Estimates of prevalence of disease and health behaviours for different areas are increasingly required for the equitable allocation of health funds according to local need and to target interventions. As stressed by Bazos et al 1 community health need assessments are ideally based on locally disaggregated (i.e. small area) health status and disease prevalence information. To estimate prevalence in different small areas, a commonly adopted approach involves synthetic estimation whereby prevalence rates for demographic subgroups of the population are obtained (e.g. from national health surveys) and an indicative rate then obtained based on the demographic composition of each area. Thus prevalence of most health conditions varies considerably with age, and often also by sex and race: so a synthetic estimate may be obtained by using age, sex and race specific prevalence rates.

However, synthetic estimates of this kind do not take account of geographic context, exemplified by interactions between demographic risk factors and geographic location, or by independent effects of geographic variables (e.g. area poverty or urbanity-rurality) on prevalence that remain even after taking account of patient level risk factors. By contrast, the multilevel prevalence model for cardiovascular outcomes proposed here as a basis for small area prevalence estimates incorporates the modifying effects of geographic context as well as patient risk factors.

In the US, a number of population health surveys are carried out and provide cumulative evidence on CVD trends and epidemiology. Thus the National Health Interview Survey (NHIS) for 2005 estimated the prevalence of cardiovascular disease (CVD) at 68 million among adults aged 18 years and over in the US, which includes coronary heart disease, hypertension, stroke, angina pectoris or heart attack. The analysis here is concerned with a positive response to one or more of three questions included in the 2005 Behavioral Risk Factor Surveillance System (BRFSS) survey; these questions encompass the different forms of CVD, namely, had the subject ever been told by a health professional that they had experienced a heart attack, or told they had undergone a stroke, or told they had CHD or angina.

The epidemiology of these conditions differ to some degree, for example in terms of male-female differentials in prevalence and incidence 2, in trends through time 3, and in ethnic group differentials. However, for these and related conditions there is evidence for a role of geographic context, in terms of wide geographic disparities by region, state and urbanity 45678. In particular, there is evidence of direct effects of area variables after controlling for person level risk factors, and evidence of interactions between place and person variables. For example, Cubbin et al 9 report higher levels of hypertension and diabetes among African American women living in socioeconomically deprived neighborhoods as against African American women from more affluent neighborhoods, after allowing for individual-level socioeconomic status, while Halverson et al 10 report local clustering of excess CVD mortality after controlling for area population composition. As for place-person interactions, Barnett et al 6 and Casper et al 11 report that ethnic disparities in CHD mortality vary by area of residence.

The prevalence model and small area prevalence estimates described here are based on around 336,000 survey responses, and on a regression analysis relating CVD status both to individual level risk factors and to county level measures of poverty and urban-rural status. The analysis further adjusts for differentiation at US state level in the impact of ethnicity on prevalence. Thus adjustment for geographic context is much more comprehensive than is possible using disease status data from the Health Survey for England where only broad regional identifiers are available – an example being the work of Congdon 12 on CHD prevalence. One goal of the analysis here is to develop prevalence estimates for micro areas, namely 32000 ZIP Code Tabulation Areas (ZCTAs) for which certain population tabulations are provided by the US Census Bureau 13. Inclusion in the prevalence model of patient risk categories such as gender and ethnicity (and interactions between them) therefore requires that such categories are available in these tabulations for micro area populations.

The regression model for prevalence includes person level attributes (age, gender, ethnicity, education level) that are known to have significant CVD risk gradients. A pronounced gradient in CVD prevalence by age is reported by Neyer et al 14; thus the MI rate among 18–44 year olds is 0.8%, among 45–64 year olds is 4.8% and among the over 65s is 12.9%. In terms of the main ethnic groups in the US (white non-hispanic, black, hispanic, other) elevated CVD mortality and morbidity for nonwhite groups are reported by Barnett et al 6 and Caspar et al 11, though ethnic differentials may to some degree express socioeconomic disadvantage. Certain subgroups such as black females, have more clearly elevated CVD prevalence 15. As to education level, Neyer et al 14 report that prevalence of one or more of an MI history or a CHD/angina history decreases with educational attainment: of persons with less than a high school diploma, 9.8% report a history of one or more of the conditions, nearly twice the proportion (5%) among college graduates. Education is interrelated with issues such as linguistic competence and health literacy that affect health status 16, and with health insurance 17.

Estimation of the three models follows the Bayesian method, whereby pre-existing knowledge regarding parameters is expressed in prior densities, and updated or posterior knowledge is obtained by combining the prior densities with the likelihood (1) of the observed data. Estimation uses iterative Monte Carlo Markov Chain sampling methods 27, as provided in the WINBUGS program 28. Goodness of fit is assessed by the Deviance Information Criterion or DIC 29, whereby the average deviance is adjusted to account for model complexity. The DIC is the average deviance plus the complexity, with lower DICs representing better fit. Summaries of parameters (means and 95% intervals) are based on the second halves of two chain runs of 5000 iterations, with dispersed initial values. Convergence was achieved in all models using Brooks-Gelman-Rubin criteria 30.

Table 2 summarises the fit of the models, while Tables 3 and 4 show gender-specific es-timates of the parameters {α_r, β_rg, η_re, κ_r, δ_ru} from the three models. The DIC criteria in Table 2 show a gain in introducing geographic contextual variables (model 2 vs model 1), and a clear gain also in making state random effects specific to ethnic groups (model 3 vs model 2).

Geographic variations in the prevalence of chronic disease partly reflect the demographic composition of area populations. However, prevalence variations may also show distinct geographic 'contextual' effects that are differentiated between ethnic and other demographic categories. Studies of cardiovascular disease in the US have found major geographic variations that do not seem to be explicable by area demography alone.

The present study has demonstrated by formal modelling methods applied to BRFSS data that improved explanation is obtained by allowing for distinct geographic effects (for counties and states) and for interaction between geographic and person variables. There are significant spatial effects (e.g. county poverty effects, state residual effects) after adjusting for CVD gradients over person level variables, namely age, education, ethnicity.

This has direct implications for an appropriate methodology to estimate prevalence at small area level, with the focus here being ZIP Code Tabulation Areas. Thus – on the basis of the model estimates in the above analysis – prevalence estimates for a ZCTA need to reflect its region of location (e.g. in a South East state as opposed to a northern or mountain state) and the poverty level of the county containing it.

In methodological terms, this paper is distinct in using a log link multilevel binary regression model that takes account of both person level risk factors and the spatial context for a major chronic disease. The use of a log link allows straightforward inferences on relative risks and potentially allows the incorporation into the model of cumulative prior evidence (e.g. on relative CVD risks over ethnic groups). Statewide contextual effects have been represented by a structured random effect, that allows for spatial correlation in unobserved risk factors but also extends to include spatially isolated areas (see Appendix 2). In an extended model (model 3) state random effects are differentiated by ethnic group, reflecting evidence from other sources that ethnic relativities are not constant geographically.

Variations and extensions to the models presented above are possible. One option is state or county averages in the person level variables such as ethnicity and education level (e.g. county percent black or county percent college graduates). This has been proposed as a way of measuring contextual effects 34, though there is likely to be a positive correlation with the already included county poverty rate. Another possibility would be a longitudinal analysis over a sequence of successive surveys, which can indicate whether gradients over person level risk factors are changing, or whether geographic variability is changing.

Let C_i, S_i and U_i denote the county, state and (county level) rural-urban category of residence for respondent i. Also let {x_i, g_i, e_i} denote the age, ethnicity and education level of respondent i. Then prevalence models are specific for gender r, and one may write prevalence model 3 (with ethnic-specific state effects) as

y_ir ~Bin(1,p_ir)

log(p_ir) = α_r + β_r[g_i] + η_r[e_i] + γ_r[g_i, x_i] + κ_rPov[C_i] + δ_r[U_i] + w_r[S_i, g_i],

where Bin(n, p) denotes the binomial density, the parameters {α, β, δ, η, κ } are fixed effects, and the parameters {γ,w} are random. This model is run separately for males and females.

Since the parameters operate on the log relative risk scale, state level relative risks by ethnic group ρ_rsg (after controlling for known person and county attributes) may be obtained by exponentiating the state effect, namely

ρ_rsg = exp(w_rsg).

Thus excess risk or unduly low risk may reflect geographic variations in prevalence that remain even after the impact of a range of important person and county attributes has been allowed for. Excess risk can be defined in terms of the 95% estimation interval for ρ_rsg being confined to values above 1.

The baseline model 1 (with person level risk factors only) is

log(p_ir) = α_r + β_r[g_i] + η_r[e_i] + γ_r[g_i,x_i].

The intermediate model (model 2), including county regression terms, and state random effects, but not including area-ethnicity interactions is

log(p_ir) = α_r + β_r[g_i] + η_r[e_i] + γ_r[g_i,x_i] + κ_rPov[C_i] + δ_r[U_i] + w_r[S_i].

Thus the unobserved state effects are assumed to be equal across ethnic groups.

For the unknown fixed effects parameters, namely {α_r, β_rg, η_re} in model 1, and {α_r, β_rg, η_re, κ_r, δ_ru} in models 2 and 3, diffuse normal priors with mean zero and variance 1000 are adopted. Corner constraints are used for the β_rg, η_re and δ_ru parameters for identifiability, namely β_r1 = η_r1 = δ_r1 = 0. To pool strength across the age pro les of different ethnic groups, a first order random walk prior is used for the G-dimensional vector γ_rx = (γ_r1x,.., γ_rGx), x = 1,.., X of age effects across G ethnic groups. This has conditional form

γ r x ~ N G ( γ r , x − 1 , Ω r − 1 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4SdC2aaSbaaSqaaiabdkhaYjabdIha4bqabaGccqGG+bGFcqWGobGtdaWgaaWcbaGaem4raCeabeaakiabcIcaOiabeo7aNnaaBaaaleaacqWGYbGCcqGGSaalcqWG4baEcqGHsislcqaIXaqmaeqaaOGaeiilaWIaeuyQdC1aa0baaSqaaiabdkhaYbqaaiabgkHiTiabigdaXaaakiabcMcaPiabcYcaSaaa@44DD@

where the G × G matrix Ωr−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuyQdC1aa0baaSqaaiabdkhaYbqaaiabgkHiTiabigdaXaaaaaa@30DA@ represents covariation between age mortality profiles of ethnic groups. The precision (inverse covariance) matrices Ω_r are assigned a Wishart prior with identity scale matrix and G degrees of freedom, namely Ω_r ~ Wish(I,G).

The 51 states in the model are the mainland US states (k = 1,.., 49) arranged alphabetically (Alabama to Wyoming, including the District of Columbia), together with Alaska and Hawaii (k = 50, 51). The presence of these two spatially isolated states complicates applications of standard approaches for spatially correlated effects, at least those based on a spatial contiguity matrix. It would still be possible to use a spatial model based on interstate distances, but this means that a spatial decay function in distance has to be specified and its parameters estimated. Here we follow the most common approach to spatial clustering, based on contiguity of areas, with a spatial effect that "should describe the fact that areas close to each other tend to behave similarly" 26.

One option that brings in all 51 states would be to follow the convolution approach of Besag et al 35 and assume there are two effects, one of which follows a conditional autoregressive scheme and applies only to the mainland states (k = 1,.., 49), while the other effect, applying to all 51 states is unstructured in the sense of not incorporating spatial structure.

Thus for states k = 1, 49 the total state effect would be

h_k + w_k,

where h_k represents spatially unstructured heterogeneity, and w_k represents a conditional autoregressive scheme based on contiguity. The suffix r for gender is omitted for simplicity. Specified conditionally on effects w_[-k] in the remaining 48 states, one has for mainland states k = 1,..,49

p(w_k|w_[-k]) ~ N(W_k, τ_w/L_k),   k = 1,.., 49

where τ_w is a variance parameter, L_k is the number of states adjacent to state k, and W_k is the average of w_m over states m = 1,.., L_k adjacent to state k. For example, W₁ (for Alabama) would be an average of the four w effects for the contiguous states, Mississippi, Georgia, Florida and Tennessee. The prior for the h_k would be over all 51 states, rather than the mainland 49 states, and typically specified as

h_k ~ N(0,τ _h)   k = 1,.., 51,

where τ_h is a variance parameter. Under this convolution approach, for states 50 and 51 (Alaska and Hawaii) the state effect would consist of h_k only.

While this approach is an option when a collection of areas includes spatial isolates, it is not used here. The problems that occur with the model (A2.1) include identifiability, since only the total h_k + w_k is identified by the data, and the heavy (i.e. non-parsimonious) parameterisation. Leroux et al 36 propose an alternative more parsimonious model that uses a single random effect, with a conditional form

w k | w [ − k ] ~ N ( λ 1 − λ + λ L k ∑ m ~ k w m , τ w 1 − λ + λ L k ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4DaC3aaSbaaSqaaiabdUgaRbqabaGccqGG8baFcqWG3bWDdaWgaaWcbaGaei4waSLaeyOeI0Iaem4AaSMaeiyxa0fabeaakiabc6ha+jabd6eaojabcIcaOKqbaoaalaaabaGaeq4UdWgabaGaeGymaeJaeyOeI0Iaeq4UdWMaey4kaSIaeq4UdWMaemitaW0aaSbaaeaacqWGRbWAaeqaaaaakmaaqafabaGaem4DaC3aaSbaaSqaaiabd2gaTbqabaaabaGaemyBa0MaeiOFa4Naem4AaSgabeqdcqGHris5aOGaeiilaWscfa4aaSaaaeaacqaHepaDdaWgaaqaaiabdEha3bqabaaabaGaeGymaeJaeyOeI0Iaeq4UdWMaey4kaSIaeq4UdWMaemitaW0aaSbaaeaacqWGRbWAaeqaaaaakiabcMcaPiabc6caUaaa@5EA6@

where m ~ k denotes states m adjacent to state k. This reduces to a purely spatial model, as in (A2.2), when λ = 1 and to pure heterogeneity (i.e. no spatial clustering) when λ = 0. The λ parameter can be estimated and provides a measure of spatial dependence actually present in the data.

Congdon 25 extends model (A2.4) to allow the spatial dependence parameters to vary by area, and a version of such an approach is used in the CVD prevalence modelling here. This extension allows spatial dependence to vary over sub-regions of the total region or nation being considered, and also allows for spatial outliers, distinct from their neighbours in terms of outcome level such as disease risk. Outliers would have relatively low λ_k values, since spatial pooling (towards the neighbourhood average) is contra-indicated by the disparity between an area's risk and that of its neighbours. By contrast, areas surrounded by areas with similar levels of the outcome would have relatively high λ_k values, since spatial pooling (towards the neighbourhood average) is supported by the data.

The conditional specification now takes the form

w k | w [ − k ] ~ N ( λ k 1 − λ k + λ k L k ∑ m ~ k λ m w m , τ w 1 − λ k + λ k L k ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6973@

This model for spatial effects adapts to spatial outliers by taking λ_k = 0, so that for the subset of areas which are not connected to other areas one has

w_k ~ N(0, τ_w).

This approach extends to a multivariate random effect w_k = (w_k1,.., w_kG) for G ethnic groups. With a uniform value of λ over areas the conditional mean under the Leroux et al 36 model is

E ( w k g | w [ − k ] ) = λ [ 1 − λ + λ L k ] ∑ m ~ k w m g , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrauKaeiikaGIaem4DaC3aaSbaaSqaaiabdUgaRjabdEgaNbqabaGccqGG8baFcqWG3bWDdaWgaaWcbaGaei4waSLaeyOeI0Iaem4AaSMaeiyxa0fabeaakiabcMcaPiabg2da9KqbaoaalaaabaGaeq4UdWgabaGaei4waSLaeGymaeJaeyOeI0Iaeq4UdWMaey4kaSIaeq4UdWMaemitaW0aaSbaaeaacqWGRbWAaeqaaiabc2faDbaakmaaqafabaGaem4DaC3aaSbaaSqaaiabd2gaTjabdEgaNbqabaaabaGaemyBa0MaeiOFa4Naem4AaSgabeqdcqGHris5aOGaeiilaWcaaa@5593@

with inverse dispersion matrix (precision matrix)

Prec(w_k|w_[-k]) = [1 - λ + λL_k]Φ,

where Φ is a symmetric matrix of dimension G. Allowing for varying spatial dependence over the entire region/nation being considered, one has

E ( w k g | w [ − k ] ) = λ k [ 1 − λ k + λ k L k ] ∑ m ~ k λ m w m g , P r e c ( w k | w [ − k ] ) = [ 1 − λ k + λ k L k ] Φ . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8146@

In the application of (A2.5) in model 2, it is assumed that 1/τ_w is gamma distributed a priori, namely 1/τ_w~Ga(1, 0.001). This is approximately equivalent to assuming 1/τ_w to be uniformly distributed while constrained to positive values. Such a choice of gamma prior for 1/τ_w follows the strategy of studies such as 35 and 37. In the application of (A2.9) in model 3, it is assumed that Φ is Wishart distributed, with G degrees of freedom and an identity scale matrix.

The varying spatial parameters in models 2 and 3 are assumed to be beta distributed

λ_k ~ Be(ν₁, ν₂)

where ν₁ and ν₂ are positive quantities equal to or exceeding 0.5. Thus ν₁ = ν₂ = 1 corresponds to a diffuse uniform prior λ_k ~ U(0, 1), while more informative priors are obtained for ν₁ > 1 and ν₂ > 1. A baseline is provided when ν₁ = ν₂ = 0.5, equivalent to a prior sample size of 1. It is assumed that ν₁ ~ U(0.5, 5) and ν₂ ~ U(0.5, 5). The average value of the λ_k over all contiguous states can be obtained as

λ_a = ν₁/(ν₁ + ν₂).

The author declares that they have no competing interests.