With increasing accessibility to Geographical Information Systems (GIS), researchers and administrators in public health are increasingly encountering areal datasets that are aggregated as case counts or rates over areal units or regions (e.g. counties, census-tracts or ZIP codes). This is common practice in public health for protecting patient privacy.

Statistical models for areal data can adjust for known causes of variability in the data and also for sparsely sampled regions by smoothing across and borrowing information from its spatial neighbors (see, e.g., Anselin, 1988; Le Sage and Pace, 2009; Banerjee et al., 2004). An especially pertinent issue is to ascertain statistically significant differences among neighboring regions, hence identifying spatial barriers or difference boundaries that delineate them. Ultimately, the underlying influences responsible for these boundaries or barriers are typically of scientific and administrative interest. This ‘boundary’ detection problem is often referred to as “wombling”, after a foundational article by Womble (1951). While statistical boundary analysis has been applied extensively to point-referenced and gridded (or lattice) data (see, e.g., Banerjee and Gelfand, 2006), formal statistical inference in areal contexts present unique challenges that we outline later.

Deterministic areal wombling is often carried out using algorithms (Jacquez and Greiling, 2003a, 2003b) that are fast and straightforward to implement but fail to account for sources of uncertainty, such as extremeness in counts and rates corresponding in thinly populated regions. Li, Banerjee and McBean (2011) proposed statistical learning for boundaries using the Bayesian Information Criterion. In hierarchical model based approaches, Lu and Carlin (2004), Lu et al. (2007) and Ma, Carlin and Banerjee (2009) investigated estimating the adjacency matrix within a hierarchical framework using priors on the edges. However, inference from these models are usually highly sensitive to prior specifications on certain parameters.

Our primary contribution is a method to deliver inference for areally aggregated health outcome data, including assessment of difference boundaries, using classes of more flexible and robust nonparametric Bayesian hierarchical models. Section 2 offers a brief exposition to models for areally referenced count data. Section 3 elucidates the key issues in areal boundary analysis and our Bayesian nonparametric modeling approaches. Sections 4 and 5 discuss, respectively, a simulation study and the analysis of a Minnesota Pneumonia & Influenza (P & I) dataset to detect spatial health barriers between neighboring counties in Minnesota. Finally, Section 6 concludes the article with an eye towards future work.

Areal data can be analyzed using Bayesian hierarchical models that incorporate geographical effects. For example, let Y_i (random) be the observed number of patients who underwent a specific preventive or clinical outcome in areal unit i, i = 1, … , n, and let E_i (fixed) be the expected number of outcomes for that unit. A commonly used likelihood is (2.1)Yi∼indPoisson(Eieμi),i=1,…,n, where μi=xi′β+ϕi represents the log-relative risk, estimates of which are often based on the departures of observed from expected counts, x_i includes explanatory, region-level covariates or predictors for region i and β are the corresponding regression coefficients.

Each φ_i represents the spatial random effect associated with region i, which is often modeled using Markov random fields (e.g. Cressie, 1993; Banerjee et al., 2004, Ch.3) that imply the following joint distribution for φ = (φ₁, φ₂, … , φ_n)′: (2.2)ϕ∼Nn(0,σ2(D−ρW)−1), where N_n denotes the n-dimensional normal distribution, D is a n×n diagonal matrix with diagonal elements m_i equal to the number of neighbors of area i, and W = {w_ij} is the adjacency matrix for the map, i.e., w_ii = 0, and w_ij = 1 if i is adjacent to j and 0 otherwise. In the joint distribution (2.2), σ² is the spatial dispersion parameter, and ρ is a spatial autocorrelation parameter. We denote this distribution concisely as CAR(ρ, σ²). A sufficient condition for D – ρW to be positive definite is that ρ ∈ (1/λ₍₁₎, 1), where λ₍₁₎ is the minimum eigenvalue of W (Banerjee et al., 2004).

The CAR model has been especially popular in Bayesian inference as its conditional specification is convenient for Gibbs sampling and MCMC schemes. The distribution in (2.2) reduces to the well-known intrinsic conditionally autoregressive (ICAR) prior if ρ = 1, or an independence model if ρ = 0. The ICAR model induces “local” smoothing by borrowing strength from the neighbors, while the independence model assumes independence of spatial rates and induces “global” smoothing. The smoothing parameter ρ in the CAR prior (2.2) controls the strength of spatial dependence among regions, though it is well-appreciated that a fairly large ρ may be required to deliver significant spatial correlation.

To evaluate our methods, we conduct a simulation study using the template of a Minnesota county map in Li et al. (2011). There are n = 87 counties in Minnesota, and 211 pairs of neighboring counties (i.e., geographical boundaries). We simulated 50 datasets on a map of Minnesota, where the state was divided into six regions. Each dataset was generated from (2.1), where μ_i was one of five different means corresponding the the five different shades mapped on Figure 4.1. The darker shades correspond to higher means. To add some irregularity, we also included one county (Sherburne county shaded white in Figure 4.1) that has all its boundaries as true difference boundaries. This resulted in “six” different clusters on the map and 47 “true difference boundaries” delineating the different clusters, i.e. these 47 boundaries are county borders that separate two areal units with substantially different means.

For this simulation experiment, since we know the true difference boundaries we can obtain the sensitivity and specificity for each of our proposed models. Sensitivity is the probability of correctly detecting a true difference boundary, while specificity is the probability of correctly labeling a geographical boundary as not a difference boundary. To be precise, for every pair of geographical neighbors (i, j), we compute the posterior probability P(φ_i ≠ φ_j ∣ Y) and choose the top T = 35, 40, 45, 50 and 55 edges with the highest posterior probabilities. As there are 47 true difference boundaries, these choices encompass settings where we could, theoretically have obtained 100% accuracy (when T = 35, 40, 45) and also where we are assured of a few false positives (when T = 50, 55).

The prior specification and computational details of the ARDP model are in the Web Supplement. The parameter α can be fixed based upon the expected number of clusters a priori. In this case, we have six clusters which would suggest a value of α around 1.25. In fact, we experimented with α ranging from 0.25 to 1.75 and obtained very robust inference. The results presented here correspond to α = 0.5, which leads to an expected number of clusters around 3, which is half of the number of true clusters. We also fixed ρ = 0.98 in the ARDP model (ICAR is inapproriate since the covariance matrix must be proper and nonsingular). Customarily, higher values of ρ (≈ 1) yield sufficient smoothing, while values lower than 0.95 tend not to (e.g., Banerjee et al., 2004). In contrast, for ARSB, we used the ICAR model (with ρ = 1) along with the sum-to-zero constraint, which yields legitimate posterior samples. We assumed a regression structure with only an intercept (i.e. x_i ≡ 1) and placed a flat prior on the corresponding β. A weakly informative prior Γ(.01, .01) is specified for the precision parameters τ_s and τ_γ . In both models, the stick-breaking prior was computed using about 15 terms.

We compare the performance of DPM, ARSB, ARDP with three existing methods: (i) the deterministic Boundary Likelihood Value (BLV) algorithm of Jacquez and Greiling (2003a, 2003b) using the BoundarySEER software (see http://www.biomedware.com) with default thresholds set from a BLV histogram, (ii) the model-based approach of Lu and Carlin (2005), which we call the “LC method”, and (iii) a class of discrete Spatial Moving Average (SMA) models outlined in Li et al. (2011). We ran these models within the R statistical software environment running 3 parallel chains for each model and dataset. Convergence was diagnosed after 12, 000 iterations of burn-in using Gelman-Rubin diagnostics and autocorrelation plots from the coda package in R. A subsequent 5, 000 × 3 = 15, 000 samples were used for posterior inference. On a workstation using a Intel dual core 4 GHz processor, each model took less than five hours of CPU time to deliver its entire inferential output for all the 50 simulated datasets.

Table 4.1 presents the average detection rates for these different methods applied to the 50 simulated datasets. The DPM and the BLV methods do not explicitly borrow strength across neighbors, while the other four methods in Table 4.1 exploit the adjacency structure of the underlying map. There seems to be little to choose between ARDP and ARSB but both methods seem to be slightly outperforming the other methods in both sensitivity and specificity under all five scenarios. In addition, while the performance of the SMA model is perhaps comparable, it is computationally onerous and less robust to prior assumptions (Li et al., 2011) than ARDP or ARSB.

The LC method is based upon a parametric CAR model that does not render itself to probabilistic boundary analysis (since P(φ_i = φ_j) will always be zero). However, one could fit parametric CAR models and use the posterior expectation of the absolute differences of the rates, i.e. E(∥η_i – η_j∥ ∣Y), where ηi=μiEi acts as a boundary difference score. Higher values will indicate spatial barriers between units i and j. The DPM, ARSB and ARDP models not only yield estimates of η_i, as in the “LC” method, but they also deliver nonzero posterior probabilities P(φ_i = φ_j ∣ Y). The LC method cannot produce these posterior probabilities. Therefore, we used the posterior expectation metric to compare its performance. The SMA model does not deliver posterior estimates of spatial effects from a single model. Hence, we exclude it from this comparison.

Table 4.2 presents the results for four of the methods. The deterministic BLV method detects 89.6% of the boundaries. The promise of our stochastic models is evident from the superior performances of the ARDP and the ARSB models. Since we know the true boundaries in Figure 4.1, we can assess the performances of these approaches in detecting the true boundaries. Using direct posterior estimates 47 difference boundaries. We find that the DPM, ARDP and the ARSB models are each able to detect about 90% of the true boundaries, which is superior to both LC and BLV. The ARDP model performs slightly better than the other two, which are approximately equally good. Using the posterior expectation metric, we again find that the proposed ARSB and ARDP models clearly outperform the LC method. The ARDP model has almost a 10% better detection rate, while the ARSB model excels by approximately 5%. Both ARDP and ARSB outperform the DPM model as well in terms of the posterior expectation metric.

We apply our method to the Minnesota Pneumonia and Influenza (P&I) diagnosis dataset. P & I rank as the eighth leading cause of death in the United States and the sixth leading cause in people over 65 years of age with Pneumonia consistently accounting for the overwhelming majority of deaths between the two. Together, they cost the U.S. economy in 2005 an estimated $40.2 billion. Identifying difference boundaries that perform well with regard to sensitivity and specificity can help identify so-called “health barriers” more accurately and buttress an active surveillance program for an influenza-like illness.

We analyze a dataset consisting of Minnesota residents above 65 years of age who were enrolled in the Medicare fee-for-service program as of December 31, 2001. The Medicare Denominator file for 2001 was used to define the cohort, which has also been used to study the impact of vaccinations on elderly Minnesota residents. The Medicare Provider Analysis and Review (MedPAR) manages patient records based on date of discharge and supplied information regarding hospitalizations resulting from P&I. Rates of P&I hospitalization are traditional measures of the impact of influenza virus in the elderly population. We identify the ‘boundaries’ that separate the more affected areas from the less affected areas.

If Y_i and O_i are the observed number of hospitalizations and the population in county i respectively, then Ei=∑k=1nYi∑k=1nOiOi is the expected number of cases (under the assumption of no spatial variation in rates), where n is the total number of counties. The choropleth map of the raw data is shown in Figure 5.2. The high valued SMR (standard mortality ratio) counties are scattered over the map, with a clump on the southwest and some isolated regions surrounded by sparsely inhabited counties that also have lower counts.

We employed the same models in Section 4to detect boundaries on the P&I hospitalization map. The same prior specification and model settings are applied here as the simulation study, except we assign α = 1, a customary choice when one does not seek a prior distribution on this parameter (Escobar and West, 1995) or has no a priori information about the number of clusters. Three parallel MCMC chains were executed on the same computing environment as described in Section 4.Convergence was diagnosed after 10, 000 iterations of burn-in using Gelman-Rubin diagnostics and autocorrelation plots and a subsequent 5, 000 × 3 = 15, 000 samples were used for posterior inference. Each model consumed less than ten minutes of CPU time to produce its entire inferential output for the Minnesota Pneumonia and Influenza dataset with very little difference between the ARSB, ARDP and the (non-spatial) DPM model.

Health administrators may prefer to use a “top bracket” of most likely difference boundaries for policy formulation. The top 50 difference boundaries detected by each model are highlighted in Figure 5.3. Table 5.3 presents a comprehensive “lookup table” containing the names of adjacent counties that have been ranked in decreasing order according to 1 – P(φ_i = φ_j ∣ Data) from the ARDP model. Instead of selecting this “bracket” arbitrarily, statisticians may prefer a threshold obtained by controlling the FDR. Setting δ = 5% yields Numbers 1-33 as difference boundaries, while setting δ = 10% detects Numbers 1-42 as difference boundaries. This table offers an easy reference for health administrators and officials to identify the more substantial spatial health barriers in the state.

About 90% of the boundaries listed in Table 5.3 are detected by all four models. As a specific example consider Cook and Koochiching county. The outcome variable in the former is substantially higher than its only neighbor, Lake, while Koochiching county is separated from all its neighbors due to its extremely high P&I SMR, even after being smoothed by the model. Among the 50 difference boundaries detected by the ARDP model, 47 are also detected by the ARSB model. The three county-pairs that went undetected by ARSB were: Goodhue and Olmsted, Freeborn and Steele, and Big Stone and Traverse. Instead, the ARSB model detected boundaries between counties Becker and Wadena, Cotton Wood and Jackson, and Cook and Lake.

The map in Figure 5.2 does not display clustering as pronounced in the simulation example. Furthermore, unlike in the simulation example, we do not know the “truth.” It does, however, reflect well on our models that the rankings in Table 5.3 are very consistent with competing methods. We already discussed the minor differences between the ARDP and the ARSB. The agreement between the ARDP and the SMA in terms of identifying the difference boundaries using FDR-based thresholds is very strong with over 95% agreement in boundary selection.

The paper presented a class of nonparametric Bayesian hierarchical models for detecting difference boundaries on maps. An advantage of the new approach is that it permits the probabilistic estimation of an edge as a difference boundary, and improves the percentage of true detection. A disadvantage is that the model cannot be easily fit into any existing commercial software. We fit these models in R (www.r-project.org), and we wish to collect these models in an R package in the near future.

The ARDP and ARSB models in conjunction with the FDR controlled threshold selection provide a major improvement over earlier work by Li et al. (2012) by circumventing the need to estimate as many models as there are geographical boundaries and by offering comprehensive model-based posterior estimates of model parameters. The latter is precluded in Li et al. (2012) who treat this as a purely hypothesis testing problem. However, issues related to optimal selection of boundaries warrants further investigation especially regarding the sensitivity of the inference to FDR-based cutoffs and to prior specifications. Further extensions can be formulated by incorporating classes of loss functions, as discussed by Müller et al. (2008), for a more comprehensive decision-theoretic framework. Such developments may, in turn, lead to more definitive conclusions regarding the performance of these models in maps that display weaker clustering patterns.

Finally, we note that both the ARDP and ARSB models render themselves to multivariate extensions, where multiple health outcomes need to be modeled jointly. Analogous to the univariate ARSB model, we can construct a multivariate areally referenced stick breaking (MARSB) model such that the sticking breaking weights p are correlated through another type of “weight” parameter that scales the V_i’s. We place a multivariate CAR (MCAR) prior on these “weight” parameters so as to capture both spatial correlation and inter-variable correlation (Jin et al., 2005, 2007). For the ARDP, the distribution of the spatial random effects can be modeled by constructing multivariate areally referenced Dirichlet processes (MARDP). These pursuits will constitute natural extensions of our current work.