arXiv subjects

Tianchen Qian

Publications and source records attributed to Tianchen Qian.

17 recordsLinked to original sources

arXiv2026

Micro-randomized Trials with Categorical Treatments and Binary Proximal Outcome: Causal Effect Estimation and Sample Size Calculation

Micro-randomized trials (MRTs) provide a framework for evaluating the marginal and moderated effects of mobile health (mHealth) interventions. In many applications, treatments take the form of categorical variables with multiple levels, such as different message contents or delivery strategies. Many scientifically meaningful longitudinal outcomes in mHealth studies are binary, such as whether a participant opens an app, engages with content, or completes a target behavior following a decision point at which treatment is randomized. This paper focuses on MRTs with categorical treatments and binary proximal outcomes. We define the causal excursion effect, propose an estimator called EMEE-catA, and derive a sample size formula for comparing categorical treatment levels that controls the type I error rate and guarantees power under working assumptions. We conduct extensive simulation studies to evaluate the operating characteristics of the proposed sample size formula, including robustness to violations of these assumptions. We further provide practical guidance for implementing the proposed approach to ensure adequate power in real-world MRTs. The methods are illustrated using data from the Drink Less MRT.

stat.ME↗

arXiv2025

Robust Estimation of Moderated Causal Excursion Odds Ratio in Micro-Randomized Trials

Micro-randomized trials (MRTs) have become increasingly popular for developing and evaluating mobile health interventions that promote healthy behaviors and manage chronic conditions. The recently proposed causal excursion effects have become the standard measure for interventions' marginal and moderated effect in MRTs. Existing methods for MRTs with binary outcomes focus on causal excursion effects on the relative risk scale. However, a causal excursion effect on the odds ratio scale is attractive for its interpretability and valid predicted probabilities, making it a valuable supplement to causal excursion relative risk. In this paper, we propose two novel estimators for the moderated causal excursion odds ratio for MRTs with longitudinal binary outcomes. When the prespecified moderator fully captures the way interventions are sequentially randomized, we propose a doubly robust estimator that remains consistent if either of two nuisance models is correctly specified. For more general settings in which treatment randomization depends on variables beyond the chosen moderator, we propose a general estimator that incorporates an association nuisance model. We further establish the general estimator's robustness to the misspecification of the association nuisance model under no causal effect, and extend the general estimator to accommodate any link functions. We establish the consistency and asymptotic normality of both estimators and demonstrate their performance through simulation studies. We apply the methods to Drink Less, a 30-day MRT for developing mobile health interventions to help reduce alcohol consumption, where the proximal outcome is whether the user opens the app in the hour following the notification.

stat.ME↗

arXiv2025

Distal Causal Excursion Effects: Modeling Long-Term Effects of Time-Varying Treatments in Micro-Randomized Trials

Micro-randomized trials (MRTs) play a crucial role in optimizing digital interventions. In an MRT, each participant is sequentially randomized among treatment options hundreds of times. While the interventions tested in MRTs target short-term behavioral responses (proximal outcomes), their ultimate goal is to drive long-term behavior change (distal outcomes). However, existing causal inference methods, such as the causal excursion effect, are limited to proximal outcomes, making it challenging to quantify the long-term impact of interventions. To address this gap, we introduce the distal causal excursion effect (DCEE), a novel estimand that quantifies the long-term effect of time-varying treatments. The DCEE contrasts distal outcomes under two excursion policies while marginalizing over most treatment assignments, enabling a parsimonious and interpretable causal model even with a large number of decision points. We propose two estimators for the DCEE -- one with cross-fitting and one without -- both robust to misspecification of the outcome model. We establish their asymptotic properties and validate their performance through simulations. We apply our method to the HeartSteps MRT to assess the impact of activity prompts on long-term habit formation. Our findings suggest that prompts delivered earlier in the study have a stronger long-term effect than those delivered later, underscoring the importance of intervention timing in behavior change. This work provides the critically needed toolkit for scientists working on digital interventions to assess long-term causal effects using MRT data.

stat.ME↗

arXiv2025

Dynamic Causal Mediation Analysis for Intensive Longitudinal Data

Intensive longitudinal data, characterized by frequent measurements across numerous time points, are increasingly common due to advances in wearable devices and mobile health technologies. We consider evaluating causal mediation pathways between time-varying exposures, time-varying mediators, and a final, distal outcome using such data. Addressing mediation questions in these settings is challenging due to numerous potential exposures, complex mediation pathways, and intermediate confounding. Existing methods, such as interventional and path-specific effects, become impractical in intensive longitudinal data. We propose novel mediation effects termed natural direct and indirect excursion effects, which quantify mediation through the most immediate mediator following each treatment time. These effects are identifiable under plausible assumptions and decompose the total excursion effect. We derive efficient influence functions and propose multiply-robust estimators for these mediation effects. The estimators are multiply-robust and accommodate flexible machine learning algorithms and optional cross-fitting. In settings where the treatment assignment mechanism is known, such as the micro-randomized trial, the estimators are doubly-robust. We establish the consistency and asymptotic normality of the proposed estimators. Our methodology is illustrated using real-world data from the HeartSteps micro-randomized trial and the SleepHealth observational study.

stat.ME↗

arXiv2025

Micro-randomized Trials with Categorical Treatments: Causal Effect Estimation and Sample Size Calculation

Micro-randomized trials (MRTs) are widely used to assess the marginal and moderated effect of mobile health (mHealth) treatments delivered via mobile devices. In many applications, the mHealth treatments are categorical with multiple levels such as different types of message contents, but existing analysis and sample size calculation methods for MRTs only focus on binary treatment options (i.e., prompt vs. no prompt). We extended the causal excursion effect definition and the weighted and centered least squares estimator to MRTs with categorical treatments. Furthermore, we developed a sample size formula for comparing categorical treatment levels, and proved the type I error and power guarantee under working assumptions. We conducted extensive simulations to assess type I error and power under assumption violations, and we provided practical guidelines for using the sample size formula to ensure adequate power in most real-world scenarios. We illustrated the proposed estimator and sample size formula using the HeartSteps MRT.

stat.ME↗

arXiv2024

Doubly Robust Estimation of Causal Excursion Effects in Micro-Randomized Trials with Missing Longitudinal Outcomes

Micro-randomized trials (MRTs) are increasingly utilized for optimizing mobile health interventions, with the causal excursion effect (CEE) as a central quantity for evaluating interventions under policies that deviate from the experimental policy. However, MRT often contains missing data due to reasons such as missed self-reports or participants not wearing sensors, which can bias CEE estimation. In this paper, we propose a two-stage, doubly robust estimator for CEE in MRTs when longitudinal outcomes are missing at random, accommodating continuous, binary, and count outcomes. Our two-stage approach allows for both parametric and nonparametric modeling options for two nuisance parameters: the missingness model and the outcome regression. We demonstrate that our estimator is doubly robust, achieving consistency and asymptotic normality if either the missingness or the outcome regression model is correctly specified. Simulation studies further validate the estimator's desirable finite-sample performance. We apply the method to HeartSteps, an MRT for developing mobile health interventions that promote physical activity.

stat.ME↗

arXiv2024

Modeling Time-Varying Effects of Mobile Health Interventions Using Longitudinal Functional Data from HeartSteps Micro-Randomized Trial

To optimize mobile health interventions and advance domain knowledge on intervention design, it is critical to understand how the intervention effect varies over time and with contextual information. This study aims to assess how a push notification suggesting physical activity influences individuals' step counts using data from the HeartSteps micro-randomized trial (MRT). The statistical challenges include the time-varying treatments and longitudinal functional step count measurements. We propose the first semiparametric causal excursion effect model with varying coefficients to model the time-varying effects within a decision point and across decision points in an MRT. The proposed model incorporates double time indices to accommodate the longitudinal functional outcome, enabling the assessment of time-varying effect moderation by contextual variables. We propose a two-stage causal effect estimator that is robust against a misspecified high-dimensional outcome regression nuisance model. We establish asymptotic theory and conduct simulation studies to validate the proposed estimator. Our analysis provides new insights into individuals' change in response profiles (such as how soon a response occurs) due to the activity suggestions, how such changes differ by the type of suggestions received, and how such changes depend on other contextual information such as being recently sedentary and the day being a weekday.

stat.AP↗

arXiv2024

Estimating Causal Effects for Binary Outcomes Using Per-Decision Inverse Probability Weighting

Micro-randomized trials are commonly conducted for optimizing mobile health interventions such as push notifications for behavior change. In analyzing such trials, causal excursion effects are often of primary interest, and their estimation typically involves inverse probability weighting (IPW). However, in a micro-randomized trial, additional treatments can often occur during the time window over which an outcome is defined, and this can greatly inflate the variance of the causal effect estimator because IPW would involve a product of numerous weights. To reduce variance and improve estimation efficiency, we propose two new estimators using a modified version of IPW, which we call "per-decision IPW". The second estimator further improves efficiency using the projection idea from the semiparametric efficiency theory. These estimators are applicable when the outcome is binary and can be expressed as the maximum of a series of sub-outcomes defined over sub-intervals of time. We establish the estimators' consistency and asymptotic normality. Through simulation studies and real data applications, we demonstrate substantial efficiency improvement of the proposed estimator over existing estimators. The new estimators can be used to improve the precision of primary and secondary analyses for micro-randomized trials with binary outcomes.

stat.ME↗

arXiv2024

Efficient and Globally Robust Causal Excursion Effect Estimation

Causal excursion effect (CEE) characterizes the effect of an intervention under policies that deviate from the experimental policy. It is widely used to study the effect of time-varying interventions that have the potential to be frequently adaptive, such as those delivered through smartphones. We study the semiparametric efficient estimation of CEE and we derive a semiparametric efficiency bound for CEE with identity or log link functions under working assumptions, in the context of micro-randomized trials. We propose a class of two-stage estimators that achieve the efficiency bound and are robust to misspecified nuisance models. In deriving the asymptotic property of the estimators, we establish a general theory for globally robust Z-estimators with either cross-fitted or non-cross-fitted nuisance parameters. We demonstrate substantial efficiency gain of the proposed estimator compared to existing ones through simulations and a real data application using the Drink Less micro-randomized trial.

stat.ME↗

arXiv2024

Incorporating nonparametric methods for estimating causal excursion effects in mobile health with zero-inflated count outcomes

In mobile health, tailoring interventions for real-time delivery is of paramount importance. Micro-randomized trials have emerged as the "gold-standard" methodology for developing such interventions. Analyzing data from these trials provides insights into the efficacy of interventions and the potential moderation by specific covariates. The "causal excursion effect", a novel class of causal estimand, addresses these inquiries. Yet, existing research mainly focuses on continuous or binary data, leaving count data largely unexplored. The current work is motivated by the Drink Less micro-randomized trial from the UK, which focuses on a zero-inflated proximal outcome, i.e., the number of screen views in the subsequent hour following the intervention decision point. To be specific, we revisit the concept of causal excursion effect, specifically for zero-inflated count outcomes, and introduce novel estimation approaches that incorporate nonparametric techniques. Bidirectional asymptotics are established for the proposed estimators. Simulation studies are conducted to evaluate the performance of the proposed methods. As an illustration, we also implement these methods to the Drink Less trial data.

stat.ME↗

arXiv2021

The Micro-Randomized Trial for Developing Digital Interventions: Experimental Design and Data Analysis Considerations

Just-in-time adaptive interventions (JITAIs) are time-varying adaptive interventions that use frequent opportunities for the intervention to be adapted--weekly, daily, or even many times a day. The micro-randomized trial (MRT) has emerged for use in informing the construction of JITAIs. MRTs can be used to address research questions about whether and under what circumstances JITAI components are effective, with the ultimate objective of developing effective and efficient JITAI. The purpose of this article is to clarify why, when, and how to use MRTs; to highlight elements that must be considered when designing and implementing an MRT; and to review primary and secondary analyses methods for MRTs. We briefly review key elements of JITAIs and discuss a variety of considerations that go into planning and designing an MRT. We provide a definition of causal excursion effects suitable for use in primary and secondary analyses of MRT data to inform JITAI development. We review the weighted and centered least-squares (WCLS) estimator which provides consistent causal excursion effect estimators from MRT data. We describe how the WCLS estimator along with associated test statistics can be obtained using standard statistical software such as R (R Core Team, 2019). Throughout we illustrate the MRT design and analyses using the HeartSteps MRT, for developing a JITAI to increase physical activity among sedentary individuals. We supplement the HeartSteps MRT with two other MRTs, SARA and BariFit, each of which highlights different research questions that can be addressed using the MRT and experimental design considerations that might arise.

stat.AP↗

arXiv2020

Estimating Time-Varying Causal Excursion Effect in Mobile Health with Binary Outcomes

Advances in wearables and digital technology now make it possible to deliver behavioral mobile health interventions to individuals in their everyday life. The micro-randomized trial (MRT) is increasingly used to provide data to inform the construction of these interventions. In an MRT, each individual is repeatedly randomized among multiple intervention options, often hundreds or even thousands of times, over the course of the trial. This work is motivated by multiple MRTs that have been conducted, or are currently in the field, in which the primary outcome is a longitudinal binary outcome. The primary aim of such MRTs is to examine whether a particular time-varying intervention has an effect on the longitudinal binary outcome, often marginally over all but a small subset of the individual's data. We propose the definition of causal excursion effect that can be used in such primary aim analysis for MRTs with binary outcomes. Under rather restrictive assumptions one can, based on existing literature, derive a semiparametric, locally efficient estimator of the causal effect. We, starting from this estimator, develop an estimator that can be used as the basis of a primary aim analysis under more plausible assumptions. Simulation studies are conducted to compare the estimators. We illustrate the developed methods using data from the MRT, BariFit. In BariFit, the goal is to support weight maintenance for individuals who received bariatric surgery.

stat.ME↗

arXiv2020

The Micro-Randomized Trial for Developing Digital Interventions: Data Analysis Methods

Although there is much excitement surrounding the use of mobile and wearable technology for the purposes of delivering interventions as people go through their day-to-day lives, data analysis methods for constructing and optimizing digital interventions lag behind. Here, we elucidate data analysis methods for primary and secondary analyses of micro-randomized trials (MRTs), an experimental design to optimize digital just-in-time adaptive interventions. We provide a definition of causal "excursion" effects suitable for use in digital intervention development. We introduce the weighted and centered least-squares (WCLS) estimator which provides consistent causal excursion effect estimators for digital interventions from MRT data. We describe how the WCLS estimator along with associated test statistics can be obtained using standard statistical software such as SAS or R. Throughout we use HeartSteps, an MRT designed to increase physical activity among sedentary individuals, to illustrate potential primary and secondary analyses.

stat.AP↗

arXiv2019

Improving Precision through Adjustment for Prognostic Variables in Group Sequential Trial Designs: Impact of Baseline Variables, Short-Term Outcomes, and Treatment Effect Heterogeneity

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 but haven't yet had their primary outcomes measured) and treatment effect heterogeneity? What is the resulting impact on power and average sample size in a group sequential design? We derive an asymptotic formula that decomposes the overall precision gain from adjusting for baseline variables and a short-term outcome into contributions from factors mentioned above, for efficient estimators in the model that only assumes randomization and independent censoring. We use our formula to approximate the precision gain from a targeted minimum loss-based estimator applied to data from a completed trial of a new surgical intervention for stroke. Our formula implies that (for an efficient estimator) adjusting for a prognostic baseline variable leads to at least as much asymptotic precision gain as adjusting for an equally prognostic short-term outcome. In many cases, such as our stroke trial application, the former leads to substantially greater precision gains than the latter. In our simulation study, we show how precision gains from adjustment can be converted into sample size reductions (even when there is no treatment effect).

stat.ME↗

arXiv2019

Deductive semiparametric estimation in Double-Sampling Designs with application to PEPFAR

Non-ignorable dropout is common in studies with long follow-up time, and it can bias study results unless handled carefully. A double-sampling design allocates additional resources to pursue a subsample of the dropouts and find out their outcomes, which can address potential biases due to non-ignorable dropout. It is desirable to construct semiparametric estimators for the double-sampling design because of their robustness properties. However, obtaining such semiparametric estimators remains a challenge due to the requirement of the analytic form of the efficient influence function (EIF), the derivation of which can be ad hoc and difficult for the double-sampling design. Recent work has shown how the derivation of EIF can be made deductive and computerizable using the functional derivative representation of the EIF in nonparametric models. This approach, however, requires deriving the mixture of a continuous distribution and a point mass, which can itself be challenging for complicated problems such as the double-sampling design. We propose semiparametric estimators for the survival probability in double-sampling designs by generalizing the deductive and computerizable estimation approach. In particular, we propose to build the semiparametric estimators based on a discretized support structure, which approximates the possibly continuous observed data distribution and circumvents the derivation of the mixture distribution. Our approach is deductive in the sense that it is expected to produce semiparametric locally efficient estimators within finite steps without knowledge of the EIF. We apply the proposed estimators to estimating the mortality rate in a double-sampling design component of the President's Emergency Plan for AIDS Relief (PEPFAR) program. We evaluate the impact of double-sampling selection criteria on the mortality rate estimates.

stat.ME↗

arXiv2019

Linear mixed models with endogenous covariates: modeling sequential treatment effects with application to a mobile health study

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 linear mixed models is problematic because potential moderators of the treatment effect are frequently endogenous---that is, may depend on prior treatment. We discuss model interpretation and biases that arise in the absence of additional assumptions when endogenous covariates are included in a linear mixed model. In particular, when there are endogenous covariates, the coefficients no longer have the customary marginal interpretation. However, these coefficients still have a conditional-on-the-random-effect interpretation. We provide an additional assumption that, if true, allows scientists to use standard software to fit linear mixed model with endogenous covariates, and person-specific predictions of effects can be provided. As an illustration, we assess the effect of activity suggestion in the HeartSteps MRT and analyze the between-person treatment effect heterogeneity.

stat.ME↗

arXiv2018

On inference validity of weighted U-statistics under data heterogeneity

Motivated by challenges on studying a new correlation measurement being popularized in evaluating online ranking algorithms' performance, this manuscript explores the validity of uncertainty assessment for weighted U-statistics. Without any commonly adopted assumption, we verify Efron's bootstrap and a new resampling procedure's inference validity. Specifically, in its full generality, our theory allows both kernels and weights asymmetric and data points not identically distributed, which are all new issues that historically have not been addressed. For achieving strict generalization, for example, we have to carefully control the order of the "degenerate" term in U-statistics which are no longer degenerate under the empirical measure for non-i.i.d. data. Our result applies to the motivating task, giving the region at which solid statistical inference can be made.

math.ST↗