Comparing the spatial attractiveness of hospitals using zero-inflated spatial models

Abstract

Policy makers increasingly rely on hospital competition to incentivize patients to choose high-value care. Amongst all possible drivers, the travel distance without any doubt is one of the most important. In this paper we propose the use of a spatial Bayesian hierarchical model to assess the impact of distance on the number of patient admissions in hospitals, and thereby, compare hospital attractiveness. To this aim a MCMC sampler has been designed. We apply our methodology to patient admissions for asthma in four hospitals located in the Hérault department of France. Results indicate that the most attractive hospital is the CHU Montpellier.

Appendix A

In this section, we give the derivation of conditional parameters distributions for the ZIP model. To obtain the derivation for the ZINB model, the distributions used in the ZIP model have to be replaced by those used in the ZINB model. As a reminder, (a) Bayes formula is

$$\begin{aligned} \pi (\theta \mid x) =\frac{\pi (x\mid \theta )\pi (\theta )}{\int \pi (x\mid \theta )\pi (\theta )\,d\theta }\propto \pi (x\mid \theta )\pi (\theta ), \end{aligned}$$

where $\pi (\theta \mid x)$ is called the posterior distribution, $\pi (\theta )$ the prior distribution and $\pi (x\mid \theta )$, the likelihood. b) and the prior distributions used in this case study for the ZIP model are

$$\begin{aligned} \begin{array}{lll} \beta _{0,h}\sim \frac{1}{10^3\sqrt{2\pi }}\exp \left\{ -~\frac{\beta _{0,h}^2}{2*10^6}\right\} &{} {}&{}\beta _{1,h}\sim \frac{1}{10^3\sqrt{2\pi }}\exp \left\{ -~\frac{\beta _{1,h}^2}{2*10^6}\right\} \\ \tau _h\sim \frac{1}{\Gamma (1)}(1/\tau _h)^{2}\exp \{-~1/\tau _h\} &{}{}&{} \omega _h\sim \frac{1}{\Gamma (1)}(1/\omega _h)^{2}\exp \{-~1/\omega _h\}\\ \theta \sim \frac{1}{10^3\sqrt{2\pi }}\exp \left\{ -~\frac{\theta ^2}{2*10^6}\right\} &{} {}&{}{} \end{array} \end{aligned}$$

Let’s call $\pi \{y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau ,\omega \}$ (with $\lambda ({\mathbf {x}})$, $\beta _0$, $\beta _1$, $\tau$ and $\omega$ as defined in Sect. 4), the joint distribution for model parameters and the data. Using the definition of conditional probabilities and assuming parameters independence within their prior distributions, we show that

$$\begin{aligned}&\pi \{y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau ,\omega \} = \pi \{y({\mathbf {x}})\mid \theta ,\lambda ({\mathbf {x}})\}\pi \{\theta \}\\&\quad \times\prod _{h=1}^{4}\pi \{\lambda _h({\mathbf {x}})\mid \beta _{0,h},\beta _{1,h},\tau _h,\omega _h\}\pi \{\beta _{0,h}\}\pi \{\beta _{1,h}\}\pi \{\tau _h\}\pi \{\omega _h\} \end{aligned}$$

(4)

Then using Bayes formula and Eq. 4, we derive the conditional distributions

$$\begin{aligned} \begin{array}{l} \pi \{\theta \mid y({\mathbf {x}}),\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau ,\omega \} \propto \pi \{y({\mathbf {x}})\mid \theta ,\lambda ({\mathbf {x}})\} \pi (\theta )\\ \pi \{\lambda _h({\mathbf {x}})\mid y({\mathbf {x}}),\lambda _{-h}({\mathbf {x}}),\theta ,\beta _0,\beta _1,\tau ,\omega \} \propto \pi \{y({\mathbf {x}})\mid \theta , \lambda ({\mathbf {x}})\}\times \\ \qquad\qquad{} \pi \{\lambda _h({\mathbf {x}})\mid \beta _{0,h},\beta _{1,h},\tau _h,\omega _h\}\\ \pi \{\beta _{0,h} \mid y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _{0,-h},\beta _1,\tau ,\omega \} \propto \pi \{\lambda _h({\mathbf {x}})\mid \beta _{0,h},\beta _{1,h},\tau _h,\omega _h\}\pi (\beta _{0,h})\\ \pi \{\beta _{1,h} \mid y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _{1,-h},\tau ,\omega \} \propto \pi \{\lambda _h({\mathbf {x}})\mid \beta _{0,h},\beta _{1,0},\tau _h,\omega _h\}\pi (\beta _{1,h})\\ \pi \{\tau _h \mid y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau _{-h},\omega \} \propto \pi \{\lambda _h({\mathbf {x}}) \mid \beta _{0,h},\beta _{1,h},\tau _h,\omega _\}\pi (\tau _h)\\ \pi \{\omega _h \mid y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau ,\omega _{-h}\} \propto \pi \{\lambda _h({\mathbf {x}})\mid \beta _{0,h},\beta _{1,h},\tau _h,\omega _h\}\pi (\omega _h) \end{array} \end{aligned}$$

which have to be used in the Gibbs sampling. To obtain the explicit form of these distributions, every term in the right members has to be replaced by its mathematical expression. Thus, let’s denote $n_{1,h}$ the number of locations where we count zero stays for a hospital h and $n_{2,h}$ the number of locations where we have at least one stay in the hospital. From Eq. 1, Sect. 3.1,

$$\begin{aligned}&\pi (y({\mathbf {x}})\mid \theta ,\lambda ({\mathbf {x}})) =\prod _{h=1}^4\prod _{i=1}^{n_{1,h}}\left[ \frac{1}{1+\exp \{-~\theta \}}+\left( 1-\frac{1}{1+\exp \{-~\theta \}}\right) \exp \{-~\lambda _h(x_i)\}\right] \nonumber \\&\quad \times ~\prod _{j=1}^{n_{2,h}}\left[ \left( 1-~\frac{1}{1+\exp \{-\theta \}}\right) \frac{\lambda _h(x_j)^{y(x_j)}}{y(x_j)\text{! }} \exp \{-~\lambda _h(x_j)\}\right] \end{aligned}$$

(5)

and by definition in Sect. 3.2

$$\begin{aligned}&\pi \{\lambda _h({\mathbf {x}})\mid \beta _{0,h},\beta _{1,h},\tau _h,\omega _h\} = (2\pi )^{-n/2}|\gamma _h({\mathbf {x}})|^{-1/2}\nonumber \\&\quad \times ~\prod _{i=1}^{n}\lambda _h^{-1}(x_i)\exp \left\{ -~\frac{1}{2}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] ^T\gamma _h({\mathbf {x}})^{-1}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] \right\} \end{aligned}$$

(6)

By replacing every term by its explicit expression, we obtain

$$\begin{aligned}&\pi (\theta \mid y({\mathbf {x}}),\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau ,\omega ) \propto \prod _{h=1}^4\prod _{i=1}^{n_{1,h}}\left[ \frac{1}{1+\exp \{-~\theta \}}+\frac{\exp \{-~\theta \}}{1+\exp \{-~\theta \}}\exp \{-~\lambda _h(x_i)\}\right] \\ {}&\quad \times \left[ \frac{\exp \{-~\theta \}}{1+\exp \{-~\theta \}}\right] ^{n_{2,h}}\exp \left\{ -~\frac{\theta ^2}{2*10^6}\right\} \\ \\&\pi (\beta _{0,h} \mid y({\mathbf {x}}),\lambda ({\mathbf {x}}),\beta _{0,-h},\beta _1,\tau ,\omega ,\theta )\propto \exp \left\{ -~\frac{1}{2}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] '\gamma _h({\mathbf {x}})^{-1}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] \right\} \\&\quad \times \exp \left\{ -~\frac{\beta _{0,h}^2}{2*10^6}\right\} \\&\pi (\beta _{1,h} \mid y({\mathbf {x}}),\lambda ({\mathbf {x}}),\beta _0,\beta _{1,-h},\tau ,\omega ,\theta )\propto \exp \left\{ -\frac{1}{2}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] '\gamma _h({\mathbf {x}})^{-1}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] \right\} \\&\quad \times \exp \left\{ -\frac{\beta _{1,h}^2}{2*10^6}\right\} \\&\pi (\tau _h \mid y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau _{-h},\omega )\propto \exp \left\{ -~\frac{1}{2}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] '\gamma _h({\mathbf {x}})^{-1}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] \right\} \\ {}&\quad \times ~|\gamma _h({\mathbf {x}})|^{-1/2}(1/\tau _h)^2\exp \{-~1/\tau _h\}\\&\pi (\omega _h \mid y({\mathbf {x}}),\theta ,\lambda ({\mathbf {x}}),\beta _0,\beta _1,\tau ,\omega _{-h})\propto \exp \left\{ -~\frac{1}{2}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] '\gamma _h({\mathbf {x}})^{-1}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] \right\} \\&\quad \times |\gamma _h({\mathbf {x}})|^{-1/2}(1/\omega _h)^2\exp \{-~1/\omega _h\} \\&\pi (\lambda _h({\mathbf {x}})\mid y({\mathbf {x}}),\theta ,\lambda _{-h}(\mathbf (x),\beta _0,\beta _1,\tau ,\omega ) \propto \exp \left\{ -~\frac{1}{2}\left[ \log \lambda _h({\mathbf {x}})-\mu _h({\mathbf {x}})\right] '\gamma _h({\mathbf {x}})^{-1}\left[ \log \lambda ({\mathbf {x}})-\mu ({\mathbf {x}})\right] \right\} \\&\quad \times ~\prod _{k=1}^{n}\lambda _h^{-1}(x_k)\prod _{j=1}^{n_{2,h}}\left[ \frac{\lambda _h(x_j)^{y(x_j)}}{y(x_j)\text{! }} \exp \{-~\lambda _h(x_j)\}\right] \\&\quad \times ~\prod _{i=1}^{n_{1,h}}\left[ \frac{1}{1+\exp \{-~\theta \}}+\frac{\exp \{-~\theta \}}{1+\exp \{-~\theta \}}\exp \{-~\lambda _h(x_i)\}\right] \end{aligned}$$