Explained Treatment Heterogeneity (ETH)

• ETH is the component of treatment-effect variation attributed to observed factors such as baseline covariates, marginal distributions, or treatment-version assignments.
• Researchers use methods like variance decomposition, sharp bounds, and plug-in estimators to quantify and test ETH in causal studies.
• ETH estimation helps design adaptive trials and refine subgroup analyses, enhancing causal inference and personalized treatment strategies.

Explained treatment heterogeneity (ETH) denotes the component of treatment-effect variation that is attributable to observed structure rather than left as residual idiosyncrasy. In the cited literature, that structure is defined in several closely related ways: by baseline covariates used to form subgroups, by the observable marginal distributions of potential outcomes, by a working model for the conditional average treatment effect (CATE), by site-level characteristics in multi-site trials, or by treatment-version assignment rules when the analyzed treatment is itself heterogeneous. ETH is therefore not a single estimand but a family of estimands and inferential targets used to quantify, test, and interpret heterogeneity in causal effects (Dai et al., 2020, Kaji et al., 2023, Li et al., 2023, Anniwaer et al., 5 Sep 2025).

1. Definitions and conceptual scope

A direct subgroup-based definition states that ETH refers to observed variation in treatment effects that is accounted for by baseline covariates used to define subgroups. In this usage, subgroup construction precedes formal testing, and ETH is established when subgroup-specific treatment effects differ beyond what can be attributed to chance (Dai et al., 2020).

A second usage treats ETH as the heterogeneity that is “explained” by the observable data and the fundamental identification constraints. In that formulation, the data identify the marginal distributions of potential outcomes but not their joint distribution, so ETH is expressed through nonparametric sharp bounds on subgroup treatment effects or on the proportion of winners. The explained component is then the range of heterogeneity consistent with the marginals, without extra assumptions (Kaji et al., 2023).

A third usage defines ETH through variance decomposition. If the CATE is $\tau(W)$, one variable-importance parameter is

$$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$

which measures how much of treatment-effect variance is explained by a targeted subset of covariates $W_s$ beyond $W_{-s}$. A scaled version, $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$, supports comparison across covariate subsets and settings (Li et al., 2023).

In semi-supervised work, ETH is the variance of a simplified working-model CATE: $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$ where $\tau^*(W)=W^\top\beta^*$ is the best $L^2$ approximation to the true CATE within the chosen working model. Here ETH quantifies how much heterogeneity is explainable by a lower-dimensional, interpretable approximation rather than by the full CATE surface (Anniwaer et al., 5 Sep 2025).

In multi-site randomized experiments, explained treatment-effect heterogeneity is the portion of the variance in site-level effects attributable to observable or unbiasedly estimable site-level characteristics, whereas unexplained heterogeneity is the residual variance after accounting for those characteristics (Chaisemartin et al., 2024).

A distinct but related usage arises when the analyzed treatment is an aggregate of multiple underlying treatment versions. In that setting, the literature separates effect heterogeneity from treatment heterogeneity and defines an explained component as the part of group differences that can be traced to differences in treatment-version assignment rather than differences in causal response to identical treatment versions (Heiler et al., 2021, Heiler et al., 2 Jul 2025).

2. Formal estimands and identification

The common starting point is the CATE,

$\tau(x)=\mathbb{E}[Y(1)-Y(0)\mid X=x].$

Related foundational summaries include the blip function

$$b_P(W)=\mathbb{E}_P[Y\mid A=1,W]-\mathbb{E}_P[Y\mid A=0,W],$$

its mean $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$0, and its variance $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$1, where VTE measures overall clinical effect heterogeneity due to observed confounders (Levy et al., 2018).

When subgroup heterogeneity is defined by untreated-outcome ranks, the subgroup treatment effect for the lowest untreated-outcome fraction $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$2 is

$$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$3

The sharp bounds

$$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$4

use only the marginal distributions. For the proportion of winners in $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$5,

$$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$6

the corresponding sharp bounds are expressed through $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$7 and $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$8, again without identifying the joint distribution of potential outcomes (Kaji et al., 2023).

For variable importance, ETH can be defined by marginalizing over a subset of covariates. If $$\psi_{2,0} = \operatorname{Var}(\tau(W)) - \operatorname{Var}(\tau_s(W)) = \mathbb{E}[\operatorname{Var}(\tau(W)\mid W_{-s})],$$9, then

$W_s$0

measures the variance explained by $W_s$1. The efficient influence curve for this parameter is

$W_s$2

which underpins targeted-learning estimation and inference (Li et al., 2023).

For multivariate continuous treatments, the generalized CATE is

$W_s$3

Total ETH is summarized by

$W_s$4

and covariate-specific explained heterogeneity is summarized by

$W_s$5

where $W_s$6 is the heterogeneity unexplained without $W_s$7 (Shin et al., 2024).

In representation-based causal characterization, the target becomes the CATE on direct effect modifiers,

$W_s$8

and ETH is captured by conditioning on a selected Markov blanket in a sufficient aligned representation: $W_s$9 This formulation makes ETH a minimal sufficient summary of effect modification under the stated assumptions (Cadei et al., 15 Jun 2026).

3. Testing whether explained heterogeneity is present

One nonparametric test of treatment-effect heterogeneity across $W_{-s}$0 strata tests

$W_{-s}$1

For strata $W_{-s}$2, the kernel

$W_{-s}$3

induces the pairwise U-statistic

$W_{-s}$4

and the overall statistic

$W_{-s}$5

Its null distribution is approximated by simulation from a multivariate normal distribution with empirically estimated covariance. The method uses only the ordering of outcomes, maintains type I error, and tends to have higher power than parametric alternatives when normality or equal-variance assumptions fail (Dai et al., 2020).

A distinct detection stage in large-scale experimentation uses a model-free omnibus test before any subgroup specification. The kurtosis-based log-variance statistic is

$W_{-s}$6

with asymptotic null $W_{-s}$7 under the constant-treatment-effect hypothesis. The same workflow compares bootstrapped F-tests and Fisher’s randomization test plus Kolmogorov–Smirnov test, and uses Benjamini–Hochberg FDR to control multiplicity across many metrics (Cai et al., 2022).

A clinical-trial workflow based on individualized treatment-effect pseudo-outcomes uses a double robust pseudo-outcome

$W_{-s}$8

then tests for dependence on covariates through a permutation-based conditional inference statistic

$W_{-s}$9

summarized by either a max-type or quadratic-type statistic. A small $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$0-value indicates explained treatment-effect heterogeneity with respect to observed baseline variables (Sechidis et al., 2 Feb 2025).

Adaptive experiments shift the question from post hoc detection to design. Response-adaptive randomization and adaptive enrichment are optimized to maximize the probability of correctly identifying the subgroup with the largest treatment effect. The large-deviation objective depends on subgroup effect differences, variances, and allocation rules, and the adaptive procedures converge to the oracle allocation under the stated conditions (Wei et al., 2023).

4. Estimation strategies and decomposition frameworks

Plug-in and targeted-learning approaches treat ETH as a functional of the data-generating law. For VTE, the plug-in estimator is

$\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$1

and CV-TMLE provides simultaneous plug-in estimates and inference for ATE and VTE by cross-validated outcome modeling followed by a targeting step that solves the efficient influence-curve equation. The cited work emphasizes that VTE lacks double robustness and may be downwardly biased when the true VTE is small or sample size is insufficient (Levy et al., 2018).

For variable-importance ETH, a TMLE updates initial estimates of $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$2, $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$3, $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$4, $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$5, and $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$6 so that the empirical mean of the efficient influence curve is near zero. The resulting estimator is a pure plug-in estimator and, by construction, respects global constraints such as nonnegativity of explained heterogeneity. This contrasts with estimating-equation methods that can yield negative finite-sample values for a variance-type parameter (Li et al., 2023).

Semi-supervised inference uses labeled data $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$7 together with unlabeled data $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$8 to estimate ETH more efficiently. The direct semi-supervised estimator combines a plug-in variance term with a debiasing term, but direct use of unlabeled data may lose efficiency under misspecification. An optimal re-weighting scheme uses weights $\psi_{3,0} = \psi_{2,0}/\psi_{1,0}$9 satisfying $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$0 and chooses them to minimize asymptotic variance. The proposed estimator has asymptotic variance no larger than that of the supervised method, which makes its use “safe” in the sense stated in the paper (Anniwaer et al., 5 Sep 2025).

In multi-site randomized experiments with many sites and few units per site, the empirical-Bayes estimator

$\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$1

subtracts within-site sampling variance to estimate between-site treatment-effect variance. Explained heterogeneity is then studied by regressing site-level effects on observed or unbiasedly estimated site-level characteristics, with explicit correction for measurement error in estimated regressors (Chaisemartin et al., 2024).

When the analyzed treatment aggregates multiple effective treatments, decomposition becomes central. One framework writes

$\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$2

where $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$3 is the effect under the actual treatment-version mix, $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$4 is the effect under the population-average mix, and $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$5 is the explained treatment heterogeneity due to nonrandom composition of treatment versions. This decomposition is estimated with double/debiased machine learning under high-dimensional confounding, many treatments, and extreme propensity scores (Heiler et al., 2021).

5. Explaining heterogeneity: variable importance, subgroup surfacing, and characterization

A major strand of the literature seeks not only to estimate heterogeneity but also to explain which variables account for it. In survival data with observational electronic health records, a pseudo-outcome-based framework estimates CATE at a fixed time point,

$\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$6

through pseudo-individualized treatment effects and weighted meta-learners such as X-, M-, DR-, D-, DEA-, and R-learners. It then computes SHAP values $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$7 for the predicted CATE and aggregates $\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$8 across subjects to form a variable-importance ranking for ETH (Bo et al., 2024).

A systematic paradigm for online experimentation separates detection, surfacing, and characterization. Surfacing uses an explained treatment-effect-variation decomposition,

$\theta_{\mathrm{ETH}} = \mathrm{Var}[\tau^*(W)],$9

to rank candidate covariates by how much HTE they explain. Characterization then moves beyond CATE by estimating conditional distributions of individualized treatment effects using matched residuals and stratified histograms (Cai et al., 2022).

In drug-development applications, covariates can be ranked by fitting conditional inference forests to the double-robust pseudo-outcome and computing permutation importance. This ranking is explicitly framed as identifying which baseline variables most strongly explain observed heterogeneity. The same workflow returns individualized treatment-effect estimates for each participant via cross-fitted regression of the pseudo-outcome on $\tau^*(W)=W^\top\beta^*$0 (Sechidis et al., 2 Feb 2025).

Variable selection can itself be used as an explanatory device. A support-vector-machine formulation with separate $\tau^*(W)=W^\top\beta^*$1 penalties on causal heterogeneity parameters and baseline adjustment parameters solves

$\tau^*(W)=W^\top\beta^*$2

Nonzero coefficients in $\tau^*(W)=W^\top\beta^*$3 identify treatment indicators or treatment-covariate interactions that explain where and for whom treatment effects differ (Imai et al., 2013).

For multivariate continuous exposures, explanation is summarized by the multivariate treatment-effect variable importance measure $\tau^*(W)=W^\top\beta^*$4, which attributes fractions of total ETH to individual covariates or grouped covariate sets. The outcome model is decomposed as

$\tau^*(W)=W^\top\beta^*$5

with $\tau^*(W)=W^\top\beta^*$6 isolating effect modification by each covariate (Shin et al., 2024).

At the most expressive end, ETH can be framed as Markov-blanket discovery in a sufficient aligned representation. Neural EXposure Interaction Search (NEXIS) iteratively tests

$\tau^*(W)=W^\top\beta^*$7

with forward inclusion and backward redundancy removal. Under principal alignment, mean faithfulness, and valid tests, the selected set recovers the principal direct HTE modifiers with asymptotic probability at least $\tau^*(W)=W^\top\beta^*$8 (Cadei et al., 15 Jun 2026).

6. Empirical findings, recurring limitations, and interpretive cautions

Applications show that ETH can be informative even when average effects are small or insignificant. In the National Supported Work Demonstration data, the U-statistic-based nonparametric test found non-significant heterogeneity by age quartiles ($\tau^*(W)=W^\top\beta^*$9) but significant heterogeneity by pre-treatment income ($L^2$0), with treatment more effective for those with zero prior income (Dai et al., 2020). In microfinance and welfare-reform applications, sharp bounds on subgroup treatment effects and winner proportions were informative even when average treatment effects were insignificant, establishing that some subgroup benefit or harm must exist regardless of the unobserved joint distribution of potential outcomes (Kaji et al., 2023).

Several studies emphasize that null marginal effects do not preclude ETH. In pragmatic gerontology trials with truncation by death, the mean CSACE among always-survivors was close to zero, but CSACE varied with baseline deprivation, living situation, mental health, and baseline quality of life, so subgroup-specific benefit and possible harm remained visible after principal-stratification modeling (Li et al., 18 Nov 2025). In tuberculosis, contextualized patient-specific models identified anemia, age of onset, and HIV as influential for treatment efficacy, illustrating a move beyond coarse subgroup analyses toward individualized context-dependent treatment-effect models (Wu et al., 2024).

The literature also warns that some reported modifiers may be artifacts. For survival outcomes in SPRINT and ACCORD, a comprehensive reanalysis concluded that many previously reported modifiers of intensive blood-pressure treatment may be spurious discoveries, especially when flexible models are insufficiently regularized or censoring is improperly modeled (Xu et al., 2022). A broad review of modern HTE methods similarly stresses that exploratory subgroup findings often disappear under rigorous inference, cross-fitting, or multiplicity control (Lipkovich et al., 2023).

Another recurrent caution is that “heterogeneity” may reflect treatment-version assignment rather than differential response to the same treatment. In Job Corps, the observed gender gap in returns was largely explained by differential selection into vocational training tracks; one decomposition attributed most of the gap to treatment heterogeneity rather than effect heterogeneity (Heiler et al., 2021, Heiler et al., 2 Jul 2025). This directly challenges the common reading of group differences in adjusted means as evidence that one group responds better to treatment.

Finally, targeted-learning work repeatedly notes that variance-based ETH parameters are statistically demanding. VTE and related ETH parameters are not doubly robust in the strong sense enjoyed by the ATE, their remainder terms depend critically on accurate CATE or outcome-model estimation, and finite-sample behavior can be poor when the true heterogeneity is small (Levy et al., 2018, Li et al., 2023). This suggests that ETH is most informative when paired with explicit identification statements, careful nuisance estimation, and formal uncertainty quantification rather than treated as a purely descriptive output.