Micro-Randomized Trials (MRTs)

• Micro-randomized trials are experimental designs where participants receive randomly assigned binary interventions at numerous decision points to assess proximal outcomes.
• They use regression-based methods and noncentral F-statistics for power analysis and sample size calculation, ensuring rigorous assessment in digital health studies.
• Applications like the HeartSteps study demonstrate MRTs’ ability to capture time-varying treatment effects and participant availability in mobile health interventions.

Micro-randomized trials (MRTs) are a class of experimental designs for digital and mobile health interventions that facilitate the empirical investigation of the proximal effects of just-in-time treatments delivered at high frequency over time. In an MRT, participants are repeatedly randomized at prespecified decision points—often hundreds or thousands of times per subject during a paper—to receive a binary intervention or control (e.g., a smartphone notification or no notification). Data from MRTs are used to estimate the immediate causal effect of interventions on proximal outcomes and to optimize adaptive treatment strategies. This entry synthesizes the foundational principles, methodological frameworks, and applied considerations for MRTs, with particular reference to their use in mobile health studies, sample size calculation, heterogeneous designs, and advanced analysis.

1. Fundamental Structure of Micro-Randomized Trials

Micro-randomized trials are defined by their use of sequential randomizations at discrete decision times, denoted $t = 1, \ldots, T$. At each decision time, a participant may be available for treatment (availability indicator $I_t = 1$), and if available, is randomized to receive a binary treatment ($A_t \in \{0, 1\}$). The principal proximal outcome, $Y_{t+1}$, is typically measured shortly after each decision time and reflects the near-term consequence of the intervention. The primary causal estimand is the time-varying, availability-restricted, proximal treatment effect: $$\beta(t) = E[Y_{t+1} | I_t = 1, A_t = 1] - E[Y_{t+1} | I_t = 1, A_t = 0]$$ This effect is well-defined only among the available subpopulation ($I_t=1$) at each decision time, and may exhibit complex temporal patterns due to habituation, changing context, or dynamic population behavior.

A regression-based working model underlies both effect estimation and sample size determination: $$E[Y_{t+1} | I_t = 1, A_t] = B_t^\top \alpha + (A_t - \rho_t) Z_t^\top \beta,$$ where $B_t$ models baseline and time trends in outcome, $Z_t$ parameterizes patterns of the treatment effect, and $\rho_t = P(A_t = 1)$ is the randomization probability.

2. Power Analysis and Sample Size Methodology

Determining the sample size necessary to achieve a target statistical power in an MRT is nontrivial due to repeated randomization, time-varying effect patterns, and treatment availability. The MRT-SS Calculator is an R Shiny application specifically designed for this purpose (Seewald et al., 2016). Its algorithmic foundation is as follows:

1. Specification of Patterns: The calculator requires explicit modeling of both time-varying proximal treatment effects and expected treatment availability, allowing realistic operationalization of longitudinal trends (e.g., constant, linear, quadratic).
2. Power Calculation: Statistical power is computed using a noncentral F-distribution. For a test statistic constructed from the least-squares estimator $\hat \beta$, the main test is:

$$N \hat \beta^\top \hat \Sigma_\beta^{-1} \hat \beta > \text{Threshold},$$

where $N$ is the planned sample size, $\hat\Sigma_\beta$ estimates the asymptotic variance (accounting for design heterogeneity), and the threshold is computed from the F-distribution, incorporating the dimensions of $\beta$ ($p$) and nuisance parameters ($q$).

1. Sample Size Formula: The minimum sample size $N$ to achieve power $1-\beta_0$ at a type I error level $\alpha_0$ is the smallest $N$ satisfying

$$F_{p, N-q-p; c_N}\left(F^{-1}_{p,N-q-p}(1-\alpha_0)\right) = 1-\beta_0,$$

where $$c_N = N d^\top (\sum_t E[I_t] \rho_t(1-\rho_t) Z_t Z_t^\top) d$$. Here, $d$ is the standardized treatment effect vector and $Z_t$ represents the time-varying effect structure.

The sample size is acutely sensitive to the number of decision points, randomization probabilities, expected availability, and the temporal pattern and magnitude of the hypothesized treatment effect.

3. Time-Varying Patterns in Effect and Availability

A defining feature of MRTs is their sensitivity to time-dependent dynamics in both treatment response and participant engagement. The MRT-SS Calculator supports parameterization of $Z_t$ (effect structure) and $E[I_t]$ (expected availability) as constant, linear, or quadratic functions of time. For instance:

• Linear effect model: $$\beta(t) = \beta_1 + (\lfloor (t-1)/5 \rfloor) \cdot \beta_2$$, suitable for daily increasing or decreasing effect.
• Quadratic availability model: accounts for non-monotonic engagement, e.g., initial adaptation followed by disengagement.

Model choice is critical because sample size calculations are invalid if the specified patterns do not reflect the actual behavioral process. For example, an overestimation of availability or failure to model declining treatment efficacy can lead to underpowered studies.

4. Case Example: HeartSteps Mobile Health Study

HeartSteps exemplifies the application of MRTs and the use of the MRT-SS Calculator:

• Design: 42 days, 5 decision times/day ($T=210$), randomization probability $\rho_t=0.4$ at each available time.
• Proximal outcome: Step count, measured after each decision time.
• Effect pattern: Treatment effect modeled as peaking mid-paper (e.g., quadratic), reflecting possible habituation.
• Availability: Pattern may also change, reflecting learning or disengagement over time.

Inputting these parameters, researchers can compute either the minimal $N$ for a target power (e.g., 80%) or, given $N$, the achieved power—accounting for design heterogeneity and subject behavior. Visualizations allow identification and rectification of implausible patterns (such as negative effects), and error checks guide re-specification.

5. Statistical Model, Test Statistic, and Formulaic Summary

The approach is built on a least-squares framework, with model: $$E[Y_{t+1} | I_t = 1, A_t] = B_t^\top \alpha + (A_t - \rho_t) Z_t^\top \beta.$$ Key formulas:

• Test Statistic:

$$N \hat \beta^\top \hat \Sigma_\beta^{-1} \hat \beta$$

compared to F-distribution-based threshold.

• Sample Size Solution:

$$F_{p, N-q-p; c_N}\left(F^{-1}_{p,N-q-p}(1-\alpha_0)\right) = 1-\beta_0$$

where

$$c_N = N d^\top (\sum_{t=1}^T E[I_t] \rho_t (1-\rho_t) Z_t Z_t^\top) d.$$

This framework elucidates how power and $N$ are controlled by the total number of decision times, the randomization and availability schedules, and the time-varying structure of the hypothesized effect.

6. Implementation, Design Guidance, and Limitations

The MRT-SS Calculator operationalizes this methodology via an R Shiny interface (https://pengliao.shinyapps.io/mrt-calculator), guiding the user through the selection of design and modeling parameters, effect parameterization, and alternative hypotheses. Error handling mitigates misconfiguration (e.g., negative effect patterns).

Considerations and Best Practices:

• Both effect and availability models must be carefully justified and, when possible, informed by pilot data or prior studies.
• Allowing different trends for effect and availability guards against design-induced Type II error.
• Visualization and feedback are integral to avoiding specification errors.
• The tool is optimized primarily for binary treatments and assumes independence of decision times, conditional on design variables.

Limitations:

• The method is not directly extensible to multi-level or categorical treatment settings, which require more general design formulas.
• Accuracy is contingent upon the correctness of functional forms for both availability and effect; deviations can impact both power and sample size estimations.

7. Summary of Key Features and Contributions

Feature	Purpose	Implementation/Role
Sequential randomization	Identifies proximal effects of interventions	Core MRT structure
Time-varying effect models	Accommodates realistic behavioral changes	$Z_t$ in regression
Flexible availability input	Reflects engagement/response dynamics	$E[I_t]$ specification
Noncentral F-based power	Accurately computes required sample size	Solver in calculator
HeartSteps illustration	Demonstrates method on real mHealth data	Example application

MRTs and the MRT-SS Calculator form a statistically rigorous foundation for the design of digital interventions, integrating time-varying modeling of treatment effects and subject availability with analytical power calculations. Their use, illustrated by the HeartSteps paper, supports efficient and ethical clinical trial planning in modern mobile health contexts (Seewald et al., 2016).