Overview of BayesianQDM

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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

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

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
)
#> [1] 0.941619

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
)
#> [1] 0.093935

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)
#> Go/NoGo/Gray Decision Probabilities (Single Binary Endpoint) 
#> ---------------------------------------------------------------- 
#>   Probability type : posterior 
#>   Design           : controlled 
#>   Threshold(s)     : TV = 0.3, MAV = 0.1 
#>   Go  threshold    : gamma_go = 0.8 
#>   NoGo threshold   : gamma_nogo = 0.2 
#>   Sample size      : n_t = 10, n_c = 10 
#>   Prior (Beta)     : a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5 
#>   Miss handling    : error_if_Miss = TRUE, Gray_inc_Miss = FALSE 
#> ---------------------------------------------------------------- 
#>       pi_t pi_c     Go   Gray   NoGo
#>  0.1500000 0.15 0.0025 0.0532 0.9443
#>  0.2333333 0.15 0.0174 0.1494 0.8332
#>  0.3166667 0.15 0.0609 0.2647 0.6744
#>  0.4000000 0.15 0.1468 0.3569 0.4963
#>  0.4833333 0.15 0.2777 0.3940 0.3283
#>  0.5666667 0.15 0.4426 0.3658 0.1915
#>  0.6500000 0.15 0.6186 0.2860 0.0955
#>  0.7333333 0.15 0.7781 0.1835 0.0384
#>  0.8166667 0.15 0.8984 0.0904 0.0112
#>  0.9000000 0.15 0.9693 0.0289 0.0018
#> ----------------------------------------------------------------
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)

Overview of BayesianQDM

Introduction

Covered Scenarios

Endpoint Type

Probability Metric

Study Design

Prior Distribution

Package Structure and Function Overview

Quick-Start Examples

Posterior Probability: Single Binary Endpoint

Posterior Probability: Single Continuous Endpoint

Go/NoGo Decision: Operating Characteristics

Optimal Threshold Search

Further Reading