Klazien de Vries, Marieke E. Timmerman, Anja F. Ernst, and Casper J. Albers write:

In psychological test norming, nonrepresentativeness in background variables in the normative sample can lead to bias in the normed score estimates. Because representativeness is difficult to establish in practice, adjustment methods are needed to combat this bias. As a candidate adjustment method, we investigated generalized additive models for location, scale, and shape with multilevel regression and poststratification (GAMLSS + MRP), the combination of MRP and continuous norming with GAMLSS. This adjustment method was then compared to current adjustment methods in continuous norming using weighted regression: GAMLSS + P (with poststratification) and cNORM + R (with raking). The results of our simulation showed that GAMLSS + MRP was generally more efficient than GAMLSS + P and cNORM + R. Furthermore, GAMLSS + MRP was better than the current methods at reducing bias in samples where the nonrepresentativeness was age-dependent. We argue that GAMLSS + MRP is a valid adjustment method in continuous norming and recommend this adjustment method to mitigate bias in nonrepresentative normative samples. To facilitate the use of GAMLSS + MRP in practice, we provide a step-wise approach for the implementation of GAMLSS + MRP. We illustrate this approach by deriving normed scores from the normative data of the third Schlichting language test.

I don’t recall how I came across this paper, and I haven’t actually read it, but I wanted to share it with you, just because it’s cool to see the different ways that multilevel regression and poststratification (MRP) can be used.

Ultimately, MRP is the inevitable consequence of three things:

1. We are interested in generalizing to populations of interest.

2. Available data are typically unrepresentative of the population. This is the case even with simple random sampling–Hello, random variation! Hello, small-area estimation!–and is even more so with selected samples, nonresponse, dropout, etc. In some settings such as medical experimentation there’s not even an attempt to get a representative sample: you’re directly aiming to include in the study the groups of people who might get the greatest benefit from the treatment.

3. When adjusting for differences between sample and population, many variables can be relevant–for example, demographic and geographical variables in a survey of people–and so simple adjustments such as raw poststratification or non-multilevel regression adjustment won’t do the job.

Put this together and you’ll want to do MRP (or, more generally, RPP). It’s not just for survey research. It comes up everywhere in statistics and machine learning, whenever there is a concern with population prediction, or generalization, or transportability, or whatever you want to call it.

It can seem like a hassle that to do this you need to know (or estimate, or postulate) a distribution of predictors in your population, but (a) this is often work that’s well worth the effort, if you really care about the population, (b) dependence of the result on the choice of population is important, and where this dependence is strong you should be aware of it, and (c) if you want to take the easy way out you can always bootstrap to get inference for the hypothetical population of which your data are considered to be a random sample.