Generalized propensity score approach to causal inference with spatial interference

Abstract

Many spatial phenomena exhibit interference, where exposures at one location may affect the response at other locations. Because interference violates the stable unit treatment value assumption, standard methods for causal inference do not apply. We propose a new causal framework to recover direct and spill-over effects in the presence of spatial interference, taking into account that exposures at nearby locations are more influential than exposures at locations further apart. Under the no unmeasured confounding assumption, we show that a generalized propensity score is sufficient to remove all measured confounding. To reduce dimensionality issues, we propose a Bayesian spline-based regression model accounting for a sufficient set of variables for the generalized propensity score. A simulation study demonstrates the accuracy and coverage properties. We apply the method to estimate the causal effect of wildland fires on air pollution in the Western United States over 2005–2018.

1 |. INTRODUCTION

Understanding spatial processes in the environmental and health sciences has taken on new importance as we grapple with emerging ecological and epidemiological issues. Much of the research in these areas are associative in nature despite the effects of interests being causal (Bind, 2019). This is not only a result of both the frequent necessity of using observational data, but also the difficulty of implementing causal inference tools on data that exhibit spatial dependence and, in particular, interference. Interference is the phenomenon in which exposures or treatments at one location may affect the response at other locations. Naturally, with spatially dependent processes, an exposure may impact the response nearby, leading to interference.

An example of spatial interference is the relationship between wildland fires and air pollution. Treating wildland fires as the exposure and pollution as the response, it is clear that the exposure can substantially impact the response at the location of exposure and at distant locations. In this example, all available data are observational, and therefore isolating average causal exposure effects requires accounting for confounding variables. Even in the ideal case where all potential confounders are observed across locations, it is unclear how to condition on these confounders without knowing their specific spatial relationships with the exposure and response. Conditioning on all locations, which is one way around this, is impractical for all but the smallest studies.

The difficulty that arises from interference in the context of spatially dependant processes is immediately apparent from the vantage of the potential outcomes framework developed by Rubin (1974). For a binary exposure without interference, there are two unit-level potential outcomes to consider. Under general exposure interference, there are $2^n$ unit-level potential outcomes to consider, where $n$ is the total number of units, because each exposure permutation across all units represents a distinct exposure. In the case of geostatistical models that contain uncountably many spatial locations, the problem becomes even more intractable. For this reason, beginning with Cox (1958), much of the causal inference literature assumes away interference. The no-interference assumption is usually invoked as part of the stable unit treatment value assumption (Rubin, 1980).

Relaxations to the no-interference assumption generally involve placing assumptions on the form of interference. Partial interference, a term coined by Sobel (2006), was the first relaxation developed, specifically for modeling vaccination exposures that are known to induce herd immunity. This assumption defines disjoint groups or clusters a priori that may exhibit interference, but precludes interference between groups. This form of interference was originally considered with experimental data by Halloran and Struchiner (1991, 1995), but expanded to nonrandomized data by Hudgens and Halloran (2008), Tchetgen and VanderWeele (2012), Liu and Hudgens (2014), and Papadogeorgou et al. (2019). The dual nature of this form of interference allows for information on both the direct exposure effects as well as the indirect or spill-over effects from interference. Additionally, the deluge of network data has resulted in a literature that allows for interference along edges of a pre-specified graph (Aronow & Samii, 2017; Athey et al., 2018; Forastiere et al., 2018, 2020; Karwa & Airoldi, 2018). In a method largely analogous to ours, Forastiere et al. (2018) use a generalized propensity score method to estimate the effect of individual exposures, as well as the effects from “neighborhood” interference-that is, interference effects from subjects connected on the prespecified graph. Aronow and Samii (2017) define an exposure mapping similar to that of Manski (2013), which allows for arbitrary interference forms.

Spatially indexed data have been analyzed using both the partial interference and network interference strategies. For naturally clustered spatial data, the partial interference assumption can be used, for example, as in Zigler and Papadogeorgou (2021), Zigler et al. (2012), and Perez-Heydrich et al. (2014). Spatial data can also be simplified to the network setting. For areal data, this often entails creating a graph with edges between neighboring units, as in Verbitsky-Savitz and Raudenbush (2012). This, however, discards information about the distance between units. Additional causal techniques for spatial data are outlined in Papadogeorgou et al. (2018), Papadogeorgou and Dominici (2020), and Schnell and Papadogeorgou (2020).

Despite these advances, there has been little exploration of strictly spatial forms of interference. To fill this gap in the literature, we propose a new framework to recover causal direct and spill-over effects in the presence of spatial interference, while taking into account the high dimensionality of the problem. We develop a generalized propensity score to account for spatial dependence in the distribution of exposure (Hirano & Imbens, 2004; Imbens, 2000; Imai & Van Dyk, 2004). To further reduce the size of the problem, we propose a model that accounts for a sufficient set of summary variables rather than the full generalized propensity score. Notably, the exposure mapping of Aronow and Samii (2017) can be given spatial form, but the mapping is assumed to be known a priori.

The proposed approach has a number of advantages over using a partial interference or network interference assumption. The partial interference assumption is only reasonable for limited cases when the data naturally cluster a significant distance apart. Moreover, the partial interference grouping must be specified a priori. The network interference assumption requires that spill-over treatments propagate along a prespecified network, which can allow for spill-over effects that are not monotonic over distance-a key feature of our spatial method. Of particular note, Forastiere et al. (2018) use a generalized propensity score similar to ours over a network. Using a “community detection” algorithm, the “neighborhood” that contributes to the spill-over effect can be assessed from the data, after which weighting between points can be given with a prespecified “exposure mapping” function. In contrast, our method simultaneously estimates the spatial range and the weighting of the nearby points, using an assumption that their spill-over effect is proportional to a kernel of distance. Finally, the highly general methodology of Aronow and Samii (2017) again assumes a known exposure mapping, which would not allow estimating the range of exposure. Our proposed method retains all spatial information, and allows for the kernel range to be estimated concurrently.

2 |. POTENTIAL OUTCOMES, INTERFERENCE, AND IDENTIFICATION

Assume that data are available at $n$ spatial locations $s\in \left\{s_1,\dots ,s_n\right\}\subset 𝒟\subset {\mathbb{R}}^2$. For spatial location $s$, define $X_s\in {\mathbb{R}}^p$ as the relevant covariates and $Y_s\in {\mathbb{R}}^1$ the response. We will consider both real-valued and binary exposures $A_s$. We use subscript ${}_{𝒟}$ to refer to the full fields of random variables, for example, $X_{𝒟}=\left\{X_s:s\in 𝒟\right\}$. Variables with subscript ${}_{-s}$ denote all locations in $𝒟$ excluding $s$. Lowercase letters refer to realizations of the variables. For notational convenience, in this section we assume a single realization of the process. We also apply this model for data with multiple time steps by assuming temporal independence and that the model has the same structure over time.

Without restrictions, the response $Y_S$ is potentially a function of $X_{𝒟}$ and $A_{𝒟}$ at all locations, greatly increasing the number of potential outcomes. To make this manageable while still taking spatial interference into account, we assume that the potential outcome $Y_s\left(a_{𝒟}\right)$ depends on exposure field $a_{𝒟}$ through two mechanisms; a direct exposure, $a_s$, and an indirect/spill-over exposure, ${\tilde{a}}_{\tau ,s}={\int }_{𝒟\setminus s}{\omega }_{\tau }\left(∥s-s^{\prime }∥\right)a_{s^{\prime }}\mathrm{d}{s}^{\prime }$, where ${\omega }_{\tau }(\cdot ):{\mathbb{R}}^{+}\mapsto \left[\mathrm{0,1}\right]$ is a kernel function with bandwidth $\tau >0$. This constitutes a general class of interference structures, although we will focus primarily on a Gaussian kernel form: ${\omega }_{\tau }\left(d\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left\{-(d/\tau {)}^2\right\}$, which decays smoothly over space. This choice of kernel, which is an assumption on how dependency decays over space, will be relaxed in Section 7. Because only finitely many locations are observed in practice, the integral form of ${\tilde{a}}_{\tau ,s}$ must be approximated with a sum. We approximate ${\tilde{a}}_{\tau ,s}$ with the form ${\tilde{a}}_{\tau ,s}={\sum }_{s^{\prime }\in \left\{s_1,\dots ,s_n\right\}\setminus s}{\omega }_{\tau }\left(∥s-s^{\prime }∥\right)a_{s^{\prime }}$.

Implicitly, we assume that for any $s$ and exposures $a_{𝒟}$ and $a_{𝒟}^{\prime },Y_s\left(a_{𝒟}\right)=Y_s\left(a_{𝒟}^{\prime }\right)$ if $a_s=a_s^{\prime }$ and ${\tilde{a}}_s={\tilde{a}}_s^{\prime }$. This simplified exposure allows us to parsimoniously define the individual potential outcomes for all possible exposure fields $a_{𝒟}$ in terms of only the local direct and spill-over exposures: $Y_S\left(a_s,{\tilde{a}}_{\tau ,s}\right)$. This is a crucial assumption about the effect of exposure, which substantially reduces the dimension of the problem. In particular, it assumes that any exposure at a given distance from location $s$ (where distance is interpretted by the kernel function) has the same effect on the response. This can be thought of as a spatial analog to the “stratified interference” assumption introduced by Hudgens and Halloran (2008), in which only the aggregated exposures within a cluster contribute to interference rather than the full set of individual exposures.

Identification of the exposure effects follows from the following assumptions:

Assumption 1 (Unconfoundedness). For all $a_{𝒟},Y_S\left(a_{𝒟}\right)=Y_s\left(a_s,{\tilde{a}}_{\tau ,s}\right)⫫A_{𝒟}\mid X_{𝒟}$.

Assumption 2 (Positivity). $\forall$ with, $\text{pr}\left(X_{𝒟}=\right)>0,\mathrm{p}\mathrm{r}\left(A_{𝒟}=a_{𝒟}\mid X_{𝒟}=\right)>0,\forall a_{𝒟}$.

Assumption 3 (Consistency). The potential outcome $Y_s\left(a_s,{\tilde{a}}_{\tau ,s}\right)=Y_s$ when $A_s=a_s,{\tilde{A}}_s={\tilde{a}}_{\tau ,s}$.

For finite $𝒟$, with only the assumptions above, exposures effects theoretically are identifiable. However, identification requires the number of repeated field observations to be at least $2^n$, which is rare. To make the situation tractable, we make one additional assumptions about our data:

Assumption 4 (Marginal Structural Model). The potential outcomes model take the form

$$Y_s\left(a_s,{\tilde{a}}_{\tau ,s}\right)={\beta }_0+{\delta }_1{a}_s+{\delta }_2{\tilde{a}}_{\tau ,s}+h{\left(X_{𝒟}\right)}_s+{ϵ}_s,$$ (1)

where $h\left(X_{𝒟}\right)$ is a general function of $X_{𝒟}$, and ${ϵ}_s$ is an error process that is independent of $A_{𝒟}$ and $X_{𝒟}$.

We will focus exclusively on i.i.d. error, however, dependence (e.g., spatial) can be incorporated into ${ϵ}_s$. Here ${\delta }_1$ and ${\delta }_2$ quantify the direct and spill-over effects of exposure, respectively. Specifically, ${\delta }_1$ and ${\delta }_2$ represent the causal effect of a unit-increase in the direct and spill-over effects:

$$\mathrm{E}\left\{Y_s\left(1,{\tilde{a}}_s\right)-Y_s\left(0,{\tilde{a}}_s\right)\right\}={\delta }_1,$$

$$\mathrm{E}\left\{Y_s\left(a_s,c+1\right)-Y_s\left(a_s,c\right)\right\}={\delta }_2.$$

Similarly, $\tau$ quantifies the range of the spill-over effect.

Under Assumptions 1–4, the coefficients in (1) are identifiable in the sense that

$$\begin{gathered} \mathrm{E}\left\{Y_s\left(a_{𝒟}\right)\mid X_{𝒟}\right\}=\mathrm{E}\left\{Y_s\left(a_{𝒟}\right)\mid X_{𝒟},A_s=a_s,\tilde{A}=\tilde{a}\right\} \\ =\mathrm{E}\left(Y_s\mid X_{𝒟},A_s=a_s,\tilde{A}=\tilde{a}\right). \end{gathered}$$ (2)

The first equality follows from Assumptions 1 and 2; the second from Assumptions 3 and 4.

It is instructive to consider the dependence that is created by Assumptions 1–4. $X_{𝒟}$ is unrestricted, and is therefore plausibly spatially correlated. Because the direct exposure mechanism is a function of $X_{𝒟},A_{𝒟}$ will likely reflect any spatial structure in $X_{𝒟}.Y_{𝒟}$ may reflect both general spatial dependence from $X_{𝒟}$ as well as any induced spatial dependence from $A_{𝒟}$.

3 |. THE GENERALIZED PROPENSITY SCORE IS A BALANCING SCORE

The identification formula (2) implies that we can estimate ${\delta }_1,{\delta }_2$, and $\tau$ using the regression model

$$Y_s={\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}+h{\left(X_{𝒟}\right)}_s+{ϵ}_s,$$

if $h(\cdot )$ is known and ${ϵ}_S$ is a mean zero error process. In most cases, though, $h\left(X_{𝒟}\right)$ is not known. The standard causal inference strategy at this point is to condition on $X_{𝒟}$ itself, if known. However, even when $X_{𝒟}$ is known, in the context of spatial analysis it is high dimensional. Specifically, for unit $s$ it does not suffice to condition on $X_s$, but requires conditioning on $X$ at all locations. With both high-dimensional confounders as well as our assumptions about the exposure mechanism, the natural path forward is to condition on the propensity of exposure (Rosenbaum & Rubin, 1983).

In a setting without interference, and thus only direct exposure effects, the standard propensity score $e_S$ for binary exposures is defined as $e_s\left(X_{𝒟}\right)=P\left(A_s=1\mid X_{𝒟}\right)$. This is easily extended to real valued exposures using the form $e_s\left(X_{𝒟}\right)=f\left(A_s=\eta \mid X_{𝒟}\right),\eta \in \mathbb{R}$. In both cases, $e_s$ simply summarizes the conditional distribution of exposure. The propensity score is an example of a balancing score: a function of the covariates that, once conditioned on, induces independence between the exposure and covariates. If all confounders are included in $X$, then $e_S$, rather than $X_{𝒟}$, may be conditioned on for unbiased exposure effects. When $X_{𝒟}$ is high dimensional, as in our motivating example, this is a substantial dimension reduction.

Under interference, with exposure components $a_s$ and ${\tilde{a}}_{\tau ,s}$, the propensity score approach can still be utilized, by defining the propensity of exposure to be a summary of the conditional distribution of $\left(A_s,{\tilde{A}}_s\right)$. To this end, we define ${\mathrm{g}}_{\tau ,s}$ to be the joint propensity of $A_s$ and ${\tilde{A}}_{\tau ,s}$:

$$g_{\tau ,s}\left(X_{𝒟}\right)=f\left(A_s=\eta ,{\tilde{A}}_{\tau ,s}=v\mid X_{𝒟}\right),\eta ,v\in \mathbb{R}.$$ (3)

We refer to the bivariate density function $g_{\tau ,s}$ as the generalized propensity score (Bia et al., 2014; Hirano and Imbens, 2004; Imbens, 2000; Imai & Van Dyk, 2004; Kluve et al., 2012; Lechner, 2001). Importantly, this form of $g_{\tau ,s}$ allows for $A_{𝒟}$ to be correlated, which may cause dependence between $A_s$ and ${\tilde{A}}_{\tau ,s}$.

The key insight is that $g_{\tau ,s}$ is a balancing score. This implies that, paired with our no unmeasured confounders assumption, the observed exposures and potential outcomes are independent conditional on $g_{\tau ,s}$. This is the strategy that we use to recover unbiased estimates of our key coefficients ${\delta }_1$ and ${\delta }_2$. Theorem 1 shows this formally, by extending the analogous result for propensity scores for continuous exposures by Hirano and Imbens (2004) to our score $g_{\tau ,s}$.

Theorem 1 ($g_{\tau ,s}$ is a balancing score). Given Assumptions 1—4, then for all locations $s$ and spill-over exposure levels $v$,

$$Y_s\left(a_{𝒟}\right)=Y_s\left(a_s,{\tilde{a}}_{\tau ,s}=\nu \right)⫫\left(A_s,{\tilde{A}}_{\tau ,s}\right)\mid g_{\tau ,s}\left(X_{𝒟}\right).$$

The proof is provided in the supporting information.

By Theorem 1, it suffices to adjust for $g_{\tau ,s}$ to remove confounding bias. That is,

$$\mathrm{E}\left\{Y_s\left(a_{𝒟}\right)\mid g_{\tau ,s}\left(X_{𝒟}\right)\right\}=\mathrm{E}\left\{Y_s\mid g_{\tau ,s}\left(X_{𝒟}\right),A_s=a_s,{\tilde{A}}_{\tau ,s}={\tilde{a}}_{\tau ,s}\right\}.$$

This suggests that we can adjust for confounding by incorporating $g_{\tau ,s}$ into the regression model.

4 |. MODELING THE GENERALIZED PROPENSITY SCORE

Estimating $g_{\tau ,s}$ is difficult. It is a bivariate distribution function over $X_{𝒟}$, and non-parametric estimation of even univariate density functions suffers from dimensionality issues. To overcome this, we make the following dimension reduction assumption.

Assumption 5 ($g_{\tau ,s}$ is a parametric distribution). $g_{\tau ,s}$ is a bivariate parametric density with parameters ${\bar{Z}}_s=\left(Z_s^{\left(1\right)},\dots ,Z_s^{\left(K\right)}\right)$ that are a functions of $\tau$ and $X_{𝒟}$.

That is, the distribution of $\left(A,{\tilde{A}}_{\tau ,\mathrm{s}}\right)$ can be completely summarized by the parameters ${\bar{Z}}_s$.

Example 1. If $A_{𝒟}$ are independent and Gaussian, then ${\tilde{A}}_{\tau ,s}$ is itself Gaussian. Setting $Z_s^1,\dots ,Z_s^4$ to be the mean and variance of both $A_s$ and ${\tilde{A}}_{\tau ,s}$ completely summarizes its distribution.

Corollary 1. Given Assumptions 1–5, then for all locations $s$

$$Y_s\left(a_{𝒟}\right)=Y_s\left(a_s,{\tilde{a}}_{\tau ,s}\right)⫫\left(A_s,{\tilde{A}}_{\tau ,s}\right)\mid {\bar{Z}}_s.$$

Corollary 1 follows immediately from Theorem 1 and states that conditioning on ${\bar{Z}}_s$ is equivalent to conditioning directly on the distribution $g_{\tau ,s}$, and so our Theorem 1 result of unconfoundedness given $g_{\tau }$ extends to the considerably more tractable situation of unconfoundedness given $\bar{Z}$.

Identification of ${\delta }_1$ and ${\delta }_2$ follows from the conditional independence in Corollary 1. Let * denote true values; variables without * being estimated values. Based on

$$\begin{gathered} \mathrm{E}\left\{Y_s\left(a_{𝒟}\right)\mid {\bar{Z}}_s\right\}={\beta }_0^{*}+{\delta }_1^{*}{a}_s+{\delta }_2^{*}{\tilde{a}}_{\tau ,s}+\mathrm{E}\left\{h{\left(X_{𝒟}\right)}_s\mid {\bar{Z}}_s\right\} \\ =\mathrm{E}\left\{Y_s\left(a_{𝒟}\right)\mid A_{𝒟}=a_{𝒟},{\bar{Z}}_s\right\} \\ =\mathrm{E}\left(Y_s\mid A_{𝒟}=a_{𝒟},{\bar{Z}}_s\right) \\ =\mathrm{E}\left({\beta }_0+{\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}\mid A_{𝒟}=a_{𝒟},{\bar{Z}}_s\right), \end{gathered}$$ (4)

we must have ${\delta }_1={\delta }_1^{*}$ and ${\delta }_2={\delta }_2^{*}$.

In practice, we will condition on the components of $g_{\tau ,s}$ using B-splines, by including them directly in the regression:

$$\begin{gathered} \mathrm{E}\left({\beta }_0+{\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}\mid A_{𝒟}=a_{𝒟},{\bar{Z}}_s\right)\approx {\beta }_0+{\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}+\mathrm{s}\mathrm{p}\mathrm{l}\left({\bar{Z}}_s\right) \\ \approx {\beta }_0+{\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}+{\mathrm{s}\mathrm{p}\mathrm{l}}_1\left(Z_s^{\left(1\right)}\right)+\cdots +{\mathrm{s}\mathrm{p}\mathrm{l}}_q\left(Z_s^{\left(K\right)}\right). \end{gathered}$$ (5)

This allows for completely arbitrary dependence between $\bar{Z}$ and $Y$. The second line of (5) implicitly assumes that the spline components enter additively, an assumption that can be tested. In the presence of nonadditivity, a tensor product of the components should be used that allows for general interactions, but at great computational cost (Wood, 2006).

5 |. BAYESIAN INFERENCE AND COMPUTATIONAL ALGORITHM

The identification results (4) and (5) allow unbiased estimation of ${\delta }_1$ and ${\delta }_2$ using a regression of the observed response onto the direct and spill-over exposures as well as the spline estimates of ${\bar{Z}}_s\left(\tau \right)$. Implementing this involves three steps: Step 1 parametrizes and estimates the propensities $g_{\tau ,s}$ of direct and spill-over exposure. Step 2 estimates a preliminary posterior for the range parameter $\tau$, which must be done in a separate step for reasons discussed below. Step 3 estimates final posterior distributions for all parameters via Markov chain Monte Carlo sampling. This is illustrated in Figure 1.

FIGURE 1.

Schematic plot for computation of the proposed method. Here $f(\cdot )$ refers to a generic probability density/mass function, and $spl$ refers to a B-spline basis

The propensities of direct exposure that are tackled in Step 1 are first estimated by regressing $A_s$ onto $X$. This requires parametrizing the form of $f\left(A\right)$, and identifying a correctly specified propensity score. The form of this score can vary in complexity. The simplest case is that of a local exposure assignment mechanism, that is, the distribution of $A_S$ is influenced by $X_S$ only. This would simply entail a regression on local covariates. A moderately complex case would allow for nearby $X$ to inform the propensity of exposure. A very general case would allow $A_{𝒟}$ to be spatially dependent, conditional on $X_{𝒟}$. That is, $A_s$ would depend directly on nearby $A$.

Estimating the spill-over propensity component of Step 1 is similar. First, a family of parametric distributions must be identified. One intuitive method of doing this is to select several candidate distributions based on the form of $A_s$, and select among them by simulating values of ${\tilde{A}}_s$. For example, if $A_S$ is binary, then the potential candidates for the distribution of $\tilde{A}$ must be nonnegative and allow for point mass at zero. Obvious contenders are zero-inflated lognormal and zero-inflated Gamma distributions. A natural way to select between them is to simulate from the estimated propensities of $A_s$, to get simulated ${\tilde{A}}_s$ values using different reasonable $\tau$. The empirical distributions of these simulated ${\tilde{A}}_s$ will often suggest one family of distributions. With a chosen distribution in hand, the parameters $\bar{Z}$ at each location can be estimated directly from the field $X_{𝒟}$ and $\tau$. Because these parameters will be conditioned on by entering into a splined regression, it is advantageous that their values have reasonable spread. To this end, one-to-one transformations of the parameters such as log and logit are helpful.

Step 2 involves identifying a plausible set of $\tau$ values to be used in Step 3. Because $\bar{Z}\left(\tau \right)$ represents a propensity score, estimating $\tau$ directly in the final model is problematic. It is clear from the definition of a propensity score that the response $Y$ should not provide any information on the propensity of exposure. However, estimating a response model that includes $\bar{Z}\left(\tau \right)$ directly does just that, since $Y$ can influence $\bar{Z}\left(\tau \right)$ through $\tau$. This problem is articulated in McCandless et al. (2010), Saarela et al. (2015), Saarela et al. (2016) Zigler et al. (2013), and Zigler (2016), while steps can be taken to mitigate feedback from $Y$ to $\bar{Z}$ issues remain.

Our solution to this issue takes inspiration from the standard two-step propensity score treatment in which propensity scores are first estimated and treated as fixed, and then conditioned on in an outcome model. Because $\tau$ is unknown, estimating $\bar{Z}\left(\tau \right)$ in advance is impossible. However, estimating the model with feedback shown below does give approximate estimates of $\tau$. From this approximate posterior of $\tau$, a set of reasonable $\tau$ values $\left({\tau }_1,\dots ,{\tau }_T\right)$ covering the plausible range of $\tau$ can be identified. Then $\bar{Z}\left({\tau }_1\right),\dots ,\bar{Z}\left({\tau }_T\right)$ can be pre-computed and conditioned on simultaneously in the response model in Step 3. Because each of these $\bar{Z}\left({\tau }_t\right)$ are computed before the response model, the feedback issue is resolved.

Therefore, in Step 2 we estimate

$$Y_s={\beta }_0+{\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}+\sum _{k=1}^K{\text{spl}}_k\left(Z_s^{\left(k\right)}\left(\tau \right)\right)+{ϵ}_s,$$ (6)

where ${ϵ}_s$ is distributed independent $\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\left(0,{\sigma }^2\right)$. An attempt to cut the feedback from $Y$ to $\bar{Z}$ is made by estimating $\tau$ in the Metropolis step using only ${\tilde{A}}_{\tau }$ while holding $\bar{Z}\left(\tau \right)$ fixed. A recommended plausible set for $\tau$ might then be $\{\widehat{\tau },\widehat{\tau }\pm 2s,\widehat{\tau }\pm 4s\}$, where $\widehat{\tau }$ and $s$ are the posterior mean and standard deviation of $\tau$ in (6).

Finally in Step 3 each fixed $\bar{Z}\left({\tau }_t\right)$ enters the final model as

$$Y_s={\beta }_0+{\delta }_1{A}_s+{\delta }_2{\tilde{A}}_{\tau ,s}+\sum _{t=1}^T\sum _{k=1}^K{\text{spl}}_{\mathit{\text{tk}}}\left(Z_s^{\left(k\right)}\left({\tau }_t\right)\right)+{ϵ}_s.$$ (7)

This model produces accurate posteriors on all variables. Although each ${\tau }_t$ is fixed within the $\bar{Z}$ terms, $\tau$ can still vary within ${\tilde{A}}_{\tau ,s}$. For the spline terms in (6)-(7), we use B-spline expansions taken at fixed intervals over the variables’ range of values (Eilers & Marx, 1996; Ngo & Wand, 2004). Before running the regression, we remove any redundant $Z_s^{\left(k\right)}$ that are collinear. Specifically, the spline columns that the R “Im” function selects to remove due to collinearity when running the analogous regression, we remove as well (R Core Team, 2018). All regression coefficients are estimated using Gibbs sampling; $\tau$, which now enters only through ${\tilde{A}}_{\tau }$, uses a Metropolis step. If Assumptions 1–5 hold, we recover unbiased estimate of the exposure effects. Comparing the forms of the assumed true model (1) and the estimated model (7) shows that we have essentially replaced the unknown $h\left(X_{𝒟}\right)$ with flexible functions of $\bar{Z}$.

6 |. SIMULATION STUDY

We examine the performance of this method using simulated data, which take inspiration from the wildfire/air pollution data in Section 7. Since we use a binary exposure in Section 7 to indicate the presence of a fire, we use $A_s\in \left\{\mathrm{0,1}\right\}$ here. In addition, we assume $A_s$ at different locations is independent conditional on local $X_s$. This precludes the more complex cases of independence conditional on $X_{𝒟}$ or conditional dependence. Doing this allows for more straightforward modeling of $g_{\tau ,s}$, as shown in 6.1.

We generate the data as follows. Fields $X_{𝒟},A_{𝒟}$, and $Y_{𝒟}$ are generated on $n^{1/2}\times n^{1/2}$ grids, with $n=\mathrm{25,100}$ on the unit square $\left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right]$. We generate $N=100$ independent repeated observations of the fields for each dataset. Thus each complete dataset involves $n\times N$ different data points. The single covariate $X_s\in {\mathbb{R}}^1$ is a mean zero, variance one, Gaussian process and with isotropic exponential covariance and spatial range 0.6. The binary direct exposure $A_s$ is determined locally and distributed independently $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}\{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left.\left(X_S-3\right)\right\}$. The spill-over exposure takes the form ${\tilde{A}}_{\tau ,s}={\sum }_{s^{\prime }}{\omega }_{\tau }\left(∥s-s^{\prime }∥\right)A_{s^{\prime }}$, with ${\omega }_{\tau }$ a Gaussian kernel with $\tau =0.3$. Several confounders $h{\left(X_{𝒟}\right)}_s$ are investigated: a weighted average $W_s$ is taken of the $X_{𝒟}$ values using a Gaussian kernel with $\tau =0.5$ and weights normalized to sum to 1. Simulations are run with $h\left(X_{𝒟}\right)$ set to $W_s,-{\left(W_s\right)}^3$, and $\mathrm{e}\mathrm{x}\mathrm{p}\left(W_S\right)$. Additionally, a final $h\left(X_{𝒟}\right)$ is included, which consists of two small Gaussian kernels $(\tau =0.15)$, that are centered two grid cells diagonally to the northeast and southwest of $s$, respectively: ${\tilde{A}}_s={\sum }_{s^{\prime }\in \left\{s_1,\dots ,S_n\right\}\setminus s^{\prime }}\left\{{\omega }_{\tau }(\parallel s-\right.\left.\left.s^{\prime }+u\parallel \right)a_{s^{\prime }}+{\omega }_{\tau }\left(∥s-s^{\prime }-u∥\right)a_{s^{\prime }}\right\}$, where $u=(1/\mathrm{9,1}/9)$ is the distance between adjacent points and ${\omega }_{\tau }$ is a Gaussian kernel. This is included to assess sensitivity to anomalous $h\left(X_{𝒟}\right)$.

Lastly, $Y_s$ follows the form of (1), with ${\beta }_0=0,{\delta }_1={\delta }_2=1$, and ${ϵ}_S$ independently distributed standard normal. Each setting is repeated 500 times.

6.1 |. Estimation

Following the three steps outlined in Section 5, we first parametrize and estimate $g_{\tau ,s}$. Because $A_s$ is assumed to be conditionally independent given $X_S$, we can estimate $Z$ components for the distributions of $A$ and $\tilde{A}$ separately. $A_s$ is binary, so we assume it has a Bernoulli distribution with the correctly specified propensity in which $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left\{\mathrm{E}\left(A_s\right)\right\}$ is affine in $X_s$. Its distribution is then captured with the standard propensity score $Z_s^{\left(1\right)}=\mathrm{p}\mathrm{r}\left(A_s=1\mid X_s\right)$. Estimates are obtained with a simple logistic regression from $A_s$ onto $X_S$, with $Z_s^{\left(1\right)}$ set to the log of the fitted values.

We then choose a parametric form for the distribution of ${\tilde{A}}_{\tau ,s}$. From our estimated $Z_s^{\left(1\right)}$, we use different plausible $\tau$ values to generate simulated A, which we then use to get an empirical distribution of simulated $\tilde{A}$. Examination of these distributions leads us to choose a zero-inflated lognormal distribution for ${\tilde{A}}_s$:

$$\mathrm{p}\mathrm{r}\left(\tilde{A}=0\mid X_s\right)=p_0,\mathrm{p}\mathrm{r}\left(\tilde{A}=v\mid \tilde{A}>0,X_s\right)=\frac{1}{v\sigma \sqrt{2\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\frac{(\mathrm{l}\mathrm{o}\mathrm{g}v-\mu {)}^2}{2{\sigma }^2}\right\}.$$

Rather than use the three parameters $p_0,\mu$, and ${\sigma }^2$ for our $Z_s^{\left(2\right)},Z_s^{\left(3\right)}$, and $Z_s^{\left(4\right)}$, we choose three more stable one-to-one transformations: $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(p_0\right),\mathrm{l}\mathrm{o}\mathrm{g}\left\{\mathrm{E}\right(\tilde{A}\left)\right\}$, and $\mathrm{l}\mathrm{o}\mathrm{g}\{\mathrm{V}\mathrm{a}\mathrm{r}(\tilde{A}\left)\right\}$.

In place of Step 2 the $\left({\tau }_1,\dots ,{\tau }_T\right)$ values used are $\left\{\mathrm{0.25,0.35,0.45,0.55}\right\}$, which surround but do not contain the true $\tau =0.3$. Rather than re-estimate these values with each simulation repetition, we use this set to ensure comparability across repetitions. Finally, Step 3 uses Markov chain Monte Carlo to estimate all variables in (7). Further details are provided in the Supporting Information.

In addition to the proposed generalized propensity score model, we estimate three comparison models: (i) the oracle model, $\mathrm{E}\left(Y_s\right)=A_s+{\tilde{A}}_{\tau ,s}+h{\left(X_{𝒟}\right)}_s$, is the true model that includes otherwise unknown $h\left(X_{𝒟}\right)$ as a covariate, (ii) the local only model, $\mathrm{E}\left(Y_s\right)=A_s+{\tilde{A}}_{\tau ,s}+{\sum }_j{\mathrm{s}\mathrm{p}\mathrm{l}}_j\left(X_s^j\right)$, conditions on local covariates using splines, and (iii) the naive model, $\mathrm{E}\left(Y_s\right)=A_s+{\tilde{A}}_{\tau ,s}$, simply regresses the outcome onto the exposures, but does not incorporate any causal conditioning.

6.2 |. Simulation results

Tables 1 and 2 show the simulation bias and coverage for the $10\times 10$ grids. The Naive model does very poorly in all scenarios, indicating substantial confounding between $A$ and $Y$. The generalized propensity score model performs substantially better than both the Local only and the Naive models, although, intuitively, the Local only model does show reasonable direct effect estimates. In most cases, the generalized propensity score model performs comparably to the Oracle model. Results for the $5\times 5$ grids are similar. A sensitivity analysis that examines robustness to incorrectly specified error and poorly fitting propensity score models is given in Web Appendix D.

TABLE 1.

Simulation bias for $10\times 10$ grids multiplied by 1000, with standard errors

$\mathit{h}{\left({\mathit{X}}_{\mathcal{D}}\right)}_{\mathit{s}}$	Model	${\mathit{\delta }}_1$	${\mathit{\delta }}_2$	$\mathit{\tau }$
$W_s$	Oracle	0.2 (1.8)	−0.7 (0.9)	0.1 (0.2)
	Generalized propensity score	1.4 (1.9)	0.5 (1)	0 (0.2)
	Local Only	2.6 (2)	−72.1 (1.1)	69.1 (0.4)
	Naive	236.7 (1.9)	52.7 (1.5)	79.2 (0.5)
$-{\left(W_s\right)}^3$	Oracle	0.1 (1.8)	−0.7 (0.9)	0.1 (0.2)
	Generalized propensity score	1.1 (1.9)	−0.1 (1)	0.1 (0.2)
	Local Only	1 (2.1)	28.5 (1.3)	−39.7 (0.4)
	Naive	−205.9 (2.8)	−104.1 (1.6)	−53.1 (0.6)
$\mathrm{e}\mathrm{x}\mathrm{p}\left(W_s\right)$	Oracle	0.3 (1.8)	−0.6 (0.9)	0.1 (0.2)
	Generalized propensity score	1.5 (1.9)	0.9 (1.1)	0.1 (0.3)
	Local Only	2.8 (2.3)	−101.4 (2.1)	120.7 (1.4)
	Naive	381.3 (2.8)	105.3 (2.3)	114.2 (1.3)
2 circles	Oracle	−0.9 (1.8)	0.5 (0.9)	0.1 (0.2)
	Generalized propensity score	−0.9 (2)	−0.1 (1.2)	0.5 (0.3)
	Local Only	0.1 (2.1)	−39 (1.2)	62.5 (0.4)
	Naive	244.1 (2)	94.1 (1.5)	72.4 (0.5)

TABLE 2.

Simulation coverage for $10\times 10$ grids, with standard errors

$\mathit{h}{\left({\mathit{X}}_{\mathcal{D}}\right)}_{\mathit{s}}$	Model	${\mathit{\delta }}_1$	${\mathit{\delta }}_2$	$\mathit{\tau }$
$W_s$	Oracle	95 (1)	94.6 (1)	93.6 (1.1)
	Generalized propensity score	93.8 (1.1)	94.4 (1)	93.6 (1.1)
	Local Only	93.2 (1.1)	8.8 (1.3)	0 (0)
	Naive	0 (0)	27.6 (2)	0 (0)
$-{\left(W_s\right)}^3$	Oracle	95.2 (1)	94.6 (1)	93.8 (1.1)
	Generalized propensity score	95.2 (1)	93.8 (1.1)	92.8 (1.2)
	Local Only	93.8 (1.1)	77.2 (1.9)	0 (0)
	Naive	2 (0.6)	6 (1.1)	0 (0)
$\mathrm{e}\mathrm{x}\mathrm{p}\left(W_s\right)$	Oracle	95.4 (0.9)	94.2 (1)	93.8 (1.1)
	Generalized propensity score	93 (1.1)	90.2 (1.3)	90.2 (1.3)
	Local Only	90.8 (1.3)	4.6 (0.9)	0 (0)
	Naive	0 (0)	8.2 (1.2)	0 (0)
2 circles	Oracle	95.6 (0.9)	95 (1)	93.4 (1.1)
	Generalized propensity score	94.6 (1)	89.8 (1.4)	90.4 (1.3)
	Local Only	95.2 (1)	47.4 (2.2)	0 (0)
	Naive	0.2 (0.2)	4 (0.9)	0 (0)

7 |. ESTIMATING THE CAUSAL EFFECT OF WILDLAND FIRES ON AIR POLLUTION

Wildland fires release harmful particles and gasses impacting air quality near the fire and downwind (Larsen et al., 2018). Fine particulate matter smaller than $2.5\mu \mathrm{m}\left({\mathrm{P}\mathrm{M}}_{2.5}\right)$ have been linked to adverse cardiorespiratory health outcomes (Brook, 2007; Corrigan et al., 2018; Dominici et al., 2006; Rappold et al., 2012; Weber et al., 2016). For these reasons, understanding the causal effect of wildland fires on air pollution across space is of significant interest. Specifically, we are interested in the time-averaged causal effect of wildfires on ambient ${\mathrm{P}\mathrm{M}}_{2.5}$ concentrations across Western United States from 2005 to 2018.

7.1 |. Data

The response $Y$ consists of 24-hour average ${\mathrm{P}\mathrm{M}}_{2.5}$ concentrations measured in $\mu \mathrm{g}/{\mathrm{m}}^3$ at 416 measurement sites, some of which are plotted in Figure 2. Observations are collected every one, three or six days depending on the station. The data are publicly available and provided by the Environmental Protection Agency (Environmental Protection Agency, 2019). For each location, the long-term mean is subtracted.

FIGURE 2.

Data snapshot on July 1, 2012. Energy Release Component (ERC) and Ignition Component (IC) are two of National Fire Danger Rating System Components; KBDI refers to the Keetch-0Byram drought index. (a) Fires are shown as cross-hatched circles and ${\mathrm{P}\mathrm{M}}_{2.5}$ locations are shown as solid circles. This figure appears in color in the electronic version of this article, and any mention of color refers to that version

The dates and locations of fires are compiled from a mix of satellite data and incident reports reported to the recently closed Geospatial Multi-Agency Coordination (GeoMAC) Wildland Fire Support program (Geospatial Multi-Agency Coordination, 2019). Moving forward these datasets will be available through the National Interagency Fire Center (National Interagency Fire Center, 2020). Because the focus of our analysis is on ${\mathrm{P}\mathrm{M}}_{2.5}$ only fires larger than 1000 acres are included in the analysis. Among the 3,930 fires, 34.8% of fires are missing either a start or end date. For these fires, we impute missing values by modeling fire duration as a linear function of log (area burned).

Finally, 11 confounders $X^1,\dots ,X^{11}$ are included in the exposure balancing score. These include the four components of the National Fire Danger Rating System (energy release component, burning index, ignition component, and spread index) that are used to monitor daily risk of fire in the United States. The other variables used in the balancing score include elevation, daily temperature, relative humidity, wind speed, precipitation level, the Keetch-Byram drought index, and the numeric day of the year. These variables are compiled and made available through the Wildland Fire Assessment System (Wildland Fire Assessment System, 2019). A snapshot of the exposure, response, energy release component, ignition component, Keetch-Byram drought index, and relative humidity are shown in Figure 2.

Our analysis treats each daily air observation as the center of a $9\times 9$ grid, with a height and width of 9 degrees latitude/longitude. For each such grid, only the center grid cell has a response $Y_S$ value. However, all 81 grid cells have covariates $X_s^j$ and direct exposure $A_s$ values. Each grid cell receives direct exposure $A_s=1$ if there was at least one fire in the cell on that particular day; 0 otherwise. Each $X_s^j$ value is taken to be the mean of the observed covariates in each cell/day combination. For cell/days with no observed values, a value is imputed from nearby cells using a kernel smoother as implemented in the “fields” R package (Nychka et al., 2014). The end result is 592,274 observed grids, each of which contains $9\times 9$ grids for $A_{𝒟}$ and $X_{𝒟}^j,j=1,\dots ,11$, as well as a centered $Y_s$ value. Finally, any grid cells whose centers extend outside of the Western United States are disregarded and excluded from analysis. In this context, the direct effect of exposure consists of the causal effect on $Y_S$ from a fire in the same grid cell $\left(A_S=1\right)$, whereas the indirect effect consists of the causal effect on $Y_S$ from $A_{s^{\prime }}$ in other cells $\left(s\ne s^{\prime }\right)$. As in the simulation study, each of these grids is treated as independent. In addition to the generalized propensity score model, we estimate a model that conditions on the local covariates only, using splines. Though pre-processing and imputing the covariates $X^{\left(j\right)}$ is inescapable given shape of the data, we do not, strictly speaking, account for this imputation (and likely smoothing) in our analysis. This potentially reduces the accuracy of our propensity score, and thereby increases the uncertainty in our final estimates. Our standard errors also do not reflect this potential uncertainty.

We use the same form of $g_{\tau ,s}$ as given in Section 6. $A_s$ at different locations are assumed to be conditionally independent given $X_s$, which allows us to estimate separate components for $A_s$ and ${\tilde{A}}_{\tau ,s}$. The propensity component $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left\{\mathrm{E}\left(A_s\right)\right\}$ is estimated as a linear model of 5-element B-splines of $X_s^1,\dots ,X_s^{11}$, and the propensity of $\tilde{A}$ is assumed to be zero-inflated lognormal. Conditioning on local $X_s$ only is justified because we posit that local $X_S$ contains the vast majority of information about the propensity of fire, with distant locations giving minimal information. Finally, we note that though the data are longitudinal, we assume i.i.d. error. Beyond including the numeric day of the year as a covariate, each $9\times 9$ grid is considered independent over time.

Before proceeding, we briefly review the key assumptions that our method makes, and why they are reasonable for this application. We assume all relevant confounders are stored in $X$. To ensure this is approximately accurate, we have included as many potential confounders as possible. Our analysis assumes the errors are independent over time and space. To satisfy this assumption, we have removed station-level means and included day of the year effects and several relevant covariates to the mean model. Finally, we make the assumption that the exposures $A$ are independent given $X_s$. This is reasonable because the chance of a fire locally is determined by the conditions locally, and not within neighboring cells. The cells used are large enough that the spread of fire from cell to cell will be rare.

7.2 |. Results

Table 3 shows the results. The causal direct effect estimate given by the generalized propensity score model is $1.03\mu \mathrm{g}/{\mathrm{m}}^3$ of ${\mathrm{P}\mathrm{M}}_{2.5}$, or 11.9% of the annual mean ${\mathrm{P}\mathrm{M}}_{2.5}$ observed throughout. The range parameter $\tau$ is estimated to be 1.53 degrees of latitude/longitude, suggesting that fires impact up to roughly 3 degrees away. The estimate of 0.13 for ${\delta }_2$ represents the height of spill-over kernel at its peak. All of ${\delta }_1,{\delta }_2$, and $\tau$ are highly significant. The estimated direct effect from the local-only model is 12% larger than the estimate from the generalized propensity score model, and the local-only model has an implausibly large and imprecise estimate of $\tau$.

TABLE 3.

Posterior mean (95% credible interval)

	Direct Effect (${\mathit{\delta }}_1$)	Spill-over Effect (${\mathit{\delta }}_2$)	Bandwidth ($\mathit{\tau }$)
Local Only	1.15 (1.05,1.25)	0.11 (0.09, 0.12)	17.37 (8.28, 42.12)
Generalized propensity score	1.03 (0.93, 1.14)	0.13 (0.03, 0.25)	1.53 (1.17, 2.88)

Figure 3 illustrates the implied causal effect of fire at different distances from the generalized propensity score model. Taking the center of a grid cell as our vantage point, the direct effect of one or more fires in the same grid cell has a time-averaged causal increase of $1.03\mu \mathrm{g}/{\mathrm{m}}^3$ of ${\mathrm{P}\mathrm{M}}_{2.5}$, which corresponds to the step from 0 to 0.5 in the east/west or north/south direction; slightly more than 0.5 when at an angle. As the fire gets progressively further away, the causal effect decays smoothly until it approaches 0 roughly 3 grid cells away. Intuitively this kernel extending out from 0 is completely determined by $\tau$ and ${\delta }_2:\tau$ corresponds to the width of the kernel; ${\delta }_2$ is the height of the kernel at its peak. Because a significant amount of the fire exposure data was imputed, the analysis was repeated without including fires that had missing start or end dates. This was done by setting $A_s=0$ for these fires. This analysis gave broadly similar results: ${\widehat{\delta }}_1=1.18\left(\mathrm{1.05,1.31}\right),{\widehat{\delta }}_2=0.19\left(\mathrm{0.07,0.34}\right),\widehat{\tau }=1.26\left(\mathrm{1.08,1.56}\right)$. The fact that the estimates for ${\delta }_1$ and ${\delta }_2$ are higher here may indicate that some fire-days in the original analysis may have been erroneously imputed, diluting the estimated exposure effects.

FIGURE 3.

Causal effect of a fire on ${\mathrm{P}\mathrm{M}}_{2.5}$ by distance, as measured in degrees of latitude and longitude. The left axis shows the raw causal increase in ${\mathrm{P}\mathrm{M}}_{2.5}$; the right axis shows this as a percentage of annual mean ${\mathrm{P}\mathrm{M}}_{2.5}$ levels

The wildfire analysis makes several simplifications that are important to consider. First, treating $A_s$ as binary sacrifices information on the number and size of fires in a given grid cell. Extending this method to incorporate information on the size of the fire would preserve information. Moreover, we assume $\tau ,{\delta }_1$, and ${\delta }_2$ are fixed, although it is possible that they naturally vary across different fires and locations. However, there is not enough information in the data to identify these differences. Additionally, we do not consider time-varying effects, as we focused on the contribution to time-averaged ${\mathrm{P}\mathrm{M}}_{2.5}$ levels. Another important simplification is the treatment of separate days as independent. There are temporal trends in the exposure, response, and covariates, and our assumption of independence may inflate the amount of information that our data appear to have. To test the sensitivity to the form of the kernel function in the definition of the indirect exposure, we refit the wildfire data with two additional kernels. However, the direct and spill-over effect estimates are similar for all kernels. These alternative results are provided in Web Appendix D.

8 |. DISCUSSION

The generalized propensity score method presented here establishes a new framework to recover causal direct and spill-over effects in the presence of spatial interference, as well as estimate the range of the interference. The inherent dimensionality issues of the problem are dealt with via a novel propensity score approach, which uses a Bayesian spline-based regression model and a dimension reduction approximation to make the problem tractable. However, there are several critical yet strong assumptions that must hold for our method to perform well. The method hinges on a correctly specified propensity score $g_{\tau ,s}\left(X_{𝒟}\right)$ as well as a correctly specified potential outcomes model in (1). This includes accommodating conditionally dependent $A_{𝒟}$, and correctly characterizing the spatial dependence on $A_s$ from nearby $X$. Moreover, the no unmeasured confounders assumption is always a strong, but necessary, assumption for causal inference on observational data. In practice, considerable effort should be made to include any potential confounders for this reason. We rely crucially on the assumption that the distribution of exposures $\left(A_s,{\tilde{A}}_s\right)$ can be encapsulated with the parameters ${\bar{Z}}_s$ of the propensity score $g_{\tau ,s}$. This will rarely be completely accurate in practice, so effort should be made to select an appropriate parametric form for $g_{\tau ,s}$. We assume that the exposure effects and range are constant across space and time, due to the difficulty of generalizing this assumption. However, allowing for spatially varying exposure effects and particularly range are a promising direction for future research.

Supplementary Material

Supplementary

Table 3: Simulation bias (standard errors) under alternative propensity score forms and correctly/incorrectly specified response form.

Table 4: Simulation coverage (standard errors) under alternative propensity score forms and correctly/incorrectly specified response form.

Table 5: Posterior mean (95% interval) exposure effects for generalized propensity score model, with alternative kernels.

Data

Data S1

DATA AVAILABILITY STATEMENT

The data that support the findings in this paper are openly available in Harvard Dataverse at https://doi.org/10.7910/DVN/11GSTG (Giffin, 2021).