A multivariate spatial mixture model for areal data: examining regional differences in standardized test scores

Summary

Researchers in the health and social sciences often wish to examine joint spatial patterns for two or more related outcomes. Examples include infant birth weight and gestational length, psychosocial and behavioral indices, and educational test scores from different cognitive domains. We propose a multivariate spatial mixture model for the joint analysis of continuous individual-level outcomes that are referenced to areal units. The responses are modeled as a finite mixture of multivariate normals, which accommodates a wide range of marginal response distributions and allows investigators to examine covariate effects within subpopulations of interest. The model has a hierarchical structure built at the individual level (i.e., individuals are nested within areal units), and thus incorporates both individual- and areal-level predictors as well as spatial random effects for each mixture component. Conditional autoregressive (CAR) priors on the random effects provide spatial smoothing and allow the shape of the multivariate distribution to vary flexibly across geographic regions. We adopt a Bayesian modeling approach and develop an efficient Markov chain Monte Carlo model fitting algorithm that relies primarily on closed-form full conditionals. We use the model to explore geographic patterns in end-of-grade math and reading test scores among school-age children in North Carolina.

1. Introduction

In 2002, the United States (U.S.) Congress enacted the No Child Left Behind (NCLB) Act requiring states to administer annual standardized tests to all students in federally funded schools (No Child Left Behind Act, 2002). In North Carolina, these tests are known as end-of-grade (EOG) tests. The EOG tests measure student performance on grade-based goals, objectives, and competencies as set forth by state-level education departments (North Carolina Department of Public Instruction, 2006). In particular, the mathematics tests measure competency in areas such as arithmetic operations, measurement, and geometry, while the reading tests measure competency in areas such as vocabulary and reading comprehension. The raw EOG scores are subsequently categorized into four achievement levels: 1) insufficient mastery; 2) inconsistent mastery; 3) consistent mastery; and 4) superior performance (North Carolina Department of Public Instruction, 2007, 2008). Results of EOG tests have important implications for both individual schools and school districts, as they may affect state and federal funding levels.

Because scores can vary across geographic regions, there has been growing interest in examining regional differences in test scores, both at the national and state level. North Carolina, like many other states, is working to close the gap between low-performing schools and those meeting NCLB standards. Despite this goal, relatively few studies have examined geographic disparities in EOG performance in an effort to identify high- and low-performing schools and school districts. In fact, we found only one related study examining gender differences in test performance across large national Census divisions (Pope and Sydnor, 2010). Thus, there remains a need for a comprehensive study of varying test performance across a refined geographic scale. By pinpointing schools that fail to meet adequate yearly standards set forth by NCLB, state and local education officials can develop targeted interventions to improve school performance in the areas of most need. Directed efforts such as these provide new opportunities to close the achievement gap in EOG test scores.

With these goals in mind, we recently conducted a study to better understand factors influencing variation in EOG scores among elementary school children from across North Carolina. As a first step, we obtained math and reading test scores for fourth graders from all 100 countries in the state following completion of the 2008 school year, the most recent year for which such data were available. The data were then geo-referenced by residential address and subsequently linked at the county level to data from the 2005–2009 American Community Survey (U.S. Census Bureau, 2010). The aims of the study were to examine statewide variation in EOG test scores and to identify individual- and county-level predictors of EOG performance.

From an analytic perspective, the EOG data posed several unique challenges. First, because math and reading scores are highly correlated measures, we needed a flexible spatial model to examine individual- and county-level factors contributing to EOG performance, while taking into account within-subject and within-county associations. We also wanted a model that could yield accurate predictions of average student performance for each county and induce spatial smoothing of predicted scores, particularly for sparsely populated counties where predictions may be less reliable. And finally, as we describe in Section 2 below, we wanted a model that was robust to region-specific departures from normality in light of the skewness observed in the data. This paper describes a novel multivariate spatial mixture model specifically designed to address these multiple aims.

Our proposed model capitalizes on recent developments in spatial modeling of multivariate, areal-referenced data, i.e., data in which the spatial units consist of aggregated regions of space such as counties. Modeling of such data typically proceeds by introducing a set of region-specific random effects, which are then linked via a multivariate conditionally autoregressive (MCAR) prior distribution (Mardia, 1988). Previous applications of joint spatial models for areal data have focused on normal responses (Gelfand and Vounatsou, 2003), count responses for disease mapping (Carlin and Banerjee, 2002; Jin et al., 2005; Zhang et al., 2009; Congdon, 2010), and categorical responses (Gelfand and Vounatsou, 2003; Wall and Liu, 2009).

In many applications, including ours, the response variables are continuous but may not be normally distributed. In such cases, mixture models can provide a flexible framework for modeling the response distribution and can improve model fit. There is a well-established literature on mixture models for non-spatial data (Mclachlan and Peel, 2000; Frühwirth-Schnatter, 2006). In the spatial setting, several authors have proposed mixture models for point-referenced data — i.e., data indexed by a set of specific geographic coordinates. Gelfand et al. (2005) used a Dirichlet process mixture model to examine precipitation measurements at fixed locations in southern France. Kottas and Sansó (2007) extended the approach by allowing the point locations to be random. Ji et al. (2009) used a similar Poisson point-process mixture model to identify cell abundance patterns from fluorescent intensity images of lymphatic tissue. For multivariate point-referenced data, Reich and Fuentes (2007) proposed a semiparametric mixture model specified through a stick-breaking process. In the areal setting, Green and Richardson (2001) and Lawson and Clark (2002) proposed univariate mixture models for mapping disease relative risks. More recently, Wall and Liu (2009) developed a spatial latent class model for multivariate binary data and modeled the latent class indictors using a multinomial probit model with spatially correlated error terms.

We extend this work by developing a multivariate spatial finite-mixture model for continuous, areal-referenced data. We introduce random effects for each mixture component, as well as for the mixing weights, to allow the shape of the multivariate response distribution to vary in flexible ways across geographic regions and covariate profiles. As such, our model provides a practical approach to spatial density estimation. By integrating across this mixture density, one can obtain region-specific inferences and model-based predictions of interest. We adopt a Bayesian inferential approach, and for posterior computation develop an efficient Markov chain Monte Carlo (MCMC) algorithm that combines closed-form Gibbs and Metropolis steps.

The remainder of the paper is organized as follows: Section 2 describes the EOG testing data; Section 3 outlines the proposed model and discusses prior specification, posterior computation, and model selection; Section 4 presents results from a simulation study highlighting important features of the model; Section 5 applies the method to the EOG data; and the final section provides a discussion and directions for future work.

2. End-of-Grade Test Data

Table 1 provides a summary of the EOG test data. For the purposes of our analysis, we restricted the sample to non-Hispanic white and non-Hispanic black students due to small sample sizes in other race and ethnicity groups and to the impact of English as a second language on early school performance. Of the 78380 students, roughly half were male, about one third were non-Hispanic black, and just over 43% received free or reduced-price lunch at school through a federal subsidy program. The math scores ranged from 319 to 373 with a median of 352, and the reading scores ranged from 313 to 370 with a median of 346. Approximately three quarters of the students achieved consistent mastery or higher on the math exam, and nearly 63% achieved consistent mastery or better on reading.

Table 1.

Summary statistics for the EOG data (N = 78380).

Variable	Median (IQR)
Math Score	352 (345, 358)
Reading Score	346 (339, 353)
County Median Household Income ($)	44319 (39676, 51110)
County Sample Size	443 (203, 874)
	n (%)
Male	39555 (50.47)
Non-Hispanic Black Race†	25219 (32.18)
Enrolled in Free or Reduced-Price Lunch Program†	33959 (43.33)
Math Achievement Level
Insufficient Mastery (Score ≤ 335)	4470 (5.70)
Inconsistent Mastery (Score 336–344)	15040 (19.19)
Consistent Mastery (Score 345≤357)	37956 (48.43)
Superior Performance (Score ≥ 358)	20914 (26.68)
Reading Achievement Level
Insufficient Mastery (Score ≤ 334)	11439 (14.59)
Inconsistent Mastery (Score 335–342)	17618 (22.48)
Consistent Mastery (Score 343–353)	30879 (39.40)
Superior Performance (Score ≥ 354)	18444 (23.53)

^†Non-Hispanic black race and free lunch enrollment coded as binary (yes/no) variables.

Figure 1 presents a bivariate histogram of the raw math and reading scores (panel a), as well as a histogram of the standardized residuals based on an ordinary least squares (OLS) regression that included as predictors gender, race, enrollment in a free- or reduced-price lunch program, and county median household income (panel b). The distribution of the residuals is skewed toward lower values, particularly for reading, and the kurtoses in both directions are slightly negative. The univariate Kolmogorov-Smirnov tests on the residuals rejected the null hypothesis of normality (p < 0.01 for both outcomes), suggesting that the bivariate response distribution might be better modeled as a low-dimensional finite mixture of normals rather than as a single bivariate normal distribution.

Fig. 1.

Bivariate histogram of (a) the raw math and reading scores and (b) OLS standardized residuals for the 2008 fourth-grade EOG test scores.

There is also substantial variation in test scores across the state. Figure 2 shows the studentized OLS residuals averaged by county for both math and reading. The spatial pattern is similar for both math and reading, with negative residuals clustering in the interior northeast, along the eastern southern border, and in the western-most counties, while pockets of positive residuals appear in the center of the state and along the southern border in the west. This pattern suggests positive spatial autocorrelation in the residuals, violating the OLS assumption of independently distributed errors. This points to the need for a model that explicitly accounts for spatial dependence, since ignoring such spatial structure could lead to biased inferences and inaccurate assessments of parameter uncertainty.

Fig. 2.

Quintiles of county-averaged (a) math and (b) reading studentized residuals for the 2008 fourth-grade EOG scores.

3. Spatial Mixture Model

3.1. Model Specification

To develop the multivariate spatial mixture model, we focus on the bivariate case. It is conceptually straight-forward to extend the approach to three or more outcomes, though computation will become more challenging.

A very general specification of the bivariate spatial mixture model can be expressed as:

$${\mathit{y}}_{ij}|{\mathit{\varphi }}_i,{\mathit{\psi }}_i~\sum _{k=1}^K{\pi }_{ijk}{N}_2({\mathit{\eta }}_{ijk},{\mathit{\Sigma }}_k){\mathit{\eta }}_{ijk}=(\begin{array}{c}\hfill {\eta }_{1ijk}\hfill \\ \hfill {\eta }_{2ijk}\hfill \end{array})={\mathit{X}}_{ij}{\mathit{\beta }}_k+{\mathit{V}}_i{\mathit{\alpha }}_k+{\mathit{\varphi }}_{ik}{\pi }_{ijk}=\frac{e^{{\mathit{x}}_{ij}^{\prime }{\mathit{\gamma }}_k+{\mathit{\upsilon }}_i^{\prime }{\mathit{\delta }}_k+{\psi }_{ik}}}{{\sum }_{h=1}^K{e}^{{\mathit{x}}_{ij}^{\prime }{\mathit{\gamma }}_h+{\mathit{\upsilon }}_i^{\prime }{\mathit{\delta }}_h+{\psi }_{ih}}},i=1,\dots ,n;j=1,\dots ,n_i;k=1,\dots K,$$ (1)

where y_ij = (y_1ij, y_2ij)′ denotes a 2×1 vector of math and reading scores for the j-th student in the i-th county; ${\mathit{X}}_{ij}=\left(\begin{array}{cc}\hfill {\mathit{x}}_{ij}^{\prime }\hfill & \hfill \mathbf{0}\hfill \\ \hfill \mathbf{0}\hfill & \hfill {\mathit{x}}_{ij}^{\prime }\hfill \end{array}\right)$ is a 2×2p matrix of subject-level covariates with corresponding 2p × 1 component-specific fixed effects ${\mathit{\beta }}_k=({\mathit{\beta }}_{1k}^{\prime },{\mathit{\beta }}_{2k}^{\prime })\prime ;{\mathit{V}}_i=\left(\begin{array}{cc}\hfill {\mathit{\upsilon }}_i^{\prime }\hfill & \hfill \mathbf{0}\hfill \\ \hfill \mathbf{0}\hfill & \hfill {\mathit{\upsilon }}_i^{\prime }\hfill \end{array}\right)$ is a 2×2r matrix of county-level covariates with corresponding 2r×1 component-specific fixed effects ${\mathit{\alpha }}_k=({\mathit{\alpha }}_{1k}^{\prime },{\mathit{\alpha }}_{2k}^{\prime })\prime ;{\mathit{\varphi }}_{ik}=({\varphi }_{1ik},{\varphi }_{2ik})\prime$ is a 2×1 vector of component-specific spatial random effects for the i-th county, with ${\mathit{\varphi }}_{\mathit{i}}=({\mathit{\varphi }}_{\mathit{i}1}^{\prime },\dots ,{\mathit{\varphi }}_{\mathit{i}\mathit{K}}^{\prime })\prime ;{\mathit{\Sigma }}_k$ is the component-k 2 × 2 variance-covariance matrix of y_ij, conditional on ϕ_ik; γ_k and δ_k are p × 1 and r × 1 vectors of mixing-weight regression parameters with γ₁ ≡ 0 and δ₁ ≡ 0 for identifiability; and ψ_ik is a spatial random effect for county i and mixing weight k, where ψ_i1 ≡ 0 and ψ_i = (ψ_i1, …, ψ_iK)′. Throughout, we assume the same set of covariates for the component means and the mixing weights, although in general this restriction is not necessary.

Model (1) is appealing because it allows the shape of the joint response distribution to change flexibly across spatial units and covariate levels. In particular, the within-component linear predictors (the η_ijk’s) permit the locations of the mixture components to vary throughout the population, while the mixing-weight parameters allow the mass of the response distribution to shift in unique ways between individuals and counties. Together, these features produce distinct response distributions for each covariate profile and areal unit, and therefore provide a general framework for multivariate spatial analysis of continuous-valued outcomes.

3.2. Prior Distributions

For parameter estimation, we adopt a fully Bayesian approach, assuming prior distributions for all model parameters. First, to allow for spatial smoothing and borrowing of information across counties, for each k, we assign component-specific conditionally autoregressive (CAR) priors (Besag, 1974; Besag et al., 1991) to the spatial random effects — a bivariate CAR prior for ϕ_ik and a univariate CAR prior for ψ_ik:

$${\mathit{\varphi }}_{ik}|{\mathit{\varphi }}_{(-ik)},{\mathbf{\Lambda }}_{ik}~{\mathbf{N}}_2({\xi }_k\sum _{l\in {\partial }_i}\frac{{\omega }_{il}}{{\omega }_{i+}}{\mathit{\varphi }}_{lk},{\mathbf{\Lambda }}_{ik}),k=1,\dots ,K$$ (2)

$${\psi }_{ik}|{\psi }_{(-ik)},{\tau }_{ik}^2~\mathrm{N}({\zeta }_k\sum _{l\in {\partial }_i}\frac{{\omega }_{il}}{{\omega }_{i+}}{\psi }_{lk},{\tau }_{ik}^2),k=2,\dots ,K,$$ (3)

where ∂_i denotes the set of neighbors for county i, ξ_k and ζ_k are spatial smoothing parameters, ω_il is an unnormalized proximity measure, ω_i+ = = Σ_l∈∂i ω_il, Λ_ik = Λ_k/w_i+ is a component-specific scaled variance-covariance matrix for ϕ_ik conditional on ϕ_(−ik), and ${\tau }_{ik}^2={\tau }_k^2/{\omega }_{i+}$ is a component-specific scaled variance parameter for ψ_ik. For the EOG study, we adopt intrinsic CAR (ICAR) priors to provide maximal smoothing of sparsely populated regions:

$${\mathit{\varphi }}_{ik}|{\mathit{\varphi }}_{(-ik)},{\mathbf{\Lambda }}_k~{\mathrm{N}}_2(\frac{1}{m_i}\sum _{l\in {\partial }_i}{\mathit{\varphi }}_{lk},\frac{1}{m_i}{\mathbf{\Lambda }}_k)$$ (4)

$${\psi }_{ik}|{\psi }_{(-ik)},{\tau }_k^2~\mathrm{N}(\frac{1}{m_i}\sum _{l\in {\partial }_i}{\psi }_{lk},{\tau }_k^2/m_i),$$ (5)

where m_i denotes the number of neighbors sharing a geographic border with county i. Following Brook’s Lemma (c.f., Banerjee et al., 2004), priors (4) and (5) give rise to improper joint distributions for ϕ_k and ψ_k:

$${\mathit{\varphi }}_k|{\mathbf{\Lambda }}_k\propto \text{exp}(-\frac{1}{2}{\mathit{\varphi }}_k^{\prime }[(\mathit{M}-\mathit{A})\otimes {\mathbf{\Lambda }}_k^{-1}]{\mathit{\varphi }}_k)$$ (6)

$${\mathit{\psi }}_k|{\tau }_k^2\propto \text{exp}(-\frac{1}{2{\tau }_k^2}{\mathit{\psi }}_k^{\prime }(\mathit{M}-\mathit{A}){\mathit{\psi }}_k),$$ (7)

where ${\mathit{\varphi }}_k=({\mathit{\varphi }}_{1k}^{\prime },\dots ,{\mathit{\varphi }}_{nk}^{\prime })\prime ,{\mathit{\psi }}_k=({\psi }_{1k},\dots ,{\psi }_{nk})\prime ,\mathit{M}=\text{diag}(m_1,\dots ,m_n)$, and A is an n × n adjacency matrix with a_ii = 0, a_il = 1 if counties i and l are neighbors, and a_il = 0 otherwise. Because (M − A) is singular, the joint distributions in (6) and (7) are over-parameterized and thus improper, although the conditional prior distributions given by equations (4) and (5) are themselves proper. Propriety of the posterior, when a fixed effect intercept is included in the model, is achieved using a sum-to-zero constraint on the spatial random effects (Banerjee et al., 2004).

To ensure a well-identified model, we assign weakly informative proper priors to the remaining model parameters. For the within-component fixed effects, we assume exchangeable normal priors: β_1k and β_2k ~ N_p(μ_β, Σ_β), α_1k and α_2k ~ N_r(μ_α, Σ_α) for k = 1, …, K. For the fixed effects within the mixing weights, γ_k and δ_k, we assign N_p(μ_γ, Σ_γ) and N_r (μ_δ, Σ_δ) priors, respectively, for k = 2, …, K. Throughout, we assume that the prior hyperparameters (μ_β, Σ_β, etc.) are identical across components, but in general this is not required. To complete the prior specification, we assign conjugate inverse-Wishart IW(κ₀, S₀) and IW(ν₀, D₀) priors respectively to Σk and Λk, and a conjugate inverse-gamma IG(g, s) prior to ${\tau }_k^2(k=2,\dots ,K)$.

Note that the proposed model accommodates a wide range of dependence structures. First, Σ_12k, the off-diagonal element of Σ_k, controls the component-specific within-subject association between outcomes. In the EOG study, for example, a positive value for Σ_12k implies that, for component k, students with high math scores also tend to have high reading scores conditional on the county-level random effects. Similarly, Λ_12k, the off-diagonal element of Λk, accounts for the component-specific, between-subject/within-region association between outcomes. In the EOG study, Λ_12k > 0 implies that, for component k, counties with higher mean math scores tend to have higher mean reading scores, adjusting for observed covariates. And finally, the CAR priors on ϕ_ik and ψ_ik capture associations between counties, implying that adjoining counties behave similarly with respect to their response distributions. Numerous submodels can be obtained by setting one or more of these association parameters to zero. For example, setting Λ_12k = 0 ∀k implies no between-subject/within-county association in responses. This is tantamount to assigning separate univariate CAR priors to ϕ_1ik and ϕ_2ik. Further restricting Σ_12k to 0 for all k would imply that there is no within-subject association between responses, and hence the outcomes are uncorrelated at all levels of the model.

3.3. Posterior Computation and Model Comparison

Posterior inference proceeds via data augmentation by introducing a discrete latent labeling variable, C_ij, that takes the value k (k = 1, …, K) with probability π_ijk defined in equation (1). Letting θ_k = {β_k,α_k,ϕ_k,Σ_k,Λ_k} denote the within-component parameters and ${\mathit{\upsilon }}_k=\{{\mathit{\gamma }}_k,{\mathit{\delta }}_k,{\mathit{\psi }}_k,{\mathit{\tau }}_k^2\}$ denote the mixing-weight parameters, the joint posterior is given by:

$$\pi ({\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_K,{\mathit{\upsilon }}_2,\dots ,{\mathit{\upsilon }}_K|\mathit{y})\propto \prod _{k=1}^K\{\prod _{i=1}^n\prod _{j=1}^{n_i}{[{\pi }_{ijk}{\mathrm{N}}_2({\mathit{y}}_{ij};{\mathit{\eta }}_{ijk},{\mathbf{\Sigma }}_k)]}^{{\mathrm{I}}_{(C_{ij}=k)}}\times \text{exp}(-\frac{1}{2}{\mathit{\varphi }}_k^{\prime }[(\mathit{M}-\mathit{A})\otimes {\mathbf{\Lambda }}_k^{-1}]{\mathit{\varphi }}_k)\pi ({\mathit{\beta }}_k)\pi ({\mathit{\alpha }}_k)\pi ({\mathit{\Sigma }}_k)\pi ({\mathbf{\Lambda }}_k)\}\times \prod _{h=2}^K\text{exp}(-\frac{1}{2{\tau }_h^2}{\psi }_h^{\prime }(\mathit{M}-\mathit{A}){\Psi }_h)\pi ({\mathit{\gamma }}_h)\pi ({\mathit{\delta }}_h)\pi ({\tau }_h^2)$$ (8)

where I(·) denotes the indicator function and the π(·)’s represent the prior distributions for their respective parameters, as described in the previous section.

For posterior computation, we propose an MCMC algorithm that combines draws from full conditionals with Metropolis-based updates. After assigning initial values to the model parameters, the algorithm iterates between the following steps:

1. For k = 2, …, K, update γ_k and δ_k using random-walk Metropolis steps;

2. For k = 2, …, K and i = 1, …, n, update ψ_ik using random-walk Metropolis;

3. For k = 2, …, K, update ${\tau }_k^2$ from its closed-form full conditional distribution;

4. For all (i, j), sample the mixture component indicators from a discrete distribution taking values {k =1, …, K} with posterior probabilities {p_ij1, …, p_ijK) as described in the Appendix;

5. For k = 1, …, K, sample the component-specific parameters β_1k, β_2k, α_1k, α_2k, Σ_k, Λ_k, and ϕ_ik (i = 1, …, n) from their closed-form full conditionals

6. At the end of each MCMC iteration, apply a sum-to-zero constraint to ϕ_k (k = 1, …, K) and ψ_k (k = 2, …, K).

Explicit details of the algorithm are provided in the Appendix. An appealing feature of the MCMC algorithm is that the within-component regression parameters and spatial effects have convenient closed-form full conditionals, leading to straightforward and efficient posterior sampling. Only the mixing-weight parameters require Metropolis-based updates.

Convergence is monitored by running multiple chains from dispersed initial values and performing standard Bayesian diagnostics, such as trace plots and evaluation of the Brooks-Gelman-Rubin statistic (Gelman et al., 2004). Careful attention to such diagnostics is especially important for complex latent variable models to ensure parameter identifiability. In our experience, the proposed MCMC algorithm is generally robust to choices of initial values, with the possible exception of the mixing parameters γ and δ, which can be slow to converge for poorly chosen starting values. One way to choose initial values for these parameters is to perform a K-level cluster analysis, fit a multinomial logit regression to the resulting cluster indicators, and use the ensuing parameter estimates as starting values.

A well-known computational challenge for Bayesian finite mixture models is “label switching” in which draws of component-specific parameters may be associated with different components labels during the course of the MCMC run. Consequently, component-specific posterior summaries that average across the draws will be invalid. As a solution, Stephens (2000) proposed a post-hoc relabeling algorithm based on a Kullback-Leibler divergence function. We apply this approach for the analysis of the EOG data described in Section 5.

For model comparison, we adopt the deviance information criterion (DIC) proposed by Spiegelhalter et al. (2002). DIC includes a goodness of fit term and also penalizes for model complexity. Models with smaller DIC are considered preferable. For the EOG application below, we apply a modified DIC recently recommended by Celeux et al. (2006) specifically for finite mixture models. This modified DIC, termed DIC3, uses the posterior predictive density of y to estimate the penalty term, and is closely related to a measure put forward by Richardson (2002) to avoid overfitting the number of mixture components.

4. Simulation Study

To test the performance of our model, we conducted a simulation study using 200 datasets generated from a basic two-component mixture model without covariates. Our aims were to evaluate MCMC performance, ensure that we obtained reasonable parameter estimates under the true model, and highlight the model as a practical approach to spatial density estimation whereby the shape of the response distribution is allowed to vary flexibly across spatial units. To emulate the EOG data, we used the North Carolina county-level adjacency matrix for the simulation. This matrix contains 512 adjacencies among the 100 North Carolina counties. For the purposes of the simulation, counties were labeled “1” to “100”. Because the (M − A) matrix in equations (6) and (7) is singular, the spatial random effects cannot be simulated directly. We therefore introduced the spatial smoothing parameters, ξ and ζ, defined in equations (2) and (3), and set them equal to 1−1E-6. We then generated spatial random effects according to joint distributions (6) and (7) augmented with the smoothing parameters. Next, for each county, we simulated 80 math and reading scores from the following two-component mixture model:

$${\mathit{y}}_{ij}|{\mathit{\varphi }}_i,{\mathit{\psi }}_i~\sum _{k=1}^2{\pi }_{ijk}{\mathrm{N}}_2[\left(\begin{array}{c}{\beta }_{10k}+{\varphi }_{1ik}\hfill \\ {\beta }_{20k}+{\varphi }_{2ik}\hfill \end{array}\right),{\mathbf{\Sigma }}_k]\text{logit}({\pi }_{ij2})={\gamma }_0+{\psi }_i,i=1,\dots ,100;j=1,\dots ,80,$$ (9)

where π_ij2 denotes the weight for the second mixture component. For k = 1, 2 we assigned independent N(0, 1000) priors to β_10k and β_20k, and IW(3, I₂) priors to Σ_k and Λ_k; for γ₀, we assigned a N(0, 1000) prior; for ϕ_ik = (ϕ_1ik, ϕ_2ik)′ and ψ_i, we assigned bivariate and univariate intrinsic CAR priors, respectively; and for τ², the conditional variance of ψ_i, we assigned an IG(0.01, 0.01) prior.

For each simulation, we ran 2000 MCMC iterations in R version 2.14 (R Development Core Team, 2011) using a burn-in of 1000, which was sufficient to ensure convergence based on standard diagnostics. To avoid label switching, we simulated extremely well-separated mixture components, effectively imposing the order constraints β₁₀₁ < β₁₀₂ and β₂₀₁ < β₂₀₂.

Table 2 presents the posterior means, averaged across the 200 simulations, along with 95% coverage rates and the true parameter values used to generate the data. The posterior estimates showed minimal bias, and the coverage rates were near the nominal values for all parameters. Figure 3 displays the bivariate densities from a randomly selected simulation study. Panel (a) shows the true (i.e., simulated) density for an “average” county in which the random effects were set to 0. Panel (b) displays the corresponding model-estimated density, and panels (c) and (d) show the estimated densities for two randomly selected counties. The white circles denote the true component means: β₀₁ = (340, 330)′ and β₀₂ = (360, 345)′. As expected, the estimated average density in panel (b) closely mirrors the true density. Panel (c) shows a county in which the mixture components diverge and there is a shift in mass toward the lower component. In panel (d), the component locations shift toward higher math and reading values and the mass is concentrated on the upper component.

Table 2.

Average posterior estimates and 95% coverage probabilities across 200 simulated datasets.

Mixture			Average Posterior
Component	Parameter	Description	True Value	Mean	95% Coverage
1	β101	Math Intercept	340	340.01	0.93
	β201	Reading Intercept	330	330.02	0.95
	Σ111	Var(y1ij|ϕ1i1)	20	20.10	0.96
	Σ121	Cov(y1ij, y2ij|ϕi1)	10	10.17	0.94
	Σ221	Var(y2ij|ϕ2i1)	36	36.34	0.95
	Λ111	Var(ϕ1i1)	9	8.77	0.96
	Λ121	Cov(ϕ1i1, ϕ2i1)	3	2.80	0.95
	Λ221	Var(ϕ2i1)	4	3.49	0.91
2	β102	Math Intercept	360	360.01	0.97
	β202	Reading Intercept	345	345.01	0.96
	Σ112	Var(y1ij|ϕ1i2)	50	50.04	0.96
	Σ122	Cov(y1ij, y2ij|ϕi2)	20	19.99	0.93
	Σ222	Var(y2ij|ϕ2i2)	40	40.10	0.96
	Λ112	Var(ϕ1i2)	4	3.53	0.93
	Λ122	Cov(ϕ1i2, ϕ2i2)	6	5.73	0.92
	Λ222	Var(ϕ2i2)	16	15.74	0.93
Mixing Weight	γ0	Mixing Weight Intercept	0.75	0.75	0.96
Parameters	τ2	Var(ψi)	1	1.00	0.94

Fig. 3.

True and estimated posterior densities from a randomly selected simulation study. Panel (a) shows the true simulated density for an “average” county (random effects set equal to zero); panel (b) depicts the corresponding model-estimated density; panel (c) presents the estimated density for a county (County 31) in which the components split apart; and panel (d) shows the estimated density for a county (County 100) in which the component locations shift toward higher values and more mass is concentrated on the upper component. White circles represent the true component means: β₀₁ = (340, 330) and β₀₂ = (360, 345)′.

5. Analysis of the EOG Test Data

Next, we fit the following two-component spatial mixture model to the EOG data:

$${\mathit{y}}_{ij}|{\mathit{\varphi }}_i,{\mathit{\psi }}_i~\sum _{k=1}^2{\pi }_{ijk}{\mathrm{N}}_2[\left(\begin{array}{c}{\eta }_{1ijk}\hfill \\ {\eta }_{2ijk}\hfill \end{array}\right),{\mathbf{\Sigma }}_k]{\eta }_{1ijk}={\beta }_{10k}+{\beta }_{11k}\times {\text{Male}}_{ij}+{\beta }_{12k}\times {\text{NHB}}_{ij}+{\beta }_{13k}\times {\text{Freelunch}}_{ij}+{\alpha }_{11k}\times {\text{Medinc}}_i+{\varphi }_{1ik}{\eta }_{2ijk}={\beta }_{20k}+{\beta }_{21k}\times {\text{Male}}_{ij}+{\beta }_{22k}\times {\text{NHB}}_{ij}+{\beta }_{23k}\times {\text{Freelunch}}_{ij}+{\alpha }_{21k}\times {\text{Medinc}}_i+{\varphi }_{2ik}\text{logit}({\pi }_{ij2})={\mathrm{\gamma }}_0+{\mathrm{\gamma }}_1\times {\text{Male}}_{ij}+{\mathrm{\gamma }}_2\times {\text{NHB}}_{ij}+{\mathrm{\gamma }}_3\times {\text{Freelunch}}_{ij}+{\delta }_1\times {\text{Medinc}}_i+{\psi }_i,i=1,\dots ,100;j=1,\dots ,n_i;k=1,2,$$ (10)

where y_ij is a vector of math and reading scores for the j-th subject in the i-th county, Male is a dichotomous indicator of male gender, NHB is a dichotomous indicator taking the value 1 if the student is non-Hispanic black and 0 if non-Hispanic white, Freelunch is a dichotomous indicator taking the value 1 if the student participated in a free or reduced-price lunch program and 0 otherwise, Medinc denotes county median household income (in $1000s), and π_ij2 denotes the weight for the second mixture component.

As in the simulation study, we assigned a bivariate CAR prior to ϕ_ik and a univariate CAR prior to ψ_i. We assumed weakly informative proper priors for all other model parameters: for β_1k = (β_10k, β_11k, β_12k, β_13k)′, β_2k = (β_20k, β_21k, β_22k, β_23k)′ and γ = (γ₀, γ₁, γ₂, γ₃)′, we assigned conjugate N₄ (0, 1000I₄) priors; for α_11k, α_21k and δ₁, we assigned N(0, 1000) priors; for Σ_k and Λ_k, we assigned IW(3, I₂) priors; and for τ² we assigned an IG(0.01, 0.01) prior. We ran two initially dispersed chains for 20000 iterations each, discarding the first 10000 as burn-in. To reduce autocorrelation, we retained every tenth iteration.

Model diagnostics indicated efficient mixing and rapid convergence of the chains. Figure (4) presents the post–burn-in trace plots for four selected model parameters: β₁₁₁, the component-1 math coefficient for male gender; γ₃, the mixing weight coefficient for Freelunch; τ², the variance of ψ_i; and Λ₂₂₂, the variance of ϕ_2i2. The chains overlapped substantially, with no evidence of label switching within individual chains, and hence Stephens’s (2000) relabeling algorithm converged rapidly. The component labels did require reordering across chains, but the proper labeling was easily identified so that the chains could be combined for summary purposes. For each of the four parameters, the Brooks-Gelman-Rubin upper credible interval was ≤ 1.21, indicating adequate convergence of the chains.

Fig. 4.

Post–burn-in MCMC trace plots for four parameters from the proposed model: (a) β₁₁₁, the component-1 math coefficient for male gender; (b) γ₃, the mixing weight coefficient for Freelunch; (c) τ², the variance of ψ_i; and (d) Λ₂₂₂, the variance of ϕ_2i2. Horizontal lines denote the posterior means from the combined chains. BGR = Brooks-Gelman-Rubin upper credible interval.

To evaluate the performance of our model, we compared our proposed model to three submodels:

1. A one-component version of the proposed spatial mixture model with a bivariate CAR prior for ϕ_i;

2. A two-component fixed-effects model excluding ϕ_ik and ψ_i; and

3. A two-component model with component-specific ϕ_ik but no ψ_i.

Table 3 provides the model comparison results for the various models. The two-component random effects models substantially outperformed submodels 1 and 2. Overall, the full model had the best performance, suggesting that incorporating the random e.ects ψi into the mixing weights provided a modest additional benefit relative to submodel 3.

Table 3.

Model comparison statistics for analysis of the EOG data.

Model Description	D̅	pD	DIC3	Δ
One-Component Model	1068553	174	1068727	—
Two-Component Fixed Effects Model	1067255	28	1067283	1444
Two-Component Model Excluding ψi	1065144	295	1065439	1844
Proposed Model	1064990	328	1065318	121

Table 4 presents the posterior means and 95% credible intervals (CrIs) for the proposed model parameters. The results suggest that there are two distinct mixing components, or “latent subpopulations” of students. Subpopulation 1 contained an estimated 58% of the overall population, and was characterized by comparatively low mean math and reading scores for the reference group (β₁₀₁ = 349.25 and β₂₀₁ = 344.50). In addition, NHB race and free/reduced lunch enrollment were associated with lower math and reading scores. Higher county median income was associated with a slight increase in reading scores (α₂₁₁ = 0.05; 95% CrI=[0.00, 0.09]). Subpopulation 2 contained the remaining 42% of the overall population and was associated with roughly a 10-point increase in scores for the reference group compared to subpopulation 1 (β₁₀₂ = 358.04 and β₂₀₂ = 354.15). In both groups, males had lower adjusted reading scores than females (e.g., β₂₁₁ = −1.74; 95% CrI=[-1.98, -1.49]); however, in subpopulation 2, males had higher adjusted math scores (β₁₁₂ = 0.91; 95% CrI=[0.71, 1.13]). These findings are consistent with previous research on gender disparities in standardized test scores, which has shown that boys tend to score lower than girls in reading but modestly higher in math and science, particularly in the upper tails of the test-score distribution (Pope and Sydnor, 2010). In terms of variability, subpopulation 1 displayed more individual-level and county-level heterogeneity than subpopulation 2. Not surprisingly, the within-subject correlations between math and reading (captured by ρ₁ and ρ₂) were moderately high for both subpopulations.

Table 4.

Posterior means and 95% credible intervals (CrIs) for the proposed model.

Mixture Component (%)	Parameter	Description	Posterior Mean	95% CrI
1 (58%)	β101	Math Intercept	349.25	(348.67, 349.77)
	β111	Male	−0.19	(−0.41, 0.05)
	β121	NHB Race	−3.12	(−3.45, −2.77)
	β131	Free/Reduced Lunch	−2.50	(−2.83, −2.18)
	α111	Median HH Income ($1000s)	0.03	(−0.01, 0.08)
	β201	Reading Intercept	344.50	(343.84, 345.07)
	β211	Male	−1.74	(−1.98, −1.49)
	β221	NHB Race	−2.95	(−3.31, −2.56)
	β231	Free/Reduced Lunch	−2.98	(−3.32, −2.62)
	α211	Median HH Income ($1000s)	0.05	(0.00, 0.09)
	Σ111	Var(y1ij|ϕ1i1)	57.58	(55.77, 59.34)
	Σ121	Cov(y1ij, y2ij|ϕ1i1)	35.26	(33.11, 37.19)
	Σ221	Var(y2ij|ϕ2i1)	67.56	(65.37, 69.63)
	ρ1	Corr(y1ij, y2ij|ϕ1i1)	0.57	(0.55, 0.58)
	Λ111	Var(ϕ1i1)	7.95	(5.36, 11.31)
	Λ121	Cov(ϕ1i1, ϕ2i1)	4.58	(2.58, 7.14)
	Λ221	Var(ϕ2iϕ2i1)	5.04	(3.09, 7.61)
2 (42%)	β102	Math Intercept	358.04	(357.69, 358.41)
	β112	Male	0.91	(0.71, 1.13)
	β122	NHB Race	−3.35	(−3.77, −2.96)
	β132	Free/Reduced Lunch	−2.71	(−3.03, −2.37)
	α112	Median HH Income ($1000s)	0.02	(−0.02, 0.05)
	β202	Reading Intercept	354.15	(353.77, 354.52)
	β212	Male	−0.37	(−0.60, −0.14)
	β222	NHB Race	−3.55	(−3.96, −3.09)
	β232	Free/Reduced Lunch	−2.73	(−3.07, −2.37)
	α212	Median HH Income ($1000s)	0.05	(0.02, 0.08)
	Σ112	Var(y1ij, |ϕ1i2)	36.08	(34.63, 37.53)
	Σ122	Cov(y1ij, y2ij|ϕi2)	21.76	(21.61, 22.93)
	Σ222	Var(y2ij|ϕ2i2)	40.52	(38.95, 42.17)
	ρ2	Corr(y1ij, y2ij|ϕi2)	0.57	(0.56, 0.58)
	Λ112	Var(ϕ1i2)	3.35	(2.00, 5.20)
	Λ122	Cov(ϕ1i2, ϕ2i2)	1.65	(0.71, 2.84)
	Λ222	Var(ϕ2i2)	1.69	(0.91, 2.84)
Mixing Weight	γ0	Mixing Weight Intercept	0.15	(−0.06, 0.37)
Parameters	γ1	Male	−0.04	(−0.13, 0.04)
	γ2	NHB Race	−1.26	(−1.39, −1.11)
	γ3	Free/Reduced Lunch	−0.89	(−1.01, −0.77)
	δ1	Median Household Income	0.02	(0.01, 0.04)
	τ2	Var(ψi)	0.43	(0.25, 0.69)

Figure 5 presents the predicted math and reading scores by county for reference group individuals (i.e., non-Hispanic white females not enrolled in a free- or reduced-price lunch program). Here, the between-county variation reflects both observed differences in county median household income as well as the latent heterogeneity captured by the spatial random effects. The spatial patterns of the predicted scores were similar to those for the OLS residuals in Figure 2, but with increased smoothing, particularly among the central “Piedmont” and southwest counties. This feature is expected, since the CAR priors act as spatial smoothers. In general, the northeastern and central Piedmont counties had higher predicted scores than those in the interior northeast, south central, and southwestern portions of the state. The predicted math scores for the reference group ranged from 352 to 356 across the state (panel a). Outlined on the map are Union County (along the southern border), which had the highest predicted math score at 356.4 [95% CrI=[355.5, 357.5]), and Avery County (in the northwest), which had the lowest predicted score at 352.1 [95% CrI=[350.2, 354.4]). There was also modest variation in the predicted reading scores across the state (panel b). The predicted reading scores ranged from 348 to 353, and showed a spatial pattern similar to the predicted math scores. Camden County, located in the northeastern corner of the state, had the highest predicted reading score (352.6, 95% CrI=[351.0, 354.3]), and Graham County in the southwest had the lowest predicted score (347.6, 95% CrI=[345.2, 349.7]).

Fig. 5.

Predicted (a) math and (b) reading scores for reference group, by county.

Figure 6 displays four model-estimated bivariate densities for the reference cohort. Panel (a) presents the density for an “average” county with household income set at the statewide median value and random effects fixed at 0. Panels (b)-(d) display the densities for three selected counties: Camden County, which had the highest predicted reading score and a top 10% math score; Bertie County, which was in the bottom 5% for both math and reading; and Orange County, which had top 5% predicted math and reading scores. The county-specific densities vary in both their location and distribution of mass relative to the average density presented in panel (a). In particular, Camden County shifted toward higher math and reading scores, reflecting the more favorable outcomes for this county, while Bertie county had a noticeable shift in mass toward the lower mixture component. Orange County, like Camden County, showed a shift in mass toward more favorable outcomes, but with a longer-tailed distribution, reflecting more heterogeneity for this county.

Fig. 6.

Estimated county-specific densities for the reference group. Panel (a) presents the estimated density for an “average” county, and panels (b)-(d) show the estimated densities for three selected counties.

By integrating across these bivariate densities, we can obtain county-specific predictions of interest. For example, to predict an individual’s joint probability of inconsistent or insufficient mastery in both the math and reading, defined as a math score < 344 and a reading score < 342 (Table 1), we evaluated the following integral:

$$(1-{\pi }_{ij2}){\int }_{-\infty }^{344}{\int }_{-\infty }^{342}{\mathrm{N}}_2({\mathit{\eta }}_{ij1},{\mathbf{\Sigma }}_1)d{y}_{2ij}d{y}_{1ij}+{\pi }_{ij2}{\int }_{-\infty }^{344}{\int }_{-\infty }^{342}{\mathrm{N}}_2({\mathit{\eta }}_{ij2},{\mathbf{\Sigma }}_2)d{y}_{2ij}d{y}_{1ij}.$$

A similar approach was used to compute additional joint probabilities of interest. Using the cut-off values defined in Table 1, we calculated four joint probabilities of policy interest: (a) the probability of inconsistent or insufficient mastery in both math and reading (labeled “low math/low reading”); (b) the probability of inconsistent or insufficient math, but consistent or superior reading (“low math/high reading”); (c) the probability of consistent or superior math, but inconsistent or insufficient reading (“high math/low reading”); and (d) the probability of consistent or superior performance on both exams (“high math/high reading”). Figure 7 displays the county-specific averages for these four joint probabilities for the reference group.

Fig. 7.

Predicted joint probabilities of (a) low math, low reading; (b) low math, high reading; (c) high math, low reading; and (d) high math, high reading for the reference group.

In general, the central Piedmont and northeastern coastal counties had more favorable outcomes than counties located in the interior northeast, along the southern border in the east, and in the north- and southwest corners of the state. The probability of low math/low reading achievement ranged from 0.04 to 0.13 (panel a), with Camden having the lowest (0.04, 95% CrI = [0.02, 0.06]) and Bertie County having the highest probability (0.13, 95% CrI=[0.09, 0.17]). These results are consistent with the predicted densities shown in Figures 6 for these counties. As panel (b) indicates, the combination of low math and high reading was a relatively rare event, with average probabilities ranging from 0.02 to 0.07 across the state. This result is not surprising, since students typically perform much better on math than on reading. One notable exception was Avery County along the central western border, which had the lowest predicted math score (c.f., Figure 5), but a reading score close to the state average at 349.40 (95% CrI=[347.67, 351.46]). High math/low reading performance was a more common phenomenon (panel c), ranging from 0.06 to 0.16 statewide, with Graham County having the highest probability (0.16, 95% CrI=[0.11, 0.22]). As panel (d) suggests, most students in the reference group demonstrated consistent or superior achievement on both exams. The county averages ranged from 0.69 in Bertie County (95% CrI=[0.65, 0.82]) to 0.87 in Camden County (95% CrI=[0.81, 0.93]), again supporting the results shown in Figures 6(b) and 6(c).

6. Discussion

We have proposed a multivariate spatial mixture model for the joint analysis of continuous, areal-referenced data. The model incorporates individual- and region-level predictors, accommodates complex dependence structures, enables investigators to examine covariate effects across subgroups of the population, and supports region-specific departures from normality through the inclusion of spatial random effects for both the location parameters and the mixing weights. By integrating across this multivariate density, one can obtain region-specific joint, marginal, and conditional inferences of interest. In the EOG study, for example, we were able to compute county-specific joint probabilities of low math/low reading performance, low math, high reading performance, etc. We specified the model within a Bayesian framework and, for posterior computation, we developed a tractable MCMC algorithm that relies in large part on easy-to-sample Gibbs steps.

Our analysis of the EOG test data is among the first to use advanced multivariate spatial modeling to examine geographic patterns in EOG scores at a refined geographic level. Through our analysis, we found that non-Hispanic black race and enrollment in subsidized lunch programs were associated with lower test scores. We also found gender gaps in EOG performance, with girls performing substantially better in reading, particularly among students with low EOG scores, and boys scoring slightly higher in math, especially among “high-achieving” students. These results are consistent with previous research on gender disparities in standardized test scores (Pope and Sydnor, 2010), which has shown that boys tend to score lower than girls on standardized reading tests, but generally perform better in math and science, particularly in the upper tails of the response distribution (i.e., among high-performing students). Together, these findings suggest the need to target two sets of gender disparities: first, in reading among students with low EOG performance and second, in math among those with high EOG scores. Future work might also examine gender-by-county interactions vis-à-vis a spatial random-coefficients model, to determine if these disparities vary by region, a finding documented at the national level in prior work (Pope and Sydnor, 2010).

Our analysis also revealed similar spatial patterns for math and reading scores, with the central Piedmont and northern coastal counties displaying higher scores on average than counties in the interior northeast, south central, and western-most portions of the state. These findings could be used to guide county-level initiatives to improve elementary education. For example, by identifying counties with comparatively low reading scores, local school boards could introduce school-based literacy programs to improve vocabulary and reading comprehension. This is especially relevant in counties such as Graham County, which had the highest rate of high math/low reading performance.

Our modeling approach should have broad applicability to other research settings. For example, the spatial mixture model could be applied in reproductive epidemiology to explore joint spatial patterns in birth outcomes and to obtain region-specific joint probabilities of low birth weight and preterm birth. The methods could also be applied in health economics to model geographic variation in medical costs and expenditures.

In our application, we considered one- and two-component mixture models, but higher-dimensional mixtures can be envisioned. For example, higher-dimensional versions of this model could look jointly at the multiple sub-domains of math and reading. Extensions to more than two outcomes are also straightforward. Future work might consider modeling the multivariate distribution nonparametrically via Dirichlet processes, as proposed by Kottas et al. (2008). In short, the multivariate mixture model described here should prove useful for the joint analysis of continuous spatially-correlated outcomes; the proposed Bayesian computational algorithm provides a practical method for fitting such models.

Appendix: MCMC Algorithm

1. Update γ_k and δ_k: The full conditional for p-dimensional vector γ_k (k = 2, …, K) is given by

   $$\pi ({\mathbf{\gamma }}_k|·)\propto \prod _{i=1}^n\prod _{j=1}^{n_i}(\frac{e^{{\mathit{x}}_{ij}^{\prime }{\mathbf{\gamma }}_k+{\mathit{\upsilon }}_i^{\prime }{\mathit{\delta }}_k+{\psi }_{ik}}}{{\sum }_{h=1}^K{e}^{{\mathit{x}}_{ij}^{\prime }{\mathbf{\gamma }}_h+{\mathit{\upsilon }}_i^{\prime }{\mathit{\delta }}_h+{\psi }_{ih}}}){\mathrm{N}}_p({\mathbf{\gamma }}_k;{\mathit{\mu }}_{\mathrm{\gamma }},{\mathbf{\Sigma }}_{\mathrm{\gamma }}),$$

   where N_p(γ_k; ·) is a p-dimensional normal distribution evaluated at γ_k. Since this full conditional does not have a closed analytic form, we update γ_k using a random-walk Metropolis algorithm based on a multivariate-t₃(s_gT_k) proposal density centered at the previous value, ${\mathrm{\gamma }}_k^{\text{old}}$, where the parameter s_g scales the covariance to achieve an optimal acceptance rates, and T_k is a component-specific scale matrix. To improve mixing, we apply the adaptive proposal approach developed by Haario et al. (2005), which uses the empirical covariance from an extended burn-in to tune T_k. A similar approach can be used to update δ_k.
2. Update ψ_ik: The full conditional for ψ_ik (i = 1, …, n; k = 2, …, K) is given by:

   $$\pi ({\psi }_{ik}|·)\propto \prod _{j=1}^{n_i}(\frac{e^{{\mathit{x}}_{ij}^{\prime }{\mathbf{\gamma }}_k+{\mathit{\upsilon }}_i^{\prime }{\mathit{\delta }}_k+{\psi }_{ik}}}{{\sum }_{h=1}^K{e}^{{\mathit{x}}_{ij}^{\prime }{\mathit{\gamma }}_h+{\mathit{\upsilon }}_i^{\prime }{\mathit{\delta }}_h{\psi }_{ih}}}\left)\mathrm{N}\right(\frac{1}{m_i}\sum _{l\in {\partial }_i}{\psi }_{lk},{\tau }_k^2/m_i),$$

   where m_i and ∂_i are defined in equation (4) of Section 3. Since this full conditional does not have a closed form, we update ψ_ik using random-walk Metropolis.

3. Update ${\tau }_k^2$: Assuming an IG(g, s) prior, draw ${\tau }_k^2$ its IG(g*, s*) full conditional, where $g*=g+(n-\omega )/2,s*=s+{\psi }_k^{\prime }(\mathit{M}-\mathit{A}){\psi }_k/2,n$ is the number of areal units, ω = max(1, number of “islands”), and (M−A) is defined in equation (6).

4. Update C_ij: For all (i, j), draw C_ij from its full conditional

   $$\pi (C_{ij}|·)=\text{Pr}(C_{ij}=k|·)=\text{Cat}(p_{ijk}),\text{where}{p}_{ijk}=\frac{{\pi }_{ijk}{\mathrm{N}}_2({\mathit{y}}_{ij};{\mathit{\eta }}_{ijk},{\mathbf{\Sigma }}_k)}{{\sum }_{h=1}^K{\pi }_{ijh}{\mathrm{N}}_2({\mathit{y}}_{ij};{\mathit{\eta }}_{ijh},{\mathbf{\Sigma }}_h)}.$$

   Here, Cat(p_ijk) denotes a “categorical” distribution taking the value k with probability p_ijk, π_ijk is the prior probability that C_ij = k, as given in equation (1) of Section 3, and N₂(y_ij; η_ijk, Σ_k) denotes the bivariate normal density from equation (1), Section 3, evaluated at y_ij.

5. Update Σ_k: Assuming an IW(κ₀, S₀) prior, update Σ_k from its IW(κ*, S*) full conditional, where κ* = κ₀ + N_k, N_k is the number of observations in component $k,{\mathit{S}}^{*}={\mathit{S}}_0+{\mathit{E}}_k^{\prime }{\mathit{E}}_k,\mathit{E}={\mathit{Y}}_k-{\mathit{\eta }}_k^{*},{\mathit{Y}}_k$ is an N_k × 2 matrix consisting of y_1ij (first column) and y_2ij (second column) response values for all (i, j) in component k, and ${\mathit{\eta }}_k^{*}$ is an N_k × 2 matrix consisting of η_1ij (first column) and η_2ij (second column) linear predictor values for all (i, j) in component k, as defined in equation (1). Use ${\mathbf{\Sigma }}_k=\left(\begin{array}{cc}\hfill {\sigma }_{1k}^2\hfill & \hfill {\rho }_k{\sigma }_{1k}{\sigma }_{2k}\hfill \\ \hfill {\rho }_k{\sigma }_{1k}{\sigma }_{2k}\hfill & \hfill {\sigma }_{2k}^2\hfill \end{array}\right)$ to obtain ${\sigma }_{1|2,k}^2=(1-{\rho }_k^2){\sigma }_{1k}^2,{\sigma }_{2|1,k}^2=(1-{\rho }_k^2){\sigma }_{2k}^2,{\beta }_{1k}^{*}={\rho }_k{\sigma }_{1k}/{\sigma }_{2k}\text{and}{\beta }_{2k}^{*}={\rho }_k{\sigma }_{2k}/{\sigma }_{1k}$.

6. Update β_1k and α_1k: We update β_1k and α_1k conditional on y_2k, where y_2k denotes the N_k × 1 vector of y_2ij response values for all (i, j) in component k. Specifically, assuming a N_p(μ_β, Σ_β) prior, update β_1k from its N_p(M_β1k, V_β1k) full conditional, where

   $${\mathit{V}}_{\beta 1k}={[{\mathbf{\sum }}_{\beta }^{-1}+{\sigma }_{1|2,k}^{-2}({\mathit{X}}_k^{\prime }{\mathit{X}}_k)]}^{-1}\text{and}{\mathit{M}}_{\beta 1k}={\mathit{V}}_{\beta 1}\{{\sum }_{\beta }^{-1}{\mu }_{\beta }+{\sigma }_{1|2,k}^{-2}{\mathit{X}}_k^{\prime }[{\mathit{y}}_{1k}-{\mathit{V}}_k{\mathit{\alpha }}_{1k}-{\mathit{\varphi }}_{1k}-{\beta }_{1k}^{*}({\mathit{y}}_{2k}-{\eta }_{2k})]\}.$$

   Here, y_1k denotes the N_k × 1 vector of y_1ij response values for all (i, j) in component k; X_k denotes the N_k × p individual-level design matrix for component k; V_k is an N_k × r county-level design matrix for component k; Φ_1k is an N_k × 1 stacked vector such that ϕ_1ik is replicated for each observation in county i and component k; η_2k is an N_k × 1 vector of η_2ij values for all (i, j) in component k, as defined in equation (1); and ${\sigma }_{1|2,k}^2$ and ${\beta }_{1k}^{*}$ are defined in Step (5). A similar set of equations can be used to update the r × 1 vector α_1k.

7. Update β_2k and α_2k: We update β_2k and α_2k conditional on y_1k. Specifically, assuming a N_p(μ_β, Σ_β) prior, update β_2k from its N_p (M_β2k, V_β2k) full conditional, where

   $${\mathit{V}}_{\beta 2k}={[{\mathbf{\sum }}_{\beta }^{-1}+{\sigma }_{2|2,k}^{-2}({\mathit{X}}_k^{\prime }{\mathit{X}}_k)]}^{-1}\text{and}{\mathit{M}}_{\beta 2k}={\mathit{V}}_{\beta 1}\{{\sum }_{\beta }^{-1}{\mathbf{\mu }}_{\beta }+{\sigma }_{2|1,k}^{-2}{\mathit{X}}_k^{\prime }[{\mathit{y}}_{2k}-{\mathit{V}}_k{\mathit{\alpha }}_{2k}-{\mathit{\varphi }}_{2k}-{\beta }_{2k}^{*}({\mathit{y}}_{1k}-{\mathit{\eta }}_{1k})]\}.$$

   Here, η_1k is an N_k × 1 vector of η_1ij values for all (i, j) in component k, as defined in equation (1) ${\sigma }_{2|1,k}^2$ and ${\beta }_{2k}^{*}$ are defined in Step (5), and all other elements are defined in a manner analogous to those in Step (6). A similar set of equations can be used to update the r × 1 vector α_2k.

8. Update ϕ_ik: For i = 1, …, n, draw the 2 × 1 vector ϕ_ik from its N₂(M_ϕ, V_ϕ) full conditional, where

   $${\mathit{V}}_{\varphi }={({\mathrm{N}}_{ik}{\mathbf{\Sigma }}_k^{-1}+m_i{\mathbf{\Lambda }}_k^{-1})}^{-1}{\mathit{M}}_{\varphi }={\mathit{V}}_{\varphi }[{\mathbf{\Sigma }}_k^{-1}{\mathit{Z}}_{ik}^{\prime }({\mathit{y}}_{ik}-{\mathit{X}}_{ik}{\mathit{\beta }}_k-{\mathit{V}}_{ik}{\mathit{\alpha }}_k)+{\mathbf{\Lambda }}_k^{-1}\sum _{l\in \partial }{\varphi }_{lk}].$$

   Here, N_ik denotes the component-k sample size for county i; m_i is the number of neighbors of county i; Z_ik is a 2N_ik × 2 matrix with alternating rows of (1, 0) and (0, 1); y_ik is a 2N_ik × 1 stacked vector consisting of alternating Y₁ and Y₂ response values for each observation in county i and component k; X_ik is a 2N_ik × 2p design matrix for county i and component k, with rows alternating between (${\mathit{x}}_{ij}^{\prime },\mathbf{0}$) and ($\mathbf{0},{\mathit{x}}_{ij}^{\prime }$) for each of the N_ik observations in county i and component k; ${\mathit{\beta }}_k={({\mathit{\beta }}_{lk}^{\prime },{\mathit{\beta }}_{2k}^{\prime })}^{\prime }$ is a 2 p × 1 vector of individual-level regression parameters; V_ik is a 2N_ik × 2r design matrix of county-level predictors; and $\mathit{\alpha }k={({\mathit{\alpha }}_{lk}^{\prime },{\mathit{\alpha }}_{2k}^{\prime })}^{\prime }$ is a corresponding 2r × 1 vector of county-level regression coefficients.

9. Update Λ_k:

   Assuming an IW(ν₀, D₀) prior, draw Λ_k from its IW(ν*, D*) full conditional, where ν* = ν + n − ω, n is the number of areal units, ω is defined in Step (3) above, and ${\mathit{D}}^{*}={\mathit{D}}_0+{\mathbf{\Phi }}_k^{{*}^{\prime }}(\mathit{M}-\mathit{A}){\mathbf{\Phi }}_k^{*}$. Here, ${\mathbf{\Phi }}_k^{*}$ denotes an n × 2 matrix with first column equal to ϕ_1k = (ϕ_11k, …, ϕ_1nk)′ and second column equal to ϕ_2k = (ϕ_21k, …, ϕ_2nk)′.