A Sparse Areal Mixed Model for Multivariate Outcomes, with an Application to Zero-Inflated Census Data

Abstract

Multivariate areal data are common in many disciplines. When fitting spatial regressions for such data, one needs to account for dependence (both among and within areal units) to ensure reliable inference for the regression coefficients. Traditional multivariate conditional autoregressive (MCAR) models offer a popular and flexible approach to modeling such data, but the MCAR models suffer from two major shortcomings: (1) bias and variance inflation due to spatial confounding, and (2) high-dimensional spatial random effects that make fully Bayesian inference for such models computationally challenging. We propose the multivariate sparse areal mixed model (MSAMM) as an alternative to the MCAR models. Since the MSAMM extends the univariate SAMM, the MSAMM alleviates spatial confounding and speeds computation by greatly reducing the dimension of the spatial random effects. We specialize the MSAMM to handle zero-inflated count data, and apply our zero-inflated model to simulated data and to a large Census dataset for the state of Iowa.

Appendix: Supplementary Materials

1.1 I Multivariate Spatial Effect Reparameterization

For the multivariate sparse areal mixed model (MSAMM), when the design matrices are the same across multivariate outcomes, i.e., X ₁ = X ₂ = ⋯ = X _J, the first and second stages can be written as

$$\displaystyle \begin{aligned} g_j\left\{\mathbb{E}\left(\boldsymbol{y}_j\mid\boldsymbol{\beta}_j,\,\boldsymbol{\delta}_{sj}\right)\right\} & = \mathbf{X}\boldsymbol{\beta}_j+\mathbf{M}\boldsymbol{\delta}_{sj}\;\;\;\;\;\;\;\;\;( j=1,\dots,J)\\ p\left(\boldsymbol{\Delta}\mid\boldsymbol{\Sigma}\right) & = \mathcal{N}\left(\mathbf{0},\,\boldsymbol{\Sigma}\otimes{\mathbf{Q}}_s^{-1}\right), \end{aligned} $$

where $\boldsymbol {\Delta }=\left (\boldsymbol {\delta }_{s1}^{\prime },\dots ,\boldsymbol {\delta }_{sJ}^{\prime }\right )'$, each δ _sj is q × 1, Σ is the J × J covariance matrix, and Q _s is the q × q spatial precision matrix.

Computation can be eased considerably as follows. Let R _s be the upper Cholesky triangle of Q _s, and let ${\mathbf {W}}_s={\mathbf {R}}_s^{-1}$ such that ${\mathbf {W}}_s{\mathbf {W}}_s^{\prime }={\mathbf {Q}}_s^{-1}$. Then, for $\boldsymbol {\Psi }=\left (\boldsymbol {\psi }_{s1}^{\prime },\dots ,\boldsymbol {\psi }_{sJ}^{\prime }\right )'$, each ψ _sj is q × 1, and $\boldsymbol {\Psi }\mid \boldsymbol {\Sigma }\sim \mathcal {N}\left (\mathbf {0},\,\boldsymbol {\Sigma }\otimes {\mathbf {I}}_q\right )$, we have that $\left ({\mathbf {I}}_J\otimes {\mathbf {W}}_s\right )\boldsymbol {\Psi }$ and Δ have the same distribution conditional on Σ. This is easy to see since $\mathbb {E}\left \{\left ({\mathbf {I}}_J\otimes {\mathbf {W}}_s\right )\boldsymbol {\Psi }\right \}=\left ({\mathbf {I}}_J\otimes {\mathbf {W}}_s\right )\mathbb {E}\left (\boldsymbol {\Psi }\right )=\mathbf {0}$ and

$$\displaystyle \begin{aligned} \text{cov}\left\{\left({\mathbf{I}}_J\otimes{\mathbf{W}}_s\right)\boldsymbol{\Psi}\right\} & = \left({\mathbf{I}}_J\otimes{\mathbf{W}}_s\right)\left(\boldsymbol{\Sigma}\otimes{\mathbf{I}}_q\right)\left({\mathbf{I}}_J\otimes{\mathbf{W}}_s\right)'\\ & = \boldsymbol{\Sigma}\otimes{\mathbf{Q}}_s^{-1}. \end{aligned} $$

Hence, the model’s first and second stages can now be written as

$$\displaystyle \begin{aligned} g_j\left\{\mathbb{E}\left(\boldsymbol{y}_j\mid\boldsymbol{\beta}_j,\,\boldsymbol{\psi}_{sj}\right)\right\} & = \mathbf{X}\boldsymbol{\beta}_j+\mathbf{M}{\mathbf{W}}_s\boldsymbol{\psi}_{sj}\;\;\;\;\;\;\;( j=1,\dots, J)\\ p\left(\boldsymbol{\Psi}\mid\boldsymbol{\Sigma}\right) & = \mathcal{N}\left(\mathbf{0},\,\boldsymbol{\Sigma}\otimes{\mathbf{I}}_q\right). \end{aligned} $$

Now suppose that X ₁≠X ₂≠⋯≠X _J. Then we have

$$\displaystyle \begin{aligned} g_j\left\{ \mathbb{E}\left(\boldsymbol{y}_j\mid\boldsymbol{\beta}_j,\,\boldsymbol{\delta}_{sj}\right)\right\} & = {\mathbf{X}}_j\boldsymbol{\beta}_j+{\mathbf{M}}_j\boldsymbol{\delta}_{sj}\\ p\left(\boldsymbol{\Delta}\mid\boldsymbol{\Sigma}\right) & = \mathcal{N}\left[\mathbf{0},\,\left\{\mathbf{R}'\left(\boldsymbol{\Sigma}^{-1}\otimes {\mathbf{I}}_q\right)\mathbf{R}\right\}^{-1}\right], \end{aligned} $$

where $\boldsymbol {\Delta }=\left (\boldsymbol {\delta }_{s1}^{\prime },\dots ,\boldsymbol {\delta }_{sJ}^{\prime }\right )'$, $\mathbf {R}=\mbox{bdiag}\left ({\mathbf {R}}_{s1},\dots , {\mathbf {R}}_{sJ}\right )$, and ${\mathbf {R}}_{sj}^{\prime }{\mathbf {R}}_{sj}={\mathbf {Q}}_{sj}$, where R _sj is the upper Cholesky triangle of Q _sj. For ease of exposition, let J = 2 (the following easily extends to the case when J > 2). The prior distribution of the spatial effects can be written

$$\displaystyle \begin{aligned} \begin{pmatrix} \boldsymbol{\delta}_{s1}\\ \boldsymbol{\delta}_{s2} \end{pmatrix}\mid\boldsymbol{\Sigma} & \;\;\sim\;\; \mathcal{N}\left[\begin{pmatrix} \mathbf{0}\\ \mathbf{0} \end{pmatrix},\,\left\{\begin{pmatrix} {\mathbf{R}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{R}}_{s2} \end{pmatrix}'\left(\boldsymbol{\Sigma}^{-1}\otimes {\mathbf{I}}_q\right)\begin{pmatrix} {\mathbf{R}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{R}}_{s2} \end{pmatrix}\right\} ^{-1}\right]\\ {} & \;\;=\;\; \mathcal{N}\left[\begin{pmatrix} \mathbf{0}\\ \mathbf{0} \end{pmatrix},\,\begin{pmatrix} {\mathbf{W}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{W}}_{s2} \end{pmatrix}\left(\boldsymbol{\Sigma}\otimes {\mathbf{I}}_q\right)\begin{pmatrix} {\mathbf{W}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{W}}_{s2} \end{pmatrix}'\right], \end{aligned} $$

where ${\mathbf {W}}_{sj}={\mathbf {R}}_{sj}^{-1}$ (j = 1, 2), and we have used the fact that $\left ({\mathbf {R}}_{sj}^{-1}\right )'=\left ({\mathbf {R}}_{sj}^{\prime }\right )^{-1}$. Now, suppose we have

$$\displaystyle \begin{aligned} \begin{pmatrix} \boldsymbol{\psi}_{s1}\\ \boldsymbol{\psi}_{s2} \end{pmatrix}\mid\boldsymbol{\Sigma}\;\;\sim\;\;\mathcal{N}\left\{ \begin{pmatrix} \mathbf{0}\\ \mathbf{0} \end{pmatrix},\,\boldsymbol{\Sigma}\otimes {\mathbf{I}}_q\right\}. \end{aligned}$$

Using basic properties of the multivariate normal distribution, we have that

$$\displaystyle \begin{aligned} \begin{pmatrix} {\mathbf{W}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{W}}_{s2} \end{pmatrix}\begin{pmatrix} \boldsymbol{\psi}_{s1}\\ \boldsymbol{\psi}_{s2} \end{pmatrix}\mid\boldsymbol{\Sigma}\;\;\sim\;\;\mathcal{N}\left\{ \begin{pmatrix} \mathbf{0}\\ \mathbf{0} \end{pmatrix},\,\begin{pmatrix} {\mathbf{W}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{W}}_{s2} \end{pmatrix}\left(\boldsymbol{\Sigma}\otimes {\mathbf{I}}_q\right)\begin{pmatrix} {\mathbf{W}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{W}}_{s2} \end{pmatrix}'\right\} . \end{aligned}$$

Then, since

$$\displaystyle \begin{aligned} \begin{pmatrix} {\mathbf{W}}_{s1} & \mathbf{0}\\ \mathbf{0} & {\mathbf{W}}_{s2} \end{pmatrix}\begin{pmatrix} \boldsymbol{\psi}_{s1}\\ \boldsymbol{\psi}_{s2} \end{pmatrix}=\begin{pmatrix} {\mathbf{W}}_{s1}\boldsymbol{\psi}_{s1}\\ {\mathbf{W}}_{s2}\boldsymbol{\psi}_{s2} \end{pmatrix}, \end{aligned}$$

we can apply a reparameterization similar to the case where design matrices are equivalent across the outcomes. Thus we can specify the first and second stages of the model as

$$\displaystyle \begin{aligned} g_j\left\{ \mathbb{E}\left(\boldsymbol{y}_j\mid\boldsymbol{\beta}_j,\,\boldsymbol{\psi}_{sj}\right)\right\} & = {\mathbf{X}}_j\boldsymbol{\beta}_j+{\mathbf{M}}_j{\mathbf{W}}_{sj}\boldsymbol{\psi}_{sj}\\ p\left(\boldsymbol{\Psi}\mid\boldsymbol{\Sigma}\right) & = \mathcal{N}\left(\mathbf{0},\,\boldsymbol{\Sigma}\otimes {\mathbf{I}}_q\right). \end{aligned} $$

1.2 I Extended Simulation Results

Table 3 provides complete results for our simulation study.

Table 3 Extended results for our simulation study