---
title: Micro-Randomized Trials (MRT)
url: https://www.emergentmind.com/topics/micro-randomized-trials-mrt
type: topic
---

# Micro-Randomized Trials (MRT)

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. MRTs 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 [2107.03544], [2005.05880], [2004.10241].

## 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) [1609.00695].
- **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 [2107.03544], [2504.15484].

## 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 [1609.00695].

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 [2504.15484].

## 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 [2107.03544], [2004.10241], [1609.00695]:
\[
\min_{\alpha, \beta} \ \frac{1}{N}\sum_{i=1}^N \sum_{t=1}^T I_{it} \left\{ Y_{i,t+1} - B_t^\top \alpha - (A_{it} - \rho_t) Z_t^\top \beta \right\}^2
\]
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 [1609.00695], [2005.05880]. 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

Source: https://www.emergentmind.com/topics/micro-randomized-trials-mrt