13.2 Cluster Randomisation, Partial Interference and Saturation Designs

Partial interference

Full network data are rare. A practical compromise is partial interference (Hudgens and Halloran): units interfere within a cluster (a village, a school) but not across clusters. Then randomisation can be done in two stages:

1. Randomise clusters to a saturation (share of members treated): 0% (pure control), 30%, 70%…
2. Within each cluster, randomise individuals to treatment at that rate.

Pure-control clusters give the outcome with no treatment anywhere nearby, which is exactly the benchmark a naive within-village comparison lacks. Baird, Bohren, McIntosh and Özler (2018) formalise these "randomised saturation" designs; Miguel and Kremer's deworming study is the famous example of large spillovers in schools.

!pip install -q numpy pandas statsmodels matplotlib
import numpy as np, pandas as pd
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
rng = np.random.default_rng(13)

A two-stage randomised vaccination design

120 villages of 50 people. Villages are randomised to 0%, 30% or 70% vaccination coverage. Vaccination cuts an individual's infection risk directly; on top of that, higher coverage in the village protects everyone (herd effect), vaccinated or not.

V, M = 120, 50
saturation = rng.permutation(np.repeat([0.0, 0.3, 0.7], V // 3))
rows = []
for v in range(V):
    s = saturation[v]
    d = rng.permutation(np.r_[np.ones(int(s * M)), np.zeros(M - int(s * M))])
    village_risk = rng.normal(0, 0.03)
    for i in range(M):
        p = 0.30 + village_risk - 0.15 * d[i] - 0.20 * s       # direct -0.15; herd effect -0.20 x coverage
        rows.append((v, s, int(d[i]), rng.binomial(1, np.clip(p, 0.01, 0.99))))
df = pd.DataFrame(rows, columns=["village", "saturation", "vaccinated", "infected"])
print(df.groupby(["saturation", "vaccinated"]).infected.agg(["mean", "size"]).round(3))

Direct, spillover and total effects

With the pure-control villages as the benchmark:

• Spillover on the untreated (at saturation s) = infection of unvaccinated people in s-villages − infection in pure-control villages.
• Direct effect (at s) = vaccinated − unvaccinated within s-villages.
• Total effect on the treated (at s) = vaccinated in s-villages − pure control.

def fit(formula, data):
    return smf.ols(formula, data=data).fit(cov_type="cluster", cov_kwds={"groups": data["village"]})

df["sat30"] = (df.saturation == 0.3).astype(int)
df["sat70"] = (df.saturation == 0.7).astype(int)
m = fit("infected ~ vaccinated + sat30 + sat70 + vaccinated:sat70", df)
print(m.params.round(3))

pure = df[df.saturation == 0].infected.mean()
for s in (0.3, 0.7):
    sub = df[df.saturation == s]
    spill = sub[sub.vaccinated == 0].infected.mean() - pure
    direct = sub[sub.vaccinated == 1].infected.mean() - sub[sub.vaccinated == 0].infected.mean()
    total = sub[sub.vaccinated == 1].infected.mean() - pure
    print(f"saturation {s:.0%}: spillover on unvaccinated {spill:+.3f} | direct {direct:+.3f} | total on vaccinated {total:+.3f}")

The within-village comparison (the "direct" column) misses the protection the unvaccinated received, which at 70% coverage is more than half the size of the direct effect. A study that randomised individuals within villages and compared them would substantially understate what the vaccine achieves at scale.

Design choices for your pilot

def one_study(V=30, M=50):
    sat = rng.permutation(np.repeat([0.0, 0.3, 0.7], V // 3)); r = []
    for v in range(V):
        d = rng.permutation(np.r_[np.ones(int(sat[v] * M)), np.zeros(M - int(sat[v] * M))]); vr = rng.normal(0, 0.03)
        p = np.clip(0.30 + vr - 0.15 * d - 0.20 * sat[v], 0.01, 0.99)
        r += [(v, sat[v], d[i], rng.binomial(1, p[i])) for i in range(M)]
    s = pd.DataFrame(r, columns=["village", "saturation", "vaccinated", "infected"])
    s = s[(s.vaccinated == 0) & s.saturation.isin([0.0, 0.7])].assign(high=lambda x: (x.saturation == 0.7).astype(int))
    return fit("infected ~ high", s).pvalues["high"] < 0.05
print(f"Power to detect the 70% spillover with 30 villages: {np.mean([one_study() for _ in range(200)]):.2f}")

Checklist and recap

• I can state partial interference and design a two-stage randomisation.
• I can estimate direct, spillover and total effects using pure-control clusters.
• I can argue which design a pilot needs, given the policy question.