In this paper, we consider an experimental setting where units enter the experiment sequentially. Our goal is to form stopping rules which lead to estimators of treatment effects with a given precision. We propose a fixed-width confidence interval design (FWCID) where the experiment terminates once a pre-specified confidence interval width is achieved. We show that under this design, the difference-in-means estimator is a consistent estimator of the average treatment effect and standard confidence intervals have asymptotic guarantees of coverage and efficiency for several versions of the design. In addition, we propose a version of the design that we call fixed power design (FPD) where a given power is asymptotically guaranteed for a given treatment effect, without the need to specify the variances of the outcomes under treatment or control. In addition, this design also gives a consistent difference-in-means estimator with correct coverage of the corresponding standard confidence interval. We complement our theoretical findings with Monte Carlo simulations where we compare our proposed designs with standard designs in the sequential experiments literature, showing that our designs outperform these designs in several important aspects. We believe our results to be relevant for many experimental settings where units enter sequentially, such as in clinical trials, as well as in online A/B tests used by the tech and e-commerce industry.
Scientific researchers utilize randomized experiments to draw casual statements. Most early studies as well as current work on experiments with sequential intervention decisions has been focusing on estimating the causal effects among sequential treatments, ignoring the non-compliance issues that experimental units might not be compliant with the treatment assignments that they were originally allocated. A series of methodologies have been developed to address the non-compliance issues in randomized experiments with time-fixed treatment. However, to our best knowledge, there is little literatu…
We propose a nonparametric sequential test that aims to address two practical problems pertinent to online randomized experiments: (i) how to do a hypothesis test for complex metrics; (ii) how to prevent type $1$ error inflation under continuous monitoring. The proposed test does not require knowledge of the underlying probability distribution generating the data. We use the bootstrap to estimate the likelihood for blocks of data followed by mixture sequential probability ratio test. We validate this procedure on data from a major online e-commerce website. We show that the proposed test contr…
We propose a dynamic allocation procedure that increases power and efficiency when measuring an average treatment effect in sequential randomized trials. Subjects arrive iteratively and are either randomized or paired via a matching criterion to a previously randomized subject and administered the alternate treatment. We develop estimators for the average treatment effect that combine information from both the matched pairs and unmatched subjects as well as an exact test. Simulations illustrate the method's higher efficiency and power over competing allocation procedures in both controlled sce…
We propose a dynamic allocation procedure that increases power and efficiency when measuring an average treatment effect in sequential randomized trials exploiting some subjects' previous assessed responses. Subjects arrive sequentially and are either randomized or paired to a previously randomized subject and administered the alternate treatment. The pairing is made via a dynamic matching criterion that iteratively learns which specific covariates are important to the response. We develop estimators for the average treatment effect as well as an exact test. We illustrate our method's increase…
A small n, sequential, multiple assignment, randomized trial (snSMART) is a small sample, two-stage design where participants receive up to two treatments sequentially, but the second treatment depends on response to the first treatment. The treatment effect of interest in an snSMART is the first-stage response rate, but outcomes from both stages can be used to obtain more information from a small sample. A novel way to incorporate the outcomes from both stages applies power prior models, in which first stage outcomes from an snSMART are regarded as the primary data and second stage outcomes a…
Comment on "Sequential validation of treatment heterogeneity" and "Comment on generic machine learning inference on heterogeneous treatment effects in randomized experiments"
In randomized controlled trials (RCTs), treatment is often assigned by stratified randomization. I show that among all stratified randomization schemes which treat all units with probability one half, a certain matched-pair design achieves the maximum statistical precision for estimating the average treatment effect (ATE). In an important special case, the optimal design pairs units according to the baseline outcome. In a simulation study based on datasets from 10 RCTs, this design lowers the standard error for the estimator of the ATE by 10% on average, and by up to 34%, relative to the origi…
Randomized experiments ensure robust causal inference that are critical to effective learning analytics research and practice. However, traditional randomized experiments, like A/B tests, are limiting in large scale digital learning environments. While traditional experiments can accurately compare two treatment options, they are less able to inform how to adapt interventions to continually meet learners' diverse needs. In this work, we introduce a trial design for developing adaptive interventions in scaled digital learning environments -- the sequential randomized trial (SRT). With the goal …
Tech companies (e.g., Google or Facebook) often use randomized online experiments and/or A/B testing primarily based on the average treatment effects to compare their new product with an old one. However, it is also critically important to detect qualitative treatment effects such that the new one may significantly outperform the existing one only under some specific circumstances. The aim of this paper is to develop a powerful testing procedure to efficiently detect such qualitative treatment effects. We propose a scalable online updating algorithm to implement our test procedure. It has thre…
Clinical trials are complex and usually involve multiple objectives such as controlling type I error rate, increasing power to detect treatment difference, assigning more patients to better treatment, and more. In literature, both response-adaptive randomization (RAR) procedures (by changing randomization procedure sequentially) and sequential monitoring (by changing analysis procedure sequentially) have been proposed to achieve these objectives to some degree. In this paper, we propose to sequentially monitor response-adaptive randomized clinical trial and study it's properties. We prove that…
In randomized trials, researchers are often interested in mediation analysis to understand how a treatment works, in particular how much of a treatment's effect is mediated by an intermediated variable and how much the treatment directly affects the outcome not through the mediator. The standard regression approach to mediation analysis assumes sequential ignorability of the mediator, that is that the mediator is effectively randomly assigned given baseline covariates and the randomized treatment. Since the experiment does not randomize the mediator, sequential ignorability is often not plausi…
Mediation analysis seeks to understand the mechanism by which a treatment affects an outcome. Count or zero-inflated count outcome are common in many studies in which mediation analysis is of interest. For example, in dental studies, outcomes such as decayed, missing and filled teeth are typically zero inflated. Existing mediation analysis approaches for count data assume sequential ignorability of the mediator. This is often not plausible because the mediator is not randomized so that there are unmeasured confounders associated with the mediator and the outcome. In this paper, we develop caus…
In a randomized control trial, the precision of an average treatment effect estimator can be improved either by collecting data on additional individuals, or by collecting additional covariates that predict the outcome variable. We propose the use of pre-experimental data such as a census, or a household survey, to inform the choice of both the sample size and the covariates to be collected. Our procedure seeks to minimize the resulting average treatment effect estimator's mean squared error, subject to the researcher's budget constraint. We rely on a modification of an orthogonal greedy algor…
In randomized trials, appropriately adjusting for baseline variables and short-term outcomes can lead to increased precision and reduced sample size. We examine the impact of such adjustment in group sequential designs, i.e., designs with preplanned interim analyses where enrollment may be stopped early for efficacy or futility. We address the following questions: how much precision gain can be obtained by appropriately adjusting for baseline variables and a short-term outcome? How is this precision gain impacted by factors such as the proportion of pipeline participants (those who enrolled bu…
To improve precision of estimation and power of testing hypothesis for an unconditional treatment effect in randomized clinical trials with binary outcomes, researchers and regulatory agencies recommend using g-computation as a reliable method of covariate adjustment. However, the practical application of g-computation is hindered by the lack of an explicit robust variance formula that can be used for different unconditional treatment effects of interest. To fill this gap, we provide explicit and robust variance estimators for g-computation estimators and demonstrate through simulations that t…
Mobile health is a rapidly developing field in which behavioral treatments are delivered to individuals via wearables or smartphones to facilitate health-related behavior change. Micro-randomized trials (MRT) are an experimental design for developing mobile health interventions. In an MRT the treatments are randomized numerous times for each individual over course of the trial. Along with assessing treatment effects, behavioral scientists aim to understand between-person heterogeneity in the treatment effect. A natural approach is the familiar linear mixed model. However, directly applying lin…
The use and development of mobile interventions are experiencing rapid growth. In "just-in-time" mobile interventions, treatments are provided via a mobile device and they are intended to help an individual make healthy decisions "in the moment," and thus have a proximal, near future impact. Currently the development of mobile interventions is proceeding at a much faster pace than that of associated data science methods. A first step toward developing data-based methods is to provide an experimental design for testing the proximal effects of these just-in-time treatments. In this paper, we pro…
Regression adjustment is broadly applied in randomized trials under the premise that it usually improves the precision of a treatment effect estimator. However, previous work has shown that this is not always true. To further understand this phenomenon, we develop a unified comparison of the asymptotic variance of a class of linear regression-adjusted estimators. Our analysis is based on the classical theory for linear regression with heteroscedastic errors and thus does not assume that the postulated linear model is correct. For a completely randomized binary treatment, we provide sufficient …
We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges, almost surely, to that of the optimum, given by the Bayes predictor. The desirable finite-sample properties of the predictor are illustrated by its performance for Markov processes. In such cases the predictor exhibits near optimal behavior even without knowing the order of …
Using observational data to estimate the effect of a treatment is a powerful tool for decision-making when randomized experiments are infeasible or costly. However, observational data often yields biased estimates of treatment effects, since treatment assignment can be confounded by unobserved variables. A remedy is offered by deconfounding methods that adjust for such unobserved confounders. In this paper, we develop the Sequential Deconfounder, a method that enables estimating individualized treatment effects over time in presence of unobserved confounders. This is the first deconfounding me…
We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the …
The rapid finding of effective therapeutics requires the efficient use of available resources in clinical trials. The use of covariate adjustment can yield statistical estimates with improved precision, resulting in a reduction in the number of participants required to draw futility or efficacy conclusions. We focus on time-to-event and ordinal outcomes. A key question for covariate adjustment in randomized studies is how to fit a model relating the outcome and the baseline covariates to maximize precision. We present a novel theoretical result establishing conditions for asymptotic normality …
Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_θ$, $θ\inΘ$, where $Θ$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $θ=θ_0$ versus a composite alternative $H_1:$ $θ>θ_0$, where $θ_0\inΘ$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(ψ,ϕ)$ with a (randomized) stopping rule $ψ$ and a (randomized) decision rule $ϕ$ let $α(ψ,ϕ)$ be the type I error probability, $\dot β_0(ψ,ϕ)$ the derivative, a…
