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Regression-Based Causal Inference for Person-Specific Effects in Within-Subject Randomization Designs

This paper proposes a regression-based causal inference framework using ordinary least squares with probability-weighted treatment indicators to estimate person-specific immediate and lag-1 effects in within-subject randomized designs, demonstrating its robustness and applicability through simulation studies and empirical examples.

Original authors: Chendong Li, Wen Luo

Published 2026-10-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Chendong Li, Wen Luo

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of psychological and behavioral research, scientists have long relied on a standard method to prove that a treatment works: the randomized controlled trial. In this classic setup, a large group of people is split into separate teams. One team receives a new intervention, like a specific therapy or a health app, while another team receives a different treatment or nothing at all. By comparing the average results of these two groups, researchers can determine if the treatment caused a change. However, this approach has a significant blind spot. It tells us what works on average for a crowd, but it often fails to reveal what works for a specific individual. A medication might help one person recover quickly while having no effect on another, yet the standard trial would simply report the average, potentially masking these vital differences. Furthermore, recruiting enough people for these large studies is often difficult, and many trials struggle to find enough participants to reach a reliable conclusion.

To solve this, researchers have turned to a different kind of experiment where the same person serves as their own control. Instead of comparing different people, scientists observe a single individual repeatedly over time, randomly switching the conditions they experience. This is known as a within-subject randomization design. Imagine a person with a chronic condition who tries a new therapy for a few days, then switches to a different one, then back again, with the order decided by chance. This allows researchers to see how that specific person reacts to each treatment in real-time. The challenge, however, has been figuring out how to analyze this complex stream of data. Because the treatments happen in a sequence, the result of one day might be influenced by the treatment from the day before, or by a slow trend in the person's behavior over time. Traditional statistical tools often struggle to untangle these immediate effects from these lingering influences, leaving researchers with descriptions of what happened but without a clear, causal proof of why it happened.

A team of researchers has now developed a new way to answer these questions using a familiar tool: regression analysis, a common statistical method used to find relationships between variables. Their work, published as a research article, creates a bridge between the rigorous logic of experimental design and the practical tools that psychologists use every day. The authors, Chendong Li and Wen Luo, propose a framework that treats the random switching of treatments not just as a schedule, but as a known mathematical pattern that can be used to isolate cause and effect. By carefully adjusting the data to account for the specific probabilities of when a treatment was assigned, they can use a standard regression model to calculate two distinct types of effects. The first is the immediate effect: how much a treatment changes a person's behavior right after it is given. The second is the lagged effect: how much a treatment from the previous day continues to influence behavior today.

The researchers tested their new method through two large-scale computer simulations, which act like virtual laboratories where they can generate thousands of fake datasets to see how their formulas perform under different conditions. In the first simulation, they modeled a scenario with three different treatment options, a common setup in modern health studies. They found that their method could accurately recover the true effects of each treatment with very little error, even when the data contained complex trends. The accuracy improved as the number of observation points increased, suggesting the method is robust for studies with longer timelines. In the second simulation, they tackled a more difficult scenario where the randomization was restricted. In many real-world studies, researchers cannot allow a person to receive the same treatment for too many days in a row because it might become boring or impractical. This restriction changes the math of the experiment. The simulations showed that without a specific adjustment for these rules, the method would produce biased results, particularly for the lagged effects. However, when the researchers added a probability adjustment to their model to account for these restrictions, the bias disappeared, and the estimates became accurate again.

To prove the method works in the real world, the team applied it to two existing datasets. The first came from a study on physical activity called HeartSteps, where a participant received random suggestions to walk or avoid sitting. Using their new framework, the researchers calculated that a walking suggestion led to an immediate increase of about 129 steps in the following 30 minutes, while an anti-sedentary suggestion led to an increase of about 143 steps. Neither of these increases was statistically significant on its own, meaning the data was not strong enough to rule out chance, but the method successfully produced clear, interpretable numbers. The second example came from a special education study involving a child with autism spectrum disorder. The child alternated between a cognitive exercise and a control activity. The analysis revealed that the cognitive exercise immediately reduced problem behaviors by nearly 19 percentage points compared to the control. The method also checked for lingering effects from the previous session but found no clear evidence that the prior treatment influenced the current day's behavior.

The study highlights that while the method is powerful, it does have limits, particularly when the data series is very short. In the simulations involving restricted randomization, the researchers found that with fewer than 30 observation points, the confidence intervals—the range of values where the true effect likely lies—were sometimes too narrow, leading to a higher risk of false alarms. This suggests that while the method is a major step forward for analyzing individual-level data, researchers must be cautious when drawing strong conclusions from very brief studies. The authors emphasize that their approach does not assume the data follows a perfect linear pattern; instead, it uses the known rules of the randomization to extract the causal signal from the noise. By connecting the design of the experiment directly to the regression analysis, they have provided a way for scientists to move beyond simple averages and understand the unique, person-specific effects of treatments. This framework offers a path forward for personalized medicine and behavioral science, allowing researchers to ask not just "does this work?" but "does this work for this person, right now, and for how long?"

Technical Summary: Regression-Based Causal Inference for Person-Specific Effects in Within-Subject Randomization Designs

Problem Statement

In psychological and behavioral research, interventions often exhibit heterogeneous effects, benefiting some individuals while having little impact on others. Traditional between-subject randomized controlled trials (RCTs) struggle to estimate these person-specific treatment effects because they average outcomes across participants, with each individual receiving only one condition. While within-subject randomization designs (e.g., Micro-Randomized Trials, N-of-1 trials, and Randomized Alternating Treatment Designs) offer a solution by repeatedly assigning conditions to the same unit over time, current analytic approaches often rely on descriptive summaries (e.g., mean differences, visual analysis) or simple statistical tests. These methods frequently fail to rigorously separate treatment effects from time trends, serial dependence, or prior treatment exposure. Furthermore, existing causal inference frameworks for these designs, such as Inverse Probability Weighting (IPW), can be computationally complex or less intuitive for applied researchers accustomed to regression tools. Additionally, standard regression-based estimators developed for unrestricted binary treatments do not directly apply to designs with multiple treatment arms or restricted randomization schemes (e.g., preventing long runs of the same condition), which are common in behavioral research.

Methodology

The authors develop a regression-based causal inference framework grounded in the potential outcomes framework. The core innovation is transforming treatment indicators based on known assignment probabilities to estimate causal effects using Ordinary Least Squares (OLS) regression, thereby retaining a design-based causal interpretation without requiring the regression model to be the true data-generating process.

1. Causal Estimands

The study defines two primary causal effects for within-subject designs:

Average Immediate Effect ( $\bar{\Delta}^{(0)}$ ): The expected difference in outcome at time $t$ between assigning treatment versus control, averaged over prior treatment histories.
Average Lag-1 Effect ( $\bar{\Delta}^{(1)}$ ): The expected difference in outcome at time $t$ attributable to the treatment assigned at time $t-1$ , holding the current treatment constant, averaged over histories and current assignments.

2. Estimation Strategy

The method involves constructing "working" OLS regressions where predictors are transformed treatment indicators.

Binary, Unrestricted Designs: Treatment indicators ( $W_t$ ) are centered by their assignment probability ( $p_t$ ) and scaled by the assignment variance ( $v_t = p_t(1-p_t)$ ). The transformed predictor is $\tilde{W}_t = (W_t - p_t) / v_t$ . The regression model is $Y_t = b_0\tilde{W}_t + b_1\tilde{W}_{t-1} + e_t$ . The coefficients are then rescaled by the inverse of the average variance to recover the causal effects.
Extension I: Multiple-Arm Interventions: For $J$ conditions, the scalar indicator is replaced by a vector of treatment indicators. These are centered and scaled using the inverse of the assignment covariance matrix ( $\Sigma_t^{-1}$ ), which accounts for the negative correlation between mutually exclusive treatment assignments. The regression estimates a vector of immediate and lag-1 effects relative to a reference condition.
Extension II: Restricted Randomization: In designs where assignment probabilities depend on history (e.g., "no more than two consecutive sessions of the same condition"), some assignments become deterministic ( $p_t = 0$ or $1$). The authors propose a probability-adjusted regression that includes the conditional probability $p_t$ (and potentially $p_{t-1}$ ) as an additional covariate to account for the dependence created by the restriction. Deterministic occasions are coded as zero in the transformed predictors but retained in the regression to maintain sample size, though they do not contribute to the randomized contrast.

3. Inference

Uncertainty is estimated using robust covariance estimators. The authors evaluate two options:

Newey–West HAC: Accounts for heteroskedasticity and serial correlation.
HC3: A heteroskedasticity-robust estimator that performs well in small samples with high leverage.
Standard errors are rescaled using the same factors applied to the point estimates to report confidence intervals on the causal-effect scale.

Key Contributions

Framework Extension: The paper extends the regression-based causal estimator (previously limited to binary, unrestricted designs) to multiple-arm interventions and restricted randomization schemes, which are prevalent in single-case and micro-randomized trials.
Probability Adjustment: It introduces a specific regression formulation for restricted randomization that includes design-based probability terms ( $p_t$ ) as covariates. This adjustment is shown to be necessary to eliminate bias in lag-1 effect estimates when assignment probabilities are history-dependent.
Integration of Tools: The approach bridges design-based causal inference with familiar OLS regression tools, making rigorous causal estimation more accessible to applied researchers without requiring the construction of complex weighted outcomes.

Simulation Results

Two Monte Carlo studies evaluated the proposed estimators:

Simulation I (Multiple-Arm): In a three-condition design, the proposed estimator demonstrated small bias for both immediate and lag-1 effects across most conditions. Bias was slightly higher in short series but decreased as the number of time points ( $T$ ) increased. Root Mean Squared Error (RMSE) decreased with longer series. Inference performance (coverage and Type I error) improved with longer series; however, HAC standard errors tended to under-cover in short series, while HC3 was more conservative.
Simulation II (Restricted Randomization):
- Bias: The non-probability-adjusted estimator showed substantial negative bias for lag-1 effects. In contrast, the probability-adjusted estimator recovered both immediate and lag-1 effects with negligible bias.
- Inference: While point estimates were unbiased, inference remained challenging in short series. HAC standard errors resulted in under-coverage and inflated Type I error rates, particularly for lag-1 effects. HC3 standard errors controlled Type I error better but were conservative (lower power).

Empirical Demonstrations

The authors applied the methods to two real-world datasets:

HeartSteps (Micro-Randomized Trial): Analyzed data from one participant to estimate the immediate and lag-1 effects of walking vs. anti-sedentary suggestions on step counts. The results provided interpretable estimates of step increases, though confidence intervals were wide and included zero.
Restricted Alternating Treatment Design (Special Education): Analyzed data from a child with autism spectrum disorder comparing a cognitive exercise to a control. The probability-adjusted estimator indicated a significant immediate reduction in problem behavior for the treatment condition but found no clear evidence of a lag-1 (residual) effect.

Significance and Claims

The paper claims to provide a methodologically rigorous yet practically accessible framework for estimating person-specific causal effects in within-subject randomized designs. By connecting familiar regression tools with design-based causal logic, the authors argue that researchers can better isolate treatment effects from time trends and serial dependence without relying on strong assumptions about the data-generating process.

The authors modestly acknowledge limitations:

Short Series Inference: Inference in short, restricted series remains difficult, with potential under-coverage of confidence intervals and inflated Type I errors, particularly for lag-1 effects.
Missing Data: The current framework assumes complete data; future work is needed to integrate missing data strategies.
Theoretical Gaps: Asymptotic theory for restricted randomization designs is still underdeveloped, and the consistency of the probability-adjusted estimator under all restricted conditions requires further theoretical derivation.
Single-Unit Focus: The current framework analyzes one unit at a time; extending it to estimate heterogeneity across multiple units is a direction for future research.

The study concludes that while the proposed estimators effectively reduce bias, researchers should interpret inference results in short, restricted designs with caution, particularly when examining lagged effects.