Background

The MRTSampleSizeBinary package provides a sample size calculator for micro-randomized trials (MRT) where the proximal outcomes are binary. This calculator can be used to either

• determine the sample size needed to ensure a specified power when testing for a nonzero marginal causal excursion effect, or

• determine the power given a sample size when testing for a nonzero marginal causal excursion effect.

MRT is an experimental design for optimizing mobile health interventions such as push notifications to increase physical activity. The calculator is based on the methods developed by Cohn et al. (2021). For a general overview of MRT, see Klasnja et al. (2015) and Liao et al. (2016). Here we briefly review the hypothesis test that the sample size is calculated for.

Let (X₁, A₁, Y₂, X₂, A₂, Y₃, …, X_m, A_m, Y_m + 1) denote the observed data from a participant in the MRT, where m denotes the total number of decision points, and for each decision point t:

• X_t denotes the time-varying covariate.

• A_t denotes the treatment assignment. We assume A_t takes value in {0, 1}: For example, if the treatment is a push notification, then A_t = 0 means the push notification is not delivered at t, and A_t = 1 means the push notification is delivered at t.

• Y_t + 1 denotes the binary proximal outcome.

• I_t denote the availability condition at t: I_t = 1 if the participant is available for treatment at t, and I_t = 0 if the participant is unavailable at t. With our notation I_t is included in X_t.

For primary analysis of an MRT, one is usually interested in testing for a marginal causal excursion effect (MEE); see Cohn et al. (2021) and Qian et al. (2020) for details. Under standard causal assumptions, MEE at decision point t can be expressed as $$ \text{MEE}(t) = \log \frac{P(Y_{t+1} = 1 \mid A_t = 1, I_t = 1)}{P(Y_{t+1} = 1 \mid A_t = 0, I_t = 1)}, $$ which is akin to a relative risk. We consider testing the following null hypothesis H₀ : MEE(t) = 0 for all 1 ≤ t ≤ m against the alternative hypothesis H₁ : MEE(t) ≠ 0 for some 1 ≤ t ≤ m.

The sample size is calculated to ensure adequate power to detect a particular target alternative H₁^target : MEE(t) = f(t)^Tβ for all 1 ≤ t ≤ m, where f(t) is a user-specified p-dimensional vector-valued function of t and β ∈ ℝ^p. For example, if the user conjectures that a likely alternative is a constant treatment effect, they can set f(t) = 1 and β will be a scalar; if the user conjectures that the treatment effect would decrease over time, they can set f(t) = (1, t)^T and β will be a vector of length 2. The user will supply f(t) and β, along with other inputs that will be detailed below, when using the calculator in this package.

Quick Start

The function to calculate the sample size is mrt_binary_ss(). The function to calculate the sample size is mrt_binary_power(). Because the syntax for their use are similar, here we illustrate the use of mrt_binary_ss().

The function mrt_binary_ss() takes in the following arguments:

• avail_pattern: A vector of length m. The t-th entry denotes the average availability at decision point t, E(I_t).

• p_t: A vector of length m. The t-th entry denotes the randomization probability at decision point t, P(A_t = 1).

• f_t and beta: They characterize the MEE under the target alternative H₁^target, where f_t is a matrix of size m × p, and beta is a vector of length p, and p is the degrees of freedom for the MEE under alternative. Specifically, under H₁^target, MEE(t) equals f_t[t, ] %*% beta for each 1 ≤ t ≤ m. Usually the first column of f_t is a column of 1’s.

• g_t and alpha: They characterize the success probability null curve E(Y_t + 1 ∣ A_t = 0, I_t = 1) for 1 ≤ t ≤ m, where g_t is a matrix of size m × q and alpha is a vector of length q, and q is the degrees of freedom for the success probability null curve. Specifically, log E(Y_t + 1 ∣ A_t = 0, I_t = 1) equals g_t[t, ] %*% alpha for each 1 ≤ t ≤ m. Usually the first column of g_t is a column of 1’s. In addition, it is required that the linear column span of g_t contains f_t[, j] * p_t for each 1 ≤ j ≤ p for the sample size calculation result to be accurate; mrt_binary_ss() checks for this and if not satisfied the function will output a warning.

• gamma: A scalar. This is the desired type I error.

• b: A scalar. This is the desired type II error. In other words, 1 − b is the desired power.

• exact: A boolean value, default to FALSE. If TRUE, outputs the resulting sample size with decimal digits. If FALSE, outputs the resulting sample size after talking the ceiling (smallest integer that is larger than or equal to the calculated sample size).

library(MRTSampleSizeBinary)

tau_t_1
#>  [1] 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

p_t_1
#>  [1] 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

f_t_1
#>       [,1] [,2]
#>  [1,]    1    1
#>  [2,]    1    2
#>  [3,]    1    3
#>  [4,]    1    4
#>  [5,]    1    5
#>  [6,]    1    6
#>  [7,]    1    7
#>  [8,]    1    8
#>  [9,]    1    9
#> [10,]    1   10

beta_1
#> [1]  0.15 -0.01

g_t_1
#>       [,1] [,2]
#>  [1,]    1    1
#>  [2,]    1    2
#>  [3,]    1    3
#>  [4,]    1    4
#>  [5,]    1    5
#>  [6,]    1    6
#>  [7,]    1    7
#>  [8,]    1    8
#>  [9,]    1    9
#> [10,]    1   10

alpha_1
#> [1] -0.2 -0.1

mrt_binary_ss(avail_pattern=tau_t_1, 
                               f_t=f_t_1, g_t=g_t_1, 
                               beta=beta_1, alpha=alpha_1,
                               p_t=p_t_1, 
                               gamma=0.05, 
                               b=.2,
                               exact=FALSE)
#> [1] 275

mrt_binary_power(avail_pattern=tau_t_1, 
                          f_t=f_t_1, g_t=g_t_1, 
                          beta=beta_1, alpha=alpha_1, 
                          p_t=p_t_1, 
                          gamma=0.05, 
                          n=275)
#> [1] 0.8015381

power_vs_n_plot(avail_pattern=tau_t_1, 
                f_t=f_t_1, g_t=g_t_1, 
                beta=beta_1, alpha=alpha_1, 
                p_t=p_t_1, 
                gamma=0.05)

power_summary(tau_t_1, f_t_1, g_t_1,beta_1, alpha_1, p_t_1, 0.05)
#>      power sample_size
#> [1,]  0.60         178
#> [2,]  0.65         198
#> [3,]  0.70         220
#> [4,]  0.75         245
#> [5,]  0.80         275
#> [6,]  0.85         311
#> [7,]  0.90         359
#> [8,]  0.95         438