THE STRATIFIED MICRO-RANDOMIZED TRIAL DESIGN: SAMPLE SIZE CONSIDERATIONS FOR TESTING NESTED CAUSAL EFFECTS OF TIME-VARYING TREATMENTS

Abstract

Technological advancements in the field of mobile devices and wearable sensors have helped overcome obstacles in the delivery of care, making it possible to deliver behavioral treatments anytime and anywhere. Here, we discuss our work on the design of a mobile health smoking cessation intervention study with the goal of assessing whether reminders, delivered at times of stress, result in a reduction/prevention of stress in the near-term, and whether this effect changes with time in study. Multiple statistical challenges arose in this effort, leading to the development of the stratified micro-randomized trial design. In these designs, each individual is randomized to treatment repeatedly at times determined by predictions of risk. These risk times may be impacted by prior treatment. We describe the statistical challenges and detail how they can be met.

Keywords: mobile health; nested causal effects; sequential randomization; stratified microrandomized trials; weighted-centered least-squares method.

Fig 1:

Illustrative example of the episodic pattern of smoothed Sense2Stop stress probabilities and its associated online classification algorithm. In the minute following t^⋆, a stress classification is made. Subsequently, all minutes from the episode beginning to episode end are given the same classification. A participant can only be available in the minute following t^⋆.

Fig 2:

Illustration of the L₂-projection of β(t; x) onto feature vector f_t. The reference distribution ${\tilde{p}}_t(1|x)$ is constant in (t, x). The feature vector is non-parametric in binary x and set within each strata to $f_t=\left(1,d_t\right)\text{or}{f}_t=\left(1,d_t,d_t^2\right)$ where d_t is equal to the number days in study; expected availability given X_t = x is constant in t time or is quadratic in t with the same average. The distribution of X_t is constant in t or is quadratic in t with the same average.

Fig 3:

Histograms of duration for pre/post-peak durations for Minnesota study. Empirical bayes pdfs for exponential (red) and weibull (black) densities are overlayed.