Parameterising the effect of a continuous treatment using average derivative effects

In this paper, we provide efficient estimators and honest confidence bands for a variety of treatment effects including local average (LATE) and local quantile treatment effects (LQTE) in data-rich environments. We can handle very many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and function-valued outcomes. Our framework covers the special case of exogenous receipt of treatment, either conditional on controls or unconditionally as in randomized control trials. In the latter case, our approach produces efficient estimators and honest bands for (functional) average treatment effects (ATE) and quantile treatment effects (QTE). To make informative inference possible, we assume that key reduced form predictive relationships are approximately sparse. This assumption allows the use of regularization and selection methods to estimate those relations, and we provide methods for post-regularization and post-selection inference that are uniformly valid (honest) across a wide-range of models. We show that a key ingredient enabling honest inference is the use of orthogonal or doubly robust moment conditions in estimating certain reduced form functional parameters. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) eligibility and participation on accumulated assets. The results on program evaluation are obtained as a consequence of more general results on honest inference in a general moment condition framework, which arises from structural equation models in econometrics. Here too the crucial ingredient is the use of orthogonal moment conditions, which can be constructed from the initial moment conditions. We provide results on honest inference for (function-valued) parameters within this general framework where any high-quality, modern machine learning methods can be used to learn the nonparametric/high-dimensional components of the model. These include a number of supporting auxilliary results that are of major independent interest: namely, we (1) prove uniform validity of a multiplier bootstrap, (2) offer a uniformly valid functional delta method, and (3) provide results for sparsity-based estimation of regression functions for function-valued outcomes.

Unequal cluster sizes are common in cluster randomized trials (CRTs). While there are a number of previous investigations studying the impact of unequal cluster sizes on the power for testing the average treatment effect in CRTs, little is known about the impact of unequal cluster sizes on the power for testing the heterogeneous treatment effect (HTE) in CRTs. In this work, we expand the sample size procedures for studying HTE in CRTs to accommodate cluster size variation under the linear mixed model framework. Through analytical derivation and graphical exploration, we show that the sample size for the HTE with an individual-level effect modifier is less affected by unequal cluster sizes than with a cluster-level effect modifier. The impact of cluster size variability jointly depends on the mean and coefficient of variation of cluster sizes, covariate intraclass correlation coefficient (ICC) and the conditional outcome ICC. In addition, we demonstrate that the HTE-motivated analysis of covariance framework can be used for analyzing the average treatment effect, and offer a more efficient sample size procedure for studying the average treatment effect adjusting for the effect modifier. We use simulations to confirm the accuracy of the proposed sample size procedures for both the average treatment effect and HTE in CRTs. Extensions to multivariate effect modifiers are provided and our procedure is illustrated in the context of the Strategies to Reduce Injuries and Develop Confidence in Elders trial.

A large part of the recent literature on program evaluation has focused on estimation of the average effect of the treatment under assumptions of unconfoundedness or ignorability following the seminal work by In many cases however, researchers are interested in the effects of programs beyond estimates of the overall average or the average for the subpopulation of treated individuals. It may be of substantive interest to investigate whether there is any subpopulation for which a program or treatment has a nonzero average effect, or whether there is heterogeneity in the effect of the treatment. The hypothesis that the average effect of the treatment is zero for all subpopulations is also important for researchers interested in assessing assumptions concerning the selection mechanism. In this paper we develop two nonparametric tests. The first test is for the null hypothesis that the treatment has a zero average effect for any subpopulation defined by covariates. The second test is for the null hypothesis that the average effect conditional on the covariates is identical for all subpopulations, in other words, that there is no heterogeneity in average treatment effects by covariates. Sacrificing some generality by focusing on these two specific null hypotheses we derive tests that are straightforward to implement.

Background Symptomatic acute occlusions of the internal carotid artery (ICA) below the circle of Willis can cause a variety of stroke symptoms, even if the major intracranial cerebral arteries remain patent; however, outcome and safety data are limited. Purpose To compare treatment effects and procedural safety of endovascular treatment (EVT) and best medical treatment (BMT) in patients with symptomatic acute occlusions of the ICA below the circle of Willis. Materials and Methods This retrospective, multicenter cohort study from 22 comprehensive stroke centers in Europe and Asia includes patients treated between January 1, 2008, and December 31, 2022. Functional (modified Rankin Scale [mRS]) and clinical (National Institutes of Health Stroke Scale [NIHSS]) outcomes, safety measures (symptomatic intracerebral hemorrhage), mortality, and procedural complications were assessed. Results A total 354 patients met the inclusion criteria (median age, 72 years [IQR, 60-81 years]; median NIHSS, 13 [IQR, 7-19]). Most frequent occlusions were in the C1 segment (243 of 354; 68.6%). Of 354 patients, 82.2% (291 patients) were administered EVT. In the overall population, favorable outcomes (mRS 0-2), mortality, and symptomatic intracerebral hemorrhage occurred in 40.6% (108 of 266 patients), 25.2% (67 of 266 patients), and 7.1% (25 of 350 patients), respectively. After adjustment, no statistically significant difference in functional outcome was observed (adjusted odds ratio [AOR], 0.82 [95% CI: 0.31, 2.12]; average treatment effect, -12.7%; P = .19) in the EVT compared with BMT group. Symptomatic intracerebral hemorrhage (average treatment effect, -0.28%; P = .95) and mortality did not differ between both groups (average treatment effect, -17.1%; P = .07). EVT resulted in complete recanalization of the occlusion in 80.9% (229 of 283) of cases. Periprocedural distal embolization occurred in 27.8% (81 of 291 patients) and was associated with poor outcomes (AOR, 0.41; 95% CI: 0.18, 0.93; P = .03). Conclusion EVT did not reveal a favorable treatment effect over BMT, and both therapies were safe. EVT had a risk for periprocedural distal embolization associated with poor outcomes. © RSNA, 2025 Supplemental material is available for this article. See also the editorial by Daou and Chaudhary in this issue.

Political scientists frequently use instrumental variables (IV) estimation to estimate the causal effect of an endogenous treatment variable. However, when the treatment effect is heterogeneous, this estimation strategy only recovers the local average treatment effect (LATE). The LATE is an average treatment effect (ATE) for a subset of the population: units that receive treatment if and only if they are induced by an exogenous IV. However, researchers may instead be interested in the ATE for the entire population of interest. In this article, we develop a simple reweighting method for estimating the ATE, shedding light on the identification challenge posed in moving from the LATE to the ATE. We apply our method to two published experiments in political science in which we demonstrate that the LATE has the potential to substantively differ from the ATE.

In this paper, we consider a switching regression model with count data outcomes, where the possible outcome differs across two alternate states and individuals endogenously select one of the states. We assume lognormal latent heterogeneity. Building on the switching regression model, we derive estimators of various treatment effects: the average treatment effect, the average treatment effect on the treated, the local average treatment effect, and the marginal treatment effect. We illustrate an application that examines the effects of public insurance on the number of doctor visits using the data employed by previous studies.

Traditional methods for evaluating a potential treatment have focused on the average treatment effect. However, there exist situations where individuals can experience significantly heterogeneous responses to a treatment. In these situations, one needs to account for the differences among individuals when estimating the treatment effect. Li et al. (2022) proposed a method based on random forest of interaction trees (RFIT) for a binary or categorical treatment variable, while incorporating the propensity score in the construction of random forest. Motivated by the need to evaluate the effect of tutoring sessions at a Math and Stat Learning Center (MSLC), we extend their approach to an ordinal treatment variable. Our approach improves upon RFIT for multiple treatments by incorporating the ordered structure of the treatment variable into the tree growing process. To illustrate the effectiveness of our proposed method, we conduct simulation studies where the results show that our proposed method has a lower mean squared error and higher optimal treatment classification, and is able to identify the most important variables that impact the treatment effect. We then apply the proposed method to estimate how the number of visits to the MSLC impacts an individual student’s probability of passing an introductory statistics course. Our results show that every student is recommended to go to the MSLC at least once and some can drastically improve their chance of passing the course by going the optimal number of times suggested by our analysis.

Loneliness is a distressing feeling that influences well-being. Immigrants' experience of acculturation to a new dominant culture places them at risk for maladaptive behaviors and daily rhythms leading to loneliness. Identifying daily-living features that causally influence loneliness is essential for developing effective preventive mental health screening. To identify the important daily living-features related to loneliness for the development of robust screening solutions using causal machine learning for health providers working with first-generation immigrants. We monitored 39 immigrants in Finland for 28 days using mobile devices and wearables under free-living conditions. Data included ecological momentary assessments of loneliness, social interactions, physical activity, sleep, and cardiac features. We estimated the average treatment effect (ATE) of each daily-living feature (treatment variable) on loneliness scores (outcome) and validated the robustness of causal estimates using three refutation techniques. Our results reveal the ATE of various daily-living features on loneliness. Features such as longer outgoing call durations (ATE = 0.197, p < 0.001), higher LF/HF ratio (ATE = 0.129, p < 0.0001), higher respiratory rate (ATE = 0.144, p < 0.001), and increased inactivity (ATE = 0.130, p < 0.001) causally increased loneliness. Conversely, certain features exhibit negative ATEs, such as higher activity calories (ATE = -0.174, p < 0.001), sleep RMSSD (ATE = -0.128, p < 0.001), longer home duration (ATE = -0.107, p < 0.001), and more sleep time (ATE = -0.103, p < 0.001) mitigated loneliness. Daily-living features, including social interactions, activity, sleep, and cardiac features, causally influence loneliness. Our findings provide a basis for loneliness screening targeting immigrant populations. Future work should refine the measurement and incorporate contextual information to establish more reliable causal links in real life.

This paper provides a review of methodological advancements in the evaluation of heterogeneous treatment effect models based on instrumental variable (IV) methods. We focus on models that achieve identification by assuming monotonicity of the treatment in the IV and analyze local average and quantile treatment effects for the subpopulation of compliers. We start with a comprehensive discussion of the binary treatment and binary IV case as for instance relevant in randomized experiments with imperfect compliance. We then review extensions to identification and estimation with covariates, multi-valued and multiple treatments and instruments, outcome attrition and measurement error, and the identification of direct and indirect treatment effects, among others. We also discuss testable implications and possible relaxations of the IV assumptions, approaches to extrapolate from local to global treatment effects, and the relationship to other IV approaches.

BackgroundOptimizing retention in human immunodeficiency virus (HIV) treatment may require sequential behavioral interventions based on patients’ response.MethodsIn a sequential multiple assignment randomized trial in Kenya, we randomly assigned adults initiating HIV treatment to standard of care (SOC), Short Message Service (SMS) messages, or conditional cash transfers (CCT). Those with retention lapse (missed a clinic visit by ≥14 days) were randomly assigned again to standard-of-care outreach (SOC-Outreach), SMS+CCT, or peer navigation. Those randomly assigned to SMS or CCT who did not lapse after 1 year were randomly assigned again to either stop or continue the initial intervention. Primary outcomes were retention in care without an initial lapse, return to the clinic among those who lapsed, and time in care; secondary outcomes included adjudicated viral suppression. Average treatment effect (ATE) was calculated using targeted maximum likelihood estimation with adjustment for baseline characteristics at randomization and certain time-varying characteristics at rerandomization.ResultsAmong 1809 participants, 79.7% of those randomly assigned to CCT (n=523/656), 71.7% to SMS (n=393/548), and 70.7% to SOC (n=428/605) were retained in care in the first year (ATE: 9.9%; 95% confidence interval [CI]: 5.4%, 14.4% and ATE: 4.2%; 95% CI: −0.7%, 9.2% for CCT and SMS compared with SOC, respectively). Among 312 participants with an initial lapse who were randomly assigned again, 69.1% who were randomly assigned to a navigator (n=76/110) returned, 69.5% randomly assigned to CCT+SMS (n=73/105) returned, and 55.7% randomly assigned to SOC-Outreach (n=54/97) returned (ATE: 14.1%; 95% CI: 0.6%, 27.6% and ATE: 11.4%; 95% CI: −2.2%, 24.9% for navigator and CCT+SMS compared with SOC-Outreach, respectively). Among participants without lapse on SMS, continuing SMS did not affect retention (n=122/180; 67.8% retained) versus stopping (n=151/209; 72.2% retained; ATE: −4.4%; 95% CI: −16.6%, 7.9%). Among participants without lapse on CCT, those continuing CCT had higher retention (n=192/230; 83.5% retained) than those stopping (n=173/287; 60.3% retained; ATE: 28.6%; 95% CI: 19.9%, 37.3%). Among 15 sequenced strategies, initial CCT, escalated to navigator if lapse occurred and continued if no lapse occurred, increased time in care (ATE: 7.2%, 95% CI: 3.7%, 10.7%) and viral suppression (ATE: 8.2%, 95% CI: 2.2%, 14.2%), the most compared with SOC throughout. Initial SMS escalated to navigator if lapse occurred, and otherwise continued, showed similar effect sizes compared with SOC throughout.ConclusionsActive interventions to prevent retention lapses followed by navigation for those who lapse and maintenance of initial intervention for those without lapse resulted in best overall retention and viral suppression among the strategies studied. Among those who remained in care, discontinuation of CCT, but not SMS, compromised retention and suppression. (Funded by National Institutes of Health grants R01 MH104123, K24 AI134413, and R01 AI074345; ClinicalTrials.gov number, NCT02338739.)

We revisit the classic semi‐parametric problem of inference on a low‐dimensional parameter θ0 in the presence of high‐dimensional nuisance parameters η0. We depart from the classical setting by allowing for η0 to be so high‐dimensional that the traditional assumptions (e.g. Donsker properties) that limit complexity of the parameter space for this object break down. To estimate η0, we consider the use of statistical or machine learning (ML) methods, which are particularly well suited to estimation in modern, very high‐dimensional cases. ML methods perform well by employing regularization to reduce variance and trading off regularization bias with overfitting in practice. However, both regularization bias and overfitting in estimating η0 cause a heavy bias in estimators of θ0 that are obtained by naively plugging ML estimators of η0 into estimating equations for θ0. This bias results in the naive estimator failing to be N−1/2 consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest θ0 can be removed by using two simple, yet critical, ingredients: (1) using Neyman‐orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ0; (2) making use of cross‐fitting, which provides an efficient form of data‐splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in an N−1/2‐neighbourhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid confidence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements, which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters, such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of the following: DML applied to learn the main regression parameter in a partially linear regression model; DML applied to learn the coefficient on an endogenous variable in a partially linear instrumental variables model; DML applied to learn the average treatment effect and the average treatment effect on the treated under unconfoundedness; DML applied to learn the local average treatment effect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.

Fixed effects estimation, with linear controls for stratum membership, is often used to estimate treatment effects when assignment propensities differ across strata. In the presence of heterogeneity in treatment effects across strata, this estimator does not target the average treatment effect, however. Indeed, the implied estimand can range anywhere from the lowest to the highest stratum-level average effect. To facilitate the interpretation of results using this approach, I establish that if stratum-level average effects are monotonic in the shares assigned to treatment, then the fixed effects estimand lies between the average treatment effect for the treated and the average treatment effect for the controls.

The local average treatment effect (LATE) may differ from the average treatment effect (ATE) when those influenced by the instrument are not representative of the overall population. Heterogeneity in treatment effects may imply that parameter estimates from 2SLS are uninformative regarding the average treatment effect, motivating a search for instruments that affect a larger share of the population. In this paper, I present and estimate a model of home production with heterogeneous costs and benefits to fertility. The results indicate that a sex-preference instrument in Taiwan produces IV estimates closer to the estimated ATE than in the United States, where sex preference is weaker.

This article reviews a set of estimands commonly used in modern applied research to operationalize causal inquiries within the Rubin Causal Model (RCM). I first introduce the basic average treatment effects (ATE, ATT, ATC) and then describe their main extensions, including local and conditional treatment effects, causal interactions, causal mediation, multivalued or continuous treatments, and longitudinal treatment effects. For each estimand, I provide a substantive explanation along with examples of research questions it can address. For the most popular estimands, I also discuss the key assumptions required for their nonparametric identification and statistical methods available for their estimation.

SummaryBackgroundWhile specialist-led early intervention services (EIS) have been shown to be effective for early-phase psychosis, the effectiveness of a non-specialist-led EIS is still unclear. The study aims to determine the effectiveness of a low-intensity case management (LICM) programme led by lay health workers for patients during early episodes of psychosis on six-month outcomes compared to usual care (UC) in Bangkok, Thailand.MethodsThis pragmatic randomised controlled trial (Lay-CARE) included patients aged ≥18 years with a history of a first or second episode of psychotic disorder residing in the Bangkok Metropolitan Region (Thammasat University hospital's catchment district). Participants were randomised to either LICM or UC group. LICM intervention consisted of family psychoeducation, facilitation of access to services, and home or telephone visits, whereas UC was the baseline of care participants were receiving. The intervention started in October 2020 and ended in May 2021. The 6-month endline assessment started in April 2021 and ended in May 2021. The primary outcome was social functioning measured by the Personal and Social Performance Scale Thai version. Secondary outcomes were symptom severity, service use and medication adherence. The treatment effect size was determined using the average treatment effect (ATE). Due to the nature of the intervention, blinding of participants and assessors was not possible. The trial was registered with the Thai Clinical Trial Registry (TCTR20210509001).Findings130 participants were randomised to LICM group and 125 to UC group. At the six-month endline, participants in LICM group demonstrated improved socially useful activities (ATE 0.06, 95% CI 0.00–0.13, p = 0.041), less aggressive or disturbing behaviour (ATE −0.17, 95% CI −0.27 to −0.06, p = 0.002), and better self-care (ATE 0.13 95% CI 0.05–0.22, p = 0.003) than participants in UC group. LICM intervention did not affect personal and social relationships, symptom severity, medication adherence, or service use.InterpretationIn low-resource settings, LICM programme can improve areas of social functioning among patients with early episodes of psychosis over six months.Funding10.13039/501100010724Health Systems Research Institute (HSRI), Thailand (Grant No. 60-087).