Hierarchical decision analysis for home radon#

Part 2 · Stage 7 · 🗳️ Data Collection & Decisions · Lesson 060 of 144 · intermediate

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Important

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A decision for every county#

Radon is a radioactive soil gas that accumulates indoors and causes lung cancer. A homeowner may measure their house (cheap, noisy) and, depending on the reading, remediate (expensive, effective). The recommendation should differ by county — some have far higher radon — but many counties have only a handful of measurements. This is the cancer-rate problem, now attached to a decision.

Hierarchical prediction first#

Model log radon concentration hierarchically: houses within counties, counties drawn from a population, with a county-level predictor (soil uranium) explaining part of the variation. Partial pooling gives every county a stable predictive distribution, and a data-poor county borrows from the rest:

\[\log y_{ij} \sim \mathrm{N}(\alpha_j + \beta \, x_{ij}, \; \sigma_y^2), \qquad \alpha_j \sim \mathrm{N}(\gamma_0 + \gamma_1 u_j, \; \sigma_\alpha^2),\]

with \(x_{ij}\) the basement indicator and \(u_j\) county uranium.

import pymc as pm
with pm.Model():
    g0, g1 = pm.Normal("g0", 0, 5), pm.Normal("g1", 0, 5)
    s_a = pm.HalfNormal("s_a", 1)
    a = pm.Normal("a", g0 + g1 * u, s_a, shape=n_counties)     # county intercepts
    b = pm.Normal("b", 0, 1)
    pm.Normal("y", a[county] + b * basement, pm.HalfNormal("s_y", 1), observed=log_radon)

Then the decision#

Remediation costs a known amount; radon exposure costs expected life-years, monetised. For each house the actions are do nothing, measure then decide, or remediate immediately. Expected utility is computed over the posterior predictive distribution of that house’s radon level — which is exactly what the hierarchical model supplies, uncertainty included.

The result is a decision rule that varies by county: in high-radon counties, remediate without measuring; in low-radon counties, do nothing; in the intermediate band — where the predictive distribution straddles the action threshold — measure first, because there the measurement changes what you would do. That middle region is precisely where the expected value of information is positive.

Why the hierarchy earns its keep#

Use each county’s raw mean and the small counties give absurd recommendations, exactly as raw cancer rates gave absurd rankings. Pool completely and every county gets the same advice, wasting the real variation. Partial pooling produces recommendations that are stable where data are thin and responsive where data are plentiful — and because the decision uses the predictive distribution for an individual house, it correctly accounts for within-county spread, not merely uncertainty about the county mean. The caveat is the usual one: the answer depends on the monetary value placed on a life-year, and that number should be stated, not smuggled.

Hint