BayesianQDM 1.0.1

• The C++ code for two continuous endpoints no longer calls std::sort() on a fixed-size array, for which GCC 16 gave a compiler warning (-Warray-bounds) during installation. The results are unchanged.

BayesianQDM 1.0.0

This version is a redesign of the package and is not backward compatible with version 0.1.0. A study is described by small specification objects (priors, data, external data, hypothetical controls and decision criteria) that are combined into a design, and three functions work with any design: qdm_decide(), qdm_oc() and qdm_calibrate(). qdm_boundary() shows the observed data that lead to each decision. Formulas and numerical methods that were wrong in version 0.1.0 have been corrected (see below), so some results differ from those of version 0.1.0. vignette("migration") shows how to translate code written for version 0.1.0.

Correspondence with version 0.1.0

Documentation, validation and computation

• Seven vignettes: an overview, one for each type of endpoint, and a guide to migrating from version 0.1.0.
• inst/validation/ contains scripts that compare the package with published and submitted results (Kang et al., 2026; Huang, Talukder, Han and Kuan, 2019; Huang et al., submitted; Yamaguchi, Homma, Takeda and Maruo, submitted; Yamaguchi et al., submitted), and inst/validation/results/summary.md reports the comparisons, with a registered reason for every difference beyond rounding, including the results that are not reproduced.
• The numerical integration and the Monte Carlo loops are written in C++ with Rcpp.

Specification functions

• Priors: prior_beta(), prior_dirichlet(), prior_nix(), prior_niw(), prior_gamma() and prior_vague().
• Data summaries: data_binary(), data_continuous(), data_binary2(), data_continuous2() and data_tte().
• Conduct of a study with a time-to-event endpoint (enrollment, dropout and time of the analysis): conduct_tte().
• External data with a power prior weight from 0 to 1: borrow().
• Hypothetical control for uncontrolled studies: hypothetical().
• Decision criteria: criterion_posterior() and criterion_predictive(). For all endpoints, the Go criterion is that the probability of the Go region is at least gamma_go and the NoGo criterion is that the probability of the NoGo region is at least gamma_nogo.
• criterion_posterior() and criterion_predictive() have the argument direction. With direction = "greater" (the default), larger effects are favorable, as in version 0.1.0. With direction = "less", smaller effects are favorable, for example for the rate of an adverse event or a hazard ratio: the Go region is that the effect is below theta_tv (or, for the predictive criterion, that the future effect is below theta_null), the NoGo region is that it is at least theta_mav (at least theta_null for the predictive criterion), and theta_tv must be smaller than theta_mav. For two endpoints, the direction can be set for each endpoint, for example an efficacy endpoint and a safety endpoint.
• qdm_regions() shows the numbering of the regions for two endpoints.

Designs and operations: one binary endpoint

• design_binary() specifies a study with one binary endpoint: sample sizes, Beta priors, external data for either group (borrow()), and a concurrent, external-only (n_c = 0 with external_c) or hypothetical (n_c = 0 with hypothetical_c) control group.
• qdm_decide() computes the probabilities of the Go and NoGo regions for observed data and returns the decision: Go, NoGo, Gray, or Miss (both criteria met).
• qdm_oc() computes the probabilities of the four decisions under scenarios for the true response rates by exact enumeration of the outcomes. Miss decisions are reported in their own column with a warning; the arguments error_if_Miss and Gray_inc_Miss of version 0.1.0 are removed.
• qdm_calibrate() chooses gamma_go and gamma_nogo as the smallest grid values for which the probabilities that the Go criterion is met under one scenario and that the NoGo criterion is met under another are at most their targets (version 0.1.0 required them to be less than the targets), and returns the design with these thresholds and its decision probabilities.
• print() methods for designs, decisions, operating characteristics and calibrations, and plot() methods (ggplot2) for operating characteristics and calibrations. Without a concurrent control group, the thresholds in the plot of operating characteristics are shifted by the mean of the distribution of the control group.
• Probabilities are no longer set to 0 below .Machine$double.eps^0.25. A probability that is below a threshold by less than 1e-10 is treated as equal to the threshold, so that exact ties are not decided by rounding errors.

Designs and operations: one continuous endpoint

• design_continuous() specifies a study with one normally distributed endpoint: sample sizes, Normal-Inverse-chi-squared (prior_nix()) or vague (prior_vague()) priors, external data for either group (borrow()), a concurrent, external-only or hypothetical control group, and the method used to compute the probabilities ("NI", "MM" or "MC").
• External data enter through the conjugate representation of the power prior. With the vague prior, the posterior distribution of the mean has a0 * n_e + n - 1 degrees of freedom, and the posterior scale uses the same denominator (version 0.1.0 used a0 * n_e + n, so the posterior did not reduce to the one without external data as a0 tends to 0).
• The posterior predictive distribution of the mean of m future subjects is t(nu_n, mu_n, sigma_n^2 * (1 / kappa_n + 1 / m)). Version 0.1.0 used the squared scale sigma_n^2 * (1 + 1 / kappa_n) / m, which treats the m future subjects as independent draws from the predictive distribution of one subject and understates the uncertainty for m of 2 or more.
• A hypothetical control (hypothetical(mean = mu0c, r = r)) gives the control mean the distribution t(nu_nt, mu0c, r * sigma_nt^2 / kappa_nt) (Kang et al., 2026, Section 4.2.4). For the predictive criterion its squared scale is r * sigma_nt^2 * (1 / kappa_nt + 1 / m_c).
• qdm_oc() and qdm_calibrate() simulate the sample means and standard deviations of nsim studies, with the same random numbers for all scenarios. With method = "MC", the probabilities of the Go and NoGo regions are estimated from the same draws, so that they add up to 1 for the predictive criterion.
• qdm_decide() has a seed argument for designs with method = "MC".

Designs and operations: two binary endpoints

• design_binary2() specifies a study with two binary endpoints: sample sizes, Dirichlet priors for the probabilities of the four combinations of responses (prior_dirichlet()), external data for either group (borrow(data_binary2(...), a0)), a concurrent, external-only or hypothetical control group, the number of Monte Carlo draws nMC, and whether the operating characteristics are computed by enumerating all outcomes (enumerate = TRUE, the default) or by simulation.
• For two endpoints, criterion_posterior() and criterion_predictive() have the argument rule. With rule = "joint" (the default), the criteria are on the probabilities of unions of regions (go_regions and nogo_regions, numbered as in qdm_regions()). With rule = "and" or rule = "or", the criteria are on each endpoint separately (the probability that the effect exceeds its TV, or is at most its MAV), with one threshold for both endpoints or one per endpoint.
• The posterior predictive probabilities of the regions are computed exactly from the joint distribution of the numbers of future responders on the two endpoints (Dirichlet-multinomial distribution) instead of by Monte Carlo simulation. A future effect equal to the null value is not counted as larger; version 0.1.0 compared differences of rounded proportions, so some exact ties were counted as larger.
• The posterior probabilities of the regions are estimated with nMC draws (default 10000) from the Dirichlet posterior distributions, computed in C++. The probabilities of all regions, and of the Go and NoGo regions, are estimated from the same draws, and qdm_decide() reports all of them. qdm_decide(), qdm_oc() and qdm_calibrate() take seed for these draws.
• qdm_oc() takes scenarios with the columns pi_t1, pi_t2, rho_t, pi_c1, pi_c2 and rho_c. With enumeration, the region probabilities are computed once for every pair of treatment and control outcomes, with the same Monte Carlo draws for all pairs, and weighted by the multinomial probabilities under each scenario. With simulation, all scenarios use the same uniform random numbers. Enumeration is limited to 1e7 pairs of outcomes (for example, 24 subjects in each group). A scenario whose correlation is outside the feasible range for its response rates is not evaluable: its probabilities are NA with a warning (version 0.1.0 stopped with an error in its Monte Carlo path). qdm_calibrate() requires feasible scenarios.
• plot() of operating characteristics for two endpoints draws a heat map of the probability of each decision over the effects on the two endpoints and marks scenarios that are not evaluable with “NE”.

Designs and operations: two continuous endpoints

• design_continuous2() specifies a study with two continuous endpoints: sample sizes, Normal-Inverse-Wishart (prior_niw()) or vague (prior_vague()) priors, external data for either group (borrow(data_continuous2(...), a0)), a concurrent, external-only or hypothetical (hypothetical(mean = c(m1, m2), r = r)) control group, the decision criterion for two endpoints (with rule as for two binary endpoints), and the method used to compute the probabilities ("MM" or "MC").
• The posterior parameters are updated with the external data (power prior) and then with the current data; the vague prior corresponds to kappa0 = 0, nu0 = -1 and Lambda0 = 0.
• The posterior predictive distribution of the mean of m future subjects has the scale matrix (1 / kappa_n + 1 / m) * Lambda_n / (nu_n - 1). Version 0.1.0 used (1 + kappa_n) / kappa_n * Lambda_n / (nu_n - 1) / m, which treats the m future subjects as independent draws from the predictive distribution of one subject.
• With method = "MM", the moment-matching approximation of the difference of two bivariate t distributions is used, and its rectangle probabilities are computed by numerical integration in C++ for real degrees of freedom (version 0.1.0 rounded the degrees of freedom to an integer for mvtnorm::pmvt()). With method = "MC", the probabilities of all regions are estimated from the same draws, and every simulated study uses its own draws (version 0.1.0 used the same draws for all simulated studies, so the Monte Carlo error did not average out).
• qdm_oc() takes scenarios with the columns mu_t1, mu_t2, sd_t1, sd_t2, rho_t, mu_c1, mu_c2, sd_c1, sd_c2 and rho_c, and simulates the sample means and the sums of squares and cross-products (Bartlett decomposition of the Wishart distribution) of nsim studies with the same random numbers for all scenarios.

Fixed control

• All design functions have the argument fixed_c for a single-arm study compared with a known value of the control parameter (a response rate, a mean, a milestone survival probability, or a median survival time for the hazard ratio; one value per endpoint for two endpoints). It requires n_c = 0 and prior_c = NULL. The effect is the treatment parameter minus the fixed value (for the hazard ratio, the treatment hazard divided by the control hazard log(2) / fixed_c), so the probabilities depend only on the posterior distribution of the treatment group: Beta probabilities (one binary endpoint), t probabilities (one continuous endpoint), probabilities of the bivariate t distribution by numerical integration for any degrees of freedom (two continuous endpoints), Monte Carlo draws from the Dirichlet posterior (posterior probabilities for two binary endpoints), Gamma probabilities (posterior probabilities for a time-to-event endpoint), and exact predictive probabilities for binary endpoints, in which a future proportion equal to the fixed value plus the null value is not counted as larger. The argument method is not used for continuous endpoints.
• Without a concurrent control group, the thresholds in the plot of operating characteristics are shifted by the fixed value.

Designs and operations: one time-to-event endpoint

• design_tte() specifies a study with one time-to-event endpoint evaluated by the survival probability at a milestone time or by the hazard ratio (see below): exponential event times with Gamma priors on the hazards (prior_gamma()), data summarized by the number of events and the total observed time (data_tte()), external data for either group through a power prior, and a concurrent, external-only, hypothetical or fixed control group. With the milestone survival, the effect is the difference of the milestone survival probabilities.
• Posterior probabilities are computed by numerical integration over the control hazard (psurvdiff()), or exactly for a fixed control. Posterior predictive probabilities are estimated by simulating nMC future studies conducted as specified by conduct_future, with hazards drawn from the posterior distributions; the milestone survival of each group in the future study is estimated at the posterior mean of the hazard given the future data and the prior of the group.
• qdm_oc() and qdm_calibrate() simulate nsim studies with piecewise uniform enrollment, piecewise exponential dropout and an analysis at a calendar time or at a given number of events among all subjects of the study (conduct_tte()), under scenarios given by the median survival times (median_t, median_c), with the same random numbers for all scenarios.
• With effect = "hazard_ratio", the effect is the hazard ratio lambda_t / lambda_c (Huang, Talukder, Han and Kuan, 2019), usually with direction = "less" in the criterion. The posterior distribution of the hazard ratio is a scaled F distribution, so posterior probabilities are exact (phazardratio()); a fixed control is given by its median survival time. Posterior predictive probabilities compare the ratio of the estimated hazards of the future study with theta_null.
• inst/validation/ reproduces the probabilities of a Go decision in Table 1 of Huang et al. (2019) (a randomized study analyzed at a given number of events) with the hazard ratio and the posterior criterion.

Decision boundaries

• qdm_boundary() returns the intervals of an observed quantity that lead to Go, NoGo, Gray and Miss decisions, for each value of a second quantity: the number of treatment responders for each number of control responders (one binary endpoint with a concurrent control group) or for each sample size (without one), the observed treatment mean for each standard deviation or sample size (one continuous endpoint), the observed treatment mean of endpoint 2 for each observed mean of endpoint 1 (two continuous endpoints), and the number of events for each total observed time (time-to-event endpoint). The intervals are also given as observed response rates or milestone survival probabilities and as observed treatment effects (differences from the control group or the fixed control value, or observed hazard ratios). Two binary endpoints are not supported.
• For observed means, the ends of the intervals are located by bisection between the values of a grid, so they are exact up to the accuracy of the probabilities, provided that a criterion changes at most once between neighboring grid values and not beyond the grid. Decisions that change twice, as with conjugate priors far from the observed data, give several intervals.
• The print() method shows the intervals as tables with one row per condition and one column per decision, for example the numbers of responders out of 20 that lead to Go (10-20), the corresponding observed response rates (50.0% to 100.0%) and the observed differences from a fixed control rate of 0.2 (0.3 to 0.8). The plot() method draws the regions of the decisions with the boundary values written on them (labels, label_size), and for one binary endpoint with a concurrent control group a secondary axis with the observed response rate of the treatment group.

Plots

• The plot() methods for operating characteristics, calibrations and decision boundaries return ggplot2 objects, which can be changed further by adding ggplot2 functions such as labs() and theme(). In all three plots, arguments set the colors of the decisions (col_go, col_nogo, col_gray and col_miss, green, red, gray and orange by default; the calibration plot uses col_go and col_nogo), the base font size (base_size, 11 by default instead of 28 in version 0.1.0) and the position of the legend (legend_position). The plots of operating characteristics and calibrations also take the width of the lines (linewidth), and the plots of operating characteristics and decision boundaries the size of the text drawn in the plot (label_size); the plot of operating characteristics also takes the size of the points (point_size). Other arguments are ignored with a warning.
• The heat maps of two endpoints shade each panel from white (probability 0) to the color of its decision (probability 1); their colors are set by the color arguments. They have no legend, since each cell shows its probability (unless cell_values = FALSE). Scenarios that are not evaluable are white and marked “NE”.
• plot() of operating characteristics for two endpoints has the argument overlay, which replaces which = "overlay" of version 0.1.0. With overlay = TRUE, a single heat map shows in each cell the decision with the highest probability, shaded in five steps of that probability, with legends for the decisions and the steps (Yamaguchi et al., submitted, Figure 2). cell_values = FALSE leaves out the probabilities written in the cells of either heat map.

Distribution functions

• pbetadiff() has the argument order (q, a_t, b_t, a_c, b_c). The range of integration is restricted to the support and split at quantiles of both Beta distributions, so concentrated distributions are integrated accurately. When a shape parameter of the control group is below 1, the integral near the unbounded end of its density is computed after a change of variables that removes the singularity. The computation is done in C++.
• pbetabinomdiff() has the argument order (q, m_t, a_t, b_t, m_c, a_c, b_c) and computes the probability mass functions on the log scale, so large numbers of future subjects no longer cause overflow. Values of q far outside [-1, 1] are handled exactly.
• ptdiff() replaces ptdiff_NI(), ptdiff_MC() and ptdiff_MM(). The numerical integration now covers the whole real line (version 0.1.0 truncated the range, with errors of up to about 3e-4 for small degrees of freedom). It integrates over the group with the smaller scale, splits the range around the peak of the density and the change of the other factor, and integrates the parts far from the origin on a logarithmic scale, so that scale ratios from 1e-9 to 1e3 and degrees of freedom from 0.5 to Inf are handled.
• rdirichlet() always returns a matrix.
• cellprob_binary2() replaces getjointbin().
• psurvdiff() computes the distribution function of the difference of two milestone survival probabilities with Gamma distributed exponential hazards, by adaptive quadrature in C++ over the hazard of one group, with the range split at quantiles of both hazards and a change of variables near 0 when a shape parameter is below 1. Each tail is computed directly, so both are accurate near 0.
• phazardratio() computes the distribution function of the ratio of two Gamma distributed hazards from the F distribution.

BayesianQDM 0.1.0

Resubmission

• Expanded the MCMC acronym to “Markov chain Monte Carlo (MCMC)” in DESCRIPTION.
• Added a reference to Kang, Yamaguchi, and Han (2026) in DESCRIPTION.
• Replaced \dontrun{} with \donttest{} in examples for getgamma2bin() and getgamma2cont().

Initial Release

Initial release providing a comprehensive Bayesian quantitative decision-making framework for clinical trials with single and two-endpoint analyses for binary and continuous outcomes.

Decision Probability Functions

• pbayesdecisionprob1bin() - Go/NoGo/Gray decision probabilities for a single binary endpoint
• pbayesdecisionprob1cont() - Go/NoGo/Gray decision probabilities for a single continuous endpoint
• pbayesdecisionprob2bin() - Go/NoGo/Gray decision probabilities for two binary endpoints
• pbayesdecisionprob2cont() - Go/NoGo/Gray decision probabilities for two continuous endpoints

S3 Methods for Decision Probability Classes

• print.pbayesdecisionprob1bin() - Print method for pbayesdecisionprob1bin objects
• print.pbayesdecisionprob1cont() - Print method for pbayesdecisionprob1cont objects
• print.pbayesdecisionprob2bin() - Print method for pbayesdecisionprob2bin objects
• print.pbayesdecisionprob2cont() - Print method for pbayesdecisionprob2cont objects
• plot.pbayesdecisionprob1bin() - Plot method for pbayesdecisionprob1bin objects; displays Go/NoGo/Gray decision probabilities as a line plot across treatment scenarios
• plot.pbayesdecisionprob1cont() - Plot method for pbayesdecisionprob1cont objects; displays Go/NoGo/Gray decision probabilities as a line plot across treatment scenarios
• plot.pbayesdecisionprob2bin() - Plot method for pbayesdecisionprob2bin objects; displays Go/NoGo/Gray decision probabilities as a tile or scatter plot over a two-dimensional treatment scenario grid
• plot.pbayesdecisionprob2cont() - Plot method for pbayesdecisionprob2cont objects; displays Go/NoGo/Gray decision probabilities as a tile or scatter plot over a two-dimensional treatment scenario grid

S3 Methods for Optimal Threshold Search Classes

• plot.getgamma1bin() - Plot method for getgamma1bin objects; displays calibration curves of marginal Go and NoGo probabilities against the threshold grid for a single binary endpoint
• plot.getgamma1cont() - Plot method for getgamma1cont objects; displays calibration curves of marginal Go and NoGo probabilities against the threshold grid for a single continuous endpoint
• plot.getgamma2bin() - Plot method for getgamma2bin objects; displays calibration curves of marginal Go and NoGo probabilities against the threshold grid for two binary endpoints
• plot.getgamma2cont() - Plot method for getgamma2cont objects; displays calibration curves of marginal Go and NoGo probabilities against the threshold grid for two continuous endpoints

Posterior and Predictive Probability Functions

• pbayespostpred1bin() - Posterior or predictive probability for a single binary endpoint
• pbayespostpred1cont() - Posterior or predictive probability for a single continuous endpoint
• pbayespostpred2bin() - Joint region probabilities for two binary endpoints
• pbayespostpred2cont() - Joint region probabilities for two continuous endpoints

Optimal Threshold Search Functions

• getgamma1bin() - Find optimal Go/NoGo thresholds for a single binary endpoint by grid search over candidate gamma values
• getgamma1cont() - Find optimal Go/NoGo thresholds for a single continuous endpoint by grid search over candidate gamma values
• getgamma2bin() - Find optimal Go/NoGo thresholds for two binary endpoints by grid search over candidate gamma value pairs
• getgamma2cont() - Find optimal Go/NoGo thresholds for two continuous endpoints by grid search over candidate gamma value pairs

Distribution Functions for Continuous Endpoints

• ptdiff_NI() - CDF for the difference of two t-distributions via numerical integration
• ptdiff_MC() - CDF for the difference of two t-distributions via Monte Carlo simulation
• ptdiff_MM() - CDF for the difference of two t-distributions via Moment-Matching approximation

Distribution Functions for Binary Endpoints

• pbetadiff() - CDF for the difference of two independent Beta distributions
• pbetabinomdiff() - Beta-binomial posterior predictive probability

Sampling Functions

• rdirichlet() - Random sampler for the Dirichlet distribution

Utility Functions

• getjointbin() - Joint binary probability from marginals and a correlation parameter
• allmultinom() - Enumerate all multinomial outcome combinations

Study Designs

• Controlled design
• Uncontrolled design (hypothetical control)
• External design with power priors

Prior Distributions

• Binary endpoints: Beta and Dirichlet conjugate priors
• Single continuous endpoints: Normal-Inverse-Chi-squared conjugate prior and vague (Jeffreys) prior
• Two continuous endpoints: Normal-Inverse-Wishart conjugate prior and vague prior

Calculation Methods for Continuous Endpoints

• NI (Numerical Integration): exact computation via adaptive quadrature
• MC (Monte Carlo): simulation-based estimation
• MM (Moment-Matching): closed-form approximation; fully vectorised and recommended for large-scale simulation studies