Data-collection models and ignorability#

Part 2 · Stage 7 · 🗳️ Data Collection & Decisions · Lesson 051 of 144 · intermediate

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Important

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When can the design be ignored?#

The previous lesson wrote the joint model \(p(y \mid \theta) \, p(I \mid y, \phi)\). Ignorability is the precise statement of when the second factor can be dropped — when the posterior obtained from modelling \(y\) alone equals the posterior obtained from modelling \(y\) and \(I\).

Two conditions#

Ignorability requires both:

Missing at random (MAR). The inclusion mechanism does not depend on the values you failed to observe, given the ones you did:

\[p(I \mid y_{\text{obs}}, y_{\text{mis}}, \phi) = p(I \mid y_{\text{obs}}, \phi).\]
Parameter distinctness. The parameters \(\theta\) of the data model and \(\phi\) of the mechanism are a priori independent, \(p(\theta, \phi) = p(\theta) p(\phi)\).

Grant both, and the mechanism contributes a factor free of \(\theta\) that cancels in the normalisation. The likelihood you wanted to use is the one you may use.

The hierarchy of mechanisms#

MCAR (missing completely at random) — \(I\) is independent of the data altogether; a simple random sample. Strong, and rarely true.
MAR — \(I\) may depend on observed quantities: sampling more heavily in some strata, or non-response predicted by recorded covariates. Ignorable, provided those quantities are in the model.
MNAR (missing not at random) — \(I\) depends on the unobserved values themselves: people with high incomes decline to state their income. Not ignorable; the mechanism must be modelled, and the answer will depend on assumptions the data cannot check.

The practical corollary#

MAR is a statement about a model, not about the world: a mechanism that is non-ignorable given no covariates can become ignorable once the variables driving selection are included as predictors. This is the single most useful fact in the chapter — it is why survey analyses must include the design variables (strata, cluster, weights’ determinants), and why omitting them silently converts an ignorable design into a biased analysis.

import pymc as pm
# Non-response depends on age and region (both recorded) -> MAR *if* they are in the model.
with pm.Model():
    beta = pm.Normal("beta", 0, 2.5, shape=X.shape[1])   # X includes age, region, strata
    pm.Bernoulli("y", logit_p=X @ beta, observed=y_obs)
    # omit those columns and the same design becomes non-ignorable

Randomisation, then, is not a ritual: assigning treatment by a coin flip makes \(I\) depend on nothing but a known probability, guaranteeing ignorability by design rather than by assumption.

Hint