Latent Variable Models for Spatial Confounding
2025-11-20
Causal Inference From Observational Data
Consider a treatment \(D\) and outcome \(Y\)
Interested in the population average treatment effect (PATE) of \(D\) on \(D\): \[E[Y | do(D=d)] - E[Y | do(D=d')]\]
- In general, the PATE is not the same as \[E[Y | D=d] - E[Y | D=d']\]
Confounders
Confounding bias
Observed data regression of \(D\) on \(Y\) fails because the distribution of \(U\) varies in the two treatment arms
We try to condition on as many observed confounders as possible to mitigate potential confounding bias
Commonly assumed that there are “no unobserved confounders” (NUC) but this is unverifiable
Unmeasured Confounding
When there are unmeasured confounders, additional assumptions are needed to identify causal effects
Sensitivity analysis: how strong would unmeasured confounding have to be to explain away the observed association? Cinelli and Hazlett (2020)
Null controls: use negative control exposures or outcomes to detect and adjust for unmeasured confounding (Shi, Miao, and Tchetgen 2020)
A Simple Example
A Simple Example
Observational data from the National Health and Nutrition Examination Study (NHANES) on alcohol consumption.
Light alcohol consumption is positively correlated with blood levels of HDL (“good cholesterol”)
Define “light alcohol consumption’’ as 1-2 alcoholic beverages per day
Non-drinkers: self-reported drinking of one drink a week or less
Control for age, gender and indicator for educational attainment
HDL and alcohol consumption
Blood mercury and alcohol consumption
Call:
lm(formula = Y[, "Methylmercury"] ~ drinking + X)
Residuals:
Min 1Q Median 3Q Max
-2.3570 -0.7363 -0.0728 0.6242 4.1127
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.442044 0.096385 4.586 4.91e-06 ***
drinking 0.364096 0.097244 3.744 0.000188 ***
Xage 0.008186 0.001536 5.330 1.14e-07 ***
Xgender -0.062664 0.052290 -1.198 0.230966
Xeduc 0.269815 0.054126 4.985 6.95e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.975 on 1434 degrees of freedom
Multiple R-squared: 0.05209, Adjusted R-squared: 0.04945
F-statistic: 19.7 on 4 and 1434 DF, p-value: 8.41e-16. . .
Residual Correlation
Pearson's product-moment correlation
data: hdl_fit$residuals and mercury_fit$residuals
t = 3.7569, df = 1437, p-value = 0.0001789
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.04718758 0.14953581
sample estimates:
cor
0.0986225
Residual correlation might be indicative of confounding bias
Multivariate Causal Inference and Unmeasured Confounding
Multiple outcomes JASA (2023)
Multiple exposures
Multiple outcomes and exposures (preprint)
This talk: spatial confounding in environmental epidemiology (preprint)
The effect of Pollution on Birth Weight
PM\(_{2.5}\) and Birth Weight
The effect of Polution on Birth Weight
Call:
lm(formula = bw_mean ~ pm25, data = mutate(pm25_data, pm25 = pm25/10))
Residuals:
Min 1Q Median 3Q Max
-501.56 -38.76 -1.35 37.71 330.60
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3337.181 2.784 1198.52 <2e-16 ***
pm25 -47.688 2.857 -16.69 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 64.38 on 8461 degrees of freedom
Multiple R-squared: 0.03187, Adjusted R-squared: 0.03175
F-statistic: 278.5 on 1 and 8461 DF, p-value: < 2.2e-16A 10 μg/m\(^3\) increase in PM\(_{2.5}\) is associated with a 48 g decrease in birth weight
The effect of Polution on Birth Weight
Outcome: 2018–2024 ZIP code–level birth counts by weight category (California Vital Data, Cal-ViDa Query Tool)
Exposure: Annual ZIP code–level PM\(_{2.5}\) concentrations, derived from high-resolution estimates (Shen et al. 2024)
Observed covariates: Maternal age, race, education, nativity, prenatal care timing, household income, geographic coordinates, and calendar year
Potential Unmeasured confounders: neighborhood deprivation, other socioeconmic factors, cultural and lifestyle factors
The effect of Polution on Birth Weight
Meta-analyses: Birth weight decreases by 16–28 g per 10 μg/m\(^3\) increase in PM\(_{2.5}\) exposure during pregnancy (Gong et al. 2022).
High between-study heterogeneity (range −79.3 g to +24.9 g)
None handles unmeasured confounders (not measurable by space)
Motivation: challenges from confounding and measurement variability call for robust causal inference
Spatial Confounding
- Spatial confounding: unmeasured confounders vary over space and are correlated with exposure
- Common to assume unmeasured confounders is a measurable function of space (Gilbert, Datta, and Ogburn 2021)
- Spatial location then can serve as a proxy for unmeasured confounders
- Nonspatial component in the exposure required
- Use Double Machine Learning (DML) to adjust for spatial location and other observed covariates (Chernozhukov et al. 2018)
Double Machine Learning
- Partially linear model: \(Y = \beta D + g(X) + \epsilon\).
- Estimate \(\mu(X) = E[Y | X]\) and \(e(X) = E[Z | X]\) using nonparametric / ML methods
- Let \(\tilde Y = Y - \hat \mu(X)\) and \(\tilde Z = Z - \hat e(X)\) and estimate causal effect by regressing \(\tilde Y\) ~ \(\tilde Z\) to get \(\hat \beta\).
Cross-fitting
We need to avoid overfitting / double-dipping using a technique analogous to cross-validation.
- Divide the data into \(K\) chunks.
- For each chunk:
- Train the outcome and treatment models on the other K-1 folds and predict \(\hat \mu(X)\) and \(\hat e(X)\) on the \(kth\) held out fold.
- Regress \(\tilde Y^{(k)} \sim \tilde Z^{(k)}\) to estimate \(\hat \beta^{(k)}\)
- Estimate \(\hat \beta\) as \(\frac{1}{K} \sum \hat \beta^{(k)}\)
Why DML alone may not be enough
In practice, confounders vary over space with idiosyncratic differences, mixing spatial and non-spatial variations
Interference: outcomes may depend on exposures at neighboring locations and past time points
DML does not explicitly address causal identifiability or omitted variable bias
We will further leverage variation over time and space to help identify causal effects in the presence of unmeasured confounding
A panel data approach
Setup: panel with \(N\) locations over \(T\) time points; exposure \(D_{it}\) and outcome \(Y_{it}\)
Latent variable modeling: residual correlations in space and time reflect unmeasured confounders
- capture long-range/global correlations
- borrow strength from sparse or irregular data
- robust to nonstationarity
A panel data approach
- Negative controls: exposures or outcomes from other locations/times serve as negative controls.
- Allow for some for some spillover / interference
- Model-agnostic: fit any spatiotemporal model for observed data and apply our method to the residuals
Spatial Causal Inference
Panel at \(N\) locations over \(T\) time points: exposure \(D_{it}\), outcome \(Y_{it}\)
Population average treatment effect (PATE) of \(D\) on \(Y\): \[\mathbb{E}\left[ Y_{it}\bigl(\mathbf d^{(1)}_{\mathcal N_{it}}\bigr) -Y_{it}\bigl(\mathbf d^{(2)}_{\mathcal N_{it}}\bigr)\right]\]
In general, PATE is not the same as \[\mathbb{E}\left[Y_{it}\mid \mathbf D_{\mathcal N_{it}} =\mathbf d^{(1)}_{\mathcal N_{it}}\right] - \mathbb{E}\left[Y_{it}\mid\mathbf D_{\mathcal N_{it}} =\mathbf d^{(2)}_{\mathcal N_{it}}\right]\]
Assumptions
Assumption (Limited interference)
For every \((i,t)\) and exposure \(\mathbf d\), \(Y_{it}(\mathbf d)=Y_{it}(\mathbf d_{\mathcal N_{it}})\), where \(\mathbf d_{\mathcal N_{it}} :=\{d_{jk}:(j,k)\in\mathcal N_{it}\}\).
Assumption (Latent positivity)
\(f_{\mathbf D_{\mathcal N_{it}}\mid \mathbf X_{it},\mathbf S_i,\mathbf U_{it}} (\mathbf d\mid \mathbf x, \mathbf s, \mathbf u) > 0\) for every \((\mathbf d, \mathbf x, \mathbf s, \mathbf u)\) in the support.
Assumption (Latent unconfoundedness)
\(Y_{it}(\mathbf d_{\mathcal N_{it}}) \perp \!\!\! \perp\mathbf D_{\mathcal N_{it}} \mid (\mathbf X_{it},\,\mathbf S_i,\,\mathbf U_{it})\) for all \(\mathbf d_{\mathcal N_{it}}\).
Structural Equation Model
Assume the exposures and outcomes are linear in a latent m-dimensional Gaussian confounder:
\[\begin{align} \mathbf U_{t} &\sim \mathcal{N}_M(0, I_M)\\ \mathbf{D}_t &= \nu(X) + B \mathbf U_{t} + \mathbf{\xi}_t \\ \mathbf{Y}_t &= g(\mathbf{D}, \mathbf{X}) + \Gamma\Sigma_{U\mid D}^{-1/2} \mathbf U_{t} + \mathbf{\epsilon}_t \end{align}\] where \(\Sigma_{U\mid D}\) are the conditional mean and covariance of unmeasured confounders
Bias is \(\Gamma\Sigma_{U\mid D}^{-1/2}E[U \mid D]\)
Structural Equation Model
The proposed model implies a factor structure: \[\begin{aligned} \operatorname{Cov}(\mathbf D_t \mid \mathbf{X}) &= BB^{\top} + \Lambda_D\\ \operatorname{Cov}(\mathbf Y_t \mid \mathbf D_t) &= \Gamma\Gamma^{\top} + \Lambda_Y \end{aligned} \]
\(\Gamma\) and \(B\) are the outcome and exposure factor loadings, respectively. Identying assumptions are well established (anderson1965statistical?).
Bounding the Bias
Proposition
Under the proposed model and assumptions on factor identifiability, the causal effect \(g(\cdot)\) is partially identified. Let \(\check \gamma_i\) be the \(i\)th row of \(\check \Gamma\). For site \(i\), the omitted variable bias for exposure vector \(\mathbf d\) is
\[
\text{Bias}(\mathbf d)_i
= \check \gamma_i \Theta \check \Sigma_{U \mid D}^{-1/2} \check B^{\top} \Sigma_{D}^{-1} \mathbf d
\in
\pm \|\check \gamma_i\|_2 \,
\bigl\|\check \Sigma_{U \mid D}^{-1/2}\check B^{\top}\Sigma_{D}^{-1}\mathbf d\bigr\|_2.
\] - \(\Theta \in \mathcal O_M\) is an orthogonal matrix.
- The interval on the right is identifiable for all \(i\).
- \(\Theta\), and hence \(g(\cdot)\), remain unidentified without further assumptions.
Limits on Interference
Assumption (Off-Neighborhood Rank — informal)
- Let \(\mathcal N_i\) be the interference neighborhood of unit \(i\) (units \(j\) such that \(\partial g_i(\mathbf D_t)/\partial D_{jt}\neq 0\)).
- There exist \(M\) “informative” units \(i_1,\dots,i_M\) such that:
- Their outcome loadings \(\{\gamma_{i_\ell}\}_{\ell=1}^M\) are linearly independent (span \(\mathbb R^M\));
- Considering only indices outside each neighborhood (\(j\notin\mathcal N_{i_\ell}\)), the exposure–confounder directions have full row rank \(M\).
In practice, \(M \ll N\) and neighborhoods are small, so this condition is mild and typically satisfied.
Identification Result
Theorem
Under the structural model and identification assumptions, the causal effect functions \(g_i\bigl((D_{jt})_{j \in \mathcal N_i}\bigr)\) are identified for all units \(i\).
Intuition:
\(N \times T\) bias matrix \(C = \Gamma \Sigma_{U \mid D}^{-1/2} B^\top \Sigma_D^{-1}\) is rank \(m\)
For \(j \notin \mathcal N_i\), any association between \(Y_{it}\) and \(D_{jt}\) reflects unmeasured confounding and identifies \(C_{ij}\).
The rank condition ensures enough entries of \(C\) are known to recover the whole matrix.
Interactive Fixed Effects Model
Closely related approach from econometrics (Bai 2009)
Treats all parameters as fixed effects, with identification in an asymptotic framework (N, P → ∞)
Do not explicitly address causal identifiability or omitted variable bias
Assume outcomes are linear in the exposures
Simulation Study
Estimators compared
DML (NUC): Adjusts for observed covariates, treating unmeasured confounding as a smooth function of space and time (Chernozhukov et al. 2018)
IFE: Interactive fixed effects estimator (Bai 2009)
FC (Proposed): Factor confounding approach, explicitly modeling latent confounders
Simulation: Comparing Estimators
Simulation: Spatial Interference
Effect of PM\(_{2.5}\) on Birth Weight
Meta-analyses: Birth weight decreases by 16–28 g per 10 μg/m\(^3\) increase in PM\(_{2.5}\) exposure during pregnancy.
Outcome: 2018–2024 ZIP code–level birth counts by weight category (California Vital Data, Cal-ViDa Query Tool)
Exposure: Annual ZIP code–level PM\(_{2.5}\) concentrations, derived from high-resolution estimates (Shen et al. 2024)
Observed confounders: Maternal age, race, education, nativity, prenatal care timing, household income, geographic coordinates, and calendar year
Effect of PM\(_{2.5}\) on Birth Weight
Validation in Causal Settings
In general, we do not know the true causal effect in observational data
Predictive accuracy is not a good substitute for causal accuracy
Negative control / placebo checks: PM\(_{2.5}\) in the year after birth should not affect birth weight (structural assumption in our model)
Robustness to covariate adjustment: if our method handles unmeasured confounding, estimates should not change much when adding or removing observed covariates
Robustness to Covariate Adjustment
Results
Takeaways
Double machine learning with spatial location as a proxy for unmeasured confounders may not fully account for confounding bias
Latent variable models can help account for unmeasured confounding
For the proposed model, with mild rank and partial interference assumptions, causal effects are identifiable and unbiased estimates can be achieved from spatiotemporal data
Future Directions
More general forms of confounding, non-linear latent variable models, etc
Analytic results much more complicated in non-linear latent variable models
Need to identify E[U | D, X]
Causal inference with tensor data (multiple outcomes / exposures across space and time)
Mixtures of multiple polutants, multiple health outcomes, etc
Acknowledgements
- Lead author, Jiaxi Wu (Former Phd student, now at Amazon)
References
Thank You!