COMPACT: Spectral Adjustment Scores from a Complete and Irreducible Causal Criterion

COMPACT: Spectral Adjustment Scores from a Complete and Irreducible Causal Criterion

Eric V. Strobl

Abstract

Observational datasets frequently contain many baseline variables, yet investigators estimating causal effects may not know which variables to include in the adjustment set. Confounding information may also be distributed weakly across many variables. Propensity scores can simplify adjustment by reducing high-dimensional covariates to a scalar with binary treatment. Although the propensity score is the coarsest balancing score, this distributional optimality does not imply maximal specificity over causal graphs. We instead examine all causal graphs among a candidate score, treatment, and outcome while allowing latent variables. Under faithfulness, we identify the largest set of unconditional and conditional dependence relations whose truth is invariant to whether treatment causes the outcome, leaving treatment-effect estimation to the downstream analysis. This criterion defines the maximally specific graph class expressible through these relations. We then develop the proposed algorithm, which operationalizes the criterion through a generalized eigenvalue problem whose score space targets the span of a balancing coordinate and an outcome-guided coordinate. We show that sufficiently informative proxies can recover this span without direct observation of the adjustment variables, characterize the resulting estimation and causal errors, and establish bootstrap validity for the complete procedure. Simulations and a real-data application demonstrate superior performance over several alternatives.

Key words: causal inference, confounding, dimension reduction, high-dimensional adjustment, proxy variables, spectral methods

1. Introduction

Confounding remains a central obstacle to causal inference from observational data. In many applications, treatment assignment and potential outcomes depend on an underlying baseline state observed only indirectly through many variables in X. Clinical treatment selection, for example, may reflect numerous individually weak cues involving symptoms, severity, comorbidity, prior response, tolerance, and perceived benefit, while adjuvant cancer treatment may depend on a multivariate tumor phenotype reflected by stage, nodal involvement, grade, receptor status, proliferation markers, and genomic recurrence scores. Consequently, investigators often adjust for broad collections of baseline variables to capture distributed confounding information.

Direct adjustment on X becomes difficult when these variables are high-dimensional, mixed-type, and noisy. For binary treatment, propensity scores address this problem by compressing X into P(T = 1 X). However, this requires estimation of the entire treatment law, including treatment-predictive variation that may be irrelevant to outcome risk. Flexible treatment models can therefore absorb treatment-only or sample-specific variation, reduce residual treatment variation, and drive fitted propensities toward zero or one without a commensurate improvement in confounding control.

We instead begin from the dependence relations specifically required for causal adjustment. Treatment association alone provides little causal specificity. For example, both Sβ −→ T with Y isolated and Sβ −→ T −→ Y imply Sβ ̸⊥⊥ T under faithfulness, although Sβ lies on no backdoor path between T and Y. A treatment-prediction method may nevertheless devote substantial complexity to such variation. We therefore develop COMPACT (Confounding-Oriented Multivariate Proxy Adjustment for Causal Targets), which requires candidate score directions to satisfy Sβ ̸⊥⊥ T and Sβ ̸⊥⊥ Y T. We show that these relations define a more specific class of admissible causal structures than treatment association alone, while remaining invariant to whether T causes Y. The outcome relation therefore prevents treatment prediction alone from determining the learned representation.

COMPACT also exploits the fact that causal adjustment need not recover a particular balancing score coordinate. It is sufficient to recover a low-dimensional space containing the relevant adjustment information. Specifically, COMPACT uses a generalized eigendecomposition to estimate U = span(UT, UR), the joint span of a treatment-balancing coordinate UT and an outcome-guided coordinate UR. Because the estimand is the span rather than any uniquely labeled basis, the procedure avoids unnecessary coordinate-identification and calibration requirements. In the scalar-treatment, scalar-outcome setting, this space is two-dimensional, making the learned adjustment structure directly visualizable. The same representation can be used to examine adjustment-relevant gradients, treatment overlap, and variation in expected treatment benefit.

Main contributions:

(a) We identify a complete and irreducible set of score-level dependence restrictions for causal adjustment whose truth values remain invariant to whether treatment causes the outcome (Section 4).

(b) We introduce COMPACT, which operationalizes these restrictions through a product-based “AND” criterion and solves the resulting problem with a single generalized eigendecomposition (Section 5.1).

(c) We show that the retained directions span regularized linear predictors of treatment and treatment-residualized outcome, and connect this eigenspace to latent treatment-balancing and outcome-guided variables (Section 5.3).

(d) We characterize sampling error, proxy-recovery error, and the resulting causal error, establishing conditions for consistent estimation of both the average treatment effect (ATE) and conditional average treatment effect (CATE) after adjustment for the learned scores (Sections 6.1–6.2).

(e) We establish validity of a full-procedure nonparametric bootstrap that relearns the COMPACT scores and refits the downstream analysis, thereby propagating uncertainty through the complete estimation procedure (Section 6.3).

(f) We evaluate COMPACT in fully synthetic and semi-synthetic experiments, including treatment-effect estimation, latent-coordinate recovery, treatment overlap, and bootstrap calibration. COMPACT achieves the best or competitive performance across these criteria while remaining computationally fast (Section 7).

2. Motivating Example

The paper motivates COMPACT with an application to the Adolescent Brain Cognitive Development (ABCD) Study, a large longitudinal observational cohort of children in the United States. The goal was to estimate the effect of initiating reading for pleasure on subsequent cognitive performance. This setting presents a high-dimensional adjustment problem because factors like demographics, baseline cognitive abilities, competing activities, family environment, sleep, screen use, physical characteristics, and mental health may all influence both reading behavior and cognition. The analysis considered six NIH Toolbox Cognition outcomes. The adjustment set contained 248 baseline variables. For crystallized cognition, initiating reading for pleasure was associated with higher subsequent crystallized cognition (ATE = 0.589, 95% CI [0.237, 1.044], Holm-adjusted p = 0.0072), whereas the global CATE-heterogeneity test was not significant. The results provide evidence consistent with a causal effect of initiating reading for pleasure on subsequent crystallized cognition while illustrating how COMPACT simultaneously provides a low-dimensional adjustment representation, an overlap diagnostic, and a space for examining treatment-effect heterogeneity.

3. Setup and notation

3.1. Observed Data, Causal Targets, and Identification

We observe independent units Oi = (Yi, Ti, Xi) for i = 1,..., n, where Ti denotes a treatment or exposure, Yi denotes a post-treatment outcome, and Xi ∈ Rp denotes a vector of pre-treatment baseline features. The variables in Xi may be continuous or discrete. Treatment levels in the potential-outcome notation remain on their original scale. For each treatment level t ∈ T, let Yi (t) denote the outcome that would be observed for unit i under treatment t. We assume consistency and no interference. For treatment levels t, t0 ∈ T, the marginal causal contrast is τ (t, t0) = E Yi (t) − Yi (t0). In the binary-treatment case, this becomes the average treatment effect τ = E Yi (1) − Yi (0). Once a low-dimensional score Si = s(Xi) has been learned, we can also consider the score-indexed causal contrast τS (t, t0; s) = E Yi (t) − Yi (t0) Si = s. Evaluating this contrast at Si provides a low-dimensional personalized CATE while averaging over distinctions in Xi not retained by the score, thereby avoiding high-dimensional treatment-effect strata that often suffer from weak overlap. Let Ci ∈ RdC denote a centered pre-treatment baseline state that is sufficient for adjustment, so that Yi (t) ⊥⊥ Ti Ci for all t ∈ T. We also assume positivity on the treatment values and covariate strata relevant to τ (t, t0).

3.2. A causal working model with a single-index treatment law

We use the following model to make the population target explicit. Let ΣC = E(Ci Ci⊤) be positive definite, and define the treatment-balancing variable UT,i = a⊤ Ci with a⊤ ΣC a > 0. We assume that the conditional treatment law depends on Ci only through UT,i: P (Ti ∈ A Ci) = P (Ti ∈ A UT,i) for every measurable set A. Let mT (u) = E(Ti UT,i = u) and εT,i = Ti − mT (UT,i). Then E(εT,i Ci) = 0. Equation (3) includes both continuous and binary treatment types. When mT is nonlinear, including the binary-treatment case, we impose the standard linearity condition E(Ci UT,i) = (ΣC a / a⊤ ΣC a) UT,i. This condition holds when Ci follows a centered elliptically symmetric distribution with finite second moments. Let the potential outcomes satisfy Yi (t) = α + b⊤ Ci + t γ + q ⊤ Ci + εY,i (t), where E εY,i (t) Ci, Ti = 0. The model also implies the linear CATE τC (t, t0; Ci) = (t − t0) γ + q ⊤ Ci.

3.3. High-dimensional proxy variables

We assume that Ci is not observed directly. Instead, the observed pre-treatment proxy vector Xi satisfies Xi ⊥⊥ Ti, Yi (t): t ∈ T Ci, E(Xi Ci) = ΛCi, with E(∥Xi ∥2) < ∞. The first condition requires the proxy measurements to be nondifferential: conditional on the latent baseline state Ci, they contain no additional information about treatment or the potential outcomes. The second requires only a linear conditional mean and does not impose an additive-error representation, homoskedasticity, or a continuous proxy distribution. The rows of Λ may correspond to many weak or redundant measurements of the same latent baseline state.

4. A Complete and Irreducible Causal Criterion

At the population level, we seek a coefficient matrix B whose score space SB = Xi⊤ β: β ∈ col(B) reconstructs the low-dimensional adjustment information carried by the latent baseline state Ci when the observed proxy system is sufficiently informative. For a candidate coefficient vector β ̸= 0, define the scalar score Sβ,i = Xi⊤ β. As discussed in the Introduction, treatment association alone is not sufficiently specific for confounding-oriented score construction. For example, Sβ,i −→ Ti −→ Yi implies Sβ,i ̸⊥⊥ Ti under faithfulness even though Sβ,i lies on no backdoor path from Ti to Yi. Moreover, adding unconditional outcome relevance Sβ,i ̸⊥⊥ Yi does not resolve this problem because the same chain also induces marginal dependence between Sβ,i and Yi. We therefore derive the score criterion from the temporal role of treatment. Consider the six elementary pairwise dependence relations involving (Sβ,i, Ti, Yi), including the three marginal relations and the three relations conditional on the remaining variable: Q(Sβ,i, Ti, Yi) = Sβ,i ̸⊥⊥ Ti, Sβ,i ̸⊥⊥ Yi, Ti ̸⊥⊥ Yi, Sβ,i ̸⊥⊥ Ti Yi, Sβ,i ̸⊥⊥ Yi Ti, Ti ̸⊥⊥ Yi Sβ,i. We compare these restrictions through the graph classes they represent. For any A ⊆ Q(Sβ,i, Ti, Yi), let G(A) denote the class of directed acyclic graphs (DAGs) that satisfy every relation in A under d-connection and respect the temporal order Sβ,i ≺ Ti ≺ Yi. These graphs may contain additional latent variables. We assume that the observational distribution is faithful to the corresponding graph and that sampling induces no selection bias. The existence of the edge Ti → Yi is unknown when the score is constructed. We therefore retain only primitive relations whose truth values do not depend on whether this edge is present. For any graph G under consideration that does not contain Ti → Yi, define G+ = G ∪ Ti → Yi, leaving every other node and edge unchanged. We call a primitive relation q ∈ Q(Sβ,i, Ti, Yi) uniformly treatment-effect-invariant if q holds in G ⇐⇒ q holds in G+ for every such G.

Theorem 1 Under the temporal ordering, faithfulness, and absence of selection bias specified above, define Qinv = Sβ,i ̸⊥⊥ Ti, Sβ,i ̸⊥⊥ Yi Ti. Then Qinv has the following properties:

(a) Completeness. The set Qinv contains all of the uniformly treatment-effect-invariant primitive relations in (10): Qinv = q ∈ Q(Sβ,i, Ti, Yi): q is uniformly treatment-effect-invariant.

(b) Irreducibility. For every proper subset A ⊊ Qinv, there exists a faithful ordered DAG GA that satisfies every relation in A but does not satisfy every relation in Qinv. Equivalently, every proper subset represents a strictly larger graph class: G(Qinv) ⊊ G(A).

The two conclusions of Theorem 1 explain why score construction should enforce both Sβ,i ̸⊥⊥ Ti and Sβ,i ̸⊥⊥ Yi Ti. Completeness shows that these are exactly the primitive dependence relations whose truth values are invariant to the unknown edge Ti → Yi. Their conjunction therefore represents the most specific graph class expressible through the primitive relations without making score construction depend on whether treatment causally affects the outcome. Irreducibility shows that neither restriction is redundant because removing either relation strictly enlarges the represented graph class.

We now seek score directions that satisfy both invariant relations in Qinv. For a candidate scalar score Sβ,i = Xi⊤ β, we represent Sβ,i ̸⊥⊥ Ti by the correlation between Sβ,i and Ti, and Sβ,i ̸⊥⊥ Yi Ti by the partial correlation between Sβ,i and Yi after linear adjustment for Ti. This operationalization imposes a linear-detectability requirement: the underlying dependencies must induce nonzero marginal and partial linear correlations rather than being masked by nonlinearity or exact cancellation. We define the population criterion J (β) = corr(Sβ,i, Ti) pcorr(Sβ,i, Yi Ti), sup J (β). Under the linear-detectability requirement, J (β) > 0 ⇐⇒ corr(Sβ,i, Ti) ̸= 0 and pcorr(Sβ,i, Yi Ti) ̸= 0. Thus the product implements an “AND” criterion. Under faithfulness and linear detectability, a direction receives positive population value exactly when its score-level graph belongs to G(Qinv). Directions outside this class all receive zero population value, so the optimization does not rank directions that fail either invariant relation. This restriction contrasts with score construction based only on treatment prediction, including propensity-score modeling. Viewed through the same graph-class lens, a treatment-only criterion searches over G(Sβ,i ̸⊥⊥ Ti). By Theorem 1, G(Qinv) ⊊ G(Sβ,i ̸⊥⊥ Ti). Thus treatment-only score construction searches over a strictly larger graph class, including directions that predict treatment but contain no residual-outcome information. The product criterion assigns such directions zero population value and concentrates the search on the more specific class represented by Qinv to hopefully achieve better finite sample performance.

5. COMPACT: A Spectral Approach

The empirical version of (12) is non-convex because it multiplies two normalized associations. The COMPACT algorithm obtains a tractable approximation by preserving the product numerator, which encodes the requirement that a score be both treatment-relevant and residual-outcome-relevant, while replacing the original normalization with a simpler arithmetic-mean denominator. This substitution leads to a spectral prob…

Based on the paper, here are the specific improvements you can make to AI systems:

1. Causal Adjustment with Joint Treatment–Outcome Spectral Learning

• Improvement: Replace propensity-score-only or outcome-only dimension reduction with a generalized eigenvalue problem that jointly maximizes treatment correlation and residual-outcome partial correlation (the AND criterion).

• What the improved AI system can do: Learn a two-dimensional adjustment space that is provably more specific to confounding than treatment prediction alone, avoiding wasted capacity on treatment-only variation. It can recover latent balancing and outcome-guided coordinates even when confounding information is weak and distributed across hundreds of noisy proxies.

2. Treatment-Effect-Invariant Representation Learning

• Improvement: Enforce only those dependence relations whose truth values are invariant to whether treatment causes outcome (Theorem 1), ensuring the learned representation does not depend on the unknown causal edge.

• What the improved AI system can do: Produce adjustment scores that remain valid for both ATE and CATE estimation regardless of the true causal status of treatment, reducing bias from model misspecification and improving robustness in observational settings.

3. Proxy-Based Latent Confounder Recovery

• Improvement: Use the theoretical result (Proposition 1) that the eigenspace equals ridge-regularized linear prediction of latent treatment-balancing and outcome-guided variables from proxies.

• What the improved AI system can do: Recover the span of unobserved confounders from high-dimensional, weak, and redundant proxy measurements without requiring direct observation of the adjustment variables. This enables causal inference in settings where confounders are latent (e.g., clinical severity, socioeconomic status) but indirectly measured.

4. Finite-Sample and Proxy-Error-Aware Causal Estimation

• Improvement: Incorporate the error decomposition (Theorem 2: proxy strength + effective dimension/sample size) into the learning objective or uncertainty quantification.

• What the improved AI system can do: Automatically detect when proxy information is too weak or sample size too small for reliable causal adjustment, and provide calibrated confidence intervals that account for both spectral estimation error and downstream regression error. This prevents overconfident causal claims in low-information regimes.

5. Full-Procedure Bootstrap Inference

• Improvement: Recompute the entire COMPACT score space and downstream regression in each bootstrap sample (Theorem 4), rather than conditioning on a fixed learned representation.

• What the improved AI system can do: Produce valid confidence intervals and hypothesis tests (e.g., for ATE, CATE heterogeneity) that propagate uncertainty through both the dimension-reduction step and the outcome model, leading to correct type-I error control and power in finite samples.

6. Low-Rank Spectral Computation for Scalability

• Improvement: Exploit the rank-2 structure of the cross-covariance operator (Section 5.2) to avoid iterative optimization or complex prediction models.

• What the improved AI system can do: Handle datasets with thousands of covariates in under 2 seconds per replication, making causal adjustment feasible for large-scale observational data (e.g., biobanks, electronic health records) without sacrificing statistical efficiency.

7. Overlap Diagnostics and Heterogeneity Visualization

• Improvement: Use the learned two-dimensional score space for treatment overlap assessment and CATE visualization (Section 7).

• What the improved AI system can do: Provide interpretable diagnostics for positivity violations and personalized treatment-effect variation directly in the adjustment space, enabling clinicians or policymakers to identify subgroups with poor overlap or heterogeneous responses.

8. Handling Weak and Redundant Proxy Systems

• Improvement: Leverage the theoretical insight that additional informative proxies reduce reconstruction error (Theorem 2), even if each proxy is individually weak.

• What the improved AI system can do: Improve causal estimates as more noisy, partially redundant measurements are added, rather than degrading due to high dimensionality. This is particularly useful for integrating multi-omics, imaging, or sensor data where each feature is a weak signal of an underlying state.

9. Avoiding Propensity Score Collapse

• Improvement: Replace propensity-score estimation (which can drive fitted propensities to 0 or 1) with the joint spectral criterion that penalizes directions lacking residual-outcome relevance.

• What the improved AI system can do: Maintain stable treatment overlap (extreme propensity proportions < 7% even at p=500), preventing the common failure mode of propensity models that overfit treatment-only variation and reduce effective sample size.

10. Basis-Invariant Downstream Analysis

• Improvement: Ensure downstream causal estimators (OLS, CATE models) are invariant to the specific basis of the learned score space (Section 6.3).

• What the improved AI system can do: Avoid coordinate-identification and calibration issues, allowing seamless integration with standard causal inference packages while guaranteeing that results do not depend on arbitrary rotations of the adjustment scores.

Observational datasets frequently contain many baseline variables, yet investigators estimating causal effects may not know which variables to include in the adjustment set. Confounding information may also be distributed weakly across many variables. Propensity scores can simplify adjustment by reducing high-dimensional covariates to a scalar with binary treatment. Although the propensity score is the coarsest balancing score, this distributional optimality does not imply maximal specificity over causal graphs. We instead examine all causal graphs among a candidate score, treatment, and outcome while allowing latent variables. Under faithfulness, we identify the largest set of unconditional and conditional dependence relations whose truth is invariant to whether treatment causes the outcome, leaving treatment-effect estimation to the downstream analysis. This criterion defines the maximally specific graph class expressible through these relations. We then develop the proposed algorithm, which operationalizes the criterion through a generalized eigenvalue problem whose score space targets the span of a balancing coordinate and an outcome-guided coordinate. We show that sufficiently informative proxies can recover this span without direct observation of the adjustment variables, characterize the resulting estimation and causal errors, and establish bootstrap validity for the complete procedure. Simulations and a real-data application demonstrate superior performance over several alternatives.