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With multiple outcomes in empirical research, a common strategy is to define a composite outcome as a weighted average of the original outcomes. However, the choices of weights are often subjective and can be controversial. We propose an inverse regression strategy for causal inference with multiple outcomes. The key idea is to regress the treatment on the outcomes, which is the inverse of the standard regression of the outcomes on the treatment. Although this strategy is simple and even counterintuitive, it has several advantages. First, testing for zero coefficients of the outcomes is equivalent to testing for the null hypothesis of zero effects, even though the inverse regression is deemed misspecified. Second, the coefficients of the outcomes provide a data-driven choice of the weights for defining a composite outcome. We also discuss the associated inference issues. Third, this strategy is applicable to general study designs. We illustrate the theory in both randomized experiments and observational studies.
Scholars of social stratification often study exposures that shape life outcomes. But some outcomes (such as wage) only exist for some people (such as those who are employed). We show how a common practice -- dropping cases with non-existent outcomes -- can obscure causal effects when a treatment affects both outcome existence and outcome values. The effects of both beneficial and harmful treatments can be underestimated. Drawing on existing approaches for principal stratification, we show how to study (1) the average effect on whether an outcome exists and (2) the average effect on the outcome among the latent subgroup whose outcome would exist in either treatment condition. To extend our approach to the selection-on-observables settings common in applied research, we develop a framework involving regression and simulation to enable principal stratification estimates that adjust for measured confounders. We illustrate through an empirical example about the effects of parenthood on labor market outcomes.
Clinical trials or studies oftentimes require long-term and/or costly follow-up of participants to evaluate a novel treatment/drug/vaccine. There has been increasing interest in the past few decades in using short-term surrogate outcomes as a replacement of the primary outcome i.e., in using the surrogate outcome, which can potentially be observed sooner, to make inference about the treatment effect on the long-term primary outcome. Very few of the available statistical methods to evaluate a surrogate are applicable to settings where both the surrogate and the primary outcome are time-to-event outcomes subject to censoring. Methods that can handle this setting tend to require parametric assumptions or be limited to assessing only the restricted mean survival time. In this paper, we propose a non-parametric approach to evaluate a censored surrogate outcome, such as time to progression, when the primary outcome is also a censored time-to-event outcome, such as time to death, and the treatment effect of interest is the difference in overall survival. Specifically, we define the proportion of the treatment effect on the primary outcome that is explained (PTE) by the censored surrogate
Regression is a fundamental tool in scientific research. Ordinary least squares (OLS), one of the most widely used regression methods, enjoys several desirable properties, including the best linear unbiased estimator (BLUE) property. It is well known that, under the assumptions of the standard model, the OLS is conditionally unbiased given the covariates, i.e., $\mathbb{E}(\widehat Y-Y\mid X=x)=0$. However, an often-overlooked property of OLS is that the prediction error is generally not unbiased conditional on the outcome, i.e., $\mathbb{E}(\widehat Y-Y\mid Y=y) eq 0$. As a consequence of minimizing mean squared error, OLS predictions are systematically shrunk toward the outcome mean, which explains the classical phenomenon of regression to the mean (RTM): large outcome values tend to be underpredicted, whereas small outcome values tend to be overpredicted. This conditional prediction bias creates a nonignorable problem for predicted outcome-based inference, where scientific inference is performed using the predicted outcome $\widehat Y$ and another variable $W$. In applications such as brain-age analysis and causal inference, we show that inference based on regression-predicted o
Parameters of interest in causal inference, such as treatment or policy effects, can often be expressed as linear functionals of an outcome regression function. Automatic debiased machine learning (AutoDML) is a unified framework for obtaining asymptotically normal estimators of such parameters, which requires estimation of both a regression function and a Riesz representer. Existing AutoDML neural network architectures, such as RieszNet and MADNet, use a shared intermediate covariate representation. However, it remains unclear whether this shared representation should be predictive of the Riesz representer or the outcome. We show that a shared representation of the covariates that preserves predictive power of the outcome while discarding information about the Riesz representer is asymptotically more efficient than the baseline AutoDML estimator that uses all covariates. Motivated by these results, we propose the outcome-adapted AutoDML estimator and establish its asymptotic behavior in a sample splitting framework. We provide a neural network implementation of the estimator that learns a sparse representation of the covariates that is predictive of the outcome but not predictive
Predicting potential and counterfactual outcomes from observational data is central to individualized decision-making, particularly in clinical settings where treatment choices must be tailored to each patient rather than guided solely by population averages. We propose PO-Flow, a continuous normalizing flow (CNF) framework for causal inference that jointly models potential outcome distributions and factual-conditioned counterfactual outcomes. Trained via flow matching, PO-Flow provides a unified approach to individualized potential outcome prediction, conditional average treatment effect estimation, and counterfactual prediction. By encoding an observed factual outcome and decoding under an alternative treatment, PO-Flow provides an encode-decode mechanism for factual-conditioned counterfactual prediction. In addition, PO-Flow supports likelihood-based evaluation of potential outcomes, enabling uncertainty-aware assessment of predictions. A supporting recovery guarantee is established under certain assumptions, and empirical results on benchmark datasets demonstrate strong performance across a range of causal inference tasks within the potential outcomes framework.
In the era of big data, secondary outcomes have become increasingly important alongside primary outcomes. These secondary outcomes, which can be derived from traditional endpoints in clinical trials, compound measures, or risk prediction scores, hold the potential to enhance the analysis of primary outcomes. Our method is motivated by the challenge of utilizing multiple secondary outcomes, such as blood biochemistry markers and urine assays, to improve the analysis of the primary outcome related to liver health. Current integration methods often fall short, as they impose strong model assumptions or require prior knowledge to construct over-identified working functions. This paper addresses these statistical challenges and potentially opens a new avenue in data integration by introducing a novel integrative learning framework that is applicable in a general setting. The proposed framework allows for the robust, data-driven integration of information from multiple secondary outcomes, promotes the development of efficient learning algorithms, and ensures optimal use of available data. Extensive simulation studies demonstrate that the proposed method significantly reduces variance in
Probabilities of causation provide explanatory information on the observed occurrence (causal necessity) and non-occurrence (causal sufficiency) of events. Here, we adapt these probabilities (probability of necessity, probability of sufficiency, and probability of necessity and sufficiency) to an important class of epidemiologic outcomes, post-infection outcomes. A defining feature of studies on these outcomes is that they account for the post-treatment variable, infection acquisition, which means that, for individuals who remain uninfected, the outcome is not defined. Following previous work by Hudgens and Halloran, we describe analyses of post-infection outcomes using the principal stratification framework, and then derive expressions for the probabilities of causation in terms of principal strata-related parameters. Finally, we show that these expressions provide insights into the contributions of different processes (absence or occurrence of infection, and disease severity), implicitly encoded in the definition of the outcome, to causation.
We investigated the possibility that a single measurement run with a definite outcome is a joint unitary evolution of all the participating systems, and measurement runs with different definite outcomes correspond to different unitary maps. With reasonable assumptions, we derived a lower bound of the dependence of the environment after measurement on the state of the system before measurement, conditioned on the same measurement outcome. An experimental test of this dependence relation can either serve as evidence that the unitary dynamics and the definite outcome in the orthodox sense cannot be true together or suggest a deviation from the von Neumann measurement model plus a "conditioning" interpretational step.
Every legal case sets a precedent by developing the law in one of the following two ways. It either expands its scope, in which case it sets positive precedent, or it narrows it, in which case it sets negative precedent. Legal outcome prediction, the prediction of positive outcome, is an increasingly popular task in AI. In contrast, we turn our focus to negative outcomes here, and introduce a new task of negative outcome prediction. We discover an asymmetry in existing models' ability to predict positive and negative outcomes. Where the state-of-the-art outcome prediction model we used predicts positive outcomes at 75.06 F1, it predicts negative outcomes at only 10.09 F1, worse than a random baseline. To address this performance gap, we develop two new models inspired by the dynamics of a court process. Our first model significantly improves positive outcome prediction score to 77.15 F1 and our second model more than doubles the negative outcome prediction performance to 24.01 F1. Despite this improvement, shifting focus to negative outcomes reveals that there is still much room for improvement for outcome prediction models.
Agents acting on our behalf in the real world (e.g. placing phone calls) must learn online from costly, often irreversible interactions rather than cheap simulator steps. Two things follow. First, deployability depends on the path, not only the outcome. An agent must respect outcome-neutral constraints such as not repeatedly calling an unresponsive user, respecting business hours, or completing required authentication constraints that outcome-based rewards cannot express, since violating them frequently improves apparent success. Second, because each interaction is expensive, the agent must learn efficiently from very few examples. Reinforcement learning from verifiable rewards (RLVR) is blind to both challenges: it optimizes solely on the outcome and wastes expensive rollouts on all-fail groups where group-relative advantage collapses to zero. Attempts to densify supervision by rewarding progress target the hard-to-verify direction. In contrast, real agentic environments can cheaply detect bad moves. Since group-relative advantage is equivalent to within-group variance, a dense signal helps only when it supplies variance the outcome lacks. A verifiable penalty on the path meets th
Causal inference in longitudinal biomedical data remains a central challenge, especially in psychiatry, where symptom heterogeneity and latent confounding frequently undermine classical estimators. Most existing methods for treatment effect estimation presuppose a fixed outcome variable and address confounding through observed covariate adjustment. However, the assumption of unconfoundedness may not hold for a fixed outcome in practice. To address this foundational limitation, we directly optimize the outcome definition to maximize causal identifiability. Our DEBIAS (Durable Effects with Backdoor-Invariant Aggregated Symptoms) algorithm learns non-negative, clinically interpretable weights for outcome aggregation, maximizing durable treatment effects and empirically minimizing both observed and latent confounding by leveraging the time-limited direct effects of prior treatments in psychiatric longitudinal data. The algorithm also furnishes an empirically verifiable test for outcome unconfoundedness. DEBIAS consistently outperforms state-of-the-art methods in recovering causal effects for clinically interpretable composite outcomes across comprehensive experiments in depression and
When estimating causal effects from observational data with numerous covariates, employing penalized covariate selection can improve the estimation efficiency. Outcome-oriented covariate selection, which involves selecting covariates related to the outcome, can enhance efficiency, even for propensity score (PS) methods. For outcome-oriented covariate selection in PS models, outcome-adaptive lasso (OAL) can be used for penalization with the oracle property. The performance of inverse propensity weighted (IPW) estimators using the OAL was shown to be superior to that of the IPW estimators using other covariate selection methods for parametric models. However, the augmented IPW (AIPW) estimator is typically employed as a doubly robust estimator for the average treatment effect, which requires both PS and outcome models. Despite this, which covariate selection method for outcome models should be combined with the OAL to form the AIPW estimator remains unclear. We evaluated the performance of the AIPW estimators using the OAL for PS models and various outcome-oriented covariate selection via penalization for outcome models. We conducted numerical experiments to evaluate the performance
An individualized treatment rule (ITR) is a decision rule that recommends treatments for patients based on their individual feature variables. In many practices, the ideal ITR for the primary outcome is also expected to cause minimal harm to other secondary outcomes. Therefore, our objective is to learn an ITR that not only maximizes the value function for the primary outcome, but also approximates the optimal rule for the secondary outcomes as closely as possible. To achieve this goal, we introduce a fusion penalty to encourage the ITRs based on different outcomes to yield similar recommendations. Two algorithms are proposed to estimate the ITR using surrogate loss functions. We prove that the agreement rate between the estimated ITR of the primary outcome and the optimal ITRs of the secondary outcomes converges to the true agreement rate faster than if the secondary outcomes are not taken into consideration. Furthermore, we derive the non-asymptotic properties of the value function and misclassification rate for the proposed method. Finally, simulation studies and a real data example are used to demonstrate the finite-sample performance of the proposed method.
Estimators that weight observed outcomes to form effect estimates have a long tradition. Their outcome weights are widely used in established procedures, such as checking covariate balance, characterizing target populations, or detecting and managing extreme weights. This paper introduces a general framework for deriving such outcome weights. It establishes when and how numerical equivalence between an original estimator representation as moment condition and a unique weighted representation can be obtained. The framework is applied to derive novel outcome weights for the six seminal instances of double machine learning and generalized random forests, while recovering existing results for other estimators as special cases. The analysis highlights that implementation choices determine (i) the availability of outcome weights and (ii) their properties. Notably, standard implementations of partially linear regression-based estimators, like causal forests, employ outcome weights that do not sum to (minus) one in the (un)treated group, not fulfilling a property often considered desirable.
Counterfactual inference aims to estimate the counterfactual outcome at the individual level given knowledge of an observed treatment and the factual outcome, with broad applications in fields such as epidemiology, econometrics, and management science. Previous methods rely on a known structural causal model (SCM) or assume the homogeneity of the exogenous variable and strict monotonicity between the outcome and exogenous variable. In this paper, we propose a principled approach for identifying and estimating the counterfactual outcome. We first introduce a simple and intuitive rank preservation assumption to identify the counterfactual outcome without relying on a known structural causal model. Building on this, we propose a novel ideal loss for theoretically unbiased learning of the counterfactual outcome and further develop a kernel-based estimator for its empirical estimation. Our theoretical analysis shows that the rank preservation assumption is not stronger than the homogeneity and strict monotonicity assumptions, and shows that the proposed ideal loss is convex, and the proposed estimator is unbiased. Extensive semi-synthetic and real-world experiments are conducted to demo
In the quest to make defensible causal claims from observational data, it is sometimes possible to leverage information from "placebo treatments" and "placebo outcomes". Existing approaches employing such information focus largely on point identification and assume (i) "perfect placebos", meaning placebo treatments have precisely zero effect on the outcome and the real treatment has precisely zero effect on a placebo outcome; and (ii) "equiconfounding", meaning that the treatment-outcome relationship where one is a placebo suffers the same amount of confounding as does the real treatment-outcome relationship, on some scale. We instead consider an omitted variable bias framework, in which users can postulate ranges of values for the degree of unequal confounding and the degree of placebo imperfection. Once postulated, these assumptions identify or bound the linear estimates of treatment effects. Our approach also does not require using both a placebo treatment and placebo outcome, as some others do. While applicable in many settings, one ubiquitous use-case for this approach is to employ pre-treatment outcomes as (perfect) placebo outcomes, as in difference-in-difference. The parall
Counterfactual decision-making in the face of uncertainty involves selecting the optimal action from several alternatives using causal reasoning. Decision-makers often rank expected potential outcomes (or their corresponding utility and desirability) to compare the preferences of candidate actions. In this paper, we study new counterfactual decision-making rules by introducing two new metrics: the probabilities of potential outcome ranking (PoR) and the probability of achieving the best potential outcome (PoB). PoR reveals the most probable ranking of potential outcomes for an individual, and PoB indicates the action most likely to yield the top-ranked outcome for an individual. We then establish identification theorems and derive bounds for these metrics, and present estimation methods. Finally, we perform numerical experiments to illustrate the finite-sample properties of the estimators and demonstrate their application to a real-world dataset.
Sibling fixed effects (FE) models are useful for estimating causal treatment effects while offsetting unobserved sibling-invariant confounding. However, treatment estimates are biased if an individual's outcome affects their sibling's outcome. We propose a robustness test for assessing the presence of outcome-to-outcome interference in linear two-sibling FE models. We regress a gain-score--the difference between siblings' continuous outcomes--on both siblings' treatments and on a pre-treatment observed FE. Under certain restrictions, the observed FE's partial regression coefficient signals the presence of outcome-to-outcome interference. Monte Carlo simulations demonstrated the robustness test under several models. We found that an observed FE signaled outcome-to-outcome spillover if it was directly associated with an sibling-invariant confounder of treatments and outcomes, directly associated with a sibling's treatment, or directly and equally associated with both siblings' outcomes. However, the robustness test collapsed if the observed FE was directly but differentially associated with siblings' outcomes or if outcomes affected siblings' treatments.
Machine learning shows promise in predicting the outcome of legal cases, but most research has concentrated on civil law cases rather than case law systems. We identified two unique challenges in making legal case outcome predictions with case law. First, it is crucial to identify relevant precedent cases that serve as fundamental evidence for judges during decision-making. Second, it is necessary to consider the evolution of legal principles over time, as early cases may adhere to different legal contexts. In this paper, we proposed a new framework named PILOT (PredictIng Legal case OuTcome) for case outcome prediction. It comprises two modules for relevant case retrieval and temporal pattern handling, respectively. To benchmark the performance of existing legal case outcome prediction models, we curated a dataset from a large-scale case law database. We demonstrate the importance of accurately identifying precedent cases and mitigating the temporal shift when making predictions for case law, as our method shows a significant improvement over the prior methods that focus on civil law case outcome predictions.