MRP with polls using simple random samples

There are two steps to estimating opinion with MRP: First, a multilevel model is fit to predict individual response to a binary question using individual-level data from a survey. Then, using the model and the joint distribution of demographic characteristics in the population from the Census, the scholar poststratifies to estimate the average response to the question of interest at the state (or other subnational) level.

The typical multilevel logistic model for MRP contains as predictors detailed demographic information about respondents, state and region indicators, and one or two state-level variables (e.g., presidential vote and religiosity). Below, I formalize a standard MRP model that could be fit from data available in a standard Gallup Poll fielded in the US during the 1980s, using the notation from Gelman and Hill (Reference Gelman and Hill2007). The outcome of interest, $ {y}_i $ , indicates an individual respondent $ i $ ’s response to a survey question. The $ \alpha $ terms are random effects corresponding to demographic or geographic groups; so $ {\alpha}_{r\left[i\right]}^{\mathrm{race}} $ indicates the random effect for the racial group $ r $ to which respondent $ i $ belongs. The state random effect $ {\alpha}_{s\left[i\right]}^{\mathrm{state}} $ is modeled as a function of the region and contextual variables included (here I use $ RepVote $ , Republican vote share in the last presidential election, and $ Relig $ , the proportion of the state that identifies as evangelical Christian or Mormon).

(1) $$ {\displaystyle \begin{array}{l}\Pr \left({y}_i=1\right)={\mathrm{logit}}^{-1}\left({\beta}^0+{\alpha}_{r\left[i\right]}^{\mathrm{race}}+{\alpha}_{g\left[i\right]}^{\mathrm{sex}}+{\alpha}_{k\left[i\right]}^{\mathrm{age}}+{\alpha}_{l\left[i\right]}^{\mathrm{educ}}+{\alpha}_{s\left[i\right]}^{\mathrm{state}}\right)\\ {}\hskip2em {\alpha}_r^{\mathrm{race}}\hskip0.35em :\hskip0.35em N\left(0,{\sigma}_{race}^2\right),\mathrm{for}\hskip0.35em r=1,\dots, 3\\ {}\hskip2em {\alpha}_g^{\mathrm{sex}}\hskip0.35em :\hskip0.35em N\left(0,{\sigma}_{sex}^2\right),\mathrm{for}\hskip0.35em g=1,2\\ {}\hskip2em {\alpha}_k^{\mathrm{age}}\hskip0.35em :\hskip0.35em N\left(0,{\sigma}_{age}^2\right),\mathrm{for}\hskip0.35em k=1,\dots, 4\\ {}\hskip1.36em {\alpha}_l^{\mathrm{educ}}\hskip0.35em :\hskip0.35em N\left(0,{\sigma}_{educ}^2\right),\mathrm{for}\hskip0.35em l=1,\dots, 4\\ {}\hskip1.24em {\alpha}_s^{\mathrm{state}}\hskip0.35em :\hskip0.35em N\left({\alpha}_{m\left[s\right]}^{\mathrm{region}}+{\beta}^{\mathrm{RepVote}}\ast {RepVote}_s+{\beta}^{\mathrm{Relig}}\ast {Relig}_s,{\sigma}_{state}^2\right),\mathrm{for}\hskip0.35em s=1,\dots, 50\\ {}\hskip1em {\alpha}_m^{\mathrm{region}}\hskip0.35em :\hskip0.35em N\left(0,{\sigma}_{region}^2\right),\mathrm{for}\hskip0.45em m=1,\dots, 8\end{array}} $$

This model allows us to predict the expected level of support for the policy $ y $ among each “type” of person in the population – that is, each of the 4,800 possible combinations of $ race\times sex\times age\times educ\times state $ . These predictions are then used to poststratify and aggregate to the state level by taking a weighted average where the weights are the share of each combination of demographic variables in the state’s population.

Estimates may be further improved by fitting a “deep” model with interactions among demographic and geographic variables in ways not captured by a simple model without interactions (Ghitza and Gelman Reference Ghitza and Gelman2013; Goplerud Reference Goplerud2024).

Cluster-sampled surveys

The above model has been developed assuming survey respondents are independently drawn from the population, as in a simple random sample. However, this assumption may not hold in many circumstances (Berinsky Reference Berinsky2017). Instead, when such sampling methods are impossible or impractical, pollsters often rely on alternative methods to produce samples that are representative of target populations. For many surveys, this target is the population of an entire country, and not any one subnational unit.

One of the most common procedures used for survey sampling is cluster sampling (also called area-probability sampling). Variations of this approach (and especially multi-stage sampling methods) were almost universally used by US polling firms from the 1950s and 1980s (Warshaw Reference Warshaw and Alvarez2016).Footnote ¹ Two major academic studies, the GSS and ANES, still use cluster-sampling to produce all or part of their samples today, as do many large multinational surveys (see, e.g., Latin American Pulic Opinion Project 2019; Asian Barometer Survey 2003). As a result, the only quality polls available to scholars in many contexts are likely to be cluster-sampled.

The aim of cluster sampling is straightforward: researchers randomly select a set of “clusters,” such as cities or neighborhoods, from which they randomly draw people to interview. This approach is appealing to survey researchers because it reduces the costs of producing a nationally representative sample for surveys that rely on in-person interviews.

As an example, consider the Gallup Poll during the late 1970s and early 1980s, on which I based the sampling algorithm in the simulation studies discussed below (Gallup Organization 1980b). First, Gallup assigned each state to a region. Within each region, geographic areas were assigned to a “size-of-community stratum” based on urban/rural status and population. These region-stratum combinations form the basis of primary sampling units (PSUs). Then, Gallup randomly selected two localities from each PSU, weighting by population, and repeated the process using progressively smaller geographies to identify a block or cluster of blocks. The resulting sample is expected to produce reasonable estimates of national opinion. A more detailed description of this procedure can be found in Supplementary Appendix A.

Case study: abortion opinion

To illustrate the problems that may arise from using MRP with cluster-sampled polls, I estimated opinion on abortion from a survey fielded by Gallup in September 1980 and downloaded from the Roper Center for Public Opinion Research (Gallup Organization 1980a). The survey is typical of the period and used cluster sampling to produce a set of respondents ( $ N=\mathrm{1,602} $ ) who were interviewed in person at their homes. I used MRP to estimate state-level opinion using responses to a question asking whether respondents “generally favor” or “generally oppose” an ban on abortion.Footnote ²

I modeled individual opinion in Stan (Carpenter et al. Reference Carpenter, Gelman, Hoffman, Lee, Goodrich, Betancourt, Brubaker, Guo, Li and Riddell2017) with random effects for race, female, the race × female interaction, age group, education, state, and region. I also included state Republican vote share and the share of the population that is evangelical Christian or Mormon as linear predictors. This specification is typical for MRP for a question about social issues like abortion. All variables in the model were used in poststratification, using weights from joint distributions of the population downloaded from IPUMS-NHGIS (Manson et al. Reference Manson, Schroeder, Van Riper, Kugler and Ruggles2021). Presidential vote data are from Leip (Reference Leip2021), and religion data are from the Churches and Church Membership in the United States study (Grammich et al. Reference Grammich, Hadaway, Houseal, Jones, Krindatch, Stanley and Taylor2019).

Figure 1 reports the results from the MRP model. The lefthand panel maps the share of each state that opposed a ban on abortion, as estimated with MRP. The righthand panel compares modeled opposition to an abortion ban with the measure of state-level abortion liberalism produced by Brace et al. (Reference Brace, Sims-Butler, Arceneaux and Johnson2002) using pooled surveys from 1974 to 1998.Footnote ³ A higher score on their scale corresponds to more liberal opinion on abortion. The MRP results are clearly only minimally correlated with the baseline estimates of abortion opinion. Similar analyses using latent policy liberalism estimates provide similar results (these can be found in Supplementary Appendix F). Many state estimates also clearly lack face validity. For example, it is unlikely that Utah would have the highest opposition to a ban on abortions in the country. Conversely, more liberal states in the Northeast and upper Midwest show surprisingly low levels of opposition to banning abortions.

Figure 1. Opposition to Abortion Ban from Traditional MRP on 1980 Gallup Poll.

Note: The lefthand panel plots opposition to a ban on abortions estimated using Traditional MRP. Darker states are more opposed to a ban. The righthand panel plots estimated opposition against state estimates of abortion liberalism from Brace et al. (Reference Brace, Sims-Butler, Arceneaux and Johnson2002). The blue curve is a least-squares regression of the relationship between the variables.

One reason for error is that state estimates may depend only on a small number of clusters from particular (non-representative) parts of the state. For example, Table 1 shows the five Utah respondents in the poll. Of the five, which all appear to come from one cluster in a city of 50,000–99,999 people, only two answered the question about an abortion ban, and both were opposed.Footnote ⁴ Respondents in the Utah cluster are more likely to be Democrats or Independents than Utahans of the era as a whole (Erikson, Wright, and McIver Reference Erikson, Wright and McIver1993). In principle, the limited nature of this Utah subsample is the type of problem MRP handles well. However, in the case of cluster sampling, the predictive model attributes the average opinion in this cluster to Utah as a whole, rather than the stratum $ \times $ region combination that it was drawn into the sample to represent.

Table 1. Utah respondents in Gallup abortion poll

Although subgroup means converge on true population means when a large number of clusters are included in a sample (Kish and Frankel Reference Kish and Frankel1974), scholars often have access to a limited number of surveys (indeed, there may be only one survey asking a particular question in the time period of interest). As a result, the relevant question for using MRP with cluster-sampled polls not whether state-level subsamples are representative of the population in expectation over repeated surveys, but rather whether the respondents from a given state in a single poll are likely to be representative. In Supplementary Appendix B, I show that state subsamples in individual polls are often not representative of their states. This problem is especially severe in states with lower populations, which are much more likely to have no respondents included in a poll, and whose subsamples are less representative of the state population (because they include fewer clusters).Footnote ⁵ It is also intuitively likely to be the case in states with more diverse and segregated populations, where any two clusters may be very different from one another. Erikson, Wright, and McIver (Reference Erikson, Wright and McIver1993) note these potential problems in discussing their decision to use disaggregated CBS/New York Times polls, rather than a cluster-sampled survey.

Simulation study 1: MRP with cluster-sampled polls

To better understand and illustrate the problems that may arise when using MRP with cluster-sampled polls, I conducted a series of simulations.

In each simulation, I generated one million “voters” distributed across 50 states (according to their actual share of the population) and seven size-of-community strata (according to a random distribution that I hold constant across simulations). Next, I randomly assigned each voter a binary demographic predictor (which I call race) distributed according to each state’s real-world white and non-white populations. I then drew a survey response for each voter such that:

(2) $$ {\displaystyle \begin{array}{l}{y}_i:\hskip0.36em \mathrm{Bern}\left({\mathrm{logit}}^{-1}\left({\alpha}_{r\left[i\right]}^{\mathrm{race}}+{\alpha}_{s\left[i\right]}^{\mathrm{state}}+{\alpha}_{u\left[i\right]}^{\mathrm{stratum}}+{\alpha}_{s\left[i\right],u\left[i\right]}^{\mathrm{state}\times \mathrm{stratum}}+{\alpha}_{m\left[i\right],u\left[i\right]}^{\mathrm{region}\times \mathrm{stratum}}\right)\right)\\ {}{\alpha}_r^{\mathrm{race}}:\hskip0.36em N\left(0,{\sigma}_{race}^2\right),\mathrm{for}\hskip0.35em r=1,2\\ {}{\alpha}_m^{\mathrm{region}}:\hskip0.36em N\left(0,{\sigma}_{region}^2\right),\mathrm{for}\hskip0.35em m=1,\dots, 8\\ {}{\alpha}_s^{\mathrm{state}}:\hskip0.36em N\left({\alpha}_{m\left[s\right]}^{\mathrm{region}}+{\beta}^{\mathrm{RepVote}}\ast {RepVote}_s,{\sigma}_{state}^2\right),\mathrm{for}\hskip0.35em s=1,\dots, 50\\ {}{\alpha}_u^{\mathrm{stratum}}:\hskip0.36em N\left(0,{\sigma}_{stratum}^2\right),\mathrm{for}\hskip0.35em u=1,\dots, 7\\ {}{\alpha}_{s,u}^{\mathrm{state}\times \mathrm{stratum}}:\hskip0.36em N\left(0,{\sigma}_{state\times stratum}^2\right),\mathrm{for}\hskip0.35em s=1,\dots, 50\hskip0.45em \mathrm{and}\hskip0.45em u=1,\dots, 7\\ {}{\alpha}_{m,u}^{\mathrm{region}\times \mathrm{stratum}}:\hskip0.36em N\left(0,{\sigma}_{region\times stratum}^2\right),\mathrm{for}\hskip0.35em m=1,\dots, 8\hskip0.45em \mathrm{and}\hskip0.45em u=1,\dots, 7\\ {}{\beta}^{\mathrm{RepVote}}:\hskip0.36em N\left(0,{\sigma}_{RepVote}^2\right),\mathrm{for}\hskip0.35em {\sigma}_{RepVote}=0.5\end{array}} $$

I drew true effect sizes for the demographic and geographic variables from a normal distribution, varying the standard deviation $ \sigma $ for one variable at a time. For each variable, I ran the simulation with $ \sigma \in \left\{\mathrm{0.1,0.75,2.0}\right\} $ , holding all other $ \sigma $ values constant at 0.1. As $ \sigma $ increases for each effect, so does the extent to which that variable independently impacts individual opinion. Finally, I use these effects to produce a probability that individual $ i $ supports a survey question and draw response $ {y}_i $ from a Bernoulli distribution.

The true data generating process for individual opinion in the simulation is based on race, state, region, and stratum, as well as the interactions between stratum $ \times $ region and stratum×state, to reflect that that rural and urban places may vary systematically in different parts of the country. $ RepVote $ is normally distributed and constrained to be modestly correlated with $ {\alpha}_s^{\mathrm{state}} $ .Footnote ⁶

With a population of one million voters in hand, I then produce two samples, each meant to mimic a standard survey of approximately 1,500 respondents.Footnote ⁷ The first is a simple random sample in which every voter has an equal probability of being selected. The second is based on the Gallup Poll cluster sampling procedure. For each stratum × region pair, I randomly select two states, weighting by their populations. From each stratum × region × state combination selected, I then sampled 14 respondents from the pool of voters. Finally, I fit both traditional and deep MRP models on both sets of polls to predict opinion using the lme4 package in R (Bates et al. Reference Bates, Mächler, Bolker and Walker2015). Deep MRP models add race $ \times $ state and race $ \times $ region random effects. I poststratified using all variables and interactions included in the model. I repeated the simulation 100 times for each combination of parameters, allowing the specific effects drawn, the voters, and the samples to vary each time.

Figure 2 reports the results of these simulations. For each variable $ j $ , I report error from the MRP model’s opinion estimates when the corresponding $ {\sigma}_j $ is set at 0.1, 0.75, or 2.0. I hold $ {\sigma}_{\neg j} $ for all other variables constant at 0.1. The lefthand column of Figure 2 reports the mean absolute error (MAE) across 100 simulations, using the observed “true” opinion value from the full pool of simulated voters. Filled circles and squares report results from using MRP with the cluster-sampled survey, while the hollow points are for the simple random sample. The righthand column reports the average correlation between between true and modeled opinion.

Figure 2. Results of Simulations: Traditional MRP with Cluster-Sampled Polls.

Note: Each point reports results for a series of 100 simulations under a given set of conditions. Simulations vary the standard deviation parameter, $ \sigma $ , for one variable’s effect on opinion. All other $ \sigma $ parameters are set to 0.1. Error bars cover results from 95% of simulations.

As I increase the magnitude of the independent effect of stratum on opinion, as well as the the interactions between stratum × state and stratum × region, the amount of error from MRP increases in the clustered random sample. While the MAE does increase slightly for the poll with a simple random sample, the increase in error is much more dramatic when cluster-sampling is used. Notably, MRP does not perform much worse under cluster-sampling when the effects for state, region, or race are large. My results also suggest that although deep models have been found to improve estimation in MRP generally, they do not seem to offer dramatic improvements when polls are cluster-sampled.

We only see divergence between cluster sampling and simple random sampling when the effect of a stratum variable (i.e., heterogeneity inside of subnational units and across the dimensions included in the sampling frame) increases. This suggests that when traditional MRP is conducted with cluster-sampled polls, to the extent that there is heterogeneity inside a state based on stratum, the model performs worse.

Approaches to MRP with cluster-sampled polls

The abortion case study and simulations above highlight two problems that may produce increased error when estimating state opinion with MRP on cluster-sampled polls.

First, clusters may not be representative of the overall population in a state. At one extreme, some states will have zero respondents despite the fact that a simple random sample might include two or three individuals in expectation. More commonly, they may have a single cluster drawn from just one (unrepresentative) community, as in the Utah case described above. In principle, this is the kind of problem MRP is designed to address (indeed, even simple random samples will produce states with very few respondents); MRP borrows strength across states and assumes that even if a state has few respondents, a reasonable prediction can be derived from the behavior of similar individuals in other states. However, if clusters are not representative of their states as a whole, this can contaminate the estimate of the state effect and thus lead to inappropriate predictions for the state as a whole.

Second, the hierarchical model typically used with traditional MRP may account for the wrong geographic variation in opinion. The traditional MRP model is inconsistent with the known data generating process for the opinion survey. Because pollsters produced the sample using important information not accounted for in the model, results may have increased error. The traditional MRP model also ignores important information that pollsters use to produce their samples. If a nationally representative survey can be produced by sampling based on stratum, then these same characteristics may be useful in accurately predicting opinion.

Because of the role that geography plays in a cluster sample, these problems make MRP particularly sensitive to geographic variation in public opinion. Stollwerk (Reference Stollwerk2017) conjectured that MRP estimates from cluster-sampled polls will be incorrect when opinion varies within states in ways that are not accounted for by demographic variables. Pollsters produce samples by randomizing not at the state level but at the region $ \times $ stratum level. As a result, the kinds of people missing from the poll may not always be well represented by those in the dataset. For example, consider the (un)representativeness of an urban neighborhood in Milwaukee being the only cluster sampled in the state of Wisconsin.

In the following sections, I propose and test two approaches to improve estimation of opinion from cluster-sampled surveys. First, I pool responses from multiple surveys. By adding new clusters, the poll in this case begins to approach a simple random sample. (Simple random samples can be thought of as clustered samples with $ N $ clusters of 1 respondent each.)

Second, I propose respecifying the predictive model in the first stage of MRP to include the geographic information that pollsters use to produce clustered random samples. This data can then be used in the poststratification step of MRP. Underpinning this approach – which I call CMRP – is the idea that scholars can improve estimates from MRP by fitting a model that accounts for the cluster-sampling procedure itself.

Simulation study: pooling

As an initial test of whether pooling can produce better estimates than using a single cluster-sampled poll, I incorporated pooling into the simulation setup described above. I found that by doubling the number of clusters in each stratum × region combination, estimation can be improved, especially when opinion varies across states or by stratum within states. This is consistent with the problems associated with unrepresentative state-level subsamples from a single poll. The simulation results are presented and discussed in Supplementary Appendix D.

However, I also find that pooling does very little to improve opinion estimates if the surveys do not increase the number of clusters. In Supplementary Appendix D, I show that doubling the sample size within clusters (analogous to pooling two surveys sampling from the same clusters) does not meaningfully improve estimates versus a traditional MRP model.

Validation: pooled surveys for presidential opinion

I confirmed the results of the simulation study using presidential election polls from 1980. Presidential elections are a useful testing ground for MRP because they offer a ground truth against which polls can be compared—the election results themselves.

Here, I use MRP to predict support for the Democratic presidential candidate in the 1968–1984 presidential elections. I use two samples: a baseline that comes from the final Gallup poll conducted in the election season, as well as a pooled sample that includes that same Gallup poll and the ANES. For each sample, I fit two models in Stan: traditional MRP, which included variables for race, sex, the race × sex interaction, age group, education, and percent evangelical or Mormon;Footnote ⁹ and a Deep MRP model adding interactions among demographic predictors and between demographic and geographic variables. I then poststratified on all included predictors using a poststratification matrix built from joint distributions of the population in the 1980 Census, obtained from IPUMS-NHGIS. The pooled models included a random effect for the survey firm, but I did not poststratify on this variable.Footnote ¹⁰ The full model specifications and details of the surveys used are in Supplementary Appendix E.

To account for variability of polls and unobserved changes in the national environment in the final weeks of the campaign (Gelman and King Reference Gelman and King1993), I report results relative to national support. I adjust both estimates and actual election results by taking the difference between state-level support and national support for the Democratic candidate.

Figure 3 reports results. The leftmost column shows the change in MAE that comes from using a pooled sample, rather than the single sample. Negative numbers reflect a reduction in MAE (i.e., more accurate estimation). The second column reports the MAE improvement as a share of the error in the traditional MRP model. The third column shows the share of states whose estimates improved when the pooled sample was used. A pooled sample reduced overall error in four of the five election years – and in the case of 1972 by more than 20%. Finally, the rightmost panel shows the average increase in the variance of state-level from using a pooled sample (versus standard MRP).Footnote ¹¹ The variance of the pooled sample is slightly larger, though not dramatically so, suggesting slightly higher uncertainty from pooling, though the point estimates themselves are more accurate on average.

Figure 3. Pooling Presidential Election Polls for MRP.

Note: Points represent the improvement observed from using a pooled sample over a single presidential election poll.

Limitations of pooling

While pooling surveys represents a promising improvement on traditional MRP in the case of cluster-sampled polls, there are two challenges that make it impractical in some situations.

First, pooling requires scholars to find multiple polls that asked the same question around the same time. This is often not possible, as many topics appear infrequently in surveys, and the exact question wording can vary widely between polling firms and even from one survey to the next. The need to collect multiple polls can also make scholars’ attempts to construct time series of opinion impossible.

Second, simulations indicate that MRP with pooled surveys works well when the number of clusters increases and not necessarily when the size of each cluster does. This makes it especially problematic that polling firms rarely change the communities from which they select respondents, particularly for face-to-face interviews. A review of memos by Gallup statisticians during the 1980s indicated that sampled areas were frequently re-used by pollsters until they had been “exhausted.”Footnote ¹² As a result, the same communities can appear repeatedly in surveys from some pollsters, and because detailed information about the exact location of respondents is generally not available, the extent of this re-use can be difficult to definitively determine. In Supplementary Appendix D, I show that increasing sample size by doubling the number respondents from each cluster does not offer the same improvements as increasing the number of clusters.

How to fit CMRP

CMRP follows a similar procedure to MRP. First, the researcher fits a multilevel model of individual opinion, incorporating the geographic units employed by the pollster to produce the sample. In the Gallup case, this would be region and size-of-community stratum. Specifically, this mirrors the standard MRP approach in eq:basicmrp above, but adds the below random effects:

(3) $$ {\displaystyle \begin{array}{l}{\alpha}_u^{\mathrm{stratum}}\sim N\left(0,{\sigma}_{\mathrm{stratum}}^2\right),\hskip0.33em \mathrm{for}\hskip0.33em u=1,\dots, 7\\ {}{\alpha}_{u,m}^{\mathrm{stratum}\times \mathrm{region}}\sim N\left(0,{\sigma}_{\mathrm{stratum}\times \mathrm{region}}^2\right),\hskip0.33em \mathrm{for}\hskip0.33em u=1,\dots, \hskip0.33em \mathrm{and}\hskip0.33em m=1,\dots, 8\\ {}{\alpha}_{u,s}^{stratum\times state}\sim N\left(0,{\sigma}_{\mathrm{stratum}\times \mathrm{state}}^2\right),\hskip0.33em \mathrm{for}\hskip0.33em u=1,\dots, 7\hskip0.33em \mathrm{and}\hskip0.33em s=1,\dots, 50\end{array}} $$

To improve estimates, we might also include interactions among demographic predictors and between geographic and demographic variables to improve estimation, which I refer to here as Deep CMRP.

Next, the researcher poststratifies to the level of the sampling unit (i.e., stratum) within each state, using joint distributions from the Census as weights. Finally, the estimates are aggregated up to the state level, again using Census data to weight. The exact steps that need to be taken to produce poststratification information from the Census vary depending on the cluster-sampling procedure used by a polling firm. In general, joint distributions of demographic variables in the population at small geographic levels (e.g., metropolitan areas, counties, cities and towns) can be downloaded from IPUMS-NHGIS. In some census years, one or two variables may not be included in joint Census tables; however, race, gender, and often age are routinely readily available. The steps I took to produce poststratification matrices used in this paper can be found in Supplementary Appendix C.

Simulation study: testing CMRP

To test CMRP, I again return to simulations. I follow the same procedure as before but do not generate pooled samples. For each sample, I fit CMRP and Deep CMRP, which adds interactions between race and all geographic variables. I also fit traditional and deep MRP models that do not adjust for clustering geography to serve as comparisons. Figure 4 reports the results of these simulations. Here, all results come from polls with clustered random samples. I report results from various CMRP methods (filled circles and squares) and the corresponding traditional MRP methods (hollow circles and squares).

Figure 4. Results of Simulations: Testing CMRP.

Note: Points report results for 100 simulations under a set of conditions. Simulations vary the standard deviation $ \sigma $ for one variable’s effect on opinion at a time. All models were performed on clustered samples. Percent Change in MAE and percent of states improving are relative measures, comparing CMRP to traditional MRP, and deep CMRP to deep MRP. Error bars cover 95% of simulations. Axis limits are constrained to preserve readability.

The leftmost column of Figure 4 reports the MAE across simulations. As the effect sizes increase for stratum × state, stratum × region, and the independent stratum effect, traditional methods perform worse, while CMRP reduces the error across all specifications. The middle column reports the difference in MAE between corresponding traditional MRP and CMRP methods as a percentage of the error in traditional MRP. A negative result means that the MAE decreases (improves) when CMRP is used. Depending on the conditions shaping opinion, using CMRP might reduce error by as much as 50% when stratum, stratum $ \times $ state, and stratum $ \times $ region effect sizes are large. The third column reports the average share of states in each simulation whose modeled estimates of opinion get closer to true opinion. Using CMRP improves the estimates of more than half of states, particularly as the effect sizes for stratum $ \times $ state, stratum × region, and stratum get larger.

CMRP with historical polls

I now turn to validating CMRP using historical cluster-sampled polls. Using CMRP and traditional MRP, I estimated state-level support for Democratic candidates in the five presidential elections from 1968 to 1984. In each case, I estimate state-level support for the Democratic candidate using the last available Gallup poll before Election Day. I fit two models: CMRP and Deep CMRP, which adds interactions among demographic predictors (e.g., $ \mathrm{race}\times \mathrm{sex}\times \mathrm{educ} $ ) and between demographic and geographic variables (e.g., $ \mathrm{race}\times \mathrm{state} $ and $ \mathrm{race}\times \mathrm{stratum} $ ), which I then compared to similar traditional MRP or deep MRP models.Footnote ¹³ As in the pooling example above, all models included variables for race, sex, the race $ \times $ sex interaction, age group, and percent evangelical or Mormon. Models for 1976–1984 included education (the requisite variables for poststratification were not available in the 1970 Census). I then poststratified on all included predictors using a matrix built from joint distributions of the population in the 1970 and 1980 Censuses, which I obtained from IPUMS-NHGIS. The full model specifications and details of the surveys used are in Supplementary Appendix E. As before, I report results relative to national support.

Figure 5 reports results. The results indicate that CMRP, on average, reduces the overall error in opinion estimates by 2.1% for the simple CMRP model and 3.3% for the deep model. These improvements are similar in magnitude to the gains from using machine learning methods in MRP, which have been tested on modern polls that use simple random samples. For example, Ornstein (Reference Ornstein2020) shows that ensembles improve MRP estimates from standard polls by approximately 2%–3%. Likewise, Goplerud (Reference Goplerud2024) finds that Bayesian additive regression trees (BART) outperform standard MRP by 4.5% and deeply specified MRP models by 2.6% in datasets with sample sizes similar to those available from Gallup. In some cases, CMRP can produce much larger improvements; using polls from the 1968 election, CMRP performs nearly 10% better than traditional MRP. As predicted in the simulation studies, CMRP and Deep CMRP appear to safeguard against the worst increases in error in the 1972 and 1976 elections. While CMRP outperforms traditional MRP on average, individual state estimates can perform worse in some rare cases, as shown in the third column.

Figure 5. Validating CMRP with Presidential Election Polls.

Note: Points represent the improvement observed from using a given CMRP method on presidential election polls, compared to an analogous traditional MRP method. Traditional MRP is used as a baseline for CMRP; Deep MRP is used as a baseline for Deep CMRP model. Detailed results are in Supplementary Appendix E.

The rightmost column presents differences in variance between estimates produced via CMRP and Traditional MRP. As in the case of pooling, there is slightly more uncertainty associated with the CMRP estimates. However, these differences are much smaller than in the case of pooling – in fact, nearly zero – suggesting that CMRP can improve the accuracy of estimates with limited reduction in precision.

I further tested CMRP for a series of specific issue questions, which are the most common context in which MRP is used but also difficult to validate as high-quality measures of “ground truth” opinion generally do not exist. In Supplementary Appendix F, I show that CMRP on average produces slight increases in the correlation between opinion and state liberalism scores by Enns and Koch (Reference Enns and Koch2013), though the improvement can be more dramatic on some issues. I also return to the abortion question above and compare it against an abortion liberalism scale, as well as a limited number of state-level public opinion polls in 10 states. I find improvements from using CMRP versus traditional MRP approaches consistent with those reported for estimates of presidential vote choice (around 4%–8% reductions in MAE, depending on the model), though I note that the underlying state polls introduce considerable noise of their own into the comparison. Finally, in Supplementary Appendix G, I tested CMRP on six issues from the 2000 ANES and compared them with the 2000 National Annenberg Election Survey (NAES). Here, I found that, on average, CMRP methods decrease MAE by 2%.