Micro-Randomized Trials (MRT)

• Micro-Randomized Trials (MRTs) are experimental designs that randomize treatment at multiple decision points to evaluate the short-term, proximal effects of just-in-time interventions.
• They are applied primarily in mobile health research, dynamically adapting behavioral interventions based on real-time context and user data.
• Statistical analysis in MRTs utilizes methods like weighted and centered least squares estimation and power analysis to determine sample sizes for reliable causal inference.

A micro-randomized trial (MRT) is an experimental design originating in mobile health research for evaluating the proximal (short-term) causal effects of "just-in-time" interventions, typically delivered via mobile or wearable technology. Unlike conventional randomized controlled trials that randomize participants once, an MRT randomizes participants at hundreds or thousands of prespecified decision points—such as several times per day—enabling granular assessment of how intervention effects vary over time and context. @@@@1@@@@ have become foundational for optimizing just-in-time adaptive interventions (JITAIs), which dynamically adapt behavioral interventions in response to real-time context, engagement, or physiological history (Qian et al., 2021, Walton et al., 2020, Qian et al., 2020).

1. Experimental Design of Micro-Randomized Trials

In an MRT, each of $N$ participants is followed over $T$ decision points. At each decision point $t = 1,\ldots,T$:

• Availability: An indicator $I_t \in \{0,1\}$ denotes if the participant is available for treatment (if $I_t=0$, no randomization occurs) (Seewald et al., 2016).
• Randomization: If available, the participant is randomized to receive treatment $A_t$, often binary ($A_t \in \{0,1\}$), with randomization probability $\rho_t = P(A_t=1)$, which may be constant or time-varying.
• Proximal Outcome: The proximal response $Y_{t+1}$ (or a lagged outcome $Y_{t,\Delta}$) is measured, capturing the immediate effect of treatment.

MRTs also accommodate multilevel categorical interventions, time-varying covariates, and may involve complex availability rules based on safety, burden, or context (Qian et al., 2021, Lin et al., 21 Apr 2025).

2. Causal Estimands: Proximal and Excursion Effects

MRTs aim to estimate proximal treatment effects: the causal effect of assigning treatment at a decision point on the immediate or near-term outcome. Formally, the proximal effect at time $t$ is

$$\beta(t) = E[Y_{t+1}|I_t=1, A_t=1] - E[Y_{t+1}|I_t=1, A_t=0]$$

This is defined only when the participant is available ($I_t=1$). For analysis, one tests hypotheses such as $H_0: \beta(t) = 0 \ \forall t$ versus alternatives where effects are nonzero for some $t$ or exhibit a prespecified trajectory (Seewald et al., 2016).

More generally, MRTs estimate causal excursion effects—the difference in potential outcomes under an "excursion" to treatment versus control at a time point, marginalized or moderated by contextual variables. In advanced settings with categorical treatments, causal estimands extend to multivariate versions (Lin et al., 21 Apr 2025).

3. Statistical Inference and Sample Size Methodology

Weighted and Centered Least Squares (WCLS) Estimation

Primary analysis in MRTs employs the weighted and centered least squares (WCLS) estimator (Qian et al., 2021, Qian et al., 2020, Seewald et al., 2016): $$\min_{\alpha, \beta} \ \frac{1}{N}\sum_{i=1}^N \sum_{t=1}^T I_{it} \left\{ Y_{i,t+1} - B_t^\top \alpha - (A_{it} - \rho_t) Z_t^\top \beta \right\}^2$$ where $B_t$ are nuisance variables (e.g., intercept, time-of-day), and $Z_t$ is a design vector encoding the hypothesized proximal effect shape (e.g., constant, linear or quadratic time trends). The centering $(A_t - \rho_t)$ renders $\hat\beta$ insensitive to misspecification of the outcome model under correct randomization.

Sample Size Determination

Power analysis in MRTs is based on a noncentral F-theoretic framework (Seewald et al., 2016, Walton et al., 2020). The required sample size $N$ for achieving power $1 - \beta_0$ at type-I error $\alpha$ is the smallest integer satisfying: $$F_{p, N - q - p; c_N}\left( F^{-1}_{p, N - q - p}(1-\alpha) \right) \geq 1 - \beta_0$$ with noncentral