In multilevel regression and poststratification (MRP), do all individual-level predictors included in the regression model need to be available in the poststratification frame? For example, if the poststratification frame provides population information only on province, age, gender, and race, is it still appropriate to include additional individual-level predictors, such as education or income, in the logistic regression model? Any help is appreciated.
Poststratification is adjusting for bias due to non-representative sampling of covariates. You can technically poststratify on any subset of the covariates to get an adjusted population estimate. Whether that will improve your estimate will depend on what the other covariates are doing and if they’ve been sampled randomly or themselves have bias.
@andrewgelman will be able to summarize the issues much better than me. I’d have to work through them all from first principles.
We discuss regression and poststratification in chapter 17 of Regression and Other Stories. The basic idea is that your outcome is y, your predictors are X, your model is p(y|X,theta), you fit the model to estimate data and then you use the fitted model to get (probabilistic) predictions of y given X. You poststratify by getting the prediction of y|X for every person in the population (if you have a “poststratification file” containing the values of X for everyone in the population) or for every type of person (if you have a “poststratification table” with the population count for every category of X). if X is discrete and the population is large, it can make sense to use a poststratification table. If X is continuous or contains lots of variables or the population is not so large, it can make sense to use a poststratification file.
In either case, the variables in X need to be observed in the data (so you can fit the model) and they need to be known in the population (so that you can construct the poststratification file or table).
If X includes variables not known in the population (we call these “non-census variables”), then you can’t poststratify on X. OK, you can. But to do so you need to first fill up the poststratification table. Let X=(X1, X2), where X1 are the variables observed in the population and X2 are the non-census variables. Then, to poststratify, you need to first estimate the distribution of X2|X1, which can be done from the data at hand. Then you “know” the joint distribution of X and you can poststratify.
Here’s an example from around 2008 where Yu-Sung and I did this so that we could religion (not measured in the U.S. Census) among our poststratification variables: https://sites.stat.columbia.edu/gelman/research/published/socsci-12-00430.pdf