## 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.