Example: reaction times and schizophrenia#

Part 5 · Stage 16 · ♾️ Mixtures & Nonparametric Bayes · Lesson 135 of 144 · advanced

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Important

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A mixture with a scientific meaning#

Not every mixture is a mere density approximation — sometimes the components are real and interpretable. A study of reaction times, comparing schizophrenic patients with a control group, provides the archetype: the mixture structure encodes a genuine hypothesis about how the mind produces the data.

The finding, and the model#

Schizophrenic patients show more variable reaction times than controls, but the excess is not a uniform slowing. The data suggest that on most trials a patient responds like a control, while on a minority of trials an attentional lapse produces a much slower response. That is a mixture within each patient: two components, a “normal” reaction-time distribution and a “delayed” one, with a patient-specific probability of lapsing.

\[y_{ij} \sim \lambda_i \, \mathrm{N}(\mu_i, \sigma^2) + (1 - \lambda_i) \, \mathrm{N}(\mu_i + \tau, \sigma^2),\]

where \(\lambda_i\) is patient \(i\)’s probability of a normal (non-lapse) response and \(\tau > 0\) the extra delay when attention lapses. The parameters carry meaning: \(\tau\) is the size of a lapse, \(\lambda_i\) how often patient \(i\) lapses.

import pymc as pm
with pm.Model():
    # hierarchical: each patient's lapse probability, pooled across patients
    lam = pm.Beta("lam", 2, 2, shape=n_patients)         # P(normal response)
    mu = pm.Normal("mu", 0, 5, shape=n_patients)
    tau = pm.HalfNormal("tau", 5)                         # lapse delay (ordered => identified)
    w = pm.math.stack([lam[pid], 1 - lam[pid]], axis=1)
    comp = [pm.Normal.dist(mu[pid], sigma), pm.Normal.dist(mu[pid] + tau, sigma)]
    pm.Mixture("y", w=w, comp_dists=comp, observed=rt)

Why this example matters#

Three points. The components are substantive, not basis elements — “normal response” and “attentional lapse” are the hypothesis, and \(\lambda_i\), \(\tau\) are the quantities of scientific interest. The hierarchy ties it to the rest of the book: lapse probabilities are pooled across patients, so a patient with few trials borrows strength. And the ordering \(\tau > 0\) identifies the components — the delayed component is, by construction, the slower one — which sidesteps the label-switching problem the next lesson tackles in general. A mixture, here, is a model of a mechanism: two cognitive states, their sizes and frequencies estimated from response times.

Hint