Calculate the operating characteristics of the 
    Bayesian Optimal Interval with Back Filling Design for dose escalation
    in early-phase oncology trials.

The backfill Bayesian optimal interval design using efficacy and toxicity outcomes for 
    dose optimization (BF-BOIN-ET) design is a novel clinical trial design to allow patients to be 
    backfilled at lower doses during a dose-finding trial while prioritizing the dose-escalation 
    cohort to explore a higher dose. The advantages compared to the other designs in terms of the 
    percentage of correct optimal dose (OD) selection, reducing the sample size, and shortening the 
    duration of the trial, in various realistic setting.

Bayes factors represent the ratio of probabilities assigned to data by competing scientific hypotheses. However, one drawback of Bayes factors is their dependence on prior specifications that define null and alternative hypotheses. Additionally, there are challenges in their computation. To address these issues, we define Bayes factor functions (BFFs) directly from common test statistics. BFFs express Bayes factors as a function of the prior densities used to define the alternative hypotheses. These prior densities are centered on standardized effects, which serve as indices for the BFF. Therefore, BFFs offer a summary of evidence in favor of alternative hypotheses that correspond to a range of scientifically interesting effect sizes. Such summaries remove the need for arbitrary thresholds to determine "statistical significance." BFFs are available in closed form and can be easily computed from z, t, chi-squared, and F statistics. They depend on hyperparameters "r" and "tau^2", which determine the shape and scale of the prior distributions defining the alternative hypotheses. Plots of BFFs versus effect size provide informative summaries of hypothesis tests that can be easily aggregated across studies.

The Bayesian Federated Inference ('BFI') method combines inference results obtained from local data sets in the separate centers. In this version of the package, the 'BFI' methodology is programmed for linear, logistic and survival regression models. For GLMs, see Jonker, Pazira and Coolen (2024) <doi:10.1002/sim.10072>; for survival models, see Pazira, Massa, Weijers, Coolen and Jonker (2025) <doi:10.48550/arXiv.2404.17464>; and for heterogeneous populations, see Jonker, Pazira and Coolen (2025) <doi:10.1017/rsm.2025.6>.