Author: Eliot Wong-Toi, Thabat Dahdoul, Tianchen Qian

# Sample Size Calculator for Micro-Randomized Trials with Binary Outcomes

#### Eliot Wong-Toi, Thabat Dahdoul, Tianchen Qian

# 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 \((X_1, A_1, Y_2, X_2, A_2, Y_3, \ldots, X_m, A_m, Y_{m+1})\) 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 \[ H_0: \text{MEE}(t) = 0 \text{ for all } 1\leq t \leq m \] against the alternative hypothesis \[ H_1: \text{MEE}(t) \neq 0 \text{ for some } 1\leq t \leq m. \]

The sample size is calculated to ensure adequate power to detect a particular target alternative \[ H_1^{\text{target}}: \text{MEE}(t) = f(t)^T \beta \text{ for all } 1\leq t \leq m, \] where \(f(t)\) is a user-specified \(p\)-dimensional vector-valued function of \(t\) and \(\beta \in \mathbb{R}^p\). For example, if the user conjectures that a likely alternative is a constant treatment effect, they can set \(f(t) = 1\) and \(\beta\) 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 \(\beta\) will be a vector of length 2. The user will supply \(f(t)\) and \(\beta\), 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 \(H_1^{\text{target}}\), where `f_t` is a matrix of size \(m \times p\), and `beta` is a vector of length \(p\), and \(p\) is the degrees of freedom for the MEE under alternative. Specifically, under \(H_1^{\text{target}}\), \(\text{MEE}(t)\) equals `f_t[t, ] %*% beta` for each \(1 \leq t \leq m\). Usually the first column of `f_t` is a column of 1’s.

- `g_t` and `alpha`: They characterize the success probability null curve \(E(Y_{t+1} \mid A_t = 0, I_t = 1)\) for \(1 \leq t \leq m\), where `g_t` is a matrix of size \(m \times q\) and `alpha` is a vector of length \(q\), and \(q\) is the degrees of freedom for the success probability null curve. Specifically, \(\log E(Y_{t+1} \mid A_t = 0, I_t = 1)\) equals `g_t[t, ] %*% alpha` for each \(1 \leq t \leq m\). 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 \(1 \leq j \leq p\) 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).

## A numerical example

We use the following numerical example to illustrate 4 functions in this package: `mrt_binary_ss()`, `mrt_binary_power()`, `power_vs_n_plot()`, and `power_summary()`. In this numerical example, the total number of decision points \(m = 10\). The degrees of freedom for the MEE and the success probability null curve are \(p = 2\) and \(q = 2\).

    library(MRTSampleSizeBinary)

`tau_t_1`: Vector of length 10 that holds the average availability at each time point. In this example the availability remains constant across decision points.

    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`: A length 10 vector of randomization probabilities for each time point. This example has the randomization probability staying constant across decision points.

    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`: A 10 by 2 matrix that defines the MEE under the target alternative hypothesis (together with `beta_1`). Each row corresponds to a decision point (in this example we have 10).

    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`: Vector that defines the MEE under the target alternative hypothesis (together with `f_t_1`).

    beta_1
    #> [1]  0.15 -0.01

`g_t_1`: A 10 by 2 matrix that defines the success probability null curve (together with `alpha_1`). As with `f_t_1`, each row corresponds to a decision point.

    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`: A length of 2 vector that defines the success probability null curve (together with `g_t_1`).

    alpha_1
    #> [1] -0.2 -0.1

### Example use of `mrt_binary_ss()`

Below we compute the required sample size for an MRT to achieve \(0.8\) power using the above numerical example. Recall that the argument `gamma` is the type I error rate, `b` is the type II error rate (\(1- \text{power}\)), and `exact` is a flag for if the function should return the exact sample size our calculator computes (this may not be an integer) or the ceiling of this number. By default `exact=FALSE`. We see that the required sample size is \(275\) individuals for this numerical example.

    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

### Example use of `mrt_binary_power()`

If the investigator would like to calculate power given a sample size and a specified significance level, then they can use the `mrt_binary_power()` function. The first seven arguments are the same as in `mrt_binary_ss()`. The final argument, `n` is the sample size (i.e., number of individuals). Notice that, as expected, the sample size (`n=275`) is the output from the previous computation and the power is very close to \(0.8\); it is not exactly 0.8 due to the rounding up of the sample size in the previous example.

    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

### Example use of `power_vs_n_plot()`

`power_vs_n_plot()` can be used to obtain a visualization of the relationship between the power and the sample size for a range of possible power and sample size values. The following example uses the default range of sample sizes to plot over, but with the additional arguments `min_n` and `max_n` the user can choose what sample size range they want to plot over.

    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)

### Example use of `power_summary()`

`power_summary()` provides a tabular way to examine the relationship between the power and the sample size. The following example presents the sample size for power ranging from \(0.6\) to \(0.95\) by increments of \(0.05\). Again, we see that a sample size of \(275\) will achieve a power of \(0.8\) in our numerical example. The user can customize the power range and increments by specifying the `power_levels` argument.

    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

# References

Cohn, Eric, Tianchen Qian, Predrag Klasnja, and Susan A Murphy. 2021. “Sample Size Considerations for Micro-Randomized Trials with Binary Proximal Outcome.” Unpublished Manuscript.
Klasnja, Predrag, Eric B. Hekler, Saul Shiffman, Audrey Boruvka, Daniel Almirall, Ambuj Tewari, and Susan A. Murphy. 2015. “Microrandomized Trials: An Experimental Design for Developing Just-in-Time Adaptive Interventions.” Health Psychology: Official Journal of the Division of Health Psychology, American Psychological Association 34S (December): 1220–28. https://doi.org/10.1037/hea0000305.
Liao, Peng, Predrag Klasnja, Ambuj Tewari, and Susan A Murphy. 2016. “Sample Size Calculations for Micro-Randomized Trials in mHealth.” Statistics in Medicine 35 (12): 1944–71.
Qian, Tianchen, Hyesun Yoo, Predrag Klasnja, Daniel Almirall, and Susan A Murphy. 2020. “Estimating Time-Varying Causal Excursion Effect in Mobile Health with Binary Outcomes.” Biometrika.