Overview of BayesianQDM"

In BayesianQDM: Bayesian Quantitative Decision-Making Framework for Binary and Continuous Endpoints

knitr::opts_chunk$set(
  collapse = TRUE,
  comment  = "#>",
  fig.path = "figures/overview-"
)
library(BayesianQDM)

Introduction

The BayesianQDM package provides a Bayesian Quantitative Decision-Making (QDM) framework for proof-of-concept (PoC) clinical trials. The framework supports a three-zone Go/NoGo/Gray decision rule, in which a Go decision recommends proceeding to the next development phase, a NoGo decision recommends stopping development, and a Gray decision indicates insufficient evidence to make a definitive recommendation.

The primary use case is the evaluation of operating characteristics for a PoC trial under a specified decision rule and sample size. Specifically, the package can answer the following practical questions:

• What is the probability of making a Go, NoGo, or Gray decision under a given true treatment effect?
• What decision thresholds best control the false Go rate and false NoGo rate simultaneously?

---

Covered Scenarios

The package supports a wide range of PoC trial configurations through the combination of the following dimensions.

Endpoint Type

• Binary endpoint: e.g., response rate, where the treatment effect is the difference in response rates $\theta = \pi_t - \pi_c$.
• Continuous endpoint: e.g., a biomarker or clinical score, where the treatment effect is the difference in means $\theta = \mu_t - \mu_c$.

Both single-endpoint and two co-primary endpoint settings are available.

Probability Metric

Two complementary metrics are implemented to quantify evidence from the PoC data:

• Posterior probability: the probability, given the observed PoC data, that the true treatment effect exceeds a clinically meaningful threshold. This metric directly reflects the evidence accumulated in the PoC trial.
• Posterior predictive probability: the probability that a future (e.g., Phase III) trial would achieve a pre-specified success criterion, given the observed PoC data. This metric is forward-looking and accounts for uncertainty in future trial outcomes.

Study Design

Three study designs are supported:

• Controlled design: a standard parallel-group trial with concurrent treatment and control groups.
• Uncontrolled design: a single-arm trial in which the control distribution is specified from prior knowledge rather than observed data.
• External design: a design that incorporates historical or external data for the treatment group, the control group, or both, via a power prior with a user-specified borrowing weight $\alpha_{0e} \in (0, 1]$.

Prior Distribution

• Vague prior (Jeffreys prior): a non-informative prior that allows the observed PoC data to dominate the posterior.
• Conjugate prior: an informative prior specified through hyperparameters, combined with the likelihood in closed form.

---

Package Structure and Function Overview

knitr::kable(data.frame(
  Function = c(
    "pbayespostpred1bin()", "pbayespostpred1cont()",
    "pbayespostpred2bin()", "pbayespostpred2cont()",
    "pbayesdecisionprob1bin()", "pbayesdecisionprob1cont()",
    "pbayesdecisionprob2bin()", "pbayesdecisionprob2cont()",
    "getgamma1bin()", "getgamma1cont()",
    "getgamma2bin()", "getgamma2cont()",
    "pbetadiff()", "pbetabinomdiff()",
    "ptdiff_NI()", "ptdiff_MC()", "ptdiff_MM()",
    "rdirichlet()",
    "getjointbin()", "allmultinom()"
  ),
  Description = c(
    "Posterior/predictive probability, single binary endpoint",
    "Posterior/predictive probability, single continuous endpoint",
    "Region probabilities, two binary endpoints",
    "Region probabilities, two continuous endpoints",
    "Go/NoGo/Gray decision probabilities, single binary endpoint",
    "Go/NoGo/Gray decision probabilities, single continuous endpoint",
    "Go/NoGo/Gray decision probabilities, two binary endpoints",
    "Go/NoGo/Gray decision probabilities, two continuous endpoints",
    "Optimal Go/NoGo thresholds, single binary endpoint",
    "Optimal Go/NoGo thresholds, single continuous endpoint",
    "Optimal Go/NoGo thresholds, two binary endpoints",
    "Optimal Go/NoGo thresholds, two continuous endpoints",
    "CDF of difference of two Beta distributions",
    "Beta-binomial posterior predictive probability",
    "CDF of difference of two t-distributions (numerical integration)",
    "CDF of difference of two t-distributions (Monte Carlo)",
    "CDF of difference of two t-distributions (moment-matching)",
    "Random sampler for the Dirichlet distribution",
    "Joint binary probability from marginals and correlation",
    "Enumerate all multinomial outcome combinations"
  )
), col.names = c("Function", "Description"))

---

Quick-Start Examples

Posterior Probability: Single Binary Endpoint

# P(pi_treat - pi_ctrl > 0.05 | data)
# Observed: 7/10 responders (treatment), 3/10 (control)
pbayespostpred1bin(
  prob = 'posterior', design = 'controlled', theta0 = 0.05,
  n_t = 10, n_c = 10, y_t = 7, y_c = 3,
  a_t = 0.5, b_t = 0.5, a_c = 0.5, b_c = 0.5,
  m_t = NULL, m_c = NULL, z = NULL,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  lower.tail = FALSE
)

Posterior Probability: Single Continuous Endpoint

# P(mu_treat - mu_ctrl > 1.0 | data) using MM method
# Observed: mean 3.5 (treatment), 1.2 (control); SD ~ 2.0
pbayespostpred1cont(
  prob = 'posterior', design = 'controlled', prior = 'vague',
  CalcMethod = 'MM', theta0 = 1.0, nMC = NULL,
  n_t = 10, n_c = 10,
  bar_y_t = 3.5, s_t = 2.0,
  bar_y_c = 1.2, s_c = 2.0,
  m_t = NULL, m_c = NULL,
  kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
  mu0_t = NULL, mu0_c = NULL, sigma0_t = NULL, sigma0_c = NULL,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, se_t = NULL, bar_ye_c = NULL, se_c = NULL
)

Go/NoGo Decision: Operating Characteristics

# Operating characteristics for single binary endpoint
oc_res <- pbayesdecisionprob1bin(
  prob      = 'posterior',
  design    = 'controlled',
  theta_TV  = 0.30,  theta_MAV = 0.10,  theta_NULL = NULL,
  gamma_go  = 0.80,  gamma_nogo = 0.20,
  pi_t      = seq(0.15, 0.9, l = 10),
  pi_c      = rep(0.15, 10),
  n_t = 10,  n_c = 10,
  a_t = 0.5, b_t = 0.5, a_c = 0.5, b_c = 0.5,
  z = NULL, m_t = NULL, m_c = NULL,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)
print(oc_res)
plot(oc_res, base_size = 20)

Optimal Threshold Search

# Find gamma_go  : smallest gamma s.t. Pr(Go)   < 0.05 under Null (pi_t = pi_c = 0.15)
# Find gamma_nogo: smallest gamma s.t. Pr(NoGo) < 0.20 under Alt  (pi_t = 0.35, pi_c = 0.15)
gg_res <- getgamma1bin(
  prob = 'posterior', design = 'controlled',
  theta_TV = 0.30, theta_MAV = 0.10, theta_NULL = NULL,
  pi_t_go = 0.15, pi_c_go = 0.15,
  pi_t_nogo = 0.35, pi_c_nogo = 0.15,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 12L, n_c = 12L,
  a_t = 0.5, b_t = 0.5, a_c = 0.5, b_c = 0.5,
  z = NULL, m_t = NULL, m_c = NULL,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)
plot(gg_res, base_size = 20)

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Further Reading

• vignette("single-binary") — detailed examples for a single binary endpoint, covering all three designs and prior specifications.
• vignette("single-continuous") — detailed examples for a single continuous endpoint, covering all three designs and calculation methods.
• vignette("two-binary") — detailed examples for two binary co-primary endpoints.
• vignette("two-continuous") — detailed examples for two continuous co-primary endpoints.

BayesianQDM documentation built on April 22, 2026, 1:09 a.m.