Author: Lukas Baumann
Date: 2024-04-09

# Introduction to baskexact

#### Lukas Baumann

#### 2024-04-09

## Scope

`baskexact` analytically calculates the operating characteristics of the power prior basket trial design (Baumann et. al, 2024) and the design of Fujikawa et al. (2020). Both designs were developed for the analysis of uncontrolled basket trials with a binary endpoint. Baskets are analysed using beta posteriors, where the posterior parameters are computed as weighted sums of the available information to share information between baskets. Currently `baskexact` supports single-stage and two-stage designs with equal sample sizes.

## Basic Usage

The first step is always to create a basket trial object using either `setupOneStageBasket` for a single-stage trial or `setupTwoStageBasket` for a two-stage trial. For example:

    library(baskexact)
    design <- setupOneStageBasket(k = 3, shape1 = 1, shape2 = 1, p0 = 0.2)

`k` is the number of baskets, `shape1` and `shape2` are the two shape parameters of the beta-prior of the response probabilities of each basket and `p0` is the response probability under the null hypothesis. Note that currently only common prior parameters and a common null response probability are supported.

The most important operating characteristics be calculated using the functions `toer` (type 1 error rate), `pow` (power) and `ecd` (expected number of correct decisions). For example:

    toer(
      design = design,
      p1 = NULL,
      n = 15,
      lambda = 0.99,
      weight_fun = weights_cpp,
      weight_params = list(a = 2, b = 2),
      results = "group"
    )

`p1` refers to the true response probabilities under which the type 1 error rate is computed. Since `p1 = NULL` is specified, the type 1 error rates under a global null hypothesis are calculated. `n` specifies the sample size per basket. `lambda` is the posterior probability cut-off to reject the null hypothesis. If the posterior probability that the response probability of the basket is larger than `p0` is larger than `lambda`, then the null hypothesis is rejected. `weight_fun` specifies which method should be used to calculate the weights. With `weights_cpp` the weights are calculated based on a response rate differences between baskets. In `weight_params` a list of parameters that further define the weights is given. See Baumann et al. (2024) for details. `results` specifies whether only the family wise type 1 error rate (option `fwer`) or also the basketwise type 1 error rates (option `group`) are calculated.

To find the probability cut-off `lambda` such that a certain FWER is maintained, use `adjust_lambda`, for example to find `lambda` such that the FWER does not exceed 2.5% (note that all hypotheses are tested one-sided):

    adjust_lambda(
      design = design,
      alpha = 0.025,
      p1 = NULL,
      n = 15,
      weight_fun = weights_cpp,
      weight_params = list(a = 2, b = 2),
      prec_digits = 4
    )

    # $lambda
    # [1] 0.991
    #
    # $toer
    # [1] 0.0231528

With `prec_digits` it is specified how many decimal places of `lambda` are considered. Use `toer` with `lambda = 0.9909` to check that 0.991 is indeed the smallest probability cut-off with four decimals with a FWER of at most 2.5%. Note that even when considering more decimal places the actual FWER will generally below the nominal level (quite substantially in some cases), since the outcome (number of responses) is discrete.

## References

Baumann, L., Sauer, L., & Kieser, M. (2024). A basket trial design based on power priors. arXiv:2309.06988. Fujikawa, K., Teramukai, S., Yokota, I., & Daimon, T. (2020). A Bayesian basket trial design that borrows information across strata based on the similarity between the posterior distributions of the response probability. Biometrical Journal, 62(2), 330-338.