 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 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.