Strategy

The answer to this problem is to create a model-based smooth density blanket for the concept of interest (primarily ideology here) such that arbitrary boundaries are immaterial. We do this with a Bayesian hierarchical spatial model that uses kriging to analyze geocoded survey responses, and then we use this analysis to forecast areal units. A kriging estimate uses an optimized linear combination of the data at observed points to fill-in unobserved points, conditional on the spatial configuration of the observed data and spatial separation between the observed and estimated points (Tam Cho and Gimpel Reference Tam Cho and Gimpel2007; Cressie and Wikle Reference Cressie and Wikle2011), thus producing a smooth density blanket over the complete geographic area of interest with lower variance than the original data. Once this density blanket exists, as determined from a posterior distribution of the chosen outcome and the subsequent smoothing, we and others can create any geographic division that a research question requires by slicing vertically such that divisions remain smooth and separated.

The idea of a smoothed density blanket is illustrated in Figure 1. The rectangular regions on the plane defined by the North–South/East–West dimensions indicate areal units of interest, which could be states, counties, etc. Two regions are highlighted in gray and black for illustration, although in our real analysis the entire United States is analyzed. Now above these two regions is the smoothed density blanket corresponding to the areal boundaries below. Notice that they are smooth at their boundary. Thus, for any region that analysts care about there is a smoothed function in the density dimension that seamlessly joints with that of its neighbors. As an analogy, consider the density blanket to be a sheet of cookie dough. Our algorithms can provide a cookie cutter in any shape that the cook wants. Of course, common “slices” will be by state, county, congressional district, or census tract. In this way, we provide a more realistic approach to modeling spatial variance for political effects.

Fig. 1

Smooth surface over two non-overlapping contiguous geographic regions

How Does Our Approach Compare With Conventional Methods?

Three approaches to measuring local-level ideology are particularly pervasive in the literature: subsetting a survey, using election returns, and post-stratification. In the subsetting approach, the researcher simply subsets a large survey by a geographic distinction of interest and aggregates the subsets of data (e.g., Erikson, Wright and McIver Reference Erikson, Wright and McIver1993). There are a couple of problems with this approach. First, the sample size can become quite small in some cases, perhaps to the point of not observing any data for some units (e.g., North Dakota and Wyoming may be states with few or no respondents in a national survey). The second issue is that survey data often are not designed to be representative of a subsample of interest. The American National Election Study, for example, stratifies on region and is therefore not necessarily representative of each state. Hence, state subsamples can be misleading.

A second approach for measuring public sentiment is to use election returns as a proxy for district ideology. The easiest way to examine this is the share of the presidential vote in a district (Erikson and Wright Reference Erikson and Wright1980; Ansolabehere, Snyder and Stewart Reference Ansolabehere, Snyder and Stewart2001). The assumption in this case is that because the candidates’ ideologies are constant nationwide, the vote share will change only in response to the median voter. Alternatively, Berry et al. (Reference Berry, Ringquist, Fording and Hanson1998) create a voting-based measure that uses votes in congressional races, but measures the ideology of both incumbents and challengers to place each district on the same scale. Although vote-based measures use abundant data that are simple to gather, there are concerns with this kind of measure as well. First, vote choice is conceptually distinct from ideology. How an individual behaves at the ballot box may not be based on general ideology, but also regional appeals, personality traits, or economic well-being, thereby inducing added measurement error. Second, the vote share in a district is based solely on who turned out to vote and does not capture the views of non-voters. Finally, vote choice alone may be a misleading measure in that it does not account for the relative dispersion of ideological positions in a district (Kernell Reference Kernell2009).

A third approach seeks to resolve the data limitations and ideological dispersion issues present with the subsetting and proxy methods. Post-stratification models individual responses to survey questions. Then, responses are predicted for every combination of covariates and weighted based on known population quantities for the region of interest (Park, Gelman and Bafumi Reference Park, Gelman and Bafumi2004; Park, Gelman and Bafumi Reference Park, Gelman and Bafumi2006; Tausanovitch and Warshaw Reference Tausanovitch and Warshaw2013). This method draws from weighting techniques also described in Gelman and Little (Reference Gelman and Little1997). Earlier work with such weighting was done by Pool, Abelson and Popkin (Reference Pool, Abelson and Popkin1965), Weber and Shaffer (Reference Weber and Shaffer1972), and Weber et al. (Reference Weber, Hopkins, Mezey and Munger1972). Jackson (Reference Jackson1989), seeking to address the constituency measurement problem, developed a technique wherein a model is fit to national-level survey data and then used to forecast public opinion in small area constituencies using average values of predictors in the constituencies.Footnote ¹ Adding these weighted predictions presents an estimate of public ideology that also includes a measure of uncertainty. By resolving many of the problems posed by other measurement strategies, post-stratification presents one of the best means of capturing public ideology by region.

However, we believe that the principle of post-stratification can be further improved upon because no current method of public opinion measurement effectively incorporates the information geography offers. Post-stratification offers a structural model for what shapes individuals’ ideological and issue preferences and the use of such a model to improve estimates of aggregate opinion. Besides observable and measured predictors that shape opinion preferences, though, there may be unobservable factors that shape individuals’ preferences as well. These may be factors such as local discourse, variation in the meaning of partisan labels in local elections, commitment to certain values, membership in labor unions, or any number of other potential factors. Normally, unobservable input variables are relegated to the disturbance term in a model. However, most models assume that each respondent’s disturbance is independent of all other respondents’ disturbances. This assumption is not fair to make.

For a theoretical background on why we expect individuals who are geographically proximate to share similar political attitudes, Gimpel and Schuknecht (Reference Gimpel and Schuknecht2003, 2–4) describe two different approaches to understanding regionalism in politics—a compositional approach and a contextual approach. First, the compositional approach maintains that political behavior bears similarity within a region because economic interests, racial origin, ethnic ancestry, religion, social structure, and other factors tend to be similar among a region’s residents (Gastil Reference Gastil1975; Garreau Reference Garreau1981; Fischer Reference Fischer1989; Lieske Reference Lieske1993). In principle, under the compositional approach, if all of these factors could be included in an empirical model, then individuals’ behavior would not show added similarities once the variables have been accounted for. However, it is often impossible to measure every relevant demographic and socioeconomic variable, or even to identify every relevant input in sufficient detail to include it in a model. (For example, would it be sufficient to know the relative importance of farming and manufacturing in an area, or are unique attributes of different crops and different products also important to regional culture?) As some relevant inputs may be overlooked or unmeasured, it is safer to assume that neighboring individuals will have a relatively similar political outlook, even holding the predictors constant.

The second approach to understanding regionalism, the contextual approach, offers the idea that citizens’ political attitudes and behaviors are influenced by political socialization and by interactions with other citizens in their social network. When citizens influence each other by discussing politics or sharing information, or when they are socialized with similar values, their opinions and actions will tend to be similar (Putnam Reference Putnam1966; Putnam Reference Putnam1993; Huckfeldt and Sprague Reference Huckfeldt and Sprague1995; DeLeon and Naff Reference DeLeon and Naff2004; Djupe and Sokhey Reference Djupe and Sokhey2011). “The first place to look for political networks is within the immediate physical proximity of each individual” (Sinclair Reference Sinclair2012, 26). Therefore, we expect under the contextual approach as well that geographically proximate individuals will have relatively similar opinions, even in a general model.

As empirical support for this point, Erikson, Wright and McIver propose that, “the unique political cultures of individual states exert an important influence on political attitudes” (Reference Erikson, Wright and McIver1993, 48). This notion builds on past work by scholars such as Elazar (Reference Elazar1966), who proposes that states can be categorized based on an individualist, moralist, or traditionalist view of government’s role. Erikson, Wright and McIver (Reference Erikson, Wright and McIver1993, 56–68) proceed to show that a higher proportion of variance in ideology and partisanship can be explained by state-level dummies than by demographic information, though state residuals also will pick up some of the effects of unmeasured individual-level variables. This result also holds in the study of urban areas, where DeLeon and Naff (Reference DeLeon and Naff2004, 703) conclude that political culture shapes the impact of identity on public opinion and political participation. It appears, then, that citizens in nearby locales do share a similar political outlook.

Our method of kriging increases the ability to reasonably capture the effects of political culture, omitted predictors, and social context by including weighted neighbors’ residuals in forecasts of public opinion. In neighboring regions, residents are likely to share a similar political culture, similar values of unobserved predictors, and similar political socialization. For example, western Kentucky and southeast Illinois are similar places that are likely to be populated with similar people, both in cultural and demographic terms. However, none of the three aforementioned approaches explicitly attempts to capture this in the measurement strategy.

Provided that geographically proximate citizens do share similar political views, then kriging will improve upon current measurement strategies in three ways. First, the kriging approach improves estimates of the structural model of individual opinion relative to methods like post-stratification. By dropping the assumption that each individual’s response is independent of all others, we allow the disturbances in individuals’ responses to survey questions to covary conditionally on how geographically proximate they are. Doing this yields more efficient parameter estimates than the wrongful assumption that the disturbances are independent. Hence, the predictive structural model ought to be more accurate.

Second, in the process of allowing spatial autocorrelation, we estimate a surface of what the spatial random effects are (see again Figure 1). By fitting this kriging surface and then using the values of this surface to predict ideology or issue opinion in a region, the added information will further raise the quality of our predictions. To the degree that unmeasured variables and political culture shape public opinion, the crafting of this surface ought to better incorporate these features into the forecast. Regardless of how small the constituency’s area is, a reasonable projection of these factors is incorporated via kriging. As the kriging surface is identified through the spatial location of each respondent, fitting this surface allows us to separate the variance not explained in the mean model into variance that spatially clusters and variance that is truly stochastic. This further reduction of variance that cannot be used in prediction improves our measures.

Third, every forecast should report a level of uncertainty in prediction. In our case, this takes the form of credible surfaces around the kriged density blanket. Predictions for larger areas (such as entire states) will be more certain than predictions for smaller areas (such as congressional or state legislative districts), and so these credible surfaces on either side of the estimate will be smaller in the former case. Naturally, the same would hold for any other method, including survey disaggregation and post-stratification, but by including the information of the spatial surface in our model and predictions, we can reduce the error of our areal predictions to be as small as possible, whereas providing accurate measures of uncertainty at any level of aggregation.

Basic Spatial Specification

Consider X^*, an n×k positive definite matrix of explanatory variables organized down columns with a leading column of ones. Denote for the ith case, x_ij is a vector of explanatory variable values, and the j index is a reminder of nesting in the jth regional group. Furthermore, partition the n×k matrix X^* into two submatrices that track the nesting distinction:

(1) $$\mathop{{{\bf X}^{{\asterisk}} }}\limits_{{n{\times}k}} =\left[ {\mathop{{\bf X}}\limits_{{n{\times}p}} \,{\vskip12pt\rot90{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}}\mathop{{\bf Z}}\limits_{{n{\times}q}} } \right],$$

where X is the matrix of variables that will have unmodeled coefficients (receiving standard point estimates) and Z the matrix of explanatory variables that will have modeled coefficients through a hierarchy and therefore will be described distributionally. Finally, specify an error structure, ϵ, containing ϵ _ij values that are all independent and identically distributed according to $$\epsilon _{{ij}} \,\sim\,{\cal N}_{n} (0,\,\sigma ^{{\rm 2}} )$$ , σ ²>0. The complete model for our outcome is now the simple form:

(2) $${\bf Y}={\bf X}\bibeta {\plus}{\bf Zu}{\plus}\epsilon ,$$

where β is a p-length vector of coefficient estimates and u the q-length vector of random effects with the assumption of multivariate normality with mean zero and positive definite variance–covariance matrix, such that ${\bf u}\,\sim\,{\cal N}_{q} (0,\Delta )$ . This produces a method of accounting for structured data through regression equations at different hierarchical levels in the data. These higher-level models are describing distributions at the level just beneath them for the coefficient that they model as if it were itself an outcome variable. In this way, multi-level models are highly symbiotic with Bayesian specifications because the focus in both settings is on making supportable distributional assumptions.

This hierarchical approach models the systematic component, but does not yet incorporate the smoothing that we require for our purposes. The latter is done with kriging through the variance component of the model. For simplicity, label μ=Xβ+Zu in (2) above and generalize the variance component according to:

(3) $$\Sigma =\sigma ^{2} {\bf H}(\phi ){\plus}\tau ^{2} {\bf I},$$

where σ ² is a weight, called the partial sill, H(ϕ) an n×n correlation matrix where the ijth value is defined by a parametric function, and τ ² another weight, called the nugget, that multiplies an n×n identity matrix (τ ²+σ ² is called the sill). In this context, ϕ is called the decay parameter, and 1/ϕ is called the range parameter. The ijth off-diagonal values of H actually could follow a variety of functional forms called covariograms (assumed spatial covariances at fixed distances), such as the wave, Gaussian, or Matérn functions, depending on what fits the data best (see Table 1 for a summary of some semivariograms). Often these parametric forms are given simple forms like exp(ϕd _ij), an exponential correlation function, where the values d _ij are coordinate-based distance measures between case i and case j, weighted by autocorrelation parameter ϕ. These terms originate from Cressie (Reference Cressie1991), and are further described in Banerjee, Carlin and Gelfand (Reference Banerjee, Carlin and Gelfand2004).

Table 1

Four Candidate Parametric Semivariograms and their Relative Fit

Note: N=21,543. Data from 2008 Cooperative Congressional Election Study. Notation from Banerjee, Carlin and Gelfand (Reference Banerjee, Carlin and Gelfand2004, table 2.2). Bayesian information criterion (BIC) formula drawn from Gujarati and Porter (Reference Gujarati and Porter2009, 494). Computed with the geoR 1.7-4 library in R 3.0.0.

The model is not yet complete, so modify it to specify a correlation at the group level:

(4) $${\bf Y}\,\mid\,\mu ,\sigma ^{2} ,\tau ^{2} ,\phi \,\sim\,{\cal N}({\bf X}\beta {\plus}{\bf Zu}{\plus}{\bf W},\tau ^{2} {\bf I}),$$

thus adding a vector of spatial random effects:

(5) $${\bf W}=\{ w(s_{1} ),w(s_{2} ),\,\,\ldots\,\,,w(s_{n} )\} ,$$

such that each Y(s_i) is conditionally independent given the w(s_i) and u_i, where s_i is a vector of location coordinates for observation i. The spatial aspect of the hierarchy is now given by:

(6) $${\bf W}\,\mid\,\sigma ^{2} ,\phi \,\sim\,{\cal N}(\bf0,\sigma ^{2} {\bf H}(\phi )),$$

where H(ϕ)_ij=ρ(ϕ,||s_i−s_j||) and ρ a spatially appropriate correlation function. The spatial density blanket is now a function of W|Y, which naturally leads to credible intervals. Putting the hierarchical component together with the kriging component, along with a multivariate normal assumption, gives the full model. Hence, we now have a complete model of individual survey respondents that accounts for the respondents’ locations in latitude and longitude. With this model in hand, we will use the technique of point-to-block realignment (spatial forecasts at new locations) to measure public sentiment for border-referenced constituencies (Banerjee, Carlin and Gelfand Reference Banerjee, Carlin and Gelfand2004, chapter 6). This model-based approach better accounts for dissimilarities across politically drawn lines and yields estimates with less measurement error as arbitrary boundaries like census tracts do not separate responses. Instead, the distance-based model recognizes shared effects by geography along with distinct effects by model hierarchy.

As a final point on background, Banerjee, Carlin and Gelfand (Reference Banerjee, Carlin and Gelfand2004) give GeoBUGS code for specifying point-referenced models similar to what we have described. However, GeoBUGS is designed as an add-on package for WinBUGS, which only works on the Windows operating system. Further, GeoBUGS has proven to be computationally inefficient for these types of models and unable to handle large data sets effectively. Hence, we currently implement this structure using R.

Model for Unobserved Spatial Points

Our fundamental goal is to model the opinions of individuals, who are located in latitude and longitude, and use the information from this model to measure the ideology of a whole constituency. In this vein, Banerjee, Carlin and Gelfand (Reference Banerjee, Carlin and Gelfand2004, 179–80) describe the method for using observations taken at various points in latitude and longitude and realigning them into measures of block-level averages. Their design assumes that the observed point-level data and the extrapolated block-level averages have a joint Gaussian distribution. If survey data include the latitude and longitude of survey respondents’ residences (or, more commonly, information like ZIP code, for which coordinates can be assigned with reasonable accuracy), then point-to-block realignment offers an opportunity for effective measurement of precincts or congressional districts.

We want to create a posterior density blanket for some outcome of interest for the entire United States that can be sliced by any districting method where the partition is available. In the case of ideology, this would be an extension of Erikson, Wright and McIver (Reference Erikson, Wright and McIver1993), where the boundaries are not required to be US states. This means that the final result will include estimates for spatial points where there is no observed data. Define now s as a set of n observed sites {s₁, s₂, … , s_n}, where Y(s) is an associated collection of outcomes Y(s)={Y(s₁), Y(s₂), … ,Y(s_n)} and X^*(s)={x^*(s₁), x^*(s₂), … ,x^*(s_n)} is a collection of hierarchical covariates at those points. Note that the X^*(s) are conditional on the level of aggregation chosen (state, county, congressional district, etc.). Now alter the linear specification in (2) to accommodate this new setting:

(7) $${\bf Y}({\bf s})=\mu ({\bf s}){\plus}\omega ({\bf s}){\plus}\epsilon ({\bf s}),$$

where μ(s)=X(s)β+Z(s)u is the mean structure based on a linear additive component, ω(s) the realizations from a mean zero stationary (usually) Gaussian spatial process that captures spatial association (includes the partial sill σ ² and the range ϕ), and ϵ(s) a regular uncorrelated disturbance term (includes the nugget effect τ ², its variance). In addition, assume that $$\omega \left( {\bi s} \right)\!\sim\!{\cal N}(\bf0,\bisigma ^{{\rm 2}} {\bf H}(\phi ))$$ and $$\epsilon \left( {\bf s} \right)\sim{\cal N}(\bf0,{\rm }\tau ^{{\rm 2}} {\bf I})$$ .

As noted, we usually specify the correlation function of ω(s) to be a function of the separation between points to give a stationary model, so that:

(8) $$\mu ({\bf s})=E[Y({\bf s})]{\rm \ and}\ f({\rm Var}({\bf s}))=f({\rm Var}({\bf s}{\plus}{\bf h})),\,{\rm for}\,{\rm any}\,{\bf h}\in \Re ^{r} ,\;r\leq n,$$

where h is called the separation vector, and f() an arbitrary user-defined function (which also just be the identity function). If this separation is purely based on distance, ||s_i−s_j||, then we get isotropy, meaning that the semivariogram function

(9) $$\gamma ({\bf h})={1 \over 2}{\rm Var}({\bf Y}({\bf s}{\plus}{\bf h}){\minus}{\bf Y}({\bf s})),$$

depends on h only through its length ||h||. In this context, the nugget can be thought of as micro-scale variability: variability at distances smaller than the smallest inter-location distance in the data. This is important as we are now making predictions about points and distances between points that are not in the original data. Figure 2 gives an example of an exponential form.

Fig. 2

Exponential semivariogram where τ²=0.1, σ ²=0.9, and ϕ=1

We clearly want the posterior distribution of the coefficient vector β for regression purposes, as well as the parameters σ ², τ ², and ϕ. However, we also want a smooth density blanket for Y(s), which requires predictions at unobserved sites, generically labeled s₀, giving Y(s₀) values at chosen locations. Kriging actually has two steps that are important here: the process of applying mathematical decision criteria, and then applying constrained optimization to get a density blanket. We consider this problem in the context of Gaussian processes, but many other distributional frameworks are available.

Optimization Criteria

The method of point-to-block realignment calls for kriging, or making forecasts of a spatially referenced variable (ideology, in our case) at new geographic locations. When choosing a kriging function with which to make forecasts, an obvious decision criterion is to find a function f(Y) that minimizes the mean squared prediction error:

$$\eqalignno{ E[(Y({\bf s}_{0} ){\minus}f({\bf Y}))^{2} \,\mid\,{\bf Y}]= \,& E[((Y({\bf s}_{0} ){\minus}E[Y({\bf s}_{0} )\,\mid\,{\bf Y}]) \cr &{\plus}(E[Y({\bf s}_{0} )\,\mid\,{\bf Y}]\,{\minus}\,f({\bf Y})))^{2} \,\mid\,{\bf Y}] \cr =\, & E[(Y({\bf s}_{0} ){\minus}E[Y({\bf s}_{0} )\,\mid\,{\bf Y}])^{2} \,\mid\,{\bf Y}] \cr & {\plus}E[(E[Y({\bf s}_{0} )\,\mid\,{\bf Y}]\,{\minus}\,f({\bf Y}))^{2} \,\mid\,{\bf Y}]. $$

However, $$E\left[ {(E[Y({\bf s}_{0} )\,\mid\,{\bf Y}]\,{\minus}\,f({\bf Y}))^{2} \,\mid\,{\bf Y}} \right]$$ has to be non-negative because of the square, so we know that

$$\eqalignno{ & E\left[ {(Y({\bf s}_{0} ){\minus}E[Y({\bf s}_{0} )\,\mid\,{\bf Y}])^{2} \,\mid\,{\bf Y}} \right]{\plus}E\left[ {(E[Y({\bf s}_{0} )\,\mid\,{\bf Y}]\,{\minus}\,f({\bf Y}))^{2} \,\mid\,{\bf Y}} \right] \cr & \geq E\left[ {(Y({\bf s}_{0} ){\minus}E[Y({\bf s}_{0} )\,\mid\,{\bf Y}])^{2} \,\mid\,{\bf Y}} \right]. $$

The minimum (as in equality, not inequality above) happens only when f(Y)=E[Y(s₀)|Y], meaning E[Y(s₀)|Y] is the function that minimizes mean squared prediction error. We know that E[Y(s₀)|Y] is the posterior mean of Y(s₀) and therefore minimizes the posterior risk with a squared error loss function. Furthermore, an estimator that minimizes the Bayesian posterior risk is one that for every $Y\in {\cal Y}$ specifies a decision rule, d(Y), that minimizes the posterior expected loss, $$E_{\pi } [{\Bbb L}(A,d({\bf Y}))]$$ , for action A. This is an overtly Bayesian statement of loss as we are averaging over uncertainty in the posterior distribution of β, not over the distribution of data since it has already been observed. The subsequent decision rule which minimizes:

(10) $$R_{B} (\bibeta ,d({\bf Y}))=\mathop{\int}_{B} \mathop{\int}_{\cal Y} {\Bbb L}(A,d({\bf Y}))dF_{{\cal Y}} (y\,\mid\,\bibeta )dF_{\pi } (\bibeta ),$$

for every $Y\in {\cal Y}$ is called optimal, and limiting the set to this choice is called a Bayes rule, denoted $$\widehat{{R_{B} }}(\beta ,d({\bf Y}))$$ here, and defined by the function minimizing value of β _A that satisfies:

(11) $$\widehat{{R_{B} }}(\beta ,d({\bf Y}))=\mathop{{\inf }}\limits_{{\beta _{A} }} \mathop{\int}_{B} R_{F} (A,d({\bf Y}))dF_{\pi } (\beta ),$$

(Gill Reference Gill2014). This is then our criteria for finding β in (4). We describe a Markov chain Monte Carlo (MCMC) process for estimating the parameters in Appendix A, and this algorithm represents a large proportion of the contribution of this work.Footnote ²

Parameter Choices

As our model assumes the observed point-level data and the extrapolated block-level averages have a joint Gaussian distribution, we get:

$$f\left( {\left( {\matrix{ {{\bf Y}_{{\bf s}} } \cr {{\bf Y}_{{\bf B}} } \cr } } \right)\,{\vskip12pt\rot90{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}}\,{\bibeta ,\sigma ^{2} ,\phi }} \right)={\cal N}\left( {\left( {\matrix{ {\mu _{{\bf s}} (\bibeta )} \cr {\mu _{{\bf B}} (\bibeta )} \cr } } \right),\sigma ^{2} \left( {\matrix{ {{\bf H}_{{\bf s}} (\phi )} & {{\bf H}_{{{\bf s},{\bf B}}} (\phi )} \cr {{\bf H}_{{{\bf s},{\bf B}}}^{T} (\phi )} & {{\bf H}_{{\bf B}} (\phi )} \cr } } \right)} \right),$$

where Y_s represents the vector of ideology among individual citizens, Y_B the vector of ideology in all block-referenced constituencies of interest, and H the correlation matrix of observations as before. Note that this presents the simplified case where there is no nugget effect (τ ²), but the result still holds if the variance–covariance terms do include a nugget, as seen in Appendix A. By standard normal theory (e.g., Ravishanker and Dey Reference Ravishanker and Dey2002), the conditional distribution of our extrapolated block averages is:

(12) $$\eqalignno{ {\bf Y}_{{\bf B}} \,\mid\,{\bf Y}_{{\bf s}} ,\bibeta ,\sigma ^{2} ,\phi \,\sim\, & {\cal N}(\mu _{{\bf B}} (\bibeta ){\plus}{\bf H}_{{{\bf s},{\bf B}}}^{T} (\phi ){\bf H}_{{\bf s}}^{{{\minus}1}} (\phi )({\bf Y}_{{\bf s}} \,{\minus}\,\mu _{{\bf s}} (\bibeta )), \cr & \sigma ^{2} [{\bf H}_{{\bf B}} (\phi ){\minus}{\bf H}_{{{\bf s},{\bf B}}}^{T} (\phi ){\bf H}_{{\bf s}}^{{{\minus}1}} (\phi ){\bf H}_{{{\bf s},{\bf B}}} (\phi )]). $$

These quantities can be estimated with Monte Carlo integration:

$$\eqalignno{ & (\hat{\mu }_{{\bf B}} (\beta ))_{k} =L_{k}^{{{\minus}1}} \mathop{\sum}\limits_\ell \mu ({\bf s}_{{k\ell }} ;\bibeta ) \cr & ({\bf \widehat{H}}_{{\bf B}} (\phi ))_{{kk'}} =L_{k}^{{{\minus}1}} L_{{k'}}^{{{\minus}1}} \mathop{\sum}\limits_\ell \mathop{\sum}\limits_{\ell '} \rho ({\bf s}_{{k\ell }} \,{\minus}\,{\bf s}_{{k'\ell '}} ;\phi ) \cr & ({\bf \widehat{H}}_{{{\bf s},{\bf B}}} (\phi ))_{{ik}} =L_{k}^{{{\minus}1}} \mathop{\sum}\limits_\ell \rho ({\bf s}_{i} {\,\minus\,}{\bf s}_{{k\ell }} ;\phi ). $$

This will allow us to forecast the average ideology with:

(13) $$\hat{\mu }_{{\bf B}} (\bibeta ){\plus}{\bf \widehat{H}}_{{{\bf s},{\bf B}}}^{T} (\phi ){\bf \widehat{H}}_{{\bf s}}^{{{\minus}1}} (\phi )({\bf Y}_{{\bf s}} \,{\minus}\,\hat{\mu }_{{\bf s}} (\bibeta )).$$

We account for the spatial element by forecasting $$\hat{Y}({\bf s}_{{k\ell }} ;\bibeta ,\sigma ^{2} ,\tau ^{2} ,\phi )$$ and using this quantity in our Monte Carlo integration. With the nugget effect, from Y(s)=μ(s)+ω(s)+ϵ(s), we also get $${\bf Y}_{{\bf s}} \sim{\cal N}(\bimu ,\,∑)$$ , where we still require ∑=σ ²H(ϕ)+τ ²I, H(ϕ)_ij=ρ(ϕ, d _ij) and d _ij=||s_i−s_j||. For this application, we selected the Gaussian semivariogram function for reasons shown below.

We want to forecast blocks using the conditional distribution in (12). However, current software is inadequate to jointly estimate the geospatial model and krige for more than a few kriged points, and there is no ability to obtain the conditional block averages. As a result, we developed our own sampler to more efficiently allow for kriging over a wide number of locations and more directly model the conditional mean of blocks (Appendix A).

Data for 2008

As our training data for this model we use the 2008 Cooperative Congressional Election Study (CCES) (Ansolabehere Reference Ansolabehere2011). These data offer 21,543 observations spread across the American states and congressional districts, and the variables are described in Appendix B. Further, the data report ZIP codes, allowing us to place respondents in geographic space.Footnote ³Figure 3 presents the geographic location of the respondents in the training data. The locations are presented on a scale of eastings and northings, which accounts for the spherical shape of the earth and places all observations on a common kilometer-referenced scale. By contrast, a degree of latitude need not constitute the same distance as a degree of longitude.Footnote ⁴

Fig. 3

Geographic locations in kilometer-referenced eastings and northings of survey respondents in the 2008 Cooperative Congressional Election Study

The CCES presents a 100-point self-reported ideology scale, which serves as the outcome of interest.Footnote ⁵ It also provides covariates for age, education, race, sex, family income, religion, employment status, and homeowner status. We also include urban–rural status as a predictor by linking respondents’ respective counties to the US Department of Agriculture’s (USDA) scale for the relative ruralness of a county (USDA 2004). Lastly, with the eastings and northings generated by ZIP code, we include a polynomial function of location as a predictor.

When forecasting for states and congressional districts, we draw most covariates by sampling from the population distribution for the ZIP code in which the kriged point falls in, thereby drawing covariates consistent with the local population. We use the complete 2000 Census data, drawing from the joint distribution of age, race, and sex, and then drawing from the conditional distribution for all other predictors given these demographics. ZIP code-referenced Census data were archived with the National Historic Geographic Information System (Minnesota Population Center 2011).Footnote ⁶ For urban–ruralness, each kriged location in space is linked to the USDA county score. Moreover, as Census does not report religion, we independently draw from each county’s distribution of religious adherents using the 2000 Religious Congregations and Membership Study (Jones et al. Reference Jones, Doty, Grammich, Horsch, Houseal, Lynn, Marcum, Sanchagrin and Taylor2002; Finke and Scheitle Reference Finke and Scheitle2005).Footnote ⁷ Thus, for a given kriged point, there is demographic information based on the ZIP code the point falls in, and we draw information on religiosity and ruralness based on the county in which the point falls.

Model Specification and Estimation

In addition to our predictors of age, education, race, sex, income, religion, employment status, homeowner status, and urban–rural status, our model of mean ideology also includes a spatial trend term. The spatial trend surface is a linear combination of polynomials of the spatial coordinates. Following the prescription of Cressie (Reference Cressie1993, 155–6), the polynomial terms are defined by the formula:

(14) $$\mathop {\mathop{\sum} \mathop{\sum} }\limits_{0\leq k{\plus}\ell \leq r} \alpha _{{k\ell }} s_{{i1}}^{k} s_{{i2}}^{\ell } ,\quad {\bf s}_{i} =(s_{{i1}} ,s_{{i2}} )',$$

where s_i is the vector location of observation i, s _i1 the easting for observation i, s _i2 the northing for observation i, α _kl the coefficient for the term where the easting is raised to the exponent k and the northing is raised to the exponent l, and r the order of the polynomial trend term. In short, for an rth-order trend surface, every product term for which the exponents on eastings and northings sum to any quantity ≤r is included. After comparing the fit of models including a constant trend (r=0) up through a fourth-order trend (r=4) using Schwarz’s Bayesian information criterion (BIC), we found that the second-order and third-order models fit the best. Owing to multicollinearity issues, simplicity in reporting, and a slightly better penalized fit, we present the results of the second-order model (r=2).Footnote ⁸

With this linear structure in place for the model of ideology’s mean, the full Bayesian model can be specified as follows:

$$\eqalignno{& \ \ \ \ \ \ \ \,{\bf Y}_{{\bf s}} \,\sim\,{\cal N}({\bf X}_{{\bf s}} \bibeta ,\Sigma ) \cr & \ \ \ \ \ \ \ \ \ \,\Sigma =\sigma ^{2} {\bf H}(\phi ){\plus}\tau ^{2} {\bf I} \cr & \ \ \ \,H(\phi )_{{ij}} =\rho (\phi ;d_{{ij}} ) \cr & \ \ \ \ \ \ \ \ \ \ \rho =\exp ({\minus}\phi ^{2} d_{{ij}}^{2} )\;{\rm (Gaussian}\,{\rm correlation}\,{\rm function)} \cr & \ \ \ \ \ \,\pi (\bibeta )\,\sim\,{\rm flat} \cr & \pi (\tau ^{2} /\sigma ^{2} )\,\sim\,{\rm Unif}(6,8) \cr & \ \ \ \ \pi (\sigma ^{2} )\,\sim\,1/\sigma ^{2} \cr & \ \ \pi (1/\phi )\,\sim\,{\rm Unif}(0,9000). $$

In this setup, rather than modeling the nugget effect (τ ²) directly, we model the relative nugget (τ ²/σ ²) or ratio of the nugget to the directly modeled partial sill. As a reasonably tight prior is required for numerical stability (Hobert and Casella Reference Hobert and Casella1996), we centered a uniform distribution around an ∼7:1 empirical ratio of these two quantities. For the partial sill (σ ²), we assume a reciprocal prior, and for the range (1/ϕ), we assume a uniform prior up to twice the maximum distance in the data. Priors on the regression coefficients are flat. The best-fitting covariance function is the Gaussian function, which will be described more in the next section, and all other model attributes are defined as before.

To estimate this model, we adapt a five-step algorithm developed by Diggle and Ribeiro (Reference Diggle and Ribeiro2002, 141).Footnote ⁹ First, we draw several values from a discrete version of the uniform priors for $${{\tau ^{2} } \over {\sigma ^{2} }}$$ and ${1 \over \phi }$ . Second, we estimate the conditional posterior distribution, $$p\left( {{{\tau ^{2} } \over {\sigma ^{2} }},\,{1 \over \phi }\,\mid\bf Y} \right)$$ by placing our draws from the discrete prior into the following formula:

(16) $$p\left( {{{\tau ^{2} } \over {\sigma ^{2} }},{1 \over \phi }\,\mid\,{\bf Y}} \right)\propto\pi \left( {{{\tau ^{2} } \over {\sigma ^{2} }}} \right)\pi \left( {{1 \over \phi }} \right)\,\mid\,V_{{\tilde{\beta }}} \,\mid\,^{{{1 \over 2}}} \left| {{\bf H}(\phi ){\plus}\left( {{{\tau ^{2} } \over {\sigma ^{2} }}} \right){\bf I}} \right|^{{{\minus}{1 \over 2}}} (\hat{\sigma }^{2} )^{{{\minus}{n \over 2}}} ,$$

where $V_{{\tilde{\beta }}} $ is the correlation matrix of the regression coefficients estimated with generalized least squares (GLS) using the current draw of 1/ϕ, n the sample size, and $$\hat{\sigma }^{2} $$ an estimate of the partial sill based on residuals drawn from the GLS coefficient estimates.Footnote ¹⁰ All other terms are defined as before. Third, we draw a single set of sample posterior values for $${{\tau ^{2} } \over {\sigma ^{2} }}$$ and ${1 \over \phi }$ from (16). Fourth, we attach the set of sampled values to $p\left( {\bibeta ,\sigma ^{2} \,\mid\,{{\tau ^{2} } \over {\sigma ^{2} }},{1 \over \phi },\bf Y} \right)$ and compute the corresponding conditional posterior distributions as:

(17) $$\eqalignno{ & \sigma ^{2} \,\mid\,{\bf Y},{{\tau ^{2} } \over {\sigma ^{2} }}^{{\asterisk}} ,{1 \over \phi }^{{\asterisk}} \,\sim\,\chi _{{ScI}}^{2} (n,\hat{\sigma }^{2} ) \cr & \bibeta \,\mid\,{\bf Y},\sigma ^{2} ,{{\tau ^{2} } \over {\sigma ^{2} }}^{{\asterisk}} ,{1 \over \phi }^{{\asterisk}} \,\sim\,{\cal N}(\tilde{\bibeta },\sigma ^{2} V_{{\tilde{\beta }}} ). $$

Here, the term χ ²Scl refers to the Gaussian-Scaled-Inverse-χ ² distribution. The terms in these equations are again drawn from the GLS estimates from the initial draw of ϕ. After taking a draw from the scaled inverse χ ² distribution for the partial sill σ ², this term is linked with the draws from the relative nugget and range terms when drawing the regression coefficients from a normal distribution. By repeating the third and fourth steps to generate a sufficiently large sample from each of the conditional posteriors, we build a sufficient Monte Carlo sample to reflect the joint posterior for the full parameter set $\left( {{{\tau ^{2} } \over {\sigma ^{2} }},{1 \over \phi },\sigma ^{2} ,\bibeta } \right)$ .