Multistage decision making: medical screening#

Part 2 · Stage 7 · 🗳️ Data Collection & Decisions · Lesson 059 of 144 · intermediate

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Important

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Decisions that buy information#

Screening is not one decision but a sequence. Test, and if the test is positive, decide whether to run a second, more invasive test; then decide whether to treat. Each stage costs something and yields information that improves the next choice. The value of a test lies not in the test but in the better decisions it enables.

Working backwards#

The solution method is backward induction. At the final stage, the treatment decision is a plain expected-utility maximisation given whatever has been learned. Step back: the value of being at that stage is the expected utility of acting optimally there. So the earlier decision — whether to test — compares the cost of testing against the improvement in the expected utility it produces:

\[\mathrm{EVSI} = \mathrm{E}_{y_{\text{new}}}\Bigl[\max_{a} \, \mathrm{E}_{\theta \mid y, y_{\text{new}}} \, U(a, \theta) \Bigr] - \max_{a} \, \mathrm{E}_{\theta \mid y} \, U(a, \theta) .\]

That quantity — the expected value of sample information — is the most one should ever pay for the test. It is necessarily non-negative: information cannot hurt a decision-maker who is free to ignore it.

The screening structure#

Screening makes the interplay of prevalence and accuracy vivid. Bayes’ rule converts a positive test into a posterior probability of disease, and when prevalence is low, even an accurate test leaves that probability modest — the record-linkage lesson again, with lives at stake.

import numpy as np
prev, sens, spec = 0.004, 0.95, 0.95            # rare disease, good test
ppv = prev * sens / (prev * sens + (1 - prev) * (1 - spec))
ppv                                              # ≈ 0.071 — most positives are false

# stage 2: treat only if expected utility of treating exceeds not treating
U_treat    = ppv * U_treat_sick + (1 - ppv) * U_treat_well     # side effects if well
U_no_treat = ppv * U_none_sick  + (1 - ppv) * U_none_well
act = "treat" if U_treat > U_no_treat else "wait"

Where it gets hard#

Three honest complications. Utilities must be commensurable — a life-year, a false-positive’s anxiety and a dollar placed on one scale, an ethical act, not a technical one. The state space grows combinatorially with stages, so realistic problems are solved by simulation rather than by exact backward induction. And the analysis assumes the model for test performance is right; a miscalibrated sensitivity propagates through every stage. Screening policy is where Bayesian decision theory is at its most useful and its most contested.

Hint