Multilevel regression with poststratification

Multilevel regression and poststratification (MRP) (sometimes called "Mister P") is a statistical technique used for correcting model estimates for known differences between a sample population (the population of the data you have), and a target population (a population you would like to estimate for). For example, Wang et. al.^[1] used survey data from Xbox gamers to predict U.S. presidential election results. The Xbox gamers were 65% 18- to 29-year-olds and 93% male, while the electorate as a whole was 19% 18- to 29-year-olds and 47% male.

The poststratification refers to the process of adjusting the estimates, essentially a weighted average of estimates from all possible combinations of attributes (in this example age and sex, though there were more). Each combination is sometimes called a "cell." The multilevel regression is used to smooth noisy estimates in the cells with too little data by using overall or nearby averages.

One application is estimating preferences in sub-regions (e.g., states, individual constituencies) based on individual-level survey data gathered at other levels of aggregation (e.g., national surveys).^[2]

The technique and its advantages[]

The technique essentially involves using data from, for example, censuses relating to various types of people corresponding to different characteristics (e.g., age, race), in a first step to estimate the relationship between those types and individual preferences (i.e., multi-level regression of the dataset). This relationship is then used in a second step to estimate the sub-regional preference based on the number of people having each type/characteristic in that sub-region (a process known as "poststratification").^[3] In this way the need to perform surveys at sub-regional level, which can be expensive and impractical in an area (e.g., a country) with many sub-regions (e.g. counties, ridings, or states), is avoided. It also avoids issues with consistency of survey when comparing different surveys performed in different areas.^[4][2] Additionally, it allows the estimating of preference within a specific locality based on a survey taken across a wider area that includes relatively few people from the locality in question, or where the sample may be highly unrepresentative.^[5]

History[]

The technique was originally developed by Gelman and T. Little in 1997,^[6] building upon ideas of Fay and Herriot^[7] and R. Little.^[8] It was subsequently expanded on by Park, Gelman, and Bafumi in 2004 and 2006. It was proposed for use in estimating US-state-level voter preference by Lax and Philips in 2009. Warshaw and Rodden subsequently proposed it for use in estimating district-level public opinion in 2012.^[2] Wang et al.^[1] subsequently used it for estimating the outcome of the 2012 US presidential election based on a survey of Xbox users, and it has also been proposed for use in the field of epidemiology.^[5]

YouGov used the technique to successfully predict the overall outcome of the 2017 UK general election,^[9] correctly predicting the result in 93% of constituencies.^[10]

Limitations and extensions[]

MRP can be extended to estimating the change of opinion over time^[4] and when used to predict elections works best when used relatively close to the polling date, after nominations have closed.^[11]

Both the "multilevel regression" and "poststratification" ideas of MRP can be generalized. Multilevel regression can be replaced by nonparametric regression^[12] or regularized prediction, and poststratification can be generalized to allow for non-census variables, i.e. poststratification totals that are estimated rather than being known.^[13]

References[]