It seems to be Mister P week here on the blog . . .
A question came in, someone was doing MRP on a political survey and wanted to adjust for political ideology, which is a variable that they can’t get poststratification data for.
Here’s what I recommended:
If a survey selects on a non-census variable such as political ideology, or if you simply wish to adjust for it because of potential nonresponse bias, my recommendation is to do MRP on all these variables.
It goes like this: suppose y is your outcome of interest, X are the census variables, and z is the additional variable, in this example it is ideology. The idea is to do MRP by fitting a multilevel regression model on y given (X, z), then poststratify based on the distribution of (X, z) in the population. The challenge is that you don’t have (X, z) in the population; you only have X. So what you do to create the poststratification distribution of (X, z) is: first, take the poststratification distribution of X (known from the census); second, estimate the population distribution of z given X (most simply by fitting a multilevel regression of z given X from your survey data, but you can also use auxiliary information if available).
Yu-Sung and I did this a few years ago in our analysis of public opinion for school vouchers, where one of our key poststratification variables was religion, which we really needed to include for our analysis but which is not on the census. To poststratify, we first modeled religion given demographics—we had several religious categories, and I think we fit a series of logistic regressions. We used these estimated conditional distributions to fill out the poststrat table and then went from there. We never wrote this up as a general method, though.