Spectral adjustment for spatial confounding

Abstract

Adjusting for an unmeasured confounder is generally an intractable problem, but in the spatial setting it may be possible under certain conditions. We derive necessary conditions on the coherence between the exposure and the unmeasured confounder that ensure the effect of exposure is estimable. We specify our model and assumptions in the spectral domain to allow for different degrees of confounding at different spatial resolutions. One assumption that ensures identifiability is that confounding present at global scales dissipates at local scales. We show that this assumption in the spectral domain is equivalent to adjusting for global-scale confounding in the spatial domain by adding a spatially smoothed version of the exposure to the mean of the response variable. Within this general framework, we propose a sequence of confounder adjustment methods that range from parametric adjustments based on the Matérn coherence function to more robust semiparametric methods that use smoothing splines. These ideas are applied to areal and geostatistical data for both simulated and real datasets.

Keywords: COVID-19; Coherence; Conditional autoregressive prior; Matérn covariance; Spatial confounding.

Fig. 1.

Example confounder adjustment for the bivariate Matérn: $X\left(s\right)$ is generated from Matérn with ${\varphi }_x=1$ and $v_x=1$ on a 50 × 50 grid with grid spacing one. The panels show the confounder adjustment $\hat{Z}\left(s\right)$ for ${\varphi }_{xz}={\varphi }_x$ and $v_{xz}=c{v}_x$ for $c\in \{1,3,5\}$. For $c=1,\hat{Z}\left(s\right)=X\left(s\right)$.

Fig. 2.

Correlations in the spectral domain for the simulation study: $\mathrm{c}\mathrm{o}\mathrm{r}\left(X_k^{*},Z_k^{*}\right)$ by ${\omega }_k$ for different kernel bandwidths $\left(\varphi \right)$ and strengths of exposure/confounder dependence $\left({\beta }_{xz}\right)$; the correlations in the spatial domain (over locations) $\mathrm{c}\mathrm{o}\mathrm{r}\left(X_i,Z_i\right)$ are 0.62 when $\varphi =1$ and ${\beta }_{xz}=\mathrm{1,0.80}$ when $\varphi =1$ and ${\beta }_{xz}=\mathrm{2,0.44}$ when $\varphi =2$ and ${\beta }_{xz}=1$, and 0.62 when $\varphi =2$ and ${\beta }_{xz}=2$.

Fig. 3.

Performance for the discrete simulation study: median (solid) and 95% confidence interval (dashed) for ${\beta }_k$ for the standard model (red), semiparametric model with the penalized complexity prior (green), and parametric model (blue) for data generated with dependence between exposure, and confounder controlled by ${\beta }_{xz}$ and kernel bandwidth $\varphi$. The black lines are the true ${\beta }_x=0.5$.

Fig. 4.

Maps for the Scotland lip cancer dataset: (a) standard mortality ratio and (b) mortality rate in the agriculture, fishing and forestry workforce.

Fig. 5.

Effect of the percentage of the workforce employed in agriculture, fishing and forestry on lip cancer in Scotland: posterior mean (solid lines) and 95% credible interval (dashed lines) of $\mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_k\right)$ for the spectral parametric model (black), the spectral semiparametric model with $L=10$ (green), a Poisson regression on the percentage of the workforce employed in agriculture, fishing and forestry (red) and a Poisson regression with residuals modelled as Leroux (blue).

Fig. 6.

(a) Average PM_2.5 ($\left(\mu \mathrm{g}/m^3\right.$) over 2000–16 and (b) the log COVID-19 mortality rate through May 12, 2020. Counties with no deaths are shaded grey.

Fig. 7.

Results for the COVID-19 example: the posterior mean (solid) and 95% credible interval (dashed) of the mortality rate ratio associated with a difference of $1\mu \mathrm{g}/m^3$ of PM_2.5, $\mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_k\right)$. Results are for (a) all $n=3109$ counties and (b) $n=1977$ counties that reported at least 10 confirmed COVID-19 deaths. The standard spatial approach refers to a regression model including all confounders and a spatial Leroux model for county random effects.