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Recent work has focused on nonparametric estimation of conditional treatment effects, but inference has remained relatively unexplored. We propose a class of nonparametric tests for both quantitative and qualitative treatment effect heterogeneity. The tests can incorporate a variety of structured assumptions on the conditional average treatment effect, allow for both continuous and discrete covariates, and do not require sample splitting. Furthermore, we show how the tests are tailored to detect alternatives where the population impact of adopting a personalized decision rule differs from using a rule that discards covariates. The proposal is thus relevant for guiding treatment policies. The utility of the proposal is borne out in simulation studies and a re-analysis of an AIDS clinical trial.

We study the problem of constructing an estimator of the average treatment effect (ATE) that exhibits doubly-robust asymptotic linearity (DRAL). This is a stronger requirement than doubly-robust consistency. A DRAL estimator can yield asymptotically valid Wald-type confidence intervals even when the propensity score or the outcome model is inconsistently estimated. On the contrary, the celebrated doubly-robust, augmented-IPW (AIPW) estimator generally requires consistent estimation of both nuisance functions for standard root-n inference. We make three main contributions. First, we propose a new hybrid class of distributions that consists of the structure-agnostic class introduced in Balakrishnan et al (2023) with additional smoothness constraints. While DRAL is generally not possible in the pure structure-agnostic class, we show that it can be attained in the new hybrid one. Second, we calculate minimax lower bounds for estimating the ATE in the new class, as well as in the pure structure-agnostic one. Third, building upon the literature on doubly-robust inference (van der Laan, 2014, Benkeser et al, 2017, Dukes et al 2021), we propose a new estimator of the ATE that enjoys DRAL. Under certain conditions, we show that its rate of convergence in the new class can be much faster than that achieved by the AIPW estimator and, in particular, matches the minimax lower bound rate, thereby establishing its optimality. Finally, we clarify the connection between DRAL estimators and those based on higher-order influence functions (Robins et al, 2017) and complement our theoretical findings with simulations., Comment: 54 pages, 2 figures

Instrumental variables (IV) are a commonly used tool to estimate causal effects from non-randomized data. An archetype of an IV is a randomized trial with non-compliance where the randomized treatment assignment serves as an IV for the non-ignorable treatment received. Under a monotonicity assumption, a valid IV non-parametrically identifies the average treatment effect among a non-identified, latent complier subgroup, whose generalizability is often under debate. In many studies, there could exist multiple versions of an IV, for instance, different nudges to take the same treatment in different study sites in a multicentre clinical trial. These different versions of an IV may result in different compliance rates and offer a unique opportunity to study IV estimates' generalizability. In this article, we introduce a novel nested IV assumption and study identification of the average treatment effect among two latent subgroups: always-compliers and switchers, who are defined based on the joint potential treatment received under two versions of a binary IV. We derive the efficient influence function for the SWitcher Average Treatment Effect (SWATE) and propose efficient estimators. We then propose formal statistical tests of the generalizability of IV estimates under the nested IV framework. We apply the proposed method to the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial and study the causal effect of colorectal cancer screening and its generalizability.

There has been a growing interest in covariate adjustment in the analysis of randomized controlled trials in past years. For instance, the U.S. Food and Drug Administration recently issued guidance that emphasizes the importance of distinguishing between conditional and marginal treatment effects. Although these effects coincide in linear models, this is not typically the case in other settings, and this distinction is often overlooked in clinical trial practice. Considering these developments, this paper provides a review of when and how to utilize covariate adjustment to enhance precision in randomized controlled trials. We describe the differences between conditional and marginal estimands and stress the necessity of aligning statistical analysis methods with the chosen estimand. Additionally, we highlight the potential misalignment of current practices in estimating marginal treatment effects. Instead, we advocate for the utilization of standardization, which can improve efficiency by leveraging the information contained in baseline covariates while remaining robust to model misspecification. Finally, we present practical considerations that have arisen in our respective consultations to further clarify the advantages and limitations of covariate adjustment.

We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning (AutoDML). These popular doubly robust or de-biased machine learning estimators combine outcome modeling with balancing weights - weights that achieve covariate balance directly in lieu of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine the coefficients from the original outcome model and coefficients from an unpenalized ordinary least squares (OLS) fit on the same data. We see that, under certain choices of regularization parameters, the augmented estimator often collapses to the OLS estimator alone; this occurs for example in a re-analysis of the Lalonde 1986 dataset. We then extend these results to specific choices of outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression. This holds numerically in finite samples and lays the groundwork for a novel analysis of undersmoothing and asymptotic rates of convergence. When the weighting model is instead lasso-penalized regression, we give closed-form expressions for special cases and demonstrate a ``double selection'' property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights.

In many clinical settings, an active-controlled trial design (e.g., a non-inferiority or superiority design) is often used to compare an experimental medicine to an active control (e.g., an FDA-approved, standard therapy). One prominent example is a recent phase 3 efficacy trial, HIV Prevention Trials Network Study 084 (HPTN 084), comparing long-acting cabotegravir, a new HIV pre-exposure prophylaxis (PrEP) agent, to the FDA-approved daily oral tenofovir disoproxil fumarate plus emtricitabine (TDF/FTC) in a population of heterosexual women in 7 African countries. One key complication of interpreting study results in an active-controlled trial like HPTN 084 is that the placebo arm is not present and the efficacy of the active control (and hence the experimental drug) compared to the placebo can only be inferred by leveraging other data sources. \bz{In this article, we study statistical inference for the intention-to-treat (ITT) effect of the active control using relevant historical placebo-controlled trials data under the potential outcomes (PO) framework}. We highlight the role of adherence and unmeasured confounding, discuss in detail identification assumptions and two modes of inference (point versus partial identification), propose estimators under identification assumptions permitting point identification, and lay out sensitivity analyses needed to relax identification assumptions. We applied our framework to estimating the intention-to-treat effect of daily oral TDF/FTC versus placebo in HPTN 084 using data from an earlier Phase 3, placebo-controlled trial of daily oral TDF/FTC (Partners PrEP).

Model-X approaches to testing conditional independence between a predictor and an outcome variable given a vector of covariates usually assume exact knowledge of the conditional distribution of the predictor given the covariates. Nevertheless, model-X methodologies are often deployed with this conditional distribution learned in sample. We investigate the consequences of this choice through the lens of the distilled conditional randomization test (dCRT). We find that Type-I error control is still possible, but only if the mean of the outcome variable given the covariates is estimated well enough. This demonstrates that the dCRT is doubly robust, and motivates a comparison to the generalized covariance measure (GCM) test, another doubly robust conditional independence test. We prove that these two tests are asymptotically equivalent, and show that the GCM test is optimal against (generalized) partially linear alternatives by leveraging semiparametric efficiency theory. In an extensive simulation study, we compare the dCRT to the GCM test. These two tests have broadly similar Type-I error and power, though dCRT can have somewhat better Type-I error control but somewhat worse power in small samples or when the response is discrete. We also find that post-lasso based test statistics (as compared to lasso based statistics) can dramatically improve Type-I error control for both methods.

We consider studies where multiple measures on an outcome variable are collected over time, but some subjects drop out before the end of follow up. Analyses of such data often proceed under either a 'last observation carried forward' or 'missing at random' assumption. We consider two alternative strategies for identification; the first is closely related to the difference-in-differences methodology in the causal inference literature. The second enables correction for violations of the parallel trend assumption, so long as one has access to a valid 'bespoke instrumental variable'. These are compared with existing approaches, first conceptually and then in an analysis of data from the Framingham Heart Study.

We link and extend two approaches to estimating time-varying treatment effects on repeated continuous outcomes--time-varying Difference in Differences (DiD; see Roth et al. (2023) and Chaisemartin et al. (2023) for reviews) and Structural Nested Mean Models (SNMMs; see Vansteelandt and Joffe (2014) for a review). In particular, we show that SNMMs, which were previously only known to be nonparametrically identified under a no unobserved confounding assumption, are also identified under a generalized version of the parallel trends assumption typically used to justify time-varying DiD methods. Because SNMMs model a broader set of causal estimands, our results allow practitioners of existing time-varying DiD approaches to address additional types of substantive questions under similar assumptions. SNMMs enable estimation of time-varying effect heterogeneity, lasting effects of a `blip' of treatment at a single time point, effects of sustained interventions (possibly on continuous or multi-dimensional treatments) when treatment repeatedly changes value in the data, controlled direct effects, effects of dynamic treatment strategies that depend on covariate history, and more. Our results also allow analysts who apply SNMMs under the no unobserved confounding assumption to estimate some of the same causal effects under alternative identifying assumptions. We provide a method for sensitivity analysis to violations of our parallel trends assumption. We further explain how to estimate optimal treatment regimes via optimal regime SNMMs under parallel trends assumptions plus an assumption that there is no effect modification by unobserved confounders. Finally, we illustrate our methods with real data applications estimating effects of Medicaid expansion on uninsurance rates, effects of floods on flood insurance take-up, and effects of sustained changes in temperature on crop yields.

Many proposals for the identification of causal effects require an instrumental variable that satisfies strong, untestable unconfoundedness and exclusion restriction assumptions. In this paper, we show how one can potentially identify causal effects under violations of these assumptions by harnessing a negative control population or outcome. This strategy allows one to leverage sup-populations for whom the exposure is degenerate, and requires that the instrument-outcome association satisfies a certain parallel trend condition. We develop the semiparametric efficiency theory for a general instrumental variable model, and obtain a multiply robust, locally efficient estimator of the average treatment effect in the treated. The utility of the estimators is demonstrated in simulation studies and an analysis of the Life Span Study., Comment: 65 pages

The problem of how to best select variables for confounding adjustment forms one of the key challenges in the evaluation of exposure effects in observational studies, and has been the subject of vigorous recent activity in causal inference. A major drawback of routine procedures is that there is no finite sample size at which they are guaranteed to deliver exposure effect estimators and associated confidence intervals with adequate performance. In this work, we will consider this problem when inferring conditional causal hazard ratios from observational studies under the assumption of no unmeasured confounding. The major complication that we face with survival data is that the key confounding variables may not be those that explain the censoring mechanism. In this paper, we overcome this problem using a novel and simple procedure that can be implemented using off-the-shelf software for penalized Cox regression. In particular, we will propose tests of the null hypothesis that the exposure has no effect on the considered survival endpoint, which are uniformly valid under standard sparsity conditions. Simulation results show that the proposed methods yield valid inferences even when covariates are high-dimensional.

Intercurrent (post-treatment) events occur frequently in randomized trials, and investigators often express interest in treatment effects that suitably take account of these events. A naive conditioning on intercurrent events does not have a straight-forward causal interpretation, and the practical relevance of other commonly used approaches is debated. In this work, we discuss how to formulate and choose an estimand, beyond the marginal intention to treat effect, from the point of view of a decision maker and drug developer. In particular, we argue that careful articulation of a practically useful research question should either reflect decision making at this point in time or future drug development. Indeed, a substantially interesting estimand is a formalization of the (plain English) description of a research question. A common feature of estimands that are practically useful is that they correspond to possibly hypothetical but well-defined interventions in identifiable (sub)populations. To illustrate our points, we consider five examples that were recently used to motivate consideration of principal stratum estimands in clinical trials. In all of these examples, we propose alternative causal estimands, such as conditional effects, sequential regime effects and separable effects, that correspond to explicit research questions of substantial interest. Certain questions require stronger assumptions for identification. However, we highlight that our proposed estimands require less stringent assumptions than estimands commonly targeted in these settings, including principal stratum effects.

A common concern when trying to draw causal inferences from observational data is that the measured covariates are insufficiently rich to account for all sources of confounding. In practice, many of the covariates may only be proxies of the latent confounding mechanism. Recent work has shown that in certain settings where the standard 'no unmeasured confounding' assumption fails, proxy variables can be leveraged to identify causal effects. Results currently exist for the total causal effect of an intervention, but little consideration has been given to learning about the direct or indirect pathways of the effect through a mediator variable. In this work, we describe three separate proximal identification results for natural direct and indirect effects in the presence of unmeasured confounding. We then develop a semiparametric framework for inference on natural (in)direct effects, which leads us to locally efficient, multiply robust estimators., Comment: 60 pages, 3 figures

Due to concerns about parametric model misspecification, there is interest in using machine learning to adjust for confounding when evaluating the causal effect of an exposure on an outcome. Unfortunately, exposure effect estimators that rely on machine learning predictions are generally subject to so-called plug-in bias, which can render naive p-values and confidence intervals invalid. Progress has been made via proposals like targeted minimum loss estimation (TMLE) and more recently double machine learning, which rely on learning the conditional mean of both the outcome and exposure. Valid inference can then be obtained so long as both predictions converge (sufficiently fast) to the truth. Focusing on partially linear regression models, we show that a specific implementation of the machine learning techniques can yield exposure effect estimators that have small bias even when one of the first-stage predictions does not converge to the truth. The resulting tests and confidence intervals are doubly robust. We also show that the proposed estimators may fail to be regular when only one nuisance parameter is consistently estimated; nevertheless, we observe in simulation studies that our proposal can lead to reduced bias and improved confidence interval coverage in moderate samples., Comment: 42 pages, 9 figures

Evaluation of treatment effects and more general estimands is typically achieved via parametric modelling, which is unsatisfactory since model misspecification is likely. Data-adaptive model building (e.g. statistical/machine learning) is commonly employed to reduce the risk of misspecification. Naive use of such methods, however, delivers estimators whose bias may shrink too slowly with sample size for inferential methods to perform well, including those based on the bootstrap. Bias arises because standard data-adaptive methods are tuned towards minimal prediction error as opposed to e.g. minimal MSE in the estimator. This may cause excess variability that is difficult to acknowledge, due to the complexity of such strategies. Building on results from non-parametric statistics, targeted learning and debiased machine learning overcome these problems by constructing estimators using the estimand's efficient influence function under the non-parametric model. These increasingly popular methodologies typically assume that the efficient influence function is given, or that the reader is familiar with its derivation. In this paper, we focus on derivation of the efficient influence function and explain how it may be used to construct statistical/machine-learning-based estimators. We discuss the requisite conditions for these estimators to perform well and use diverse examples to convey the broad applicability of the theory.

The analysis of randomized trials with time-to-event endpoints is nearly always plagued by the problem of censoring. As the censoring mechanism is usually unknown, analyses typically employ the assumption of non-informative censoring. While this assumption usually becomes more plausible as more baseline covariates are being adjusted for, such adjustment also raises concerns. Pre-specification of which covariates will be adjusted for (and how) is difficult, thus prompting the use of data-driven variable selection procedures, which may impede valid inferences to be drawn. The adjustment for covariates moreover adds concerns about model misspecification, and the fact that each change in adjustment set, also changes the censoring assumption and the treatment effect estimand. In this paper, we discuss these concerns and propose a simple variable selection strategy that aims to produce a valid test of the null in large samples. The proposal can be implemented using off-the-shelf software for (penalized) Cox regression, and is empirically found to work well in simulation studies and real data analyses.

Inference for the parameters indexing generalised linear models is routinely based on the assumption that the model is correct and a priori specified. This is unsatisfactory because the chosen model is usually the result of a data-adaptive model selection process, which may induce excess uncertainty that is not usually acknowledged. Moreover, the assumptions encoded in the chosen model rarely represent some a priori known, ground truth, making standard inferences prone to bias, but also failing to give a pure reflection of the information that is contained in the data. Inspired by developments on assumption-free inference for so-called projection parameters, we here propose novel nonparametric definitions of main effect estimands and effect modification estimands. These reduce to standard main effect and effect modification parameters in generalised linear models when these models are correctly specified, but have the advantage that they continue to capture respectively the primary (conditional) association between two variables, or the degree to which two variables interact (in a statistical sense) in their effect on outcome, even when these models are misspecified. We achieve an assumption-lean inference for these estimands (and thus for the underlying regression parameters) by deriving their influence curve under the nonparametric model and invoking flexible data-adaptive (e.g., machine learning) procedures., Comment: 37 pages, 2 tables

Eliminating the effect of confounding in observational studies typically involves fitting a model for an outcome adjusted for covariates. When, as often, these covariates are high-dimensional, this necessitates the use of sparse estimators such as the Lasso, or other regularisation approaches. Naiive use of such estimators yields confidence intervals for the conditional treatment effect parameter that are not uniformly valid. Moreover, as the number of covariates grows with sample size, correctly specifying a model for the outcome is non-trivial. In this work, we deal with both of these concerns simultaneously, delivering confidence intervals for conditional treatment effects that are uniformly valid, regardless of whether the outcome model is correct. This is done by incorporating an additional model for the treatment-selection mechanism. When both models are correctly specified, we can weaken the standard conditions on model sparsity. Our procedure extends to multivariate treatment effect parameters and complex longitudinal settings., Comment: 40 pages, 4 tables

After variable selection, standard inferential procedures for regression parameters may not be uniformly valid; there is no finite-sample size at which a standard test is guaranteed to approximately attain its nominal size. This problem is exacerbated in high-dimensional settings, where variable selection becomes unavoidable. This has prompted a flurry of activity in developing uniformly valid hypothesis tests for a low-dimensional regression parameter (e.g. the causal effect of an exposure A on an outcome Y) in high-dimensional models. So far there has been limited focus on model misspecification, although this is inevitable in high-dimensional settings. We propose tests of the null that are uniformly valid under sparsity conditions weaker than those typically invoked in the literature, assuming working models for the exposure and outcome are both correctly specified. When one of the models is misspecified, by amending the procedure for estimating the nuisance parameters, our tests continue to be valid; hence they are doubly robust. Our proposals are straightforward to implement using existing software for penalized maximum likelihood estimation and do not require sample-splitting. We illustrate them in simulations and an analysis of data obtained from the Ghent University Intensive Care Unit., Comment: 53 pages, 5 tables