I recognize that propensity scores are often used for causal inference. Just to clarify from the outset, that's not what I'm interested in here.
Instead, I'm looking at using propensity scores to match observations from a probability sample with observations from a nonprobability sample so that I can assign weights to the nonprobability sample in the hopes that the re-weighted nonprobability sample will more closely represent the target population. As I understand it, this is relatively common practices for web surveys, etc. and there's extensive literature on the topic.
Where I'm confused is trying to get my head around how to use the survey design weights from the probability sample in the calculation of propensity scores. This article by Valliant and Dever (2011) discusses the issue of weights but I don't entirely follow the conversation about weights.
Could anyone explain what I need to do here and perhaps also how to do it in R, preferably with the MatchIt package? Here are a few of my thoughts.
If I used something like nearest neighbor, I could just match the observations and then following the advice in the article above under "Mechanics of Estimating Propensity Scores,"(b) by dividing the observations into quantiles of equal size (utilizing propensity scores) and then weight the probabilistic observations in each quantile by taking the average probability for all the observations in the quantile.
However, this would ignore the survey weights of the probability sample. I could perhaps utilize those weights after the fixing the quantiles by taking the average survey weights for each probability observation in quantile and multiplying this by the initial weight for the quantile. Probably doesn't make sense but I'm just thinking out loud at this point.
Perhaps the MatchIt package or another package in R facilitates the utilization of survey weights in the calculation of the propensity scores (and not just by adding the weights in as a variable).
Any direction would be most appreciated!