Spatial Quantile Treatment Effects (SQTE)

• SQTE are location-specific causal contrasts that quantify treatment-induced shifts in conditional outcome quantiles by comparing treated and control groups.
• They employ semiparametric deep spatial quantile regression using spline mixtures and multi-resolution spatial basis functions to model full conditional distributions.
• The framework mitigates hidden spatial confounding through neighborhood adjustments, ensuring robust estimation of spatially heterogeneous quantile treatment effects.

Spatial Quantile Treatment Effects (SQTE) are location-specific causal contrasts that describe how a treatment shifts the conditional outcome distribution across quantiles and across space. In the formulation introduced in “Causal Spatial Quantile Regression” (Gong et al., 2 Sep 2025), SQTE are defined for spatially indexed observational data in the Rubin–Neyman potential-outcomes framework, with outcome $Y$, binary treatment $T$, covariates $X$, and spatial location $s=(s_1,s_2)^T$. The central estimand is the spatially varying $\tau$-th quantile treatment effect

$$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$

where $q^1(\tau\mid X,s)$ and $q^0(\tau\mid X,s)$ are the conditional $\tau$-quantiles of the potential outcomes under treatment and control, respectively. This construction generalizes standard quantile treatment effects by explicitly allowing causal effects to vary with location, and it generalizes mean-based spatial causal estimands by targeting the full conditional outcome distribution rather than only its expectation (Gong et al., 2 Sep 2025).

1. Conceptual definition and estimands

The spatial causal setup uses observed data

$\mathcal{O}=(Y,T,X^T,s^T)^T,$

with $T$0, $T$1, $T$2, and $T$3. The observed conditional quantile function is written as

$T$4

where $T$5 is the conditional CDF of $T$6 given $T$7. For the two treatment regimes,

$T$8

The SQTE estimand is then

$T$9

or equivalently

$X$0

for a probability measure $X$1 on $X$2 (Gong et al., 2 Sep 2025).

A second object is the spatially averaged QTE,

$X$3

which averages the location-specific distributional effect over the spatial domain. This separation between $X$4 and $X$5 is substantive: the former is a local causal surface over quantile level and geography, whereas the latter is a global summary.

SQTE differ from standard non-spatial QTE because they retain explicit dependence on $X$6. Standard QTE typically take forms such as

$X$7

without spatial indexing. SQTE also differ from the spatial average treatment effect

$X$8

because ATE targets a mean contrast, whereas SQTE target contrasts at each quantile of the outcome distribution. In the simulation design of (Gong et al., 2 Sep 2025), the true heterogeneous SQTE is

$X$9

which makes the dependence on both longitude and quantile explicit.

2. Identification, causal assumptions, and the role of space

Identification in the proposed framework rests on four stated assumptions. The first is SUTVA, defined as one version of the treatment and no interference. In particular, spatial interference is ruled out: a unit’s outcome at location $s=(s_1,s_2)^T$0 depends only on its own treatment, not on neighbors’ treatments. The second is consistency, so that if $s=(s_1,s_2)^T$1, then $s=(s_1,s_2)^T$2. The third is ignorability, namely treatment assignment is independent of the potential outcomes given the confounders. The fourth is positivity: $s=(s_1,s_2)^T$3 for all $s=(s_1,s_2)^T$4 in the support of $s=(s_1,s_2)^T$5 (Gong et al., 2 Sep 2025).

Under these assumptions, the conditional quantile functions $s=(s_1,s_2)^T$6 are identified from observed data, and so is

$s=(s_1,s_2)^T$7

The object is therefore a location-specific, covariate-averaged quantile treatment effect: at each spatial location, it describes how treatment shifts the entire conditional outcome distribution across quantiles.

The treatment of space in this framework is simultaneously structural and restrictive. It is structural because spatial location enters the conditional distribution directly and is allowed to modify the causal effect surface. It is restrictive because the identification argument excludes spatial interference. This is a central point of interpretation: the framework addresses spatial heterogeneity of causal effects, not causal spillovers across locations. A common misconception is to treat any spatial causal model as automatically accommodating interference. In the SQTE formulation of (Gong et al., 2 Sep 2025), that is not the case.

The paper nevertheless recognizes that ignorability is “usually violated and untestable” in observational settings, especially when unobserved variables are spatially correlated. It therefore distinguishes between identification under no hidden confounders and estimation under possible spatial hidden confounding. This distinction is methodologically important: SQTE are identified under standard causal assumptions, but the estimation strategy is explicitly designed to mitigate violations generated by unobserved spatial structure (Gong et al., 2 Sep 2025).

3. Semiparametric deep spatial quantile regression

The proposed estimator is built on a semiparametric neural network-based conditional quantile regression model. Rather than fitting separate quantile regressions at each $s=(s_1,s_2)^T$8, the method models the full conditional density and CDF: $s=(s_1,s_2)^T$9

$\tau$0

where $\tau$1 are second-order $\tau$2-spline basis functions on $\tau$3, $\tau$4 are the corresponding second-order $\tau$5-splines, and the mixture weights satisfy nonnegativity and simplex constraints. The conditional quantile function is then obtained by inversion,

$\tau$6

This representation is associated in the paper with SPQR and is combined with DeepKriging-style spatial features (Gong et al., 2 Sep 2025).

The neural network maps $\tau$7 and spatial basis inputs to the mixture weights through an $\tau$8-layer feed-forward architecture with softmax output,

$\tau$9

Spatial information is enriched by multi-resolution radial basis functions $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$0 constructed from compactly supported Wendland basis functions,

$$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$1

At resolutions $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$2, these bases are formed around grid nodes $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$3 with bandwidths $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$4, and concatenated to the treatment-covariate inputs. The resulting model uses $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$5, which permits multi-resolution spatial structure in the conditional quantile surface (Gong et al., 2 Sep 2025).

SQTE estimation proceeds by plug-in. After fitting the model and obtaining $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$6, one computes

$$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$7

then averages their difference at each location: $$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$8 and finally averages over locations to obtain

$$\Delta(\tau,s)=\mathbb{E}_{X}\{q^1(\tau\mid X,s)-q^0(\tau\mid X,s)\},$$9

The paper does not present formal asymptotic theory for $q^1(\tau\mid X,s)$0; uncertainty is instead evaluated through repeated simulations and bootstrap-based confidence intervals in the empirical application (Gong et al., 2 Sep 2025).

A notable feature of this construction is that quantiles are induced by a proper conditional CDF representation. This contrasts with classical quantile regression workflows that solve separate check-loss problems at each quantile. In non-spatial linear quantile regression, inference is often complicated by the need to estimate conditional densities at the quantile. “Quantile-Regression Inference With Adaptive Control of Size” develops a variance estimator for Wald-type tests whose size distortion decays at the same rate as an infeasible benchmark with known conditional densities (Escanciano et al., 2018). That result is not an SQTE estimator, but it is directly relevant to downstream heterogeneity testing when spatial QTE are represented in linear quantile models.

4. Spatial hidden confounding and neighborhood adjustment

A major concern in spatial causal inference is unobserved spatial confounding: latent spatial variables may affect both treatment assignment and outcome, thereby violating ignorability. In the hidden-confounding simulation scenario of (Gong et al., 2 Sep 2025), the paper introduces hidden variables $q^1(\tau\mid X,s)$1 that affect both the outcome quantile and the treatment probability. For example, treatment is assigned according to

$q^1(\tau\mid X,s)$2

while the outcome quantile includes hidden terms

$q^1(\tau\mid X,s)$3

When these variables are unobserved, residuals and estimated SQTE may exhibit structured spatial bias (Gong et al., 2 Sep 2025).

To address this, the paper proposes a neighborhood spatial confounding adjustment. The key assumption is that hidden confounders are smoother in space than the treatment. If they vary slowly, then within a sufficiently small neighborhood around a target location they are approximately constant. A local fit can then absorb the hidden confounder as a neighborhood-specific intercept or random effect. Operationally, the procedure chooses a central location and a radius containing approximately a fixed proportion of observations, subsets the data to that neighborhood, fits the same deep spatial quantile model locally, and then uses the estimated coefficients or weights from the local fit to construct predictions under $q^1(\tau\mid X,s)$4 and $q^1(\tau\mid X,s)$5 across the entire spatial domain (Gong et al., 2 Sep 2025).

The effect of this adjustment is heterogeneous across model classes. In Scenario 3, it reduces log(RMISE) for predicted responses and SQTE in models without spatial basis functions, and yields more spatially homogeneous and smaller RMISE maps. By contrast, when the model already includes rich multi-resolution spatial basis functions, additional local adjustment provides limited gains because the network can already absorb much of the spatial structure. The resulting interpretation is precise: the neighborhood adjustment is particularly useful when the base regression model is not highly spatially flexible, and less crucial when rich spatial features are already present (Gong et al., 2 Sep 2025).

This suggests a broader methodological point. Spatial confounding can be attacked either by making the outcome model highly expressive in spatial coordinates and basis functions, or by introducing localized estimation that treats slowly varying latent structure as approximately constant. The paper shows these are not identical strategies, but they may become close substitutes when spatial representation is sufficiently rich.

5. Simulation evidence and substantive empirical findings

The simulation study in (Gong et al., 2 Sep 2025) uses a $q^1(\tau\mid X,s)$6 location grid on $q^1(\tau\mid X,s)$7, with $q^1(\tau\mid X,s)$8 observations at each location and six observed covariates, three non-spatial and three spatial Gaussian processes. It evaluates five model specifications ranging from covariates only to covariates plus multi-resolution spatial basis functions with up to three resolutions. Performance is assessed by root mean integrated squared error (RMISE) for predicted responses and estimated SQTE, together with spatial RMISE maps at $q^1(\tau\mid X,s)$9.

Across the unconfounded and observed-confounding scenarios, models with spatial basis functions perform substantially better for prediction than models without them, reflecting the importance of spatial random effects. SQTE estimation exhibits small RMISE overall, with better performance around $q^0(\tau\mid X,s)$0 and larger errors in the tails. In the observed-confounding case, directly including the confounders $q^0(\tau\mid X,s)$1 and $q^0(\tau\mid X,s)$2 is sufficient for performance similar to the unconfounded case, which the paper interprets as evidence that the neural network regression can absorb confounding when all relevant covariates are observed (Gong et al., 2 Sep 2025).

In the hidden-confounding scenario, the challenge is sharper. Without adjustment, RMISE for both prediction and SQTE is larger than in the first two scenarios. Model 5, which uses the richest spatial basis expansion, recovers SQTE almost as well as in the unconfounded case, while other models exhibit larger SQTE errors. With neighborhood adjustment, Models 1 and 2 improve notably, and their spatial error patterns become more homogeneous. Overall, the simulations support the claim that the deep spatial quantile regression plus plug-in SQTE estimator can accurately recover spatially heterogeneous distributional treatment effects, including under hidden spatial confounding when spatial structure is modeled effectively (Gong et al., 2 Sep 2025).

The empirical application studies maternal smoking and newborn birth weight in North Carolina using births from 1988–2002 for first-time white mothers, with approximately $q^0(\tau\mid X,s)$3 observations across 691 distinct ZIP code centroids. The outcome is birth weight in grams, and the treatment is maternal smoking during pregnancy. For each model, the paper estimates location-specific SQTE and a spatially averaged QTE. The main result is stable across models and quantiles: maternal smoking has a negative effect on birth weight, and the negative effect is more pronounced in lower quantiles, particularly around $q^0(\tau\mid X,s)$4. The mapped estimates of $q^0(\tau\mid X,s)$5 further show that negative effects are widespread but vary in intensity across regions; under richer spatial modeling, southern and northeastern North Carolina exhibit less homogeneous patterns, though the estimated effects remain negative (Gong et al., 2 Sep 2025).

A common misunderstanding in interpreting quantile effects is to equate stronger effects in lower quantiles with larger effects for a fixed subgroup of “low-outcome” individuals. The paper’s result is distributional rather than rank-invariant in that sense: it shows larger treatment-induced shifts in lower conditional birth-weight quantiles, not permanent individual rank membership. The practical implication drawn in the paper is that interventions to reduce maternal smoking could yield the largest gains in regions and quantiles where estimated SQTE are most negative.

6. Relation to adjacent literatures, alternative formulations, and limitations

The SQTE framework in (Gong et al., 2 Sep 2025) is positioned at the intersection of four literatures: causal quantile treatment effects, spatial causal inference, spatial quantile regression, and machine learning for heterogeneous treatment effects. The paper characterizes itself as the first integrated framework for causal spatial quantile regression, combining SPQR-type mixture-spline quantile modeling, DeepKriging-style spatial basis functions, and a neighborhood spatial confounding adjustment (Gong et al., 2 Sep 2025).

Several adjacent contributions clarify what SQTE are and are not. “A Unified Approach for Estimating Various Treatment Effects in Causal Inference” formulates ATE, QTE, and DTE as statistical functionals of counterfactual CDFs and shows how IPW-based CDF estimators can be plugged into these functionals (Wu et al., 28 Mar 2025). That framework is not spatial by default, but it directly suggests a conditional-CDF route to spatial QTE by replacing global counterfactual distributions with location-specific or spatially smoothed counterfactual CDFs. This suggests a functional interpretation of SQTE that is complementary to the conditional quantile regression route of (Gong et al., 2 Sep 2025).

A different neighboring line is dynamic spatial quantile modeling without explicit causal identification. “Bayesian inference for dynamic spatial quantile models with interactive effects” specifies conditional quantiles in a spatial panel with contemporaneous spatial lags, lagged spatial lags, dynamics, and interactive effects (Ando et al., 2 Mar 2025). The paper does not define treatment effects formally, but it shows how a regressor such as a Covid dummy can be interpreted as a quantile-specific, unit-specific effect that propagates through spatial and temporal multipliers. This is not the same estimand as the location-specific covariate-averaged SQTE of (Gong et al., 2 Sep 2025), but it is closely related in spirit and useful for panel settings with strong cross-sectional dependence.

An even more distinct approach appears in “Heterogeneous Treatment Effects for Networks, Panels, and other Outcome Matrices,” which introduces spectral treatment effects as a matrix analog of QTE based on eigenvalue differences of treatment-specific outcome matrices (Auerbach et al., 2022). These spectral treatment effects are designed for matrix-valued outcomes with spillovers and interference. They are therefore conceptually different from SQTE as defined in (Gong et al., 2 Sep 2025), which assume no interference and scalar outcomes, but they show that “quantile-like” causal heterogeneity in spatial or network settings can be formalized in ways other than location-indexed scalar quantiles.

The limitations of the current SQTE framework are explicit. The paper provides no formal asymptotic theory for the plug-in estimator. Inference is empirical rather than fully semiparametric. SUTVA and no interference may be questionable in many spatial applications. The neighborhood adjustment relies on hidden confounders being smoother than the treatment; if hidden confounders vary at similar or finer spatial scales, effectiveness may be limited. Estimation accuracy deteriorates in extreme quantiles, as in standard quantile regression. Computationally, large neural networks with spline mixtures and spatial basis expansions are intensive, and performance depends on choices such as the number of spline components $q^0(\tau\mid X,s)$6, network depth $q^0(\tau\mid X,s)$7, and spatial resolutions $q^0(\tau\mid X,s)$8 (Gong et al., 2 Sep 2025).

Taken together, these points locate SQTE as a specific causal estimand and estimation program rather than a generic label for any spatially heterogeneous distributional effect. In the current literature, the term most precisely refers to location-specific, covariate-averaged treatment effects on conditional outcome quantiles under a no-interference potential-outcomes framework, estimated in (Gong et al., 2 Sep 2025) through semiparametric deep spatial quantile regression with optional local adjustment for smooth hidden spatial confounding.