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Bayesian Posterior or Posterior Predictive Probability for a Single Binary Endpoint

Source: R/pbayespostpred1bin.R
pbayespostpred1bin.Rd

Computes the Bayesian posterior probability or posterior predictive probability for binary-outcome clinical trials under a beta-binomial conjugate model. The function supports controlled, uncontrolled, and external designs, with optional incorporation of external data through power priors. Vector inputs for y_t and y_c are supported for efficient batch processing (e.g., across all possible trial outcomes in pbayesdecisionprob1bin).

Usage

pbayespostpred1bin(
  prob = "posterior",
  design = "controlled",
  theta0,
  n_t,
  n_c,
  y_t,
  y_c = NULL,
  a_t,
  a_c,
  b_t,
  b_c,
  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 = TRUE
)

Arguments

prob

A character string specifying the probability type. Must be 'posterior' or 'predictive'.

design

A character string specifying the trial design. Must be 'controlled', 'uncontrolled', or 'external'.

theta0

A numeric scalar in (-1, 1) giving the threshold for the treatment effect (difference in response rates).

n_t

A positive integer giving the number of patients in the treatment group in the proof-of-concept (PoC) trial.

n_c

A positive integer giving the number of patients in the control group in the PoC trial. For design = 'uncontrolled', this is the hypothetical control sample size (required for consistency with other designs).

y_t

A non-negative integer or integer vector giving the observed number of responders in the treatment group (must satisfy 0 <= y_t <= n_t).

y_c

A non-negative integer or integer vector giving the number of responders in the control group (must satisfy 0 <= y_c <= n_c). Set to NULL for design = 'uncontrolled' and use z instead. When provided as a vector, must have the same length as y_t.

a_t

A positive numeric scalar giving the first shape parameter (alpha) of the prior Beta distribution for the treatment group.

a_c

A positive numeric scalar giving the first shape parameter (alpha) of the prior Beta distribution for the control group.

b_t

A positive numeric scalar giving the second shape parameter (beta) of the prior Beta distribution for the treatment group.

b_c

A positive numeric scalar giving the second shape parameter (beta) of the prior Beta distribution for the control group.

m_t

A positive integer giving the number of patients in the treatment group for the future trial. Required when prob = 'predictive'; otherwise set to NULL.

m_c

A positive integer giving the number of patients in the control group for the future trial. Required when prob = 'predictive'; otherwise set to NULL.

z

A non-negative integer giving the hypothetical number of responders in the control group. Required when design = 'uncontrolled'; otherwise set to NULL. When used, y_c should be NULL.

ne_t

A positive integer giving the number of patients in the treatment group of the external data set. Required when design = 'external' and external treatment data are available; otherwise set to NULL.

ne_c

A positive integer giving the number of patients in the control group of the external data set. Required when design = 'external' and external control data are available; otherwise set to NULL.

ye_t

A non-negative integer giving the number of responders in the treatment group of the external data set. Required when design = 'external'; otherwise set to NULL.

ye_c

A non-negative integer giving the number of responders in the control group of the external data set. Required when design = 'external'; otherwise set to NULL.

alpha0e_t

A numeric scalar in (0, 1] giving the power prior weight for the external treatment data. Required when external treatment data are used; otherwise set to NULL.

alpha0e_c

A numeric scalar in (0, 1] giving the power prior weight for the external control data. Required when external control data are used; otherwise set to NULL.

lower.tail

A logical scalar; if TRUE (default), the function returns \(P(\mathrm{effect} \le \theta_0)\), otherwise \(P(\mathrm{effect} > \theta_0)\).

Value

A numeric scalar or vector in [0, 1]. When y_t and y_c are vectors of length \(n\), a vector of length \(n\) is returned.

Details

Posterior shape parameters are computed from the beta-binomial conjugate model:

  • Prior: \(\pi_j \sim \mathrm{Beta}(a_j, b_j)\).

  • Posterior: \(\pi_j \mid y_j \sim \mathrm{Beta}(a_j + y_j, b_j + n_j - y_j)\).

For design = 'external', external data are incorporated through the power prior, which inflates the prior by the weighted external sufficient statistics: $$(\alpha_j^*, \beta_j^*) = (a_j + ae_j \cdot ye_j, b_j + ae_j \cdot (ne_j - ye_j)).$$

For design = 'uncontrolled', the hypothetical control value z is used in place of y_c, and n_c acts as the hypothetical control sample size.

The final probability is obtained by calling pbetadiff for prob = 'posterior' or pbetabinomdiff for prob = 'predictive', applied element-wise via mapply.

Examples

# Example 1: Controlled design - posterior probability
pbayespostpred1bin(
  prob = 'posterior', design = 'controlled', theta0 = 0.15,
  n_t = 12, n_c = 15, y_t = 7, y_c = 5,
  a_t = 0.5, a_c = 0.5, b_t = 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] 0.6860634

# Example 2: Uncontrolled design - posterior probability
pbayespostpred1bin(
  prob = 'posterior', design = 'uncontrolled', theta0 = 0.20,
  n_t = 20, n_c = 20, y_t = 12, y_c = NULL,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  m_t = NULL, m_c = NULL, z = 3,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL, lower.tail = FALSE
)
#> [1] 0.952403

# Example 3: External design - posterior probability
pbayespostpred1bin(
  prob = 'posterior', design = 'external', theta0 = 0.15,
  n_t = 12, n_c = 15, y_t = 7, y_c = 9,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  m_t = NULL, m_c = NULL, z = NULL,
  ne_t = 12, ne_c = 12, ye_t = 6, ye_c = 6, alpha0e_t = 0.5, alpha0e_c = 0.5,
  lower.tail = FALSE
)
#> [1] 0.1399401

# Example 4: Controlled design - posterior predictive probability
pbayespostpred1bin(
  prob = 'predictive', design = 'controlled', theta0 = 0.10,
  n_t = 12, n_c = 15, y_t = 7, y_c = 5,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  m_t = 30, m_c = 30, z = NULL,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL, lower.tail = FALSE
)
#> [1] 0.710533

# Example 5: Uncontrolled design - posterior predictive probability
pbayespostpred1bin(
  prob = 'predictive', design = 'uncontrolled', theta0 = 0.20,
  n_t = 20, n_c = 20, y_t = 12, y_c = NULL,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  m_t = 30, m_c = 30, z = 3,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL, lower.tail = FALSE
)
#> [1] 0.8874071

# Example 6: External design - posterior predictive probability
pbayespostpred1bin(
  prob = 'predictive', design = 'external', theta0 = 0.15,
  n_t = 12, n_c = 15, y_t = 7, y_c = 9,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  m_t = 30, m_c = 30, z = NULL,
  ne_t = 12, ne_c = 12, ye_t = 6, ye_c = 6, alpha0e_t = 0.5, alpha0e_c = 0.5,
  lower.tail = FALSE
)
#> [1] 0.2027863