Village Survey with a Missing Stratum
A national statistics office runs a household income survey in a district of 240 villages. Villages are stratified by remoteness into three strata: accessible (150 villages), moderate (60), and remote (30). The plan is a two-stage design: sample villages with probability proportional to size within stratum, then 20 households per village. Fieldwork finishes, but the remote stratum returns only 4 of the 10 planned villages because two were inaccessible and four refused. Response rates in the other strata are 92% and 88%. The published estimate must cover the whole district, and the remote stratum is known from census auxiliaries to hold about 9% of the district's households with markedly lower mean income.
1. Explain the effect of the shortfall in the remote stratum on the district estimate and on its variance.
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The district estimate is a weighted combination of stratum means, with weights proportional to household counts. If the remote stratum is represented by only 4 villages instead of 10, the remote mean is estimated from a smaller and possibly unrepresentative set of villages. Because the four missing villages are precisely the inaccessible and refusing ones, the nonresponse is not missing at random: inaccessible villages are likely poorer, so the remote mean is probably biased upward.
The bias propagates into the district estimate with weight 0.09. If the remote mean is overstated by, say, 15%, the district estimate is overstated by about 0.09 × 0.15 ≈ 1.4% - small but real, and in a direction that flatters the district.
The variance is affected more sharply. The remote stratum contributes a between-village variance term divided by the number of sampled villages; with 4 instead of 10, that component roughly doubles. Because the stratum is small in population but heterogeneous, its contribution to the district variance can be disproportionate. The published standard error will be understated if it is computed from the achieved sample as though 4 were the planned number, because the design effect assumed 10.
2. Recommend how to produce a defensible district estimate and an honest measure of uncertainty.
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Three steps. First, do not silently reweight as if the four villages were a random subset. Instead, model the nonresponse: use census auxiliaries (remoteness, household count, access to services) to compare responding and nonresponding villages, and if the responders look systematically better-off, apply a nonresponse adjustment within the remote stratum, calibrated to the known village count of 30.
Second, produce the estimate with a design-based variance that reflects the achieved sample size in each stratum, and add a nonresponse variance component for the remote stratum. A replicate-weighting approach - jackknife or bootstrap at the village level - handles both the unequal selection probabilities and the shortfall more gracefully than a closed-form formula.
Third, publish a sensitivity range. Report the headline estimate under the standard adjustment, and a second estimate under a pessimistic assumption that the missing villages are, say, 20% poorer than responders. If the two estimates differ materially, say so. The district total should carry a wide interval that honestly reflects the 4-of-10 shortfall, and the methodology note should state the refusal pattern explicitly rather than burying it.