Bringing spatial confounding into the causal inferential fold

Abstract

Spatial patterning of environmental hazards often leads to concerns about spatial confounding: that the exposures we study share similar spatial distributions with other causes of disease. Recent efforts to address spatial confounding have approached it using clever specification of spatial models, or models that adjust for aspects of spatial location itself. In the article by Li et al. (Am J Epidemiol. XXXX;XXX(XX):XXXX–XXXX)), the authors describe and demonstrate several models for addressing spatial confounding for binary exposures. These important results demonstrate an aspect of environmental exposures that should concern all environmental epidemiologists: inadequate adjustment for spatial confounding can increase, rather than decrease, bias from spatial confounding. In this commentary, we disagree with some terminology and enthusiastically agree with the importance of the problem and the utility of the approaches described by these authors. Causal inference in environmental epidemiology is fraught with many challenges, and Li et al. give hope for progress on one of the lesser understood, yet potentially ubiquitous, problems: spatial confounding.

Environmental exposures often share similar sources or fates, such that geographic location can yield information about multiple health hazards. For example, proximity to coal fired power plants is associated with higher ambient levels of multiple air pollutants including particulate matter, sulfur dioxide, and nitrogen oxides, and down-river distance from a mining operations can predict sediment levels of multiple toxic metals.^{1, 2} We refer to the link between location and exposure as “spatial patterning” so that spatially-patterned exposures have variability attributable to location. When studying health effects of spatially-patterned environmental exposures, correlation among pollutants leads to concern about potential confounding from spatially-patterned co-exposures.³ This type of confounding been given its own distinct term: spatial confounding.⁴

The recent interest in spatial confounding from a causal inferential perspective comes from at least one conceptualization of the problem. From the causal diagram in Fig. 1, (where we ignore measured confounders) it is apparent that, for estimating an effect of some exposure X on some health outcome Y, the path through the unmeasured, spatially-patterned set of confounders U can be blocked by conditioning on spatial location (S). That is, under Fig. 1, unmeasured confounding by a potentially large set of environmental exposures can be addressed simply by adjusting for participant location at the time of exposure.

Figure 1:

A directed acyclic graph for the problem of estimating an effect of X on Y, subject to unmeasured confounding through some variable U, where U and X are both affected by the S node, representing spatial location.

However, even under this (overly) simple view, non-parametric adjustment for spatial location will generally violate positivity assumptions because participants often have unique spatial locations. Positivity assumptions can sometimes be replaced by model specification assumptions, raising the possibility of controlling confounding by parametric adjustment. Thus, the field of spatial modeling, which has long studied smooth, parametric models for spatial patterns, gives some hope for addressing spatial confounding. Within generalized linear models (GLMs), spatial patterning in health outcomes can be addressed in the “mean-model-side” by adding, for example, a flexible regression spline on spatial location variables (e.g. latitude and longitude). Alternatively, spatial patterning can be addressed on the “error-side” by employing a residual covariance matrix that allows correlated observations under some assumed spatial structure.

In the article by Li et al., the authors compare a series of approaches to cope with spatial confounding.⁵ The methods address an important methodologic gap in the field: a favored method for addressing spatial confounding, spatial+, relies on a two-stage approach that is only possible with continuous exposures.⁶ The authors adapt and test, via simulations, a number of different approaches that could potentially fill a similar gap for binary or categorical exposures. These methods include unpenalized and penalized regression splines on location, spatial-distance-based matching, and various combinations of these approaches. Throughout their simulations, they utilize “scale” (error-side) and “penalty” (mean-model-side) parameters that characterize the “wiggliness” of the association between spatial location and the outcome, given other variables in the model (here we note that the distinction between error-side and mean-model-side is only for convenient parametric representation because they can coincide in large samples⁷). A spatial pattern is wigglier if it displays more variation within a smaller spatial range (e.g. there is a rapid change in a variable over small distances). Echoing findings from others,⁶ the authors found that more flexible approaches (those that were closest to non-parametric adjustment while remaining feasible) were able to control spatial confounding without having direct measurements of the spatial confounder.

Aside: is spatial confounding just confounding?

It is worth noting briefly that the term “spatial confounding” has been defined in multiple ways, including some that conflict with our usage. As noted by Gilbert and Ogburn, a method called “restricted spatial regression,” was specifically developed for the nominal problem of spatial confounding. However, this approach utilizes a spatial random effect that is, by design, not associated with the exposure of interest and therefore unable to address unmeasured confounding by spatially-patterned variables.⁷ Notably, this method was inspired by an approach to collinearity in which the affect of the confounder is entirely attributed to the exposure, thus not addressing confounding bias for the exposure. It is worth noting that Li et al. (and others⁸) use a definition of spatial confounding that conforms with modern epidemiologic definitions.⁵ Thus, in a meaningful sense, “spatial confounding” is just “confounding,” and we believe the literature on spatial confounding will benefit from more engagement with the tools of potential outcomes-based causal inference for addressing confounding.

What is bias exacerbation?

Not all methods for spatial confounding successfully reduced bias. The authors use the term “bias exacerbation” to describe a phenomenon where, upon attempting to adjust for spatial location, bias is larger for adjusted relative to unadjusted estimates of exposure effects. In the spatial confounding literature, has been previously noted and has more recently become referred to as “bias amplification”.^{4, 9}

Bias amplification is a well-studied phenomenon in the instrumental variable literature. Using Pearl’s description,¹⁰ this bias amplification occurs when adjusting for a pure instrument (a variable that is only associated with the outcome through the exposure) in analyses of exposure-disease relationships that are already subject to unmeasured confounding (e.g. Fig. 2). This phenomenon called bias amplification because the bias of the instrument-adjusted model can be given as $B_U/(1-{\alpha }_I^2)$, where B_U is bias due to unmeasured confounding and the amplification term $(1-{\alpha }_I^2)$, reflects the strength of the relationship between the instrument (I) and the exposure. The amplification term is bound between zero and one, so adjustment for an instrument always increases the magnitude of bias from unmeasured confounding.

Figure 2:

A path-labeled directed acyclic graph demonstrating conditions under which bias amplification can arise for an effect of X on Y, subject to unmeasured confounding by U, where Z is an instrument for exposure

Does spatial confounding put us at risk of bias amplification?

In the authors’ simulations, some methods for adjusting for space increased bias relative to unadjusted models. Without even considering causal structure, these results seem consistent with bias amplification. However, a (naïve) look at the structure of Fig. 1 makes attribution to bias amplification seem confusing. If all backdoor paths on the DAG through unmeasured variables U can be blocked by conditioning on spatial location S, then bias amplification would not seem to apply because even though there is an unmeasured confounder, it is no longer a confounder once we have already conditioned on the measured variable S, so there is no bias to amplify.

The nuance we are missing, however, is how the data are generated, as shown in Figure 1 of Li et al.⁵ First, they simulated a “spatial confounder,” and then they simulated a “spatial instrument” that only played a role in the exposure. While both variables ostensibly relate to spatial locations, they are nonetheless uncorrelated with each other. For those with some training in simulation methods, this method of generating data may seem foreign. It works by generating variables in relation to space via “fields” rather than as direct functions of specific locations, so spatial location can be related to multiple factors without those factors being related to each other. The data generated by this simulation method are easy to relate to the real world: imagine a study in which the exposure of interest is the level of arsenic in tap water, and we were interested in its effects on ischemic heart disease incidence. A spatial instrument for arsenic levels could be bedrock composition (unrelated to the outcome) and a spatial confounder could be exposure to agricultural runoff (which could include other environmental causes of disease, such as pesticides). A priori, we would expect that these two variables would be uncorrelated with each other. However, adjusting for spatial location will induce adjustment for the spatial confounder as well as the spatial instrument because individuals who live at proximal locations will tend have similar values for each. As noted by a reviewer, adjustment would also be induced for spatial mediators, if they exist, further complicating inference.

Because adjustment for space will typically require a statistical model, model specification must be considered. The authors show that if we include the true spatial confounder in a correct model, then bias is eliminated. Indeed, were the authors to additionally include the true spatial instrument in the model, bias would still be eliminated because there would be no unmeasured confounding to amplify. Thus, the bias seen by the authors results from residual unmeasured confounding under model misspecification. Bias amplification from partial adjustment of the spatial instrument could explain why the authors, in some scenarios, see increased bias relative to a model that does not adjust for spatial location. Alternatively, the higher bias may be purely due to model misspecification: Ogburn and Vanderweele showed that mis-specifying a continuous confounder can mean bias will not be reduced by adjustment, and they cite some examples in which bias increases from model misspecification alone.¹¹

We do not think that bias amplification (or exacerbation) is a wholly appropriate term for the phenomenon because it seems to be solvable with a sufficiently flexible model. The authors’ findings support the idea that more flexible, fixed degree of freedom splines (which can be used in a GLM) are sufficient to reduce bias in most cases. We are making a narrow argument over terminology, but it is helpful to define the problem as misspecification, rather than bias amplification, because the tools for misspecification are the ones used to solve the problem. It also re-emphasizes the importance of thoughtful modeling of spatially-patterned phenomena and demonstrates the care that must be taken.

Li et al. note that flexible models do not appear sufficient for controlling spatial confounding for continuous exposures, especially if they are purely spatial (i.e. exposures that vary across space but are identical for all observations at a given location). This is evidenced by prior findings and additional simulations in which they dichotomize a continuous exposure during analysis. However, a key difference between binary exposures and continuous (or dichotomized continuous) exposures is that binary spatial exposures will inherently have non-spatial variability. Given location, under positivity (so that either exposure level is possible at every location), each exposure constitutes a random draw from a Bernoulli distribution and can be treated as though it were randomized, given other confounders. In contrast, a purely spatial, continuous exposure can only take on one possible value at each location, which would violate positivity. As Gilbert and Ogburn have noted, bias in such cases results from model specification because, in the case of non-positivity, inference is based solely on the model.⁷ Indeed, they note that an exposure must have non-spatial variability for positivity to be met.

We should also note that, outside the problems studied by the authors, bias amplification is a useful concept to understand for spatial confounding. Bias amplification can occur when adjusting for an instrument in the presence of an unmeasured confounder. Thus, spatial modeling, which can adjust for a spatial instrument of exposure, can amplify bias from non-spatially-patterned, unmeasured confounders. Conversely, if spatial confounding is present and we do not address it, adjustment for strong predictors of the exposure (mistaking them for confounders) will amplify the spatial confounding bias.

To avoid non-positivity issues arising from purely spatial exposures, it may be useful to instead consider adjustment for a larger dimensional set of co-exposures using methods from the mixtures literature, provided that they are spatially-patterned confounders. For example, say the exposure of interest X has strong spatial patterning such that adjusting for space would induce non-positivity, but spatial confounding is mostly due to co-exposures, Z, then adjustment for Z may be preferable if, at a given set of values for Z, there is variation in X. Gilbert and Ogburn demonstrated a “shift intervention” estimator that relaxes positivity assumptions if there is some non-spatial variation in X.⁷ Previously, we have applied such estimators to exposure mixtures to estimate effects of very small changes in one exposure, holding multiple others constant.¹² Other methods from the mixtures literature focus on the joint effects of X and Z, so that positivity need not exist for X, conditional on Z.^13–15

Conclusion

Regardless of how we label the phenomenon, the authors have given a wonderful example of the downsides of haphazardly applying spatial models to address the potential for spatial confounding. There are clear tradeoffs in doing so that can be addressed via clever model specification and subject matter knowledge. From this paper we take the optimistic lesson that, with the proper expertise, there is still some hope of usefully addressing the potentially ubiquitous problem of spatially-patterned unmeasured confounders.

Abbreviations:

DAG

directed acyclic graph

GLM

generalized linear model