Stratified Income Survey with Unequal Nonresponse
A household income survey stratifies a city into 7,200 urban and 4,800 rural households. A sample of 400 urban and 300 rural households is drawn; 320 urban and 180 rural households respond. Among respondents, mean monthly income is 52,000 urban and 31,000 rural (local currency). A junior analyst reports the unweighted respondent mean, 44,440, as the city estimate. The survey director notes that rural response is only 60% versus 80% urban and that rural nonrespondents are believed to have lower incomes than rural respondents. The sample design weights are N_h/n_h. The director asks whether the estimate is biased, what weighting should be used, and how much the unequal weighting costs in effective sample size. No post-stratification has been applied. The survey has no auxiliary income variable.
1. Compute the design-weighted estimate under within-stratum MAR and compare it with the analyst's unweighted figure.
Read the worked answer
The unweighted mean 44,440 equals (320×52,000 + 180×31,000)/500. It overweights urban households: urban is 60% of the population but 64% of respondents. Under within-stratum MAR, the design-weighted estimate is 0.6×52,000 + 0.4×31,000 = 43,600, using the population stratum shares 7,200/12,000 and 4,800/12,000. The difference is 840, so the analyst's figure is biased upward by about 840 under this assumption.
2. What is the likely direction of residual nonresponse bias after weighting, and what should be reported?
Read the worked answer
Rural response is 60% versus 80% urban, and rural nonrespondents are believed to have lower incomes. Even the weighted respondent mean uses only respondents, so if rural nonrespondents are poorer than rural respondents, the weighted estimate 43,600 is still too high. The bias is in the same direction as the unweighted error but not necessarily the same magnitude. Post-stratification or nonresponse weighting within stratum can adjust for observed response rates, but not for unobserved differences. Report the direction of likely bias and avoid claiming precision that the response pattern does not support.
3. Compute the approximate design effect due to unequal weighting and interpret it.
Read the worked answer
Respondent weights are 7,200/320 = 22.5 urban and 4,800/180 = 26.667 rural. Sum of weights = 12,000. Sum of squared weights = 320(22.5^2) + 180(26.667^2) = 162,000 + 128,000 = 290,000. Effective sample size = (12,000^2)/290,000 = 496.6. Design effect from weighting = 500/496.6 = 1.007. Unequal weighting costs little here; the larger issue is nonresponse bias, not variance inflation.