Modeling State Legislative Elections

We want to build a reasonable but simple demographic model of voter turnout and preference which we can use to estimate the outcome of an election in a state legislative district. We want to build something we can apply fairly easily to any state. And, since this is a redistricting year, we want something that we can apply to newly drawn districts as well as existing ones.

One possibility is that we work bottom-up, using voting precincts as building blocks:

1. Get the geographic boundary and election results for each precinct, e.g., from openprecincts.
2. Build a demographic profile of the precinct using overlaps¹ of census “blocks” or “block groups”².
3. Use the total number of votes and votes cast for the Democratic candidate in each precinct to infer a demographic model of turnout and voter preference in the entire set of precincts.
4. Apply that model to the demographics of a SLD to generate a rough estimate of the likely election result.

For a given SLD (or set of SLD’s), what precincts do we include in the model? In order to keep things simple we want a model that covers multiple districts. We could model using every precinct in the country or at least the state. Using every precinct in the country is hard: some of that data is unavailable and there are a lot of precincts (about 175,000 of them)! Using data only from the state risks being too influenced by the local election history.

So instead, we work top-down from national-level data:

1. Use a national survey of voter turnout, e.g., the Current Population Survey Voter Registration Supplement (CPSVRS), and/or the Cooperative Election Survey (CES, formerly the CCES) and a national survey of voter preference, like the CES, to build demographically stratified turnout and preference data at the state or congressional-district (CD) level.
2. Use that data to infer a demographic model of turnout and voter preference, possibly with state or CD-level effects.
3. Apply³ that model to the demographics of a SLD to generate a rough estimate of the likely election result.

This approach might be too general: voters in different regions might not be well described by a national model. The data we use is organized by CD, so we can add some geographic specificity to the model. Even better, we can compare the quality of models with different levels of geographic detail.

This data is considerably simpler to work with and more comprehensive precinct-level data is not available for all states and all election years. But we remain interested in the bottom-up approach as well, and we might implement some version of it for comparison.

Some details:

1. For turnout, we have the CPSVRS and/or the CES. The CPSVRS is self-reported whereas the CES validates the turnout data via voter files. All other things equal, we’d prefer the validated data. But there’s some evidence that the validation process used by the CES introduces bias because it tends to miss people who move between elections, and they are disproportionately likely to be young and poor. Because it the more used source, we use the CPSVRS as a source of turnout data, though we also run the models using the CES as a turnout source to see if it makes a large difference. People tend to over-report their own turnout, which presents a problem for non-validated sources. So we use some standard adjustments to the turnout data, first suggested by Hur & Achen, which adjusts the turnout probabilities from CPSVRS so that the actual recorded total turnout matches the CPSVRS post-stratified on the geography in question. The Hur & Achen paper doesn’t address how to re-weight among various demographic groups within the same geography. For this we follow the procedure outlined (in a slightly different context) on pages 9-10 of this paper by Ghitza and Gelman.

2. We choose congressional districts (CD’s) as our basic geographic unit for constructing the model of turnout and voter preference. We use CD’s as our modeling units for two reasons: CD’s are a core interest at BlueRipple politics and they are the smallest geography easily available in all the required data. We can get more fine grained census data for demographics but the turnout and voter preference data that we use (the CPSVRS and CES) is not available at the SLD level.

For each of the 436 districts (435 CDs plus DC) in the U.S., we have:

• The CPSVRS, containing self-reported voter turnout for about 1% of eligible voters in each district, broken down by sex (female or male), education (non-college-grad or college-grad) and race (Black, Latinx, Asian, white-non-Latinx, other).

• The CES, containing validated turnout and voter preference–specifically the political party of the voter’s choice for congressional representative, broken down by the same demographic categories.

• Population density, computed by aggregating data from the Census Bureau’s American Community Survey (ACS) at the Public-Use-Microdata-Area (PUMA) level.

For each State Legislative District (SLD) we have data from the ACS, which we aggregate from the block-group level using areal interpolation. Our shapefiles for the SLDs and the census block-groups come from the Census Bureau. The result of this aggregation is a breakdown of the citizens in each district by the same demographic variables as the CPSVRS and CES data, as well as an estimate of population density in the SLD.

Modeling proceeds as follows:

1. We combine the CPSVRS, CES, and CD-level population density into one data-set, with rows for each combination of demographic variables within each CD. That’s 2 x 2 x 5 x 436 = 8720 rows, each with turnout data from both CPSVRS and CES, party preference from the CES and population density from the ACS.

2. We model turnout and vote choice as binomially distributed with probabilities determined by the various demographic parameters. That is, given a district, \(d\), with population density \(\rho\), and the subset of people with sex \(s\), education-level \(e\) and race \(r\), we assume that the turnout (\(T\)) of eligible voters (\(E\)) and the votes for the Democratic candidate (\(D\)), out of all validated voters (\(V\)), both follow binomial distributions:

   \(T^{(d)}_{ser}(\rho) \thicksim B\big(E^{(d)}_{ser}(\rho);t^{(d)}_{ser}(\rho)\big)\)

   \(D^{(d)}_{ser}(\rho) \thicksim B\big(V^{(d)}_{ser}(\rho);p^{(d)}_{ser}(\rho)\big)\)

   where \(B(n;p)\) is the distribution of successful outcomes from \(n\)-trials with fixed probability of success \(p\). NB: If we were using the same data for turnout and preference, \(T^{(d)}_{ser}(\rho)\) would be the same as \(V^{(d)}_{ser}(\rho)\), but since we are using different data sets, we need to model them separately.

   The probabilities \(t\) and \(p\) must be between 0 and 1 and are modeled via logistic functions of the parameters so that the parameters themselves are unconstrained. This allows the fitting to proceed more easily. The logistic then maps the unconstrained sum of the parameters to a probability.

   \(\begin{equation} t^{(d)}_{ser}(\rho) = \textrm{logit}(X\cdot\tilde{\rho} + A_s + B_e + C_r + D_{S(d)}) \end{equation}\)

   \(\begin{equation} p^{(d)}_{ser}(\rho) = \textrm{logit}(x\cdot\tilde{\rho} + \alpha_s + \beta_e + \gamma_r + \delta_{S(d)}) \end{equation}\)

   where \(S(d)\) is the state in which the district \(d\) is located, and \(\tilde{\rho} = \textrm{log}(\rho) - \textrm{mean}(\textrm{log}(\rho))\)

   • \(X\) and \(x\) are single coefficients, given broad, normal, zero-centered priors.

   • \(A_s\), \(B_e\), \(\alpha_s\), and \(\beta_e\) are constrained to be \(\pm\) one number each, since the categories are binary. E.g., \(B_\textrm{college-grad} = -B_\textrm{non-grad}\). They are given broad, normal, zero-centered priors.

   • \(C_r\) and \(\gamma_r\) represent one number per race category and are partially pooled: we add hyper-parameters representing the mean value among all races and the variance of the specific race coefficients around that mean. The mean acts like the constant term in the fit, setting an average turnout/preference among all people, and the variance adjusts how much the race-specific levels of turnout/D-preference are derived from each racial group separately.

   • \(D_s\) and \(\delta_s\) take on one value per state and are also partially pooled. Here we assume zero mean and use one hyper-parameter for the variance, allowing the fit to determine how much the state-by-state variance is determined by the demographics of the state vs. something intrinsic to the state itself, an issue we looked at in some detail in a previous post.

3. We use Haskell to parse the data, reorganize it and produce code for Stan, which then runs the Hamiltonian Monte Carlo to estimate the posterior distribution of all the parameters and thus posterior distributions of \(t^{(d)}_{ser}(\rho)\) and \(p^{(d)}_{ser}(\rho)\).

4. Using those distributions and the breakdown of the demographics of each SLD, which amounts to a population density (\(\rho*\)), and numbers, \(N_{ser}\) of eligible voters for each combination of sex, education-level and race, we post-stratify to get a distribution of votes, \(V\), and democratic votes, \(D\):

   \(\begin{equation} V = \sum_{ser} N_{ser} t_{ser}(\rho*) \end{equation}\)

   \(\begin{equation} D = \sum_{ser} N_{ser} t_{ser}(\rho*) p_{ser}(\rho*) \end{equation}\)

   where it’s understood that we are setting the state \(S=\textrm{Virginia}\). From here it’s straightforward to compute the distribution of democratic vote share \(D/V\).

One advantage of estimating these things via Monte Carlo, is that we compute each quantity, including the post-stratified ones, on each Monte Carlo iteration. So rather than just getting an estimate of the value of each quantity, we get some large number of samples from its posterior distribution. So we have good diagnostics around these quantities: we can see if we’ve done enough sampling for their distributions to have converged and we can extract informative confidence intervals–even of derived quantities like the post-stratifications–rather than just crude estimates of standard deviation as you might get from various methods which estimate parameters via optimization of the maximum-likelihood function.