= Solution
This is a family-level <cluster-randomized trial>, but there are only about 1.1 contributing children per family. Most families therefore contribute a single eligible child, leaving little opportunity for within-family <intraclass correlation> to add much uncertainty. The equal-size exchangeable approximation has <design effect> $1+(m-1)\rho$, with cluster size $m$ and correlation $\rho$; when $m$ is close to one, this is close to one. A multilevel <Poisson regression> can consequently give nearly the same standard error as the independent-child working analysis.
Agreement is compatible with limited within-family dependence and little change from covariate adjustment; it does not prove all dependence absent. Mean family size alone does not determine the exact <design effect> for highly unequal cluster sizes.
Larger differences would be expected with \b[larger families and stronger within-family correlation], especially with unequal family sizes. Pronounced between-family heterogeneity, recurrent attendances concentrated in particular children, or important imbalanced covariates can also create <overdispersion> or change the adjusted comparison.
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