Design and analysis of one- and two-stage binomial
    clinical phase II trials using Bayes factors. Implements Bayes factors
    for point-null and directional hypotheses, predictive densities under
    different hypotheses, and power and sample size calibration. Both one-arm
    trials with only a single treatment arm and two-arm trials with treatment
    and control arm are implemented for the one- and two-stage designs.

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

Implements the Backfill Bayesian Optimal Interval Design (BF-BOIN-ET), 
             a novel clinical trial methodology for dose optimization that simultaneously consider both efficacy 
             and toxicity outcome as described in (Takeda et al  (2025) <doi:10.1002/pst.2470>).
             The package has been extended to include a seamless two-stage phase I/II trial design with backfill 
             and joint efficacy and toxicity monitoring as described in (Takeda et al (2026) <doi:10.1002/pst.70092>).

Implements the distance measure for mixed-scale variables proposed by 
    Buttler and Fickel (1995), based on normalized mean pairwise distances 
    (Gini mean difference), and an R2 statistic to assess clustering quality.

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.