Survey Statistics: poststratification without population level information

Poststratification uses population data on X to estimate E(Y) via E(E(Y | X, R = 1)), where R = 1 are survey respondents who provide Y and X. When the inner expectation “E” is estimated via Multilevel Regression, this is called MRP. The outer “E” needs p(X), population data on X. Sometimes we have to estimate the population distributions. We’ve seen a few examples: “2 flavors of calibration”: Say we have p(X), but we also need p(Z | X), the population distribution of another variable Z. We can estimate p(Z | X, R = 1) using survey data, but nonresponse could make this unreliable. Say we have population data on aggregates p(Z) (e.g. from census tables), then we can logit-shift to anchor to the population aggregate. See Kuriwaki et al. 2024. “MRPW”: Say we have p(X), but we also need p(W | X), the population distribution of the survey weights W. We can estimate p(W | X, R = 1) using survey data. Then because we assume survey weights are proportional to inverse probability of response…