What belongs where? Variable selection for zero-inflated count models with an application to the demand for health care

Abstract

This paper develops a Bayesian spike and slab model for zero-inflated count models which are commonly used in health economics. We account for model uncertainty and allow for model averaging in situations with many potential regressors. The proposed techniques are applied to a German data set analyzing the demand for health care. An accompanying package for the free statistical software environment R is provided.

Appendix: Details of the sampling algorithm

The proposed Gibbs sampling algorithm consists of the following steps:

• 1. Sample $(\alpha ,\varvec{\beta }^{\prime })^{\prime }$ from $\varphi (\alpha ,\varvec{\beta }|\varvec{\eta }^*,\sigma ^2,\varvec{\tau }^\beta , \varvec{\kappa }^\beta )=\mathrm N [(\alpha ,\varvec{\beta }^{\prime })^{\prime }|\varvec{\mu },\varvec{\varSigma }]$ with variance

  $$\begin{aligned} \varvec{\varSigma }=\left(\varvec{B}^{-1}+\frac{1}{\sigma ^2}\sum _{i=1} ^N{\tilde{{\varvec{x}}}}_i{\tilde{{\varvec{x}}}}_i^{\prime }\right)^{-1} \end{aligned}$$

  (20)

  and mean

  $$\begin{aligned} \varvec{\mu }=\varvec{\varSigma }_{\varvec{\beta }}\left(\frac{1}{\sigma ^2} \sum _{i=1}^N{\tilde{{\varvec{x}}}}_i\eta _i^*\right), \end{aligned}$$

  (21)

  where $\varvec{B}\equiv \mathrm diag (a_0,\tau _1^\beta \kappa _1^\beta , \ldots ,\tau _K^\beta \kappa _K^\beta )$ and ${\tilde{{\varvec{x}}}}_i=(1,{\varvec{{x}}}_i^{\prime })^{\prime }$.

• 2. Draw $\sigma ^2$ from $\varphi (\sigma ^2|\varvec{\eta }^*,\alpha ,\varvec{\beta }) =\text{ Inv-Gamma}\left(e_0+\frac{N}{2},f_0+\frac{\sum _{i=1} ^N(\eta _i^*-\alpha -{\varvec{{x}}}_i^{\prime }\varvec{\beta })^2}{2}\right)$.

• 3. Sample $(\gamma ,\varvec{\delta }^{\prime })^{\prime }$ from $\varphi (\gamma ,\varvec{\delta }|\varvec{d}^*,\varvec{\tau }^\delta ,\varvec{\kappa }^\delta ) =\mathrm N [(\gamma ,\varvec{\delta }^{\prime })^{\prime }|\varvec{\mu },\varvec{\varSigma }]$ with variance

  $$\begin{aligned} \varvec{\varSigma }=\left(\varvec{D}^{-1}+\sum _{i=1}^N{\tilde{{\varvec{x}}}}_i{\varvec{{x}}}_i^{\prime }\right)^{-1} \end{aligned}$$

  (22)

  and mean

  $$\begin{aligned} \varvec{\mu }=\varvec{\varSigma }_{\varvec{\delta }}\left(\sum _{i=1}^N{\tilde{{\varvec{x}}}}_id_i^*\right), \end{aligned}$$

  (23)

  where $\varvec{D}\equiv \mathrm diag (c_0,\tau _1^\delta \kappa _1^\delta ,\ldots , \tau _K^\delta \kappa _K^\delta )$ and ${\tilde{{\varvec{x}}}}_i=(1,{\varvec{{x}}}_i^{\prime })^{\prime }$.

• 4. For $i=1,\ldots ,N$ sample $(\eta _i^*,d_i^*)$: If $y_i=0$, first draw $\eta _i^*$ from

  $$\begin{aligned} \varphi (\eta _i^*|\alpha ,\varvec{\beta },\sigma ^{2},\gamma ,\varvec{\delta })&= \left\{ 1-\Phi (\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta })+\exp [-\exp (\eta _i^*)] \Phi (\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta })\right\} \nonumber \\&\times \mathrm N (\eta ^*_i|\alpha +{\varvec{{x}}}_i^{\prime }\varvec{\beta },\sigma ^2), \end{aligned}$$

  (24)

  where $\Phi (\cdot )$ denotes the standard-normal CDF. Second, draw $d_i^*$ from

  $$\begin{aligned} \varphi (d_i^*|\eta _i^*,\alpha ,\varvec{\beta },\sigma ^{2},\gamma ,\varvec{\delta })&= \frac{\rho _0}{\rho _0+\rho _1}\mathrm TN ^-(\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta },1,0)\nonumber \\&+\frac{\rho _1}{\rho _0+\rho _1}\mathrm TN ^+(\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta },1,0) \end{aligned}$$

  (25)

  with $\rho _0=1-\Phi (\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta })$ and $\rho _1= \Phi (\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta })\exp [-\exp (\eta _i^*)]$. $\mathrm TN ^-(\mu ,\sigma ^2,a)$ and $\mathrm TN ^+(\mu ,\sigma ^2,a)$ denote Normal distributions with mean $\mu$ and variance $\sigma ^2$ that are truncated at the right and at the left at $a$, respectively. If $y_i>0$, first draw $\eta _i^*$ from

  $$\begin{aligned} \varphi (\eta _i^*|\alpha ,\varvec{\beta },\sigma ^{-2},\gamma ,\varvec{\delta }) = \frac{\exp [-\exp (\eta ^*_i)]\exp (\eta ^*_iy_i)}{y_i!} \; \mathrm N (\eta ^*_i|\alpha +{\varvec{{x}}}_i^{\prime }\varvec{\beta },\sigma ^2). \end{aligned}$$

  (26)

  Second, draw $d_i^*$ from $\varphi (d_i^*|\gamma ,\varvec{\delta })=\mathrm TN ^+(\gamma +{\varvec{{x}}}_i^{\prime }\varvec{\delta },1,0)$.

• 5. For $k\!=\!1,\ldots ,K$ sample $\kappa _k^\beta$ from $\varphi (\kappa _k^\beta |\beta _k,\tau _k^\beta )\!=\!\text{ Inv-Gamma} \left(\!g_0^\beta \!+\!\frac{1}{2},h_0^\beta \!+\!\frac{\beta _k^2}{2\tau _k^\beta } \!\right)$.

• 6. For $k=1,\ldots ,K$ sample $\tau _k^\beta$ from

  $$\begin{aligned} \varphi (\tau _k^\beta |\beta _k,\kappa _k^\beta ,\omega ^\beta )= \frac{\omega _{0k}}{\omega _{0k}+\omega _{1k}}\delta _{\nu ^\beta _0} +\frac{\omega _{1k}}{\omega _{0k}+\omega _{1k}}\delta _1 \end{aligned}$$

  with

  $$\begin{aligned} \omega _{0k}=(1-\omega ^\beta )(\nu _0^\beta )^{-\frac{1}{2}} \exp \left(-\frac{\beta ^2_k}{2\nu _0^\beta \kappa _k^\beta }\right) \end{aligned}$$

  and

  $$\begin{aligned} \omega _{1k}=\omega ^\beta \exp \left(-\frac{\beta ^2_k}{2\kappa _k^\beta }\right). \end{aligned}$$

• 7. Sample $\omega ^\beta$ from $\varphi (\omega ^\beta |\varvec{\tau }^\beta )=\mathrm Beta \left(1+\#\{k:\tau _k^\beta =1\} ,1+\#\{k:\tau _k^\beta =\nu ^\beta _0\}\right).$

• 8. For $k\!=\!1,\ldots ,K$ sample $\kappa _k^\delta$ from $\varphi (\kappa _k^\delta |\delta _k,\tau _k^\delta )\!=\!\text{ Inv-Gamma} \left(g_0^\delta +\frac{1}{2},h_0^\delta +\frac{\delta _k^2}{2\tau _k^\delta }\right)$.

• 9. For $k=1,\ldots ,K$ sample $\tau _k^\delta$ from

  $$\begin{aligned} \varphi (\tau _k^\delta |\delta _k,\kappa _k^\delta ,\omega ^\delta ) =\frac{\omega _{0k}}{\omega _{0k}+\omega _{1k}}\delta _{\nu ^\delta _0} +\frac{\omega _{1k}}{\omega _{0k}+\omega _{1k}}\delta _1 \end{aligned}$$

  with

  $$\begin{aligned} \omega _{0k}=(1-\omega ^\delta )(\nu _0^\delta )^{-\frac{1}{2}} \exp \left(-\frac{\delta ^2_k}{2\nu _0^\delta \kappa _k^\delta }\right) \end{aligned}$$

  and

  $$\begin{aligned} \omega _{1k}=\omega ^\delta \exp \left(-\frac{\delta ^2_k}{2\kappa _k^\delta }\right). \end{aligned}$$

• 10. Sample $\omega ^\delta$ from $\varphi (\omega ^\delta |\varvec{\tau }^\delta )=\mathrm Beta \left(1+\#\{k:\tau _k^\delta =1\},1+\#\{k:\tau _k^\delta =\nu ^\delta _0\}\right).$