------------------

Where the censoring **time** is fixed in advance, or depends only on observed quantities, the mechanism
is ignorable in the technical sense — the censored contributions must appear in the likelihood, but no
separate model for :math:`\phi` is needed. When censoring depends on the **unobserved** value itself —
patients withdrawing *because* they are deteriorating — the mechanism is **MNAR**, and the analysis
must model why they left. That is not a computational difficulty but an **identification** one: the
data are silent, and the answer will move with the assumption. Report the sensitivity.

.. hint::

   **Related lessons:** :doc:`Bayesian inference requires a model for data collection <050-bayesian-inference-requires-a-model-for-data-collection>`  ·  :doc:`Data-collection models and ignorability <051-data-collection-models-and-ignorability>`  ·  :doc:`Missing values with counted data <122-missing-values-with-counted-data>`  ·  :doc:`Multiple imputation <119-multiple-imputation>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/11/censoring-and-truncation/ <https://insightful-data-lab.com/2025/11/11/censoring-and-truncation/>`__ (insightful-data-lab.com).

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