Censoring and truncation#

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

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Important

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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 \(> c\). 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:

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

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

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

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:

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 \(\phi\) 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