Urban Form, Heart Disease, and Geography: A Case Study in Composite Index Formation and Bayesian Spatial Modeling

Abstract

Recent studies indicate a relationship between measures of urban form as applied to urban and suburban areas, and obesity, a risk factor for heart disease. Measures of urban form for exurban and rural areas are considerably scarce; such measures could prove useful in measuring relationships between urban form and both mortality and morbidity in such areas. In modeling area-level mortality, geographic relationships between counties warrant consideration because geographically adjacent areas tend to have more in common than areas farther from each other. We modify county-level indices of urban form found in the literature so that they can be applied to exurban and rural counties. We then use these indices in a Bayesian spatial model that accounts for spatial autocorrelation to determine if there is a relationship between such measures and cardiovascular disease mortality for white males age 35 and older for the time period 1999–2001. Issues related to the formation and usefulness of the indices, and issues related to the spatial model, are discussed. Maps of observed and expected relative risk of mortality are presented.

Appendices

Appendix A

This study used 1990 Census data but 1999–2001 mortality data. Therefore changes in county boundaries 1990 and 2000 (U.S. Census Bureau 2001–2005) are accounted for as follows: In Montana the portion of Yellowstone National Park inside of Montana was divided between Gallatin and Park Counties; therefore, Gallatin and Park Counties are pooled for this study. In Virginia South Boston town (a county equivalent) was merged into Halifax County; for this study South Boston and Halifax County are pooled. In Alaska Denali borough was formed from parts of Yukon-Koyukak and Southeast Fairbanks boroughs; for this study the areas for Yukon-Koyukak, Southeast Fairbanks, and Denali boroughs are merged. Also in Alaska, Skagway-Yakutat-Angoon borough was divided into Skagway-Hoonan-Angoon borough and Yakutat Borough; for this study the boroughs are merged. Annexations of portions of a county into another county that did not dissolve a county were not accounted for.

Appendix B

For our model the priors are: $1/\sigma_u^2, 1/\sigma_v^2 \sim \Upgamma (0.5,\;1/0.0005),$ and all $\beta^{\prime}s\sim N(0,\;1/0.00001),$ where $\Upgamma \left(\alpha,\varepsilon \right)$ is a Gamma function with shape parameters α and ɛ and N(μ, σ²) is a Normal distribution with mean μ and variance σ².

Appendix C

For each sprawl index WINBUGS 1.4 (The BUGS Project 2004; Spiegelhalter et al. 2003) was used for running three independent Markov Chains. First, for each of our two urban form indices (density and road accessibility), model (1) without spatial autocorrelation and unstructured variation components (e.g., without the terms u _i and v _k ) was run in PROC GENMOD in SAS (SAS Institute 1999b, pp. 1365–1464) to obtain maximum likelihood estimates of the coefficients. Initial values for the three chains were those estimates plus 4, 0 and −4 standard deviations. Initial values for $1/\sigma_u^2$ and $1/\sigma_v^2$ were taken as 0.001, 1,000 (mean of the prior distribution) and 7,000, respectively. Initial values for u _i and v _k were all set at 0.

Time series Gelman-Rubin diagnostic graphs and trace graphs were used to check convergence of relative risk estimates and parameter estimates for all three chains (Spiegelhalter et al. 2003). After 10,000 iterations the traces of the parameter estimates had good mixing around a common value, with varying degrees of white noise around those values. Gelman–Rubin graphs were convergent and stable. Chains were examined for $1/\sigma_u^2, 1/\sigma_v^2, \sigma_u$ and σ _v ; these chains also mixed well. Hence, after 10,000 iterations convergence of all parameters was concluded. An additional 30,000 iterations were then taken for each chain, but to reduce autocorrelation every third iteration was kept for later calculations. As a result, a total of 30,000 iterations (10,000 from each chain) were used to obtain parameter estimates.

Convergence of parameter estimates was checked according to the “Checking convergence” section of the WINBUGS 1.4 (Spiegelhalter et al. 2003) manual. The chains clearly appeared to be overlapping one another, and parameter estimates look stable. Posterior graphs had the desired bell shape.

The Brooks and Gelman (1998) version of the Gelman–Rubin convergence statistic (GR) as given in WINBUGS 1.4 was used for iterations 5,001–10,000. Let X be the width of the middle 80% of the parameter estimates of all three chains pooled together, and let Y be the average of the widths of the middle 80% of the parameter estimates for each of the three chains individually. Then GR = X/Y. Here, we want GR to converge close to 1 and both X and Y to converge to some number. Numerical values for X, Y and GR for later iterations are found in Table C1. The X and Y columns are numerically close to each other and the GR values are nearly equal to one.

Table C1 Summary of Gelman–Rubin statistics for convergence

Finally, we calculated parameter estimates for a variety of prior distributions to determine whether said estimates were sensitive to the choice of the prior. We found the parameter estimates to be very similar for all our choices.

Appendix D

This discussion of CAR and ICAR follows much of Wakefield et al. (2001, p. 110ff). We first define the CAR model. Define $N_n \left({\mathbf{0}}_{{\mathbf{n}}}, \;\sigma^{2}\Upsigma \right)$ as an n-dimensional normal distribution with n × n positive definite (i.e., the matrix has an inverse) correlation matrix Σ and parameter σ², and let ${\mathbf {Q}} = \Sigma^{-1}$ have elements $Q_{id},\, i,d=1,\ldots,n.$ The general CAR model can be written as (Besag and Kooperburg 1995)

$$U_i |\left({U_d =u_d,\;d\neq i} \right)\sim N\left(\sum\limits_{d=1}^n {M_{id} u_d},\sigma_u^2 V_{ii} \right),$$

(5)

where

• $M_{id} =\left\{\begin{array}{ll} \left. -Q_{id} \right/ Q_{ii} & \hbox{if}\;i\neq d \\ 0 & \hbox{if}\;i=d \\ \end{array} \right.,$

• $\sigma_u^2$ is a measure of overall variance of the u _i ’s and

• V _ii = 1/Q _ii .

For a complete derivation of the above relationships see Wakefield et al. (2001, pp. 124–125). Since Q is symmetric $M_{id} V_{dd} =M_{di} V_{ii}.$ In matrix form the correlation matrix Σ is:

$$\Upsigma =\hbox{Q}^{-1}=\hbox{V}^{-1}\hbox{(I}-\hbox{M)},$$

(6)

where:

• V is a matrix with elements V _ii ,i = 1,…,n and 0 otherwise, and

• M is the matrix of spatial weights M _id .

If the matrix Q has an inverse than the matrix ${\mathbf{U}} =\left[ u_1,\;u_2,\ldots,\;u_n \right]^{T}$ has distribution ${{\mathbf{U}}} \sim N_n \left(\hbox{0}_{\rm n},\,\sigma_u^2 \left(\hbox{I}-\hbox{M} \right)^{-1}\hbox{V}\right).$

To obtain the ICAR model (4) from the general CAR model of (5) set V _ii = 1/a _i and M _id = w _id /a _i . Here Q does not have an inverse. Proof: Each row i of the matrix I–M has a solitary 1 on the diagonal, a _i elements with value −1/a _i , and the remaining elements equal to zero. Since the sum of the elements on each row all equal zero the matrix Q has rank n − 1 <  n, and so is not full rank and is not invertible.

Appendix E

We first define the DIC and then derive the DIC for the model $F_1 =\ln (\mu_k)=\ln (e_k)+u_i +v_k.$ where i and k are defined as in (1). Recall that after 10,000 “burn-in” iterations obtain convergence we obtained 30,000 more iterations for the parameter estimates. The Log Likelihood LL is

$$LL = LL \left(\user2{Y},\left\{\hat {\user2{Y}}|\varpi \right\} \right){\mathbf =} \sum {\left({Y_k \log \hat {Y}_k -\hat {Y}_k -Y_k !} \right)},$$

(7)

where $\user2{Y} = \{Y_k \}$ are the observed counts and $\left\{{\hat {{Y}}|\varpi} \right\} = \{\hat {{Y}}_k \}$ is the set of predicted counts derived from the set of parameter estimates $\varpi$ (Dobson 2002, p. 76). Using (7), the DIC (Spiegelhalter et al. 2002, 2003) is:

$$ \hbox {DIC} = 2\overline{\left(-2 \user2{LL}\left(\user2{Y},\left\{\user2{{\hat {Y}}}|\varpi \right\} \right) \right)} -\left(-2 \user2 {LL}\left(\user2 {Y},\left\{\user2{\hat {Y}}|\bar{\varpi} \right\} \right) \right),$$

(8)

where:

• $\overline {-2 \user2 {LL}\left(\user2 {Y},\left\{\user2{{\hat {Y}}}|\varpi \right\} \right)}$ is the mean of the 30,000 individual iterations of $-2\user2 {LL}\left(\user2{Y},\left\{\user2{{\hat {Y}}}|\varpi \right\} \right)$ and

• $\overline \varpi$ is the posterior mean of the parameter estimates from the 30,000 iterations.

To calculate the DIC for model F ₁, follow these three steps. First, calculate:

$$\overline{-2\user2{LL}\left(\user2{Y},\left\{\user2{{\hat {Y}}}|\varpi \right\} \right)} = -2\sum\limits_{l=1}^{30000} {\;\sum\limits_{k=1}^{18822} \left(\left(Y_k \log \hat {Y}_k^{(l)} -\hat {Y}_k^{(l)} -Y_k ! \right) \right)}/30000,$$

(9)

where

• $l =\hbox{iteration number }(l = 1,2,\ldots,30,000.$ taken after the 10,000 iterations for convergence),

• $\hat {Y}_k^{(l)} = \hbox{the expected number of deaths predicted via the}\,\,l\hbox{th iteration from model }F_1 : \hat {{Y}}_k^{(l)} =e_k\,{\bullet}\,\hbox{exp} (\hat {u}_i^{(l)} +\hat {v}_k^{(l)})),$

• $\hat {u}_i^{(l)} = \hbox{the spatial autocorrelation component predicted via the}\,\,l\hbox{th iteration,}$

• $\hat {{v}}_k^{(l)} = \hbox{the unstructured variation component predicted via the}\,\,l\hbox{th iteration,}$

• $\varpi =\left\{\hat {u}_i^{(l)},\hat {v}_k^{(l)} \right\},$ the set of parameter estimates for iteration k, and

• $\left\{\user2{\hat {Y}}|\varpi \right\} = \hbox{the set of individual}\,\,\hat {Y}_k^{(l)} \hbox{s}.$

Second, calculate

$$-2\user2{LL}\left(\user2{Y},\left\{\user2{{\hat {Y}}}|\bar{\varpi} \right\} \right)= -2\sum\limits_{k=1}^{18822} {\left(Y_k \log \hat {Y}_k^{\bullet} -\hat {Y}_k^{\bullet} -Y_k ! \right)},$$

(10)

where

• $\hat {Y}_k^{\bullet} =\hbox{the estimated number of deaths using}\,\,\bar{\varpi} : \hat {Y}_k^{\bullet} =e_k\,{\bullet}\,\hbox{exp} (\bar{\hat {u}}_i^ +\bar{\hat {v}}_k),$

• $\bar{{\hat {{u}}}}_i =\hbox{the mean of the }\hat {u}_i^{(l)} \hbox{'s}, \bar{\hat {u}}_i =\left. \sum\limits_{l=1}^{30000} \hat {u}_i^{(l)} \right/ 30000,$

• $\bar{\hat {v}}_k =\hbox{the mean of the }\hat {v}_k^{(l)} \hbox{'s}, \bar{\hat {v}}_k =\left. \sum\limits_{l=1}^{30000} \hat {v}_k^{(l)} \right/ 30000,$

• $\bar{\varpi} =\left\{\bar{\hat {u}}_i,\bar{\hat {v}}_k \right\},$ the set of means of the 30,000 parameter estimates, and

• $\left\{\user2{\hat {Y}}|\bar{\varpi} \right\}= \hbox{the set of individual} \hat {Y}_k^{\bullet} \hbox{s}.$

Finally, substitute the results of Eqs. 9 and 10 into Eq. 8 to find the DIC.

Appendix F

SAR models are similar to the AR(1) model in time series. Note that in our model (1) we set $\varepsilon_k =u_i +v_k$ with structured variation u _i independent of unstructured variation v _k . As this is not possible with the SAR model we default to the error term ɛ _k .

For our conditions (1) we can write a SAR model as:

$$\left\{\begin{aligned} {\mathbf{Y}}\varvec{\sim} {\mathbf{Poisson}}\left(\varvec{\lambda} \right) \\ \hbox{ln} (\varvec{\lambda})=\varvec{\alpha} +\hbox{ln} ({\mathbf{e}})+{\mathbf{X}}\varvec{\beta} +\varvec{\varepsilon} \\ \varvec{\varepsilon} =\rho {\mathbf{S}}\varvec{\varepsilon} +\varvec{\eta} \end{aligned} \right.,$$

(11)

where

• Y is the matrix of observed mortality counts as per (1),

• $\varvec{\lambda} =[\lambda_1 \ldots \lambda_{18822}]^{T}, \lambda_k =e_k \bullet\hbox{exp} \left(\alpha +{\mathbf{x}}_{{\mathbf{k}}}^{{\mathbf{T}}} \varvec{\beta} \right)$ for each k with remaining variables defined as per (1), and α is a constant,

• $\varvec{\varepsilon} =[\varepsilon_1 \ldots \varepsilon_{18822} ]^{T}=$ regression error terms,

• ρ is a measure of spatial correlation (−1≤ ρ ≤ 1),

• S is an 18,822 ×  18,822 neighborhood (spatial weighting) matrix with zeroes on the diagonals (not necessarily symmetric), standardized so the row sums add to one, and

• $\varvec{\eta} =[\eta_1 \ldots \eta_{18822} ]^{T}, \eta_k \sim N(0,\sigma^{2}).$

It follows from (10) that $\varvec{\varepsilon} =({\mathbf{I-}}\rho {\mathbf{S}})^{-1}\varvec{\eta}$ and so the covariance matrix Σ for the SAR model is $\Sigma =\sigma^{2}{\mathbf{(I}}-\varvec{\rho} {\mathbf{S)}}^{{\mathbf{-1}}}{\mathbf{(I}}-\varvec{\rho} {\mathbf{S}}^{{\mathbf{T}}}{\mathbf{)}}^{{\mathbf{-1}}}.$

If desired SAR can also use nearest neighbor weighting: Set s _id = w _id /a _i with w _id ,  a _i defined as in (4)

Briefly comparing and contrasting the CAR and SAR models (Arab et al. 2007; Cressie 1993; Whittle 1954; Bao no date):

• CAR sets specifications for the Y _k s conditionally, while SAR does so simultaneously.

• Spatial weighting matrices do not have to be symmetric in the SAR model but do in the CAR model.

• A SAR model can always be restated as a CAR model, but not vice versa.

• The SAR model and the CAR model are the same if and only if the covariance matrices are the same.

• The CAR model is more computationally efficient than the SAR model because the matrix I–M in the CAR model is symmetric while the matrix ${\mathbf{I}}-\varvec{\rho}{\mathbf{S}}$ in the SAR model is not.

• In some cases the spatial weights in the SAR model may not be identifiable.

• Parameter estimates for the SAR model are statistically not consistent. That is, for increasing sample sizes the parameter estimates may not converge with high probability to the actual parameter.

• CAR gives the best (that is, the minimum mean squared prediction error) estimates of Y _k based on all the other $Y_l^{\prime}s,\;l\neq k.$

• If it makes more sense to specify the model conditionally, or if there is a symmetric structure in the correlation matrix, use a CAR model.