Evaluating the treatment effects has become an important topic for many applications. However, most existing literature focuses mainly on the average treatment effects. When the individual effects are heavy-tailed or have outlier values, not only may the average effect not be appropriate for summarizing the treatment effects, but also the conventional inference for it can be sensitive and possibly invalid due to poor large-sample approximations. In this paper we focus on quantiles of individual effects, which can be more robust measures of treatment effects in the presence of extreme individual effects. Moreover, our inference for quantiles of individual effects are purely randomization-based, which avoids any distributional assumption on the units. We first consider inference for stratified randomized experiments, extending the recent work of Caughey et al. (2021). The calculation of valid $p$-values for testing null hypotheses on quantiles of individual effects involves linear integer programming, which is generally NP hard. To overcome this issue, we propose a greedy algorithm with a certain optimal transformation, which has much lower computational cost, still leads to valid $p$-values and is less conservative than the usual relaxation by dropping the integer constraint. We then extend our approach to matched observational studies and propose sensitivity analysis to investigate to what extent our inference on quantiles of individual effects is robust to unmeasured confounding. Both the randomization inference and sensitivity analysis are simultaneously valid for all quantiles of individual effects, which are actually free lunches added to the conventional analysis assuming constant effects. Furthermore, the inference results can be easily visualized and interpreted.
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Individualized treatment rules, cornerstones of precision medicine, inform patient treatment decisions with the goal of optimizing patient outcomes. These rules are generally unknown functions of patients' pre-treatment covariates, meaning they must be estimated from clinical or observational study data. Myriad methods have been developed to learn these rules, and these procedures are demonstrably successful in traditional asymptotic settings with moderate number of covariates. The finite-sample performance of these methods in high-dimensional covariate settings, which are increasingly the norm in modern clinical trials, has not been well characterized, however. We perform a comprehensive comparison of state-of-the-art individualized treatment rule estimators, assessing performance on the basis of the estimators' accuracy, interpretability, and computational efficacy. Sixteen data-generating processes with continuous outcomes and binary treatment assignments are considered, reflecting a diversity of randomized and observational studies. We summarize our findings and provide succinct advice to practitioners needing to estimate individualized treatment rules in high dimensions. All code is made publicly available, facilitating modifications and extensions to our simulation study. A novel pre-treatment covariate filtering procedure is also proposed and is shown to improve estimators' accuracy and interpretability.
Recent years have seen tremendous advances in the theory and application of sequential experiments. While these experiments are not always designed with hypothesis testing in mind, researchers may still be interested in performing tests after the experiment is completed. The purpose of this paper is to aid in the development of optimal tests for sequential experiments by analyzing their asymptotic properties. Our key finding is that the asymptotic power function of any test can be matched by a test in a limit experiment where a Gaussian process is observed for each treatment, and inference is made for the drifts of these processes. This result has important implications, including a powerful sufficiency result: any candidate test only needs to rely on a fixed set of statistics, regardless of the type of sequential experiment. These statistics are the number of times each treatment has been sampled by the end of the experiment, along with final value of the score (for parametric models) or efficient influence function (for non-parametric models) process for each treatment. We then characterize asymptotically optimal tests under various restrictions such as unbiasedness, \alpha-spending constraints etc. Finally, we apply our our results to three key classes of sequential experiments: costly sampling, group sequential trials, and bandit experiments, and show how optimal inference can be conducted in these scenarios.
This paper studies inference in two-stage randomized experiments under covariate-adaptive randomization. In the initial stage of this experimental design, clusters (e.g., households, schools, or graph partitions) are stratified and randomly assigned to control or treatment groups based on cluster-level covariates. Subsequently, an independent second-stage design is carried out, wherein units within each treated cluster are further stratified and randomly assigned to either control or treatment groups, based on individual-level covariates. Under the homogeneous partial interference assumption, I establish conditions under which the proposed difference-in-"average of averages" estimators are consistent and asymptotically normal for the corresponding average primary and spillover effects and develop consistent estimators of their asymptotic variances. Combining these results establishes the asymptotic validity of tests based on these estimators. My findings suggest that ignoring covariate information in the design stage can result in efficiency loss, and commonly used inference methods that ignore or improperly use covariate information can lead to either conservative or invalid inference. Finally, I apply these results to studying optimal use of covariate information under covariate-adaptive randomization in large samples, and demonstrate that a specific generalized matched-pair design achieves minimum asymptotic variance for each proposed estimator. The practical relevance of the theoretical results is illustrated through a simulation study and an empirical application.
Treatment effect heterogeneity is of major interest in economics, but its assessment is often hindered by the fundamental lack of identification of the individual treatment effects. For example, we may want to assess the effect of insurance on the health of otherwise unhealthy individuals, but it is infeasible to insure only the unhealthy, and thus the causal effects for those are not identified. Or, we may be interested in the shares of winners from a minimum wage increase, while without observing the counterfactual, the winners are not identified. Such heterogeneity is often assessed by quantile treatment effects, which do not come with clear interpretation and the takeaway can sometimes be equivocal. We show that, with the quantiles of the treated and control outcomes, the ranges of these quantities are identified and can be informative even when the average treatment effects are not significant. Two applications illustrate how these ranges can inform us about heterogeneity of the treatment effects.
In biomedical studies, estimating drug effects on chronic diseases requires a long follow-up period, which is difficult to meet in randomized clinical trials (RCTs). The use of a short-term surrogate to replace the long-term outcome for assessing the drug effect relies on stringent assumptions that empirical studies often fail to satisfy. Motivated by a kidney disease study, we investigate the drug effects on long-term outcomes by combining an RCT without observation of long-term outcome and an observational study in which the long-term outcome is observed but unmeasured confounding may exist. Under a mean exchangeability assumption weaker than the previous literature, we identify the average treatment effects in the RCT and derive the associated efficient influence function and semiparametric efficiency bound. Furthermore, we propose a locally efficient doubly robust estimator and an inverse probability weighted (IPW) estimator. The former attains the semiparametric efficiency bound if all the working models are correctly specified. The latter has a simpler form and requires much fewer model specifications. The IPW estimator using estimated propensity scores is more efficient than that using true propensity scores and achieves the semiparametric efficient bound in the case of discrete covariates and surrogates with finite support. Both estimators are shown to be consistent and asymptotically normally distributed. Extensive simulations are conducted to evaluate the finite-sample performance of the proposed estimators. We apply the proposed methods to estimate the efficacy of oral hydroxychloroquine on renal failure in a real-world data analysis.
We develop a new model for spatial random field reconstruction of a binary-valued spatial phenomenon. In our model, sensors are deployed in a wireless sensor network across a large geographical region. Each sensor measures a non-Gaussian inhomogeneous temporal process which depends on the spatial phenomenon. Two types of sensors are employed: one collects point observations at specific time points, while the other collects integral observations over time intervals. Subsequently, the sensors transmit these time-series observations to a Fusion Center (FC), and the FC infers the spatial phenomenon from these observations. We show that the resulting posterior predictive distribution is intractable and develop a tractable two-step procedure to perform inference. Firstly, we develop algorithms to perform approximate Likelihood Ratio Tests on the time-series observations, compressing them to a single bit for both point sensors and integral sensors. Secondly, once the compressed observations are transmitted to the FC, we utilize a Spatial Best Linear Unbiased Estimator (S-BLUE) to reconstruct the binary spatial random field at any desired spatial location. The performance of the proposed approach is studied using simulation. We further illustrate the effectiveness of our method using a weather dataset from the National Environment Agency (NEA) of Singapore with fields including temperature and relative humidity.
Heterogeneity and comorbidity are two interwoven challenges associated with various healthcare problems that greatly hampered research on developing effective treatment and understanding of the underlying neurobiological mechanism. Very few studies have been conducted to investigate heterogeneous causal effects (HCEs) in graphical contexts due to the lack of statistical methods. To characterize this heterogeneity, we first conceptualize heterogeneous causal graphs (HCGs) by generalizing the causal graphical model with confounder-based interactions and multiple mediators. Such confounders with an interaction with the treatment are known as moderators. This allows us to flexibly produce HCGs given different moderators and explicitly characterize HCEs from the treatment or potential mediators on the outcome. We establish the theoretical forms of HCEs and derive their properties at the individual level in both linear and nonlinear models. An interactive structural learning is developed to estimate the complex HCGs and HCEs with confidence intervals provided. Our method is empirically justified by extensive simulations and its practical usefulness is illustrated by exploring causality among psychiatric disorders for trauma survivors.
Statisticians show growing interest in estimating and analyzing heterogeneity in causal effects in observational studies. However, there usually exists a trade-off between accuracy and interpretability for developing a desirable estimator for treatment effects, especially in the case when there are a large number of features in estimation. To make efforts to address the issue, we propose a score-based framework for estimating the Conditional Average Treatment Effect (CATE) function in this paper. The framework integrates two components: (i) leverage the joint use of propensity and prognostic scores in a matching algorithm to obtain a proxy of the heterogeneous treatment effects for each observation, (ii) utilize non-parametric regression trees to construct an estimator for the CATE function conditioning on the two scores. The method naturally stratifies treatment effects into subgroups over a 2d grid whose axis are the propensity and prognostic scores. We conduct benchmark experiments on multiple simulated data and demonstrate clear advantages of the proposed estimator over state of the art methods. We also evaluate empirical performance in real-life settings, using two observational data from a clinical trial and a complex social survey, and interpret policy implications following the numerical results.
We use the lens of weak signal asymptotics to study a class of sequentially randomized experiments, including those that arise in solving multi-armed bandit problems. In an experiment with $n$ time steps, we let the mean reward gaps between actions scale to the order $1/\sqrt{n}$ so as to preserve the difficulty of the learning task as $n$ grows. In this regime, we show that the sample paths of a class of sequentially randomized experiments -- adapted to this scaling regime and with arm selection probabilities that vary continuously with state -- converge weakly to a diffusion limit, given as the solution to a stochastic differential equation. The diffusion limit enables us to derive refined, instance-specific characterization of stochastic dynamics, and to obtain several insights on the regret and belief evolution of a number of sequential experiments including Thompson sampling (but not UCB, which does not satisfy our continuity assumption). We show that all sequential experiments whose randomization probabilities have a Lipschitz-continuous dependence on the observed data suffer from sub-optimal regret performance when the reward gaps are relatively large. Conversely, we find that a version of Thompson sampling with an asymptotically uninformative prior variance achieves near-optimal instance-specific regret scaling, including with large reward gaps, but these good regret properties come at the cost of highly unstable posterior beliefs.
Analyzing observational data from multiple sources can be useful for increasing statistical power to detect a treatment effect; however, practical constraints such as privacy considerations may restrict individual-level information sharing across data sets. This paper develops federated methods that only utilize summary-level information from heterogeneous data sets. Our federated methods provide doubly-robust point estimates of treatment effects as well as variance estimates. We derive the asymptotic distributions of our federated estimators, which are shown to be asymptotically equivalent to the corresponding estimators from the combined, individual-level data. We show that to achieve these properties, federated methods should be adjusted based on conditions such as whether models are correctly specified and stable across heterogeneous data sets.
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