# Bayesian Response-Adaptive Design Analysis with BRADA

#### Riko Kelter

# Introduction

The getting started vignette illustrated the basic features of the `brada` package. In this vignette, we illustrate how to monitor a running trial with the `brada` package.

Note that there are more vignettes which illustrate

- how to apply and calibrate the predictive evidence value design with the `brada` package. This vignette is hosted at the Open Science Foundation.
- how to monitor a running clinical trial with a binary endpoint by means of the `brada` package

# Monitoring a trial

To apply the package, first, load the package:

    library(brada)

Monitoring a trial with the `brada` package is straightforward through the `monitor` function. Suppose we have analyzed and calibrated a design according to our requirements, and end up with the following design:

    design = brada(Nmax = 30, batchsize = 5, nInit = 10,
                       p_true = 0.4, p0 = 0.4, p1 = 0.4,
                       nsim = 100,
                       theta_T = 0.90, theta_L = 0.1, theta_U = 1,
                       method = "PP",
                       cores = 2)

Now, suppose the trial is performed and the first ten patients show the response pattern \((0,1,0,0,0,0,0,1,0,0)\), where \(1\) encodes a response and \(0\) no response. Thus, there are \(2\) responses out of `nInit=10` observations. To check whether the trial can be stopped for futility or efficacy based on `theta_L=0.1` and `theta_U=1`, we run the `monitor` function as follows:

    monitor(design, obs = c(0,1,0,0,0,0,0,1,0,0))

    ## --------- BRADA TRIAL MONITORING ---------
    ## Primary endpoint: binary
    ## Test of H_0: p <= 0.4 against H_1: p > 0.4
    ## Trial design: Predictive probability design
    ## Maximum sample size: 30
    ## First interim analysis at: 10
    ## Interim analyses after each 5 observations
    ## Last interim analysis at: 25 observations
    ## -----------------------------------------
    ## Current trial size: 10 patients
    ## --------------- RESULTS -----------------

    ## Predictive probability of trial success: 0.00768
    ##  Futility threshold: 0.1
    ## Decision: Stop for futility

Thus, the results indicate that we should stop for efficacy. This is intuitively in agreement with the notion that \(2\) responses out of \(10\) observations are quite unlikely if \(H_1:p>0.4\) would hold.

Note that it is not important which value the `p_true` or `nsim` arguments had in the `brada` call which returned the object `design`. We could also have simulated data under `p_true=0.2` and `nsim=3000` or some other values, the monitor function only takes the `brada` object and applies the design specified in the `method` argument of the object, in this case, the predictive probability design. All necessary arguments are identified by the `monitor` function automatically. The predictive evidence value design can be monitored analogue, for details on the design and its calibration see the Open Science Foundation.

## References

Berry, S. M. (2011). Bayesian Adaptive Methods for Clinical Trials. CRC Press.

Kelter, R. (2022). The Evidence Interval and the Bayesian Evidence Value - On a unified theory for Bayesian hypothesis testing and interval estimation. British Journal of Mathematical and Statistical Psychology (2022). https://doi.org/10.1111/bmsp.12267

Kelter, R. (2021b). Bayesian Hodges-Lehmann tests for statistical equivalence in the two-sample setting: Power analysis, type I error rates and equivalence boundary selection in biomedical research. BMC Medical Research Methodology, 21(171). https://doi.org/10.1186/s12874-021-01341-7

Kelter, R. (2021a). fbst: An R package for the Full Bayesian Significance Test for testing a sharp null hypothesis against its alternative via the e-value. Behav Res (2021). https://doi.org/10.3758/s13428-021-01613-6

Kelter, R. (2020). Analysis of Bayesian posterior significance and effect size indices for the two-sample t-test to support reproducible medical research. BMC Medical Research Methodology, 20(88). https://doi.org/https://doi.org/10.1186/s12874-020-00968-2

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Pereira, C. A. d. B., & Stern, J. M. (2020). The e-value: a fully Bayesian significance measure for precise statistical hypotheses and its research program. São Paulo Journal of Mathematical Sciences, 1–19. https://doi.org/10.1007/s40863-020-00171-7

Rosner, G. L. (2021). Bayesian Thinking in Biostatistics. Chapman and Hall/CRC.

Rosner, G. L. (2020). Bayesian Adaptive Designs in Drug Development. In E. Lesaffre, G. Baio, & B. Boulanger (Eds.), Bayesian Methods in Pharmaceutical Research (pp. 161–184). CRC Press.

Rouder, Jeffrey N., Paul L. Speckman, Dongchu Sun, Richard D. Morey, and Geoffrey Iverson. 2009. “Bayesian t tests for accepting and rejecting the null hypothesis.” Psychonomic Bulletin and Review 16 (2): 225–37. https://doi.org/10.3758/PBR.16.2.225