Enumerate All Multinomial Count Vectors for a Bivariate Binary Outcome

Description

Generates the complete set of non-negative integer vectors (x_{00}, x_{01}, x_{10}, x_{11}) satisfying x_{00} + x_{01} + x_{10} + x_{11} = n_j, where n_j is the sample size of group j \in \{t, c\}. Each row of the returned matrix corresponds to one possible realisation of the aggregated bivariate binary counts. This enumeration is a prerequisite for exact operating characteristic assessment in pbayesdecisionprob2bin.

Usage

allmultinom(n)

Arguments

Details

The number of non-negative integer solutions to x_{00} + x_{01} + x_{10} + x_{11} = n_j is the stars-and-bars count

\binom{n_j + 3}{3} = \frac{(n_j + 1)(n_j + 2)(n_j + 3)}{6}.

For example, n_j = 10 yields 286 rows and n_j = 20 yields 1771 rows. The matrix is pre-allocated to avoid repeated memory reallocation, making the function efficient for the sample sizes typical in rare-disease proof-of-concept studies.

This function is an internal computational building block used by pbayesdecisionprob2bin to enumerate the sample space over which multinomial probabilities and decision indicators are summed when computing exact operating characteristics.

Value

An integer matrix with \binom{n_j + 3}{3} rows and 4 columns named x00, x01, x10, x11. Each row is a distinct non-negative integer solution to x_{00} + x_{01} + x_{10} + x_{11} = n_j. Rows are ordered by x00 (ascending), then x10 (ascending), then x01 (ascending).

Examples

# Example 1: n = 2 (smallest non-trivial case)
allmultinom(2)

# Example 2: n = 10 (typical rare-disease PoC group size)
mat <- allmultinom(10)
nrow(mat)              # Should be choose(13, 3) = 286
all(rowSums(mat) == 10)  # Every row must sum to n

# Example 3: n = 0 (edge case - only the all-zero row)
allmultinom(0)

# Example 4: Verify column names
colnames(allmultinom(5))

# Example 5: Row counts match the stars-and-bars formula
n <- 15L
mat <- allmultinom(n)
nrow(mat) == choose(n + 3L, 3L)  # Should be TRUE

Find Optimal Go/NoGo Thresholds for a Single Binary Endpoint

Description

Computes the optimal Go threshold \gamma_{\mathrm{go}} and NoGo threshold \gamma_{\mathrm{nogo}} for a single binary endpoint by searching over a grid of candidate values. The two thresholds are calibrated independently under separate scenarios:

• \gamma_{\mathrm{go}} is the smallest value in gamma_grid such that the marginal Go probability \Pr(g_{\mathrm{Go}} \ge \gamma_{\mathrm{go}}) is strictly less than target_go under the Go-calibration scenario (pi_t_go, pi_c_go); typically the Null scenario.

• \gamma_{\mathrm{nogo}} is the smallest value in gamma_grid such that the marginal NoGo probability \Pr(g_{\mathrm{NoGo}} \ge \gamma_{\mathrm{nogo}}) is strictly less than target_nogo under the NoGo-calibration scenario (pi_t_nogo, pi_c_nogo); typically the Alternative scenario.

Here g_{\mathrm{Go}} = P(\theta > \theta_{\mathrm{TV}} \mid y_t, y_c) and g_{\mathrm{NoGo}} = P(\theta \le \theta_{\mathrm{MAV}} \mid y_t, y_c) for prob = 'posterior', consistent with the decision rule in pbayesdecisionprob1bin.

Usage

getgamma1bin(
  prob = "posterior",
  design = "controlled",
  theta_TV = NULL,
  theta_MAV = NULL,
  theta_NULL = NULL,
  pi_t_go,
  pi_c_go = NULL,
  pi_t_nogo,
  pi_c_nogo = NULL,
  target_go,
  target_nogo,
  n_t,
  n_c,
  a_t,
  a_c,
  b_t,
  b_c,
  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,
  gamma_grid = seq(0.01, 0.99, by = 0.01)
)

Arguments

Details

The function uses a two-stage precompute-then-sweep strategy:

1. Precomputation: All possible outcome pairs (y_t, y_c) are enumerated. For each pair, pbayespostpred1bin computes g_{\mathrm{Go}} (lower.tail = FALSE at theta_TV) and g_{\mathrm{NoGo}} (lower.tail = TRUE at theta_MAV). This step is independent of \gamma.

2. Gamma sweep: Marginal probabilities are computed as weighted sums of binary indicators over the grid: \Pr(\mathrm{Go}) uses w_go (weights under pi_t_go, pi_c_go) and the indicator g_{\mathrm{Go}} \ge \gamma; \Pr(\mathrm{NoGo}) uses w_nogo (weights under pi_t_nogo, pi_c_nogo) and the indicator g_{\mathrm{NoGo}} \ge \gamma.

Both \Pr(\mathrm{Go}) and \Pr(\mathrm{NoGo}) are monotone non-increasing functions of \gamma. The optimal \gamma_{\mathrm{go}} is the smallest grid value crossing below target_go. The optimal \gamma_{\mathrm{nogo}} is also the smallest grid value crossing below target_nogo: a smaller \gamma_{\mathrm{nogo}} makes NoGo harder to trigger (more permissive), so this is the least restrictive threshold that still controls the false NoGo rate.

Value

A list of class getgamma1bin with the following elements:

gamma_go

Optimal Go threshold: the smallest value in gamma_grid for which \Pr(\mathrm{Go}) < \code{target\_go} under the Go-calibration scenario. NA if no such value exists.

gamma_nogo

Optimal NoGo threshold: the smallest value in gamma_grid for which \Pr(\mathrm{NoGo}) < \code{target\_nogo} under the NoGo-calibration scenario. NA if no such value exists.

PrGo_opt

Marginal \Pr(g_{\mathrm{Go}} \ge \gamma_{\mathrm{go}}) at the optimal \gamma_{\mathrm{go}} under the Go-calibration scenario. NA if gamma_go is NA.

PrNoGo_opt

Marginal \Pr(g_{\mathrm{NoGo}} \ge \gamma_{\mathrm{nogo}}) at the optimal \gamma_{\mathrm{nogo}} under the NoGo-calibration scenario. NA if gamma_nogo is NA.

target_go

The value of target_go supplied by the user.

target_nogo

The value of target_nogo supplied by the user.

grid_results

A data frame with columns gamma_grid, PrGo_grid (marginal Go probability under the Go-calibration scenario), and PrNoGo_grid (marginal NoGo probability under the NoGo-calibration scenario).

Examples

# Example 1: Controlled design, posterior probability
# gamma_go  : smallest gamma s.t. Pr(Go)   < 0.05 under Null (pi_t = pi_c = 0.15)
# gamma_nogo: largest  gamma s.t. Pr(NoGo) < 0.20 under Alt  (pi_t = 0.35, pi_c = 0.15)
getgamma1bin(
  prob = 'posterior', design = 'controlled',
  theta_TV = 0.20, theta_MAV = 0.05, 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, a_c = 0.5, b_t = 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
)

# Example 2: Uncontrolled design, posterior probability
getgamma1bin(
  prob = 'posterior', design = 'uncontrolled',
  theta_TV = 0.20, theta_MAV = 0.05, theta_NULL = NULL,
  pi_t_go = 0.15, pi_c_go = NULL,
  pi_t_nogo = 0.35, pi_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 12L, n_c = 12L,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = 3L, m_t = NULL, m_c = NULL,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 3: External design, posterior probability
getgamma1bin(
  prob = 'posterior', design = 'external',
  theta_TV = 0.20, theta_MAV = 0.05, 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, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = NULL, m_t = NULL, m_c = NULL,
  ne_t = 15L, ne_c = 15L, ye_t = 6L, ye_c = 4L,
  alpha0e_t = 0.5, alpha0e_c = 0.5
)

# Example 4: Controlled design, predictive probability
getgamma1bin(
  prob = 'predictive', design = 'controlled',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 0.10,
  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, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = NULL, m_t = 30L, m_c = 30L,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 5: Uncontrolled design, predictive probability
getgamma1bin(
  prob = 'predictive', design = 'uncontrolled',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 0.10,
  pi_t_go = 0.15, pi_c_go = NULL,
  pi_t_nogo = 0.35, pi_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 12L, n_c = 12L,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = 3L, m_t = 30L, m_c = 30L,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 6: External design, predictive probability
getgamma1bin(
  prob = 'predictive', design = 'external',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 0.10,
  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, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = NULL, m_t = 30L, m_c = 30L,
  ne_t = 15L, ne_c = 15L, ye_t = 6L, ye_c = 4L,
  alpha0e_t = 0.5, alpha0e_c = 0.5
)

Find Optimal Go/NoGo Thresholds for a Single Continuous Endpoint

Description

Computes the optimal Go threshold \gamma_{\mathrm{go}} and NoGo threshold \gamma_{\mathrm{nogo}} for a single continuous endpoint by searching over a grid of candidate values. The two thresholds are calibrated independently under separate scenarios:

• \gamma_{\mathrm{go}} is the smallest value in gamma_grid such that the marginal Go probability \Pr(g_{\mathrm{Go}} \ge \gamma_{\mathrm{go}}) is strictly less than target_go under the Go-calibration scenario (mu_t_go, mu_c_go, sigma_t_go, sigma_c_go); typically the Null scenario.

• \gamma_{\mathrm{nogo}} is the smallest value in gamma_grid such that the marginal NoGo probability \Pr(g_{\mathrm{NoGo}} \ge \gamma_{\mathrm{nogo}}) is strictly less than target_nogo under the NoGo-calibration scenario (mu_t_nogo, mu_c_nogo, sigma_t_nogo, sigma_c_nogo); typically the Alternative scenario.

Here g_{\mathrm{Go}} = P(\theta > \theta_{\mathrm{TV}} \mid \bar{y}_t, s_t, \bar{y}_c, s_c) and g_{\mathrm{NoGo}} = P(\theta \le \theta_{\mathrm{MAV}} \mid \bar{y}_t, s_t, \bar{y}_c, s_c) for prob = 'posterior', consistent with the decision rule in pbayesdecisionprob1cont.

Usage

getgamma1cont(
  nsim,
  prob = "posterior",
  design = "controlled",
  prior = "vague",
  CalcMethod = "NI",
  theta_TV = NULL,
  theta_MAV = NULL,
  theta_NULL = NULL,
  nMC = NULL,
  mu_t_go,
  mu_c_go = NULL,
  sigma_t_go,
  sigma_c_go = NULL,
  mu_t_nogo,
  mu_c_nogo = NULL,
  sigma_t_nogo,
  sigma_c_nogo = NULL,
  target_go,
  target_nogo,
  n_t,
  n_c = NULL,
  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,
  bar_ye_c = NULL,
  se_t = NULL,
  se_c = NULL,
  gamma_grid = seq(0.01, 0.99, by = 0.01),
  seed
)

Arguments

Details

The function uses a two-stage simulate-then-sweep strategy:

1. Simulation: nsim datasets are generated independently for each calibration scenario. For the Go-calibration scenario, standardised residuals are shifted by mu_t_go and mu_c_go; for the NoGo-calibration scenario, by mu_t_nogo and mu_c_nogo. pbayespostpred1cont is called once per scenario to obtain g_{\mathrm{Go}} (lower.tail = FALSE at theta_TV) and g_{\mathrm{NoGo}} (lower.tail = TRUE at theta_MAV).

2. Gamma sweep: Marginal probabilities are estimated as the proportion of simulated datasets satisfying the respective indicator: g_{\mathrm{Go}} \ge \gamma for PrGo_grid, and g_{\mathrm{NoGo}} \ge \gamma for PrNoGo_grid. Both are monotone non-increasing in \gamma.

The optimal \gamma_{\mathrm{go}} and \gamma_{\mathrm{nogo}} are each the smallest grid value crossing below the respective target probability.

Value

A list of class getgamma1cont with the following elements:

gamma_go

Optimal Go threshold: the smallest value in gamma_grid for which \Pr(\mathrm{Go}) < \code{target\_go} under the Go-calibration scenario. NA if no such value exists.

gamma_nogo

Optimal NoGo threshold: the smallest value in gamma_grid for which \Pr(\mathrm{NoGo}) < \code{target\_nogo} under the NoGo-calibration scenario. NA if no such value exists.

PrGo_opt

Marginal \Pr(g_{\mathrm{Go}} \ge \gamma_{\mathrm{go}}) at the optimal \gamma_{\mathrm{go}} under the Go-calibration scenario. NA if gamma_go is NA.

PrNoGo_opt

Marginal \Pr(g_{\mathrm{NoGo}} \ge \gamma_{\mathrm{nogo}}) at the optimal \gamma_{\mathrm{nogo}} under the NoGo-calibration scenario. NA if gamma_nogo is NA.

target_go

The value of target_go supplied by the user.

target_nogo

The value of target_nogo supplied by the user.

grid_results

A data frame with columns gamma_grid, PrGo_grid (marginal Go probability under the Go-calibration scenario), and PrNoGo_grid (marginal NoGo probability under the NoGo-calibration scenario).

Examples

# Example 1: Controlled design, vague prior, posterior probability
# gamma_go  : smallest gamma s.t. Pr(Go)   < 0.05 under Null (mu_t = mu_c = 1.0)
# gamma_nogo: smallest gamma s.t. Pr(NoGo) < 0.20 under Alt  (mu_t = 2.5, mu_c = 1.0)
getgamma1cont(
  nsim = 1000L, prob = 'posterior', design = 'controlled',
  prior = 'vague', CalcMethod = 'MM',
  theta_TV = 1.5, theta_MAV = 0.0, theta_NULL = NULL, nMC = NULL,
  mu_t_go = 1.0, mu_c_go = 1.0, sigma_t_go = 2.0, sigma_c_go = 2.0,
  mu_t_nogo = 2.5, mu_c_nogo = 1.0, sigma_t_nogo = 2.0, sigma_c_nogo = 2.0,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 15L, n_c = 15L, 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, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  gamma_grid = seq(0.01, 0.99, by = 0.01), seed = 1L
)

# Example 2: Uncontrolled design, N-Inv-Chisq prior, posterior probability
getgamma1cont(
  nsim = 1000L, prob = 'posterior', design = 'uncontrolled',
  prior = 'N-Inv-Chisq', CalcMethod = 'NI',
  theta_TV = 1.0, theta_MAV = 0.0, theta_NULL = NULL, nMC = NULL,
  mu_t_go = 1.5, mu_c_go = NULL, sigma_t_go = 1.5, sigma_c_go = NULL,
  mu_t_nogo = 3.0, mu_c_nogo = NULL, sigma_t_nogo = 1.5, sigma_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 20L, n_c = NULL, m_t = NULL, m_c = NULL,
  kappa0_t = 2, kappa0_c = NULL, nu0_t = 5, nu0_c = NULL,
  mu0_t = 3.0, mu0_c = 1.5, sigma0_t = 1.5, sigma0_c = NULL,
  r = 1.0, ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  gamma_grid = seq(0.01, 0.99, by = 0.01), seed = 2L
)

# Example 3: External design, vague prior, posterior probability
getgamma1cont(
  nsim = 1000L, prob = 'posterior', design = 'external',
  prior = 'vague', CalcMethod = 'MM',
  theta_TV = 1.0, theta_MAV = 0.0, theta_NULL = NULL, nMC = NULL,
  mu_t_go = 1.0, mu_c_go = 1.0, sigma_t_go = 1.5, sigma_c_go = 1.5,
  mu_t_nogo = 2.5, mu_c_nogo = 1.0, sigma_t_nogo = 1.5, sigma_c_nogo = 1.5,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 15L, n_c = 15L, 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 = 20L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = 0.0, se_t = NULL, se_c = 1.5,
  gamma_grid = seq(0.01, 0.99, by = 0.01), seed = 4L
)

# Example 4: Controlled design, vague prior, predictive probability
getgamma1cont(
  nsim = 1000L, prob = 'predictive', design = 'controlled',
  prior = 'vague', CalcMethod = 'MM',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 1.0, nMC = NULL,
  mu_t_go = 1.0, mu_c_go = 1.0, sigma_t_go = 2.0, sigma_c_go = 2.0,
  mu_t_nogo = 2.5, mu_c_nogo = 1.0, sigma_t_nogo = 2.0, sigma_c_nogo = 2.0,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 15L, n_c = 15L, m_t = 50L, m_c = 50L,
  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, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  gamma_grid = seq(0.01, 0.99, by = 0.01), seed = 3L
)

# Example 5: Uncontrolled design, vague prior, predictive probability
getgamma1cont(
  nsim = 1000L, prob = 'predictive', design = 'uncontrolled',
  prior = 'vague', CalcMethod = 'MM',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 1.0, nMC = NULL,
  mu_t_go = 1.0, mu_c_go = NULL, sigma_t_go = 2.0, sigma_c_go = NULL,
  mu_t_nogo = 2.5, mu_c_nogo = NULL, sigma_t_nogo = 2.0, sigma_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 15L, n_c = NULL, m_t = 50L, m_c = 50L,
  kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
  mu0_t = NULL, mu0_c = 0.0, sigma0_t = NULL, sigma0_c = NULL,
  r = 1.0, ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  gamma_grid = seq(0.01, 0.99, by = 0.01), seed = 5L
)

# Example 6: External design, vague prior, predictive probability
getgamma1cont(
  nsim = 1000L, prob = 'predictive', design = 'external',
  prior = 'vague', CalcMethod = 'MM',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 1.0, nMC = NULL,
  mu_t_go = 1.0, mu_c_go = 1.0, sigma_t_go = 1.5, sigma_c_go = 1.5,
  mu_t_nogo = 2.5, mu_c_nogo = 1.0, sigma_t_nogo = 1.5, sigma_c_nogo = 1.5,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 15L, n_c = 15L, m_t = 50L, m_c = 50L,
  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 = 20L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = 0.0, se_t = NULL, se_c = 1.5,
  gamma_grid = seq(0.01, 0.99, by = 0.01), seed = 6L
)

Find Optimal Go/NoGo Thresholds for Two Binary Endpoints

Description

Computes the optimal Go threshold \gamma_{\mathrm{go}} and NoGo threshold \gamma_{\mathrm{nogo}} for two binary endpoints by searching over a two-dimensional grid of candidate value pairs. The two thresholds are calibrated independently under separate scenarios:

• \gamma_{\mathrm{go}} is the smallest value in gamma_go_grid such that the worst-case marginal Go probability over all \gamma_{\mathrm{nogo}} in gamma_nogo_grid is strictly less than target_go under the Go-calibration scenario (pi_t1_go, pi_t2_go, rho_t_go, pi_c1_go, pi_c2_go, rho_c_go); typically the Null scenario.

• \gamma_{\mathrm{nogo}} is the smallest value in gamma_nogo_grid such that the worst-case marginal NoGo probability over all \gamma_{\mathrm{go}} in gamma_go_grid is strictly less than target_nogo under the NoGo-calibration scenario (pi_t1_nogo, pi_t2_nogo, rho_t_nogo, pi_c1_nogo, pi_c2_nogo, rho_c_nogo); typically the Alternative scenario.

Usage

getgamma2bin(
  prob = "posterior",
  design = "controlled",
  GoRegions,
  NoGoRegions,
  pi_t1_go,
  pi_t2_go,
  rho_t_go,
  pi_c1_go = NULL,
  pi_c2_go = NULL,
  rho_c_go = NULL,
  pi_t1_nogo,
  pi_t2_nogo,
  rho_t_nogo,
  pi_c1_nogo = NULL,
  pi_c2_nogo = NULL,
  rho_c_nogo = NULL,
  target_go,
  target_nogo,
  n_t,
  n_c,
  a_t_00,
  a_t_01,
  a_t_10,
  a_t_11,
  a_c_00,
  a_c_01,
  a_c_10,
  a_c_11,
  theta_TV1 = NULL,
  theta_MAV1 = NULL,
  theta_TV2 = NULL,
  theta_MAV2 = NULL,
  theta_NULL1 = NULL,
  theta_NULL2 = NULL,
  m_t = NULL,
  m_c = NULL,
  z00 = NULL,
  z01 = NULL,
  z10 = NULL,
  z11 = NULL,
  xe_t_00 = NULL,
  xe_t_01 = NULL,
  xe_t_10 = NULL,
  xe_t_11 = NULL,
  xe_c_00 = NULL,
  xe_c_01 = NULL,
  xe_c_10 = NULL,
  xe_c_11 = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  nMC = 1000L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)

Arguments

Details

The function uses the same two-stage precompute-then-sweep strategy as pbayesdecisionprob2bin.

Stage 1 (precomputation): pbayespostpred2bin is called for every possible multinomial outcome combination (x_t, x_c) enumerated by allmultinom. The resulting region probability vector is summed over GoRegions and NoGoRegions to obtain \hat{g}_{Go,ij} and \hat{g}_{NoGo,ij}. These are independent of the calibration scenario; only the multinomial weights differ.

Stage 2 (gamma sweep): For each pair (\gamma_{\mathrm{go}}, \gamma_{\mathrm{nogo}}) in the two-dimensional grid, operating characteristics are computed separately under each calibration scenario using the respective multinomial weights:

\Pr(\mathrm{Go}) = \sum_{i,j} w_{ij}^{(\mathrm{go})} \mathbf{1}\!\left[\hat{g}_{Go,ij} \ge \gamma_{\mathrm{go}},\; \hat{g}_{NoGo,ij} < \gamma_{\mathrm{nogo}}\right]

\Pr(\mathrm{NoGo}) = \sum_{i,j} w_{ij}^{(\mathrm{nogo})} \mathbf{1}\!\left[\hat{g}_{NoGo,ij} \ge \gamma_{\mathrm{nogo}},\; \hat{g}_{Go,ij} < \gamma_{\mathrm{go}}\right]

Stage 3 (optimal threshold selection): For each candidate \gamma_{\mathrm{go}}, the worst-case \Pr(\mathrm{Go}) over all \gamma_{\mathrm{nogo}} in gamma_nogo_grid is computed; the optimal \gamma_{\mathrm{go}} is the smallest grid value for which this worst-case probability is less than target_go. Analogously, the optimal \gamma_{\mathrm{nogo}} is the smallest grid value for which the worst-case \Pr(\mathrm{NoGo}) is less than target_nogo.

Value

A list of class getgamma2bin with the following elements:

gamma_go

Optimal Go threshold: the smallest value in gamma_go_grid for which the marginal \Pr(\mathrm{Go}) < \code{target\_go} under the Go-calibration scenario. NA if no such value exists.

gamma_nogo

Optimal NoGo threshold: the smallest value in gamma_nogo_grid for which the marginal \Pr(\mathrm{NoGo}) < \code{target\_nogo} under the NoGo-calibration scenario. NA if no such value exists.

PrGo_opt

Marginal \Pr(\mathrm{Go}) at gamma_go under the Go-calibration scenario. NA if gamma_go is NA.

PrNoGo_opt

Marginal \Pr(\mathrm{NoGo}) at gamma_nogo under the NoGo-calibration scenario. NA if gamma_nogo is NA.

target_go

The value of target_go supplied by the user.

target_nogo

The value of target_nogo supplied by the user.

grid_results

A data frame with columns gamma_grid, PrGo_grid (marginal Go probability under the Go-calibration scenario), and PrNoGo_grid (marginal NoGo probability under the NoGo-calibration scenario).

Examples

# Example 1: Controlled design, posterior probability
# gamma_go  : smallest gamma_go   s.t. max_{gamma_nogo} Pr(Go)   < 0.05 under Null
# gamma_nogo: smallest gamma_nogo s.t. max_{gamma_go}   Pr(NoGo) < 0.20 under Alt

getgamma2bin(
  prob = 'posterior', design = 'controlled',
  GoRegions = 1L, NoGoRegions = 9L,
  pi_t1_go = 0.15, pi_t2_go = 0.20, rho_t_go = 0.0,
  pi_c1_go = 0.15, pi_c2_go = 0.20, rho_c_go = 0.0,
  pi_t1_nogo = 0.35, pi_t2_nogo = 0.40, rho_t_nogo = 0.0,
  pi_c1_nogo = 0.15, pi_c2_nogo = 0.20, rho_c_nogo = 0.0,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 7L, n_c = 7L,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  theta_TV1   = 0.15, theta_MAV1 = 0.10,
  theta_TV2   = 0.15, theta_MAV2 = 0.10,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  m_t = NULL, m_c = NULL,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)


# Example 2: Uncontrolled design, posterior probability

getgamma2bin(
  prob = 'posterior', design = 'uncontrolled',
  GoRegions = 1L, NoGoRegions = 9L,
  pi_t1_go = 0.15, pi_t2_go = 0.20, rho_t_go = 0.0,
  pi_c1_go = NULL, pi_c2_go = NULL, rho_c_go = NULL,
  pi_t1_nogo = 0.35, pi_t2_nogo = 0.40, rho_t_nogo = 0.0,
  pi_c1_nogo = NULL, pi_c2_nogo = NULL, rho_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 7L, n_c = 7L,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  theta_TV1   = 0.15, theta_MAV1 = 0.10,
  theta_TV2   = 0.15, theta_MAV2 = 0.10,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  m_t = NULL, m_c = NULL,
  z00 = 3L, z01 = 2L, z10 = 3L, z11 = 2L,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)


# Example 3: External design, posterior probability

getgamma2bin(
  prob = 'posterior', design = 'external',
  GoRegions = 1L, NoGoRegions = 9L,
  pi_t1_go = 0.15, pi_t2_go = 0.20, rho_t_go = 0.0,
  pi_c1_go = 0.15, pi_c2_go = 0.20, rho_c_go = 0.0,
  pi_t1_nogo = 0.35, pi_t2_nogo = 0.40, rho_t_nogo = 0.0,
  pi_c1_nogo = 0.15, pi_c2_nogo = 0.20, rho_c_nogo = 0.0,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 7L, n_c = 7L,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  theta_TV1   = 0.15, theta_MAV1 = 0.10,
  theta_TV2   = 0.15, theta_MAV2 = 0.10,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  m_t = NULL, m_c = NULL,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = 3L, xe_c_01 = 2L, xe_c_10 = 3L, xe_c_11 = 2L,
  alpha0e_t = NULL, alpha0e_c = 0.5,
  nMC = 100L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)


# Example 4: Controlled design, predictive probability

getgamma2bin(
  prob = 'predictive', design = 'controlled',
  GoRegions = 1L, NoGoRegions = 4L,
  pi_t1_go = 0.15, pi_t2_go = 0.20, rho_t_go = 0.0,
  pi_c1_go = 0.15, pi_c2_go = 0.20, rho_c_go = 0.0,
  pi_t1_nogo = 0.35, pi_t2_nogo = 0.40, rho_t_nogo = 0.0,
  pi_c1_nogo = 0.15, pi_c2_nogo = 0.20, rho_c_nogo = 0.0,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 7L, n_c = 7L,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  theta_TV1   = NULL, theta_MAV1 = NULL,
  theta_TV2   = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.10, theta_NULL2 = 0.10,
  m_t = 5L, m_c = 5L,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)


# Example 5: Uncontrolled design, predictive probability

getgamma2bin(
  prob = 'predictive', design = 'uncontrolled',
  GoRegions = 1L, NoGoRegions = 4L,
  pi_t1_go = 0.15, pi_t2_go = 0.20, rho_t_go = 0.0,
  pi_c1_go = NULL, pi_c2_go = NULL, rho_c_go = NULL,
  pi_t1_nogo = 0.35, pi_t2_nogo = 0.40, rho_t_nogo = 0.0,
  pi_c1_nogo = NULL, pi_c2_nogo = NULL, rho_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 7L, n_c = 7L,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  theta_TV1   = NULL, theta_MAV1 = NULL,
  theta_TV2   = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.10, theta_NULL2 = 0.10,
  m_t = 5L, m_c = 5L,
  z00 = 3L, z01 = 2L, z10 = 3L, z11 = 2L,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)


# Example 6: External design, predictive probability

getgamma2bin(
  prob = 'predictive', design = 'external',
  GoRegions = 1L, NoGoRegions = 4L,
  pi_t1_go = 0.15, pi_t2_go = 0.20, rho_t_go = 0.0,
  pi_c1_go = 0.15, pi_c2_go = 0.20, rho_c_go = 0.0,
  pi_t1_nogo = 0.35, pi_t2_nogo = 0.40, rho_t_nogo = 0.0,
  pi_c1_nogo = 0.15, pi_c2_nogo = 0.20, rho_c_nogo = 0.0,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 7L, n_c = 7L,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  theta_TV1   = NULL, theta_MAV1 = NULL,
  theta_TV2   = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.10, theta_NULL2 = 0.10,
  m_t = 5L, m_c = 5L,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = 3L, xe_t_01 = 2L, xe_t_10 = 3L, xe_t_11 = 2L,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = 0.5, alpha0e_c = NULL,
  nMC = 100L,
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01)
)

Find Optimal Go/NoGo Thresholds for Two Continuous Endpoints

Description

Computes the optimal Go threshold \gamma_{\mathrm{go}} and NoGo threshold \gamma_{\mathrm{nogo}} for two continuous endpoints by searching independently over candidate threshold grids. The two thresholds are calibrated marginally under separate scenarios:

• \gamma_{\mathrm{go}} is the smallest value in gamma_go_grid such that the marginal Go probability is strictly less than target_go under the Go-calibration scenario (mu_t_go, Sigma_t_go, mu_c_go, Sigma_c_go); typically the Null scenario.

• \gamma_{\mathrm{nogo}} is the smallest value in gamma_nogo_grid such that the marginal NoGo probability is strictly less than target_nogo under the NoGo-calibration scenario (mu_t_nogo, Sigma_t_nogo, mu_c_nogo, Sigma_c_nogo); typically the Alternative scenario.

Usage

getgamma2cont(
  nsim = 10000L,
  prob = "posterior",
  design = "controlled",
  prior = "vague",
  GoRegions,
  NoGoRegions,
  mu_t_go,
  Sigma_t_go,
  mu_c_go = NULL,
  Sigma_c_go = NULL,
  mu_t_nogo,
  Sigma_t_nogo,
  mu_c_nogo = NULL,
  Sigma_c_nogo = NULL,
  target_go,
  target_nogo,
  n_t,
  n_c = NULL,
  theta_TV1 = NULL,
  theta_MAV1 = NULL,
  theta_TV2 = NULL,
  theta_MAV2 = NULL,
  theta_NULL1 = NULL,
  theta_NULL2 = NULL,
  m_t = NULL,
  m_c = NULL,
  kappa0_t = NULL,
  nu0_t = NULL,
  mu0_t = NULL,
  Lambda0_t = NULL,
  kappa0_c = NULL,
  nu0_c = NULL,
  mu0_c = NULL,
  Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL,
  ne_c = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  bar_ye_t = NULL,
  bar_ye_c = NULL,
  se_t = NULL,
  se_c = NULL,
  nMC = NULL,
  CalcMethod = "MC",
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed
)

Arguments

Details

The function uses a two-stage simulate-then-sweep strategy:

Stage 1 (simulation and precomputation): nsim bivariate datasets are generated independently for each calibration scenario. For the Go-calibration scenario, datasets are drawn from N_2(\mu_{t,\mathrm{go}}, \Sigma_{t,\mathrm{go}}) (and N_2(\mu_{c,\mathrm{go}}, \Sigma_{c,\mathrm{go}}) for controlled/external designs); for the NoGo-calibration scenario, the corresponding _nogo parameters are used. pbayespostpred2cont is called once per scenario in vectorised mode to return an nsim \times 9 matrix of region probabilities. The probabilities are summed over GoRegions (for the Go scenario) and NoGoRegions (for the NoGo scenario) to obtain \hat{g}_{Go,i} and \hat{g}_{NoGo,i}, independent of the decision thresholds.

Stage 2 (gamma sweep): For each pair (\gamma_{\mathrm{go}}, \gamma_{\mathrm{nogo}}) in the two-dimensional grid, operating characteristics are computed separately under each calibration scenario:

\Pr(\mathrm{Go}) = \frac{1}{n_{\mathrm{sim}}} \sum_{i=1}^{n_{\mathrm{sim}}} \mathbf{1}\!\left[\hat{g}_{Go,i} \ge \gamma_{\mathrm{go}},\; \hat{g}_{NoGo,i} < \gamma_{\mathrm{nogo}}\right]

\Pr(\mathrm{NoGo}) = \frac{1}{n_{\mathrm{sim}}} \sum_{i=1}^{n_{\mathrm{sim}}} \mathbf{1}\!\left[\hat{g}_{NoGo,i} \ge \gamma_{\mathrm{nogo}},\; \hat{g}_{Go,i} < \gamma_{\mathrm{go}}\right]

Stage 3 (optimal threshold selection): For each candidate \gamma_{\mathrm{go}}, the worst-case \Pr(\mathrm{Go}) over all \gamma_{\mathrm{nogo}} in gamma_nogo_grid is computed; the optimal \gamma_{\mathrm{go}} is the smallest grid value for which this worst-case probability is less than target_go. Analogously, the optimal \gamma_{\mathrm{nogo}} is the smallest grid value for which the worst-case \Pr(\mathrm{NoGo}) is less than target_nogo.

Value

A list of class getgamma2cont with the following elements:

gamma_go

Optimal Go threshold: the smallest value in gamma_go_grid for which the marginal \Pr(\mathrm{Go}) < \code{target\_go} under the Go-calibration scenario. NA if no such value exists.

gamma_nogo

Optimal NoGo threshold: the smallest value in gamma_nogo_grid for which the marginal \Pr(\mathrm{NoGo}) < \code{target\_nogo} under the NoGo-calibration scenario. NA if no such value exists.

PrGo_opt

Marginal \Pr(\mathrm{Go}) at gamma_go under the Go-calibration scenario. NA if gamma_go is NA.

PrNoGo_opt

Marginal \Pr(\mathrm{NoGo}) at gamma_nogo under the NoGo-calibration scenario. NA if gamma_nogo is NA.

target_go

The value of target_go supplied by the user.

target_nogo

The value of target_nogo supplied by the user.

grid_results

A data frame with columns gamma_grid, PrGo_grid (marginal Go probability under the Go-calibration scenario), and PrNoGo_grid (marginal NoGo probability under the NoGo-calibration scenario).

Examples

# Example 1: Controlled design, posterior probability, vague prior
# gamma_go  : smallest gamma_go   s.t. max_{gamma_nogo} Pr(Go)   < 0.05 under Null
# gamma_nogo: smallest gamma_nogo s.t. max_{gamma_go}   Pr(NoGo) < 0.20 under Alt

Sigma_null <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
Sigma_alt  <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
getgamma2cont(
  nsim = 1000L, prob = 'posterior', design = 'controlled',
  prior = 'vague',
  GoRegions = 1L, NoGoRegions = 9L,
  mu_t_go = c(-5.0, 0.0), Sigma_t_go = Sigma_null,
  mu_c_go = c(-10.0, -1.0), Sigma_c_go = Sigma_null,
  mu_t_nogo = c(5.0, 1.0), Sigma_t_nogo = Sigma_alt,
  mu_c_nogo = c(-10.0, -1.0), Sigma_c_nogo = Sigma_alt,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 30L, n_c = 30L,
  theta_TV1 = 10.0, theta_MAV1 = 5.0,
  theta_TV2 = 2.0,  theta_MAV2 = 1.0,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  m_t = NULL, m_c = NULL,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = NULL, Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed = 1L
)


# Example 2: Uncontrolled design, posterior probability, vague prior

Sigma_null <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
Sigma_alt  <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
getgamma2cont(
  nsim = 1000L, prob = 'posterior', design = 'uncontrolled',
  prior = 'vague',
  GoRegions = 1L, NoGoRegions = 9L,
  mu_t_go = c(-5.0, 0.0), Sigma_t_go = Sigma_null,
  mu_c_go = NULL, Sigma_c_go = NULL,
  mu_t_nogo = c(5.0, 1.0), Sigma_t_nogo = Sigma_alt,
  mu_c_nogo = NULL, Sigma_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 30L, n_c = NULL,
  theta_TV1 = 10.0, theta_MAV1 = 5.0,
  theta_TV2 = 2.0,  theta_MAV2 = 1.0,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  m_t = NULL, m_c = NULL,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = c(-10.0, -1.0), Lambda0_c = NULL,
  r = 1.0,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = NULL, CalcMethod = 'MM',
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed = 1L
)


# Example 3: External design (control only), posterior probability, NIW prior

Sigma  <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
Lambda <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
se_c   <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
getgamma2cont(
  nsim = 1000L, prob = 'posterior', design = 'external',
  prior = 'N-Inv-Wishart',
  GoRegions = 1L, NoGoRegions = 9L,
  mu_t_go = c(-5.0, 0.0), Sigma_t_go = Sigma,
  mu_c_go = c(-10.0, -1.0), Sigma_c_go = Sigma,
  mu_t_nogo = c(5.0, 1.0), Sigma_t_nogo = Sigma,
  mu_c_nogo = c(-10.0, -1.0), Sigma_c_nogo = Sigma,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 30L, n_c = 30L,
  theta_TV1 = 10.0, theta_MAV1 = 5.0,
  theta_TV2 = 2.0,  theta_MAV2 = 1.0,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  m_t = NULL, m_c = NULL,
  kappa0_t = 0.1, nu0_t = 4.0, mu0_t = c(0.0, 1.0),  Lambda0_t = Lambda,
  kappa0_c = 0.1, nu0_c = 4.0, mu0_c = c(-10.0, -1.0), Lambda0_c = Lambda,
  r = NULL,
  ne_t = NULL, ne_c = 10L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = c(-10.0, -1.0), se_t = NULL, se_c = se_c,
  nMC = 500L, CalcMethod = 'MC',
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed = 1L
)


# Example 4: Controlled design, predictive probability, vague prior

Sigma_null <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
Sigma_alt  <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
getgamma2cont(
  nsim = 1000L, prob = 'predictive', design = 'controlled',
  prior = 'vague',
  GoRegions = 1L, NoGoRegions = 4L,
  mu_t_go = c(-5.0, 0.0), Sigma_t_go = Sigma_null,
  mu_c_go = c(-10.0, -1.0), Sigma_c_go = Sigma_null,
  mu_t_nogo = c(5.0, 1.0), Sigma_t_nogo = Sigma_alt,
  mu_c_nogo = c(-10.0, -1.0), Sigma_c_nogo = Sigma_alt,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 30L, n_c = 30L,
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 5.0, theta_NULL2 = 1.0,
  m_t = 100L, m_c = 100L,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = NULL, Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed = 1L
)


# Example 5: Uncontrolled design, predictive probability, vague prior

Sigma_null <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
Sigma_alt  <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
getgamma2cont(
  nsim = 1000L, prob = 'predictive', design = 'uncontrolled',
  prior = 'vague',
  GoRegions = 1L, NoGoRegions = 4L,
  mu_t_go = c(-5.0, 0.0), Sigma_t_go = Sigma_null,
  mu_c_go = NULL, Sigma_c_go = NULL,
  mu_t_nogo = c(5.0, 1.0), Sigma_t_nogo = Sigma_alt,
  mu_c_nogo = NULL, Sigma_c_nogo = NULL,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 30L, n_c = NULL,
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 5.0, theta_NULL2 = 1.0,
  m_t = 100L, m_c = 100L,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = c(-10.0, -1.0), Lambda0_c = NULL,
  r = 1.0,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed = 1L
)


# Example 6: External design (control only), predictive probability, NIW prior

Sigma  <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
Lambda <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
se_c   <- matrix(c(6400.0, 15.0, 15.0, 36.0), 2, 2)
getgamma2cont(
  nsim = 1000L, prob = 'predictive', design = 'external',
  prior = 'N-Inv-Wishart',
  GoRegions = 1L, NoGoRegions = 4L,
  mu_t_go = c(-5.0, 0.0), Sigma_t_go = Sigma,
  mu_c_go = c(-10.0, -1.0), Sigma_c_go = Sigma,
  mu_t_nogo = c(5.0, 1.0), Sigma_t_nogo = Sigma,
  mu_c_nogo = c(-10.0, -1.0), Sigma_c_nogo = Sigma,
  target_go = 0.05, target_nogo = 0.20,
  n_t = 30L, n_c = 30L,
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 5.0, theta_NULL2 = 1.0,
  m_t = 100L, m_c = 100L,
  kappa0_t = 0.1, nu0_t = 4.0, mu0_t = c(0.0, 1.0),  Lambda0_t = Lambda,
  kappa0_c = 0.1, nu0_c = 4.0, mu0_c = c(-10.0, -1.0), Lambda0_c = Lambda,
  r = NULL,
  ne_t = NULL, ne_c = 10L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = c(-10.0, -1.0), se_t = NULL, se_c = se_c,
  nMC = 500L, CalcMethod = 'MC',
  gamma_go_grid = seq(0.01, 0.99, by = 0.01),
  gamma_nogo_grid = seq(0.01, 0.99, by = 0.01),
  seed = 1L
)

Compute Joint Bivariate Binary Probabilities from Marginal Rates and Correlation

Description

Converts the parameterisation (\pi_{j1}, \pi_{j2}, \rho_j) into the four-cell joint probability vector (p_{j,00}, p_{j,01}, p_{j,10}, p_{j,11}) for a bivariate binary outcome in group j \in \{t, c\}. The conversion uses the standard moment-matching identity for the Pearson correlation of two Bernoulli variables, and checks that the requested correlation is feasible for the supplied marginal rates.

Usage

getjointbin(pi1, pi2, rho, tol = 1e-10)

Arguments

Details

For a bivariate binary outcome (Y_{i1}, Y_{i2}) of patient i (i = 1, \ldots, n_j) in group j with marginal success probabilities \pi_{j1} = \Pr(Y_{i1} = 1) and \pi_{j2} = \Pr(Y_{i2} = 1), the Pearson correlation is

\rho_j = \frac{p_{j,11} - \pi_{j1} \pi_{j2}} {\sqrt{\pi_{j1}(1 - \pi_{j1})\, \pi_{j2}(1 - \pi_{j2})}}.

Solving for p_{j,11} gives

p_{j,11} = \rho_j \sqrt{\pi_{j1}(1-\pi_{j1})\,\pi_{j2}(1-\pi_{j2})} + \pi_{j1} \pi_{j2},

from which the remaining probabilities follow:

p_{j,10} = \pi_{j1} - p_{j,11}, \quad p_{j,01} = \pi_{j2} - p_{j,11}, \quad p_{j,00} = 1 - p_{j,10} - p_{j,01} - p_{j,11}.

For all four cell probabilities to lie in [0, 1], the correlation must satisfy

\rho_{\min} = \frac{\max(0,\; \pi_{j1} + \pi_{j2} - 1) - \pi_{j1}\pi_{j2}} {\sqrt{\pi_{j1}(1-\pi_{j1})\,\pi_{j2}(1-\pi_{j2})}} \;\le\; \rho_j \;\le\; \frac{\min(\pi_{j1}, \pi_{j2}) - \pi_{j1}\pi_{j2}} {\sqrt{\pi_{j1}(1-\pi_{j1})\,\pi_{j2}(1-\pi_{j2})}} = \rho_{\max}.

The function raises an error if rho falls outside this range (subject to tol).

Value

A named numeric vector of length 4 with elements p00, p01, p10, p11, where p_lm = Pr(Endpoint 1 = l, Endpoint 2 = m) for l, m \in \{0, 1\}. All elements are non-negative and sum to 1.

Examples

# Example 1: Independent endpoints (rho = 0)
getjointbin(pi1 = 0.3, pi2 = 0.4, rho = 0.0)

# Example 2: Positive correlation
getjointbin(pi1 = 0.3, pi2 = 0.4, rho = 0.3)

# Example 3: Negative correlation
getjointbin(pi1 = 0.3, pi2 = 0.4, rho = -0.2)

# Example 4: Verify cell probabilities sum to 1
p <- getjointbin(pi1 = 0.25, pi2 = 0.35, rho = 0.1)
sum(p)  # Should be 1

# Example 5: Verify marginal recovery
p <- getjointbin(pi1 = 0.25, pi2 = 0.35, rho = 0.1)
p["p10"] + p["p11"]  # Should equal pi1 = 0.25
p["p01"] + p["p11"]  # Should equal pi2 = 0.35

Go/NoGo/Gray Decision Probabilities for a Clinical Trial with a Single Binary Endpoint

Description

Evaluates operating characteristics (Go, NoGo, Gray probabilities) for binary-outcome clinical trials under the Bayesian framework by enumerating all possible trial outcomes. The function supports controlled, uncontrolled, and external designs.

Usage

pbayesdecisionprob1bin(
  prob = "posterior",
  design = "controlled",
  theta_TV = NULL,
  theta_MAV = NULL,
  theta_NULL = NULL,
  gamma_go,
  gamma_nogo,
  pi_t,
  pi_c = NULL,
  n_t,
  n_c,
  a_t,
  a_c,
  b_t,
  b_c,
  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,
  error_if_Miss = TRUE,
  Gray_inc_Miss = FALSE
)

Arguments

Details

Operating characteristics are computed by exact enumeration:

1. All possible outcome pairs (y_t, y_c) with y_t \in \{0,\ldots,n_t\} and y_c \in \{0,\ldots,n_c\} (or fixed at z for uncontrolled) are evaluated.

2. For each pair, pbayespostpred1bin computes the posterior or predictive probability at both thresholds (TV/MAV or NULL).

3. Outcomes are classified into Go, NoGo, Miss, or Gray:

   • Go: P(\mathrm{Go}) \ge \gamma_1 AND P(\mathrm{NoGo}) < \gamma_2

   • NoGo: P(\mathrm{Go}) < \gamma_1 AND P(\mathrm{NoGo}) \ge \gamma_2

   • Miss: both Go and NoGo criteria met simultaneously

   • Gray: neither Go nor NoGo criteria met

4. Each outcome is weighted by its binomial probability under the true rates.

Value

A data frame with one row per pi_t scenario and columns:

pi_t

True treatment response probability.

pi_c

True control response probability (omitted for uncontrolled design).

Go

Probability of making a Go decision.

Gray

Probability of making a Gray (inconclusive) decision.

NoGo

Probability of making a NoGo decision.

Miss

(Optional) Probability where Go and NoGo criteria are simultaneously met. Included when error_if_Miss = FALSE and Gray_inc_Miss = FALSE.

The returned object has S3 class pbayesdecisionprob1bin with an associated print method.

Examples

# Example 1: Controlled design with posterior probability
pbayesdecisionprob1bin(
  prob = 'posterior', design = 'controlled',
  theta_TV = 0.4, theta_MAV = 0.2, theta_NULL = NULL,
  gamma_go = 0.8, gamma_nogo = 0.2,
  pi_t = c(0.2, 0.4, 0.6, 0.8), pi_c = rep(0.2, 4),
  n_t = 12, n_c = 12,
  a_t = 0.5, a_c = 0.5, b_t = 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,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)

# Example 2: Uncontrolled design with hypothetical control
pbayesdecisionprob1bin(
  prob = 'posterior', design = 'uncontrolled',
  theta_TV = 0.30, theta_MAV = 0.15, theta_NULL = NULL,
  gamma_go = 0.75, gamma_nogo = 0.25,
  pi_t = c(0.3, 0.5, 0.7), pi_c = NULL,
  n_t = 15, n_c = 15,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = 5, m_t = NULL, m_c = NULL,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)

# Example 3: External design with 50 percent power prior borrowing
pbayesdecisionprob1bin(
  prob = 'posterior', design = 'external',
  theta_TV = 0.4, theta_MAV = 0.2, theta_NULL = NULL,
  gamma_go = 0.8, gamma_nogo = 0.2,
  pi_t = c(0.2, 0.4, 0.6, 0.8), pi_c = rep(0.2, 4),
  n_t = 12, n_c = 12,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = NULL, m_t = NULL, m_c = NULL,
  ne_t = 15, ne_c = 15, ye_t = 6, ye_c = 4, alpha0e_t = 0.5, alpha0e_c = 0.5,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)

# Example 4: Posterior predictive probability for controlled design
pbayesdecisionprob1bin(
  prob = 'predictive', design = 'controlled',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 0,
  gamma_go = 0.9, gamma_nogo = 0.3,
  pi_t = c(0.2, 0.4, 0.6, 0.8), pi_c = rep(0.2, 4),
  n_t = 12, n_c = 12,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = NULL, m_t = 30, m_c = 30,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)

# Example 5: Uncontrolled design with posterior predictive probability
pbayesdecisionprob1bin(
  prob = 'predictive', design = 'uncontrolled',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 0,
  gamma_go = 0.75, gamma_nogo = 0.25,
  pi_t = c(0.3, 0.5, 0.7), pi_c = NULL,
  n_t = 15, n_c = 15,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = 5, m_t = 30, m_c = 30,
  ne_t = NULL, ne_c = NULL, ye_t = NULL, ye_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)

# Example 6: External design with posterior predictive probability
pbayesdecisionprob1bin(
  prob = 'predictive', design = 'external',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 0,
  gamma_go = 0.9, gamma_nogo = 0.3,
  pi_t = c(0.2, 0.4, 0.6, 0.8), pi_c = rep(0.2, 4),
  n_t = 12, n_c = 12,
  a_t = 0.5, a_c = 0.5, b_t = 0.5, b_c = 0.5,
  z = NULL, m_t = 30, m_c = 30,
  ne_t = 15, ne_c = 15, ye_t = 6, ye_c = 4, alpha0e_t = 0.5, alpha0e_c = 0.5,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)

Bayesian Go/NoGo/Gray Decision Probabilities for Single Continuous Endpoint

Description

Evaluates Go/NoGo/Gray decision probabilities for a single continuous endpoint via Monte Carlo simulation. Supports controlled (parallel control), uncontrolled (single-arm with informative priors), and external (power prior borrowing) designs with both posterior and predictive probability approaches.

Usage

pbayesdecisionprob1cont(
  nsim,
  prob,
  design,
  prior,
  CalcMethod,
  theta_TV = NULL,
  theta_MAV = NULL,
  theta_NULL = NULL,
  nMC = NULL,
  gamma_go,
  gamma_nogo,
  n_t,
  n_c = NULL,
  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,
  mu_t,
  mu_c = NULL,
  sigma_t,
  sigma_c = NULL,
  r = NULL,
  ne_t = NULL,
  ne_c = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  bar_ye_t = NULL,
  bar_ye_c = NULL,
  se_t = NULL,
  se_c = NULL,
  error_if_Miss = TRUE,
  Gray_inc_Miss = FALSE,
  seed
)

Arguments

Details

The function performs Monte Carlo simulation to evaluate operating characteristics by:

• Generating random trial data based on specified true parameters

• Computing posterior or predictive probabilities for each simulated trial

• Classifying each trial as Go, NoGo, or Gray based on decision thresholds

Posterior parameter calculations (mu.t1, mu.t2, sd.t1, sd.t2, nu.t1, nu.t2) are fully vectorized over the nsim simulated datasets. pbayespostpred1cont is called twice (once per threshold), with each call receiving vectors of length nsim and returning a vector of nsim probabilities - no inner loop over simulation replicates is required.

For external designs, power priors are incorporated using exact conjugate representation:

• Power priors for normal data are mathematically equivalent to Normal-Inverse-Chi-squared distributions

• This enables closed-form computation without MCMC sampling

• Alpha parameters control the degree of borrowing (0 = no borrowing, 1 = full borrowing)

Decision rules:

• Go: gGo >= gamma_go and gNoGo < gamma_nogo

• NoGo: gGo < gamma_go and gNoGo >= gamma_nogo

• Gray: neither Go nor NoGo criterion is met

• Miss: both Go and NoGo criteria are met simultaneously

Handling Miss probability:

• When error_if_Miss = TRUE (default): Function stops with error if Miss probability > 0, prompting reconsideration of thresholds

• When error_if_Miss = FALSE and Gray_inc_Miss = TRUE: Miss probability is added to Gray probability

• When error_if_Miss = FALSE and Gray_inc_Miss = FALSE: Miss probability is reported as a separate category

The function can be used for:

• Controlled design: Two-arm randomized trial

• Uncontrolled design: Single-arm trial with informative priors (historical control)

• External design: Incorporating historical data through power priors

Value

A data frame containing the true means for both groups and the Go, NoGo, and Gray probabilities. When error_if_Miss = FALSE and Gray_inc_Miss = FALSE, Miss probability is also included as a separate column. For uncontrolled design, only mu_t is included (not mu_c).

Examples

# Example 1: Controlled design with vague prior and NI method
# (default: error_if_Miss = TRUE, Gray_inc_Miss = FALSE)
pbayesdecisionprob1cont(
  nsim = 100, prob = 'posterior', design = 'controlled', prior = 'vague', CalcMethod = 'NI',
  theta_TV = 1.5, theta_MAV = -0.5, theta_NULL = NULL,
  nMC = NULL, gamma_go = 0.7, gamma_nogo = 0.2,
  n_t = 15, n_c = 15, 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,
  mu_t = 3, mu_c = 1, sigma_t = 1.2, sigma_c = 1.1,
  r = NULL, ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 2
)

# Example 2: Uncontrolled design with informative prior
pbayesdecisionprob1cont(
  nsim = 100, prob = 'posterior', design = 'uncontrolled', prior = 'N-Inv-Chisq', CalcMethod = 'NI',
  theta_TV = 1.0, theta_MAV = 0.0, theta_NULL = NULL,
  nMC = NULL, gamma_go = 0.8, gamma_nogo = 0.2,
  n_t = 20, n_c = NULL, m_t = NULL, m_c = NULL,
  kappa0_t = 2, kappa0_c = NULL, nu0_t = 5, nu0_c = NULL,
  mu0_t = 3.0, mu0_c = 1.5, sigma0_t = 1.5, sigma0_c = NULL,
  mu_t = 3.5, mu_c = NULL, sigma_t = 1.3, sigma_c = NULL,
  r = 1, ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 3
)

# Example 3: External design with control data using MM approximation
pbayesdecisionprob1cont(
  nsim = 100, prob = 'posterior', design = 'external', prior = 'vague', CalcMethod = 'MM',
  theta_TV = 1.0, theta_MAV = 0.0, theta_NULL = NULL,
  nMC = NULL, gamma_go = 0.8, gamma_nogo = 0.2,
  n_t = 12, n_c = 12, 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,
  mu_t = 2, mu_c = 0, sigma_t = 1, sigma_c = 1,
  r = NULL, ne_t = NULL, ne_c = 20, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = 0, se_t = NULL, se_c = 1,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 4
)

# Example 4: Controlled design with predictive probability
pbayesdecisionprob1cont(
  nsim = 100, prob = 'predictive', design = 'controlled', prior = 'N-Inv-Chisq', CalcMethod = 'NI',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 2.0,
  nMC = NULL, gamma_go = 0.75, gamma_nogo = 0.35,
  n_t = 15, n_c = 15, m_t = 50, m_c = 50,
  kappa0_t = 3, kappa0_c = 3, nu0_t = 4, nu0_c = 4,
  mu0_t = 3.5, mu0_c = 1.5, sigma0_t = 1.5, sigma0_c = 1.5,
  mu_t = 3.2, mu_c = 1.3, sigma_t = 1.4, sigma_c = 1.2,
  r = NULL, ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 5
)

# Example 5: Uncontrolled design with predictive probability
pbayesdecisionprob1cont(
  nsim = 100, prob = 'predictive', design = 'uncontrolled', prior = 'vague', CalcMethod = 'NI',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 1.0,
  nMC = NULL, gamma_go = 0.75, gamma_nogo = 0.35,
  n_t = 20, n_c = NULL, m_t = 40, m_c = 40,
  kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
  mu0_t = NULL, mu0_c = 1.5, sigma0_t = NULL, sigma0_c = NULL,
  mu_t = 3, mu_c = NULL, sigma_t = 1.3, sigma_c = NULL,
  r = 1, ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 9
)

# Example 6: External design with predictive probability using MC method
pbayesdecisionprob1cont(
  nsim = 100, prob = 'predictive', design = 'external', prior = 'vague', CalcMethod = 'MC',
  theta_TV = NULL, theta_MAV = NULL, theta_NULL = 1.5,
  nMC = 5000, gamma_go = 0.7, gamma_nogo = 0.4,
  n_t = 12, n_c = 12, m_t = 30, m_c = 30,
  kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
  mu0_t = NULL, mu0_c = NULL, sigma0_t = NULL, sigma0_c = NULL,
  mu_t = 2.5, mu_c = 1.0, sigma_t = 1.3, sigma_c = 1.1,
  r = NULL, ne_t = 15, ne_c = 18, alpha0e_t = 0.6, alpha0e_c = 0.7,
  bar_ye_t = 2.3, bar_ye_c = 0.9, se_t = 1.2, se_c = 1.0,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE, seed = 6
)

Go/NoGo/Gray Decision Probabilities for a Clinical Trial with Two Binary Endpoints

Description

Evaluates operating characteristics (Go, NoGo, Gray probabilities) for clinical trials with two binary endpoints under the Bayesian framework. The function supports controlled, uncontrolled, and external designs, and uses both posterior probability and posterior predictive probability criteria.

Usage

pbayesdecisionprob2bin(
  nsim = 10000L,
  prob = "posterior",
  design = "controlled",
  GoRegions,
  NoGoRegions,
  gamma_go,
  gamma_nogo,
  pi_t1,
  pi_t2,
  rho_t,
  pi_c1 = NULL,
  pi_c2 = NULL,
  rho_c = NULL,
  n_t,
  n_c = NULL,
  a_t_00,
  a_t_01,
  a_t_10,
  a_t_11,
  a_c_00,
  a_c_01,
  a_c_10,
  a_c_11,
  m_t = NULL,
  m_c = NULL,
  theta_TV1 = NULL,
  theta_MAV1 = NULL,
  theta_TV2 = NULL,
  theta_MAV2 = NULL,
  theta_NULL1 = NULL,
  theta_NULL2 = NULL,
  z00 = NULL,
  z01 = NULL,
  z10 = NULL,
  z11 = NULL,
  xe_t_00 = NULL,
  xe_t_01 = NULL,
  xe_t_10 = NULL,
  xe_t_11 = NULL,
  xe_c_00 = NULL,
  xe_c_01 = NULL,
  xe_c_10 = NULL,
  xe_c_11 = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  nMC = 10000L,
  CalcMethod = "Exact",
  error_if_Miss = TRUE,
  Gray_inc_Miss = FALSE
)

Arguments

Details

Computation proceeds in two stages for efficiency. In Stage 1, posterior or predictive probabilities are precomputed for every possible multinomial count combination (x_t, x_c). In Stage 2, operating characteristics for each scenario are obtained by weighting the Stage 1 decisions by their multinomial probabilities under the true parameters, avoiding repeated Monte Carlo sampling per scenario.

Two-stage computation.

Stage 1 (precomputation): All possible multinomial count combinations x_t are enumerated using allmultinom. For design = 'controlled' or 'external', all combinations x_c are also enumerated and the Go/NoGo probability matrix PrGo[i, j] has dimensions n_t \times n_c. For design = 'uncontrolled', the control distribution is fixed as \mathrm{Dir}(\alpha_{2,**} + z_{**}) and does not depend on any observed control counts; the probability matrix therefore has dimensions n_t \times 1, eliminating the \binom{n_2+3}{3}-row control enumeration and the associated Gamma sampling. This stage is independent of the true scenario parameters.

Stage 2 (scenario evaluation): For each scenario (\pi_{t1}, \pi_{t2}, \rho_t, \pi_{c1}, \pi_{c2}, \rho_c), getjointbin converts the marginal rates and correlation into the four-cell probability vector p_k. The multinomial probability mass function weights the Stage 1 decisions:

\Pr(\mathrm{Go} \mid \Theta) = \sum_{i,j} \mathbf{1}(\mathrm{Go}_{ij}) \cdot P(x_{t,i} \mid n_1, p_t) \cdot P(x_{c,j} \mid n_2, p_c),

where \mathbf{1}(\mathrm{Go}_{ij}) is the Go indicator from Stage 1. Stage 2 requires no additional Monte Carlo and is fast even for large scenario grids.

Decision categories.

• Go: sum of Go region probabilities \ge \gamma_1 AND sum of NoGo region probabilities < \gamma_2.

• NoGo: sum of Go region probabilities < \gamma_1 AND sum of NoGo region probabilities \ge \gamma_2.

• Miss: both Go and NoGo criteria satisfied simultaneously.

• Gray: neither Go nor NoGo criterion satisfied.

Value

A data frame with one row per scenario and columns:

pi_t1

True treatment response probability for Endpoint 1.

pi_t2

True treatment response probability for Endpoint 2.

rho_t

True within-arm correlation in the treatment arm.

pi_c1

True control response probability for Endpoint 1 (omitted when design = 'uncontrolled').

pi_c2

True control response probability for Endpoint 2 (omitted when design = 'uncontrolled').

rho_c

True within-arm correlation in the control arm (omitted when design = 'uncontrolled').

Go

Probability of making a Go decision.

Gray

Probability of making a Gray (inconclusive) decision.

NoGo

Probability of making a NoGo decision.

Miss

Probability where Go and NoGo criteria are simultaneously met. Included when error_if_Miss = FALSE and Gray_inc_Miss = FALSE.

The returned object has S3 class pbayesdecisionprob2bin with an associated print method.

Examples

# Example 1: Posterior probability, controlled design (rho > 0)
# Explores the impact of positive within-arm endpoint correlation.
pbayesdecisionprob2bin(
  prob        = 'posterior',
  design      = 'controlled',
  GoRegions   = 1L,
  NoGoRegions = 9L,
  gamma_go    = 0.52,
  gamma_nogo  = 0.52,
  pi_t1       = c(0.15, 0.25, 0.30, 0.40),
  pi_t2       = c(0.20, 0.30, 0.35, 0.45),
  rho_t       = rep(0.6, 4),
  pi_c1       = rep(0.15, 4),
  pi_c2       = rep(0.20, 4),
  rho_c       = rep(0.6, 4),
  n_t = 10, n_c = 10,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  m_t = NULL, m_c = NULL,
  theta_TV1   = 0.15, theta_MAV1 = 0.10,
  theta_TV2   = 0.15, theta_MAV2 = 0.10,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE
)

# Example 2: Posterior probability, uncontrolled design
# Hypothetical control specified via pseudo-counts.
pbayesdecisionprob2bin(
  prob        = 'posterior',
  design      = 'uncontrolled',
  GoRegions   = 1L,
  NoGoRegions = 9L,
  gamma_go    = 0.27,
  gamma_nogo  = 0.36,
  pi_t1       = c(0.15, 0.30, 0.40),
  pi_t2       = c(0.20, 0.35, 0.45),
  rho_t       = rep(0.2, 3),
  pi_c1       = NULL,
  pi_c2       = NULL,
  rho_c       = NULL,
  n_t = 10, n_c = NULL,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  m_t = NULL, m_c = NULL,
  theta_TV1   = 0.15, theta_MAV1 = 0.10,
  theta_TV2   = 0.15, theta_MAV2 = 0.10,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  z00 = 2L, z01 = 1L, z10 = 2L, z11 = 1L,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE
)

# Example 3: Posterior probability, external control design
# External control data incorporated via power prior (alpha0e_c = 0.5).
pbayesdecisionprob2bin(
  prob        = 'posterior',
  design      = 'external',
  GoRegions   = 1L,
  NoGoRegions = 9L,
  gamma_go    = 0.27,
  gamma_nogo  = 0.36,
  pi_t1       = c(0.15, 0.25, 0.30, 0.40),
  pi_t2       = c(0.20, 0.30, 0.35, 0.45),
  rho_t       = rep(0.0, 4),
  pi_c1       = rep(0.15, 4),
  pi_c2       = rep(0.20, 4),
  rho_c       = rep(0.0, 4),
  n_t = 10, n_c = 10,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  m_t = NULL, m_c = NULL,
  theta_TV1   = 0.15, theta_MAV1 = 0.10,
  theta_TV2   = 0.15, theta_MAV2 = 0.10,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = 4L,  xe_c_01 = 2L,  xe_c_10 = 3L,  xe_c_11 = 1L,
  alpha0e_t = NULL, alpha0e_c = 0.5,
  nMC = 100,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE
)

# Example 4: Predictive probability, controlled design
# Future trial has m_t = m_c = 30 patients per arm.
pbayesdecisionprob2bin(
  prob        = 'predictive',
  design      = 'controlled',
  GoRegions   = 1L,
  NoGoRegions = 4L,
  gamma_go    = 0.60,
  gamma_nogo  = 0.80,
  pi_t1       = c(0.15, 0.25, 0.30, 0.40),
  pi_t2       = c(0.20, 0.30, 0.35, 0.45),
  rho_t       = rep(0.3, 4),
  pi_c1       = rep(0.15, 4),
  pi_c2       = rep(0.20, 4),
  rho_c       = rep(0.3, 4),
  n_t = 10, n_c = 10,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  m_t = 30L, m_c = 30L,
  theta_TV1   = NULL, theta_MAV1 = NULL,
  theta_TV2   = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.10, theta_NULL2 = 0.10,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE
)

# Example 5: Predictive probability, uncontrolled design
# Hypothetical control specified via pseudo-counts; future trial m_t = m_c = 30.
pbayesdecisionprob2bin(
  prob        = 'predictive',
  design      = 'uncontrolled',
  GoRegions   = 1L,
  NoGoRegions = 4L,
  gamma_go    = 0.60,
  gamma_nogo  = 0.80,
  pi_t1       = c(0.15, 0.30, 0.40),
  pi_t2       = c(0.20, 0.35, 0.45),
  rho_t       = rep(0.7, 3),
  pi_c1       = rep(0.15, 3),
  pi_c2       = rep(0.20, 3),
  rho_c       = rep(0.7, 3),
  n_t = 10, n_c = 10,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  m_t = 30L, m_c = 30L,
  theta_TV1   = NULL, theta_MAV1 = NULL,
  theta_TV2   = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.10, theta_NULL2 = 0.10,
  z00 = 2L, z01 = 1L, z10 = 2L, z11 = 1L,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL,
  nMC = 100,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE
)

# Example 6: Predictive probability, external treatment design
# External treatment data incorporated via power prior (alpha0e_t = 0.5).
pbayesdecisionprob2bin(
  prob        = 'predictive',
  design      = 'external',
  GoRegions   = 1L,
  NoGoRegions = 4L,
  gamma_go    = 0.60,
  gamma_nogo  = 0.80,
  pi_t1       = c(0.15, 0.25, 0.30, 0.40),
  pi_t2       = c(0.20, 0.30, 0.35, 0.45),
  rho_t       = rep(0.5, 4),
  pi_c1       = rep(0.15, 4),
  pi_c2       = rep(0.20, 4),
  rho_c       = rep(0.5, 4),
  n_t = 10, n_c = 10,
  a_t_00 = 0.25, a_t_01 = 0.25, a_t_10 = 0.25, a_t_11 = 0.25,
  a_c_00 = 0.25, a_c_01 = 0.25, a_c_10 = 0.25, a_c_11 = 0.25,
  m_t = 30L, m_c = 30L,
  theta_TV1   = NULL, theta_MAV1 = NULL,
  theta_TV2   = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.10, theta_NULL2 = 0.10,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = 3L, xe_t_01 = 2L, xe_t_10 = 3L, xe_t_11 = 2L,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = 0.5, alpha0e_c = NULL,
  nMC = 100,
  error_if_Miss = FALSE, Gray_inc_Miss = FALSE
)

Go/NoGo/Gray Decision Probabilities for Two Continuous Endpoints

Description

Computes the operating characteristics (Go, NoGo, Gray, and optionally Miss probabilities) for a two-continuous-endpoint Bayesian Go/NoGo decision framework by Monte Carlo simulation over treatment scenarios.

Usage

pbayesdecisionprob2cont(
  nsim,
  prob,
  design,
  prior,
  GoRegions,
  NoGoRegions,
  gamma_go,
  gamma_nogo,
  theta_TV1 = NULL,
  theta_MAV1 = NULL,
  theta_TV2 = NULL,
  theta_MAV2 = NULL,
  theta_NULL1 = NULL,
  theta_NULL2 = NULL,
  n_t,
  n_c = NULL,
  m_t = NULL,
  m_c = NULL,
  mu_t,
  Sigma_t,
  mu_c = NULL,
  Sigma_c = NULL,
  kappa0_t = NULL,
  nu0_t = NULL,
  mu0_t = NULL,
  Lambda0_t = NULL,
  kappa0_c = NULL,
  nu0_c = NULL,
  mu0_c = NULL,
  Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL,
  ne_c = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  bar_ye_t = NULL,
  bar_ye_c = NULL,
  se_t = NULL,
  se_c = NULL,
  nMC = NULL,
  CalcMethod = "MC",
  error_if_Miss = TRUE,
  Gray_inc_Miss = FALSE,
  seed
)

Arguments

Details

The function follows the same structure as pbayesdecisionprob1cont:

1. For each scenario s, nsim datasets are simulated by generating treatment (and control) observations from N_2(\mu_k^{(s)}, \Sigma_k). To minimise overhead, raw standardised residuals are generated once (scenario- invariant) and shifted by the scenario mean.

2. All nsim simulated sufficient statistics (\bar{y}_{1,i}, S_{1,i}) (and (\bar{y}_{2,i}, S_{2,i}) for controlled/external designs) are passed to pbayespostpred2cont in a single vectorised call, returning an nsim \times n_{\rm regions} matrix of region probabilities.

3. Go/NoGo/Miss probabilities are obtained as the column means of indicator matrices derived from the region probability matrix.

Value

A data frame of class c("pbayesdecisionprob2cont", "data.frame") with columns for the scenario parameters and Go, NoGo, Gray (and optionally Miss) probabilities. All input parameters are attached as attributes.

Examples

# Example 1: Controlled design, posterior probability, vague prior
Sigma <- matrix(c(4.0, 0.8, 0.8, 1.0), 2, 2)
pbayesdecisionprob2cont(
  nsim = 100L, prob = 'posterior', design = 'controlled',
  prior = 'vague',
  GoRegions = 1L, NoGoRegions = 9L,
  gamma_go = 0.8, gamma_nogo = 0.8,
  theta_TV1 = 1.5, theta_MAV1 = 0.5,
  theta_TV2 = 1.0, theta_MAV2 = 0.3,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  n_t = 20L, n_c = 20L, m_t = NULL, m_c = NULL,
  mu_t = rbind(c(1.0, 0.5), c(2.5, 1.5), c(4.0, 2.5)),
  Sigma_t = Sigma,
  mu_c = rbind(c(0.0, 0.0), c(0.0, 0.0), c(0.0, 0.0)),
  Sigma_c = Sigma,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = NULL, Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 1L
)

# Example 2: Uncontrolled design, posterior probability, NIW prior
Sigma <- matrix(c(4.0, 0.8, 0.8, 1.0), 2, 2)
L0    <- matrix(c(8.0, 0.0, 0.0, 2.0), 2, 2)
pbayesdecisionprob2cont(
  nsim = 100L, prob = 'posterior', design = 'uncontrolled',
  prior = 'N-Inv-Wishart',
  GoRegions = 1L, NoGoRegions = 9L,
  gamma_go = 0.8, gamma_nogo = 0.8,
  theta_TV1 = 1.5, theta_MAV1 = 0.5,
  theta_TV2 = 1.0, theta_MAV2 = 0.3,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  n_t = 20L, n_c = NULL, m_t = NULL, m_c = NULL,
  mu_t = rbind(c(1.0, 0.5), c(2.5, 1.5), c(4.0, 2.5)),
  Sigma_t = Sigma,
  mu_c = rbind(c(0.0, 0.0), c(0.0, 0.0), c(0.0, 0.0)),
  Sigma_c = Sigma,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = c(0.0, 0.0), Lambda0_c = NULL,
  r = 1.0,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 3L
)

# Example 3: External design (control only), posterior probability, NIW prior
Sigma  <- matrix(c(4.0, 0.8, 0.8, 1.0), 2, 2)
L0     <- matrix(c(8.0, 0.0, 0.0, 2.0), 2, 2)
se_mat <- matrix(c(7.0, 1.2, 1.2, 1.8), 2, 2)
pbayesdecisionprob2cont(
  nsim = 100L, prob = 'posterior', design = 'external',
  prior = 'N-Inv-Wishart',
  GoRegions = 1L, NoGoRegions = 9L,
  gamma_go = 0.8, gamma_nogo = 0.8,
  theta_TV1 = 1.5, theta_MAV1 = 0.5,
  theta_TV2 = 1.0, theta_MAV2 = 0.3,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  n_t = 20L, n_c = 20L, m_t = NULL, m_c = NULL,
  mu_t = rbind(c(1.0, 0.5), c(2.5, 1.5), c(4.0, 2.5)),
  Sigma_t = Sigma,
  mu_c = rbind(c(0.0, 0.0), c(0.0, 0.0), c(0.0, 0.0)),
  Sigma_c = Sigma,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = 2.0, nu0_c = 5.0, mu0_c = c(0.0, 0.0), Lambda0_c = L0,
  r = NULL,
  ne_t = NULL, ne_c = 15L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = c(0.2, 0.1), se_t = NULL, se_c = se_mat,
  nMC = 500L, CalcMethod = 'MC',
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 5L
)

# Example 4: Controlled design, predictive probability, NIW prior
Sigma <- matrix(c(4.0, 0.8, 0.8, 1.0), 2, 2)
L0    <- matrix(c(8.0, 0.0, 0.0, 2.0), 2, 2)
pbayesdecisionprob2cont(
  nsim = 100L, prob = 'predictive', design = 'controlled',
  prior = 'N-Inv-Wishart',
  GoRegions = 1L, NoGoRegions = 4L,
  gamma_go = 0.8, gamma_nogo = 0.8,
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.5, theta_NULL2 = 0.3,
  n_t = 20L, n_c = 20L, m_t = 60L, m_c = 60L,
  mu_t = rbind(c(1.0, 0.5), c(2.5, 1.5), c(4.0, 2.5)),
  Sigma_t = Sigma,
  mu_c = rbind(c(0.0, 0.0), c(0.0, 0.0), c(0.0, 0.0)),
  Sigma_c = Sigma,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = 2.0, nu0_c = 5.0, mu0_c = c(0.0, 0.0), Lambda0_c = L0,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 4L
)

# Example 5: Uncontrolled design, predictive probability, vague prior
Sigma <- matrix(c(4.0, 0.8, 0.8, 1.0), 2, 2)
pbayesdecisionprob2cont(
  nsim = 100L, prob = 'predictive', design = 'uncontrolled',
  prior = 'vague',
  GoRegions = 1L, NoGoRegions = 4L,
  gamma_go = 0.8, gamma_nogo = 0.8,
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.5, theta_NULL2 = 0.3,
  n_t = 20L, n_c = NULL, m_t = 60L, m_c = 60L,
  mu_t = rbind(c(1.0, 0.5), c(2.5, 1.5), c(4.0, 2.5)),
  Sigma_t = Sigma,
  mu_c = NULL,
  Sigma_c = NULL,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = c(0.0, 0.0), Lambda0_c = NULL,
  r = 1.0,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 500L, CalcMethod = 'MC',
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 8L
)

# Example 6: External design (control only), predictive probability, NIW prior
Sigma  <- matrix(c(4.0, 0.8, 0.8, 1.0), 2, 2)
L0     <- matrix(c(8.0, 0.0, 0.0, 2.0), 2, 2)
se_mat <- matrix(c(7.0, 1.2, 1.2, 1.8), 2, 2)
pbayesdecisionprob2cont(
  nsim = 100L, prob = 'predictive', design = 'external',
  prior = 'N-Inv-Wishart',
  GoRegions = 1L, NoGoRegions = 4L,
  gamma_go = 0.8, gamma_nogo = 0.8,
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.5, theta_NULL2 = 0.3,
  n_t = 20L, n_c = 20L, m_t = 60L, m_c = 60L,
  mu_t = rbind(c(1.0, 0.5), c(2.5, 1.5), c(4.0, 2.5)),
  Sigma_t = Sigma,
  mu_c = rbind(c(0.0, 0.0), c(0.0, 0.0), c(0.0, 0.0)),
  Sigma_c = Sigma,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = 2.0, nu0_c = 5.0, mu0_c = c(0.0, 0.0), Lambda0_c = L0,
  r = NULL,
  ne_t = NULL, ne_c = 15L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = c(0.2, 0.1), se_t = NULL, se_c = se_mat,
  nMC = 500L, CalcMethod = 'MC',
  error_if_Miss = TRUE, Gray_inc_Miss = FALSE, seed = 9L
)

Bayesian Posterior or Posterior Predictive Probability for a Single Binary Endpoint

Description

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

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.

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.

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
)

# 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
)

# 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
)

# 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
)

# 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
)

# 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
)

Bayesian Posterior or Posterior Predictive Probability for a Single Continuous Endpoint

Description

Computes the Bayesian posterior probability or posterior predictive probability for continuous-outcome clinical trials under a Normal-Inverse-Chi-squared (or vague Jeffreys) conjugate model. The function supports controlled, uncontrolled, and external designs, with optional incorporation of external data through power priors. Vector inputs for bar_y_t, s_t, bar_y_c, and s_c are supported for efficient batch processing (e.g., across simulation replicates in pbayesdecisionprob1cont).

Usage

pbayespostpred1cont(
  prob = "posterior",
  design = "controlled",
  prior = "vague",
  CalcMethod = "NI",
  theta0,
  nMC = NULL,
  n_t,
  n_c = NULL,
  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,
  bar_y_t,
  bar_y_c = NULL,
  s_t,
  s_c = NULL,
  r = NULL,
  ne_t = NULL,
  ne_c = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  bar_ye_t = NULL,
  bar_ye_c = NULL,
  se_t = NULL,
  se_c = NULL,
  lower.tail = TRUE
)

Arguments

Details

Under the Normal-Inverse-Chi-squared (N-Inv-ChiSq) conjugate model, the marginal posterior/predictive distribution of the group mean follows a non-standardised t-distribution, parameterised by a location \mu_j, a scale \sigma_j, and degrees of freedom \nu_j that depend on the design and probability type. The final probability is computed by one of three methods:

• NI: exact numerical integration via ptdiff_NI.

• MC: Monte Carlo simulation via ptdiff_MC.

• MM: Moment-Matching approximation via ptdiff_MM.

Performance note: CalcMethod = 'NI' calls integrate() once per element of the input vector, which is prohibitively slow for large vectors (e.g., nsim x n_scenarios). CalcMethod = 'MC' creates an nMC x n matrix internally, consuming O(nMC \cdot n) memory. For large-scale simulation use CalcMethod = 'MM', which is fully vectorised and exact in the normal limit.

Value

A numeric scalar or vector in [0, 1]. When the input data parameters (bar_y_t, etc.) are vectors of length n, a vector of length n is returned.

Examples

# Example 1: Controlled design - posterior probability with N-Inv-Chisq
pbayespostpred1cont(
  prob = 'posterior', design = 'controlled', prior = 'N-Inv-Chisq',
  CalcMethod = 'NI', theta0 = 2, n_t = 12, n_c = 12,
  kappa0_t = 5, kappa0_c = 5, nu0_t = 5, nu0_c = 5,
  mu0_t = 5, mu0_c = 5, sigma0_t = sqrt(5), sigma0_c = sqrt(5),
  bar_y_t = 2, bar_y_c = 0, s_t = 1, s_c = 1,
  m_t = NULL, m_c = NULL, r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  lower.tail = FALSE
)

# Example 2: Uncontrolled design - posterior probability with vague prior
pbayespostpred1cont(
  prob = 'posterior', design = 'uncontrolled', prior = 'vague',
  CalcMethod = 'MM', theta0 = 1.5, n_t = 15, n_c = NULL,
  bar_y_t = 3.5, bar_y_c = NULL, s_t = 1.2, s_c = NULL,
  mu0_c = 1.5, r = 1.2,
  kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
  mu0_t = NULL, sigma0_t = NULL, sigma0_c = NULL,
  m_t = NULL, m_c = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  lower.tail = FALSE
)

# Example 3: External design - posterior probability
pbayespostpred1cont(
  prob = 'posterior', design = 'external', prior = 'N-Inv-Chisq',
  CalcMethod = 'MM', theta0 = 2, n_t = 12, n_c = 12,
  kappa0_t = 5, kappa0_c = 5, nu0_t = 5, nu0_c = 5,
  mu0_t = 5, mu0_c = 5, sigma0_t = sqrt(5), sigma0_c = sqrt(5),
  bar_y_t = 2.5, bar_y_c = 1, s_t = 1.1, s_c = 0.9,
  m_t = NULL, m_c = NULL, r = NULL,
  ne_t = 10, ne_c = 10, alpha0e_t = 0.5, alpha0e_c = 0.5,
  bar_ye_t = 2, bar_ye_c = 0.5, se_t = 1, se_c = 0.8,
  lower.tail = FALSE
)

# Example 4: Controlled design - posterior predictive probability
pbayespostpred1cont(
  prob = 'predictive', design = 'controlled', prior = 'N-Inv-Chisq',
  CalcMethod = 'MM', theta0 = 1, n_t = 12, n_c = 12,
  kappa0_t = 5, kappa0_c = 5, nu0_t = 5, nu0_c = 5,
  mu0_t = 5, mu0_c = 5, sigma0_t = sqrt(5), sigma0_c = sqrt(5),
  bar_y_t = 2, bar_y_c = 0, s_t = 1, s_c = 1,
  m_t = 20, m_c = 20, r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  lower.tail = FALSE
)

# Example 5: Uncontrolled design - posterior predictive probability
pbayespostpred1cont(
  prob = 'predictive', design = 'uncontrolled', prior = 'vague',
  CalcMethod = 'MM', theta0 = 1.5, n_t = 15, n_c = NULL,
  bar_y_t = 3.5, bar_y_c = NULL, s_t = 1.2, s_c = NULL,
  mu0_c = 1.5, r = 1.2,
  kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
  mu0_t = NULL, sigma0_t = NULL, sigma0_c = NULL,
  m_t = 20, m_c = 20,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  lower.tail = FALSE
)

# Example 6: External design - posterior predictive probability
pbayespostpred1cont(
  prob = 'predictive', design = 'external', prior = 'N-Inv-Chisq',
  CalcMethod = 'MM', theta0 = 1, n_t = 12, n_c = 12,
  kappa0_t = 5, kappa0_c = 5, nu0_t = 5, nu0_c = 5,
  mu0_t = 5, mu0_c = 5, sigma0_t = sqrt(5), sigma0_c = sqrt(5),
  bar_y_t = 2.5, bar_y_c = 1, s_t = 1.1, s_c = 0.9,
  m_t = 20, m_c = 20, r = NULL,
  ne_t = 10, ne_c = 10, alpha0e_t = 0.5, alpha0e_c = 0.5,
  bar_ye_t = 2, bar_ye_c = 0.5, se_t = 1, se_c = 0.8,
  lower.tail = FALSE
)

Bayesian Posterior or Posterior Predictive Probability for Two Binary Endpoints

Description

Computes the Bayesian posterior probability or posterior predictive probability for clinical trials with two binary endpoints under a Dirichlet-multinomial conjugate model. The function returns probabilities for nine decision regions (posterior) or four decision regions (predictive) defined by target values (TV) and minimum acceptable values (MAV) for both endpoints. Three study designs are supported: controlled, uncontrolled (hypothetical control), and external (power prior).

Usage

pbayespostpred2bin(
  prob = "posterior",
  design = "controlled",
  theta_TV1 = NULL,
  theta_MAV1 = NULL,
  theta_TV2 = NULL,
  theta_MAV2 = NULL,
  theta_NULL1 = NULL,
  theta_NULL2 = NULL,
  x_t_00,
  x_t_01,
  x_t_10,
  x_t_11,
  x_c_00 = NULL,
  x_c_01 = NULL,
  x_c_10 = NULL,
  x_c_11 = NULL,
  a_t_00,
  a_t_01,
  a_t_10,
  a_t_11,
  a_c_00,
  a_c_01,
  a_c_10,
  a_c_11,
  m_t = NULL,
  m_c = NULL,
  z00 = NULL,
  z01 = NULL,
  z10 = NULL,
  z11 = NULL,
  xe_t_00 = NULL,
  xe_t_01 = NULL,
  xe_t_10 = NULL,
  xe_t_11 = NULL,
  xe_c_00 = NULL,
  xe_c_01 = NULL,
  xe_c_10 = NULL,
  xe_c_11 = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  nMC = 10000L
)

Arguments

Details

Model. The four response categories are ordered as (0, 0), (0, 1), (1, 0), (1, 1). For each group j, the observed count vector follows a multinomial distribution, and a conjugate Dirichlet prior is placed on the cell probability vector p_j:

p_j \sim \mathrm{Dir}(\alpha_{j,00}, \alpha_{j,01}, \alpha_{j,10}, \alpha_{j,11}).

The posterior is

p_j \mid x_j \sim \mathrm{Dir}(\alpha_{j,00} + x_{j,00}, \alpha_{j,01} + x_{j,01}, \alpha_{j,10} + x_{j,10}, \alpha_{j,11} + x_{j,11}).

Marginal response rates are \pi_{j1} = p_{j,10} + p_{j,11} (Endpoint 1) and \pi_{j2} = p_{j,01} + p_{j,11} (Endpoint 2). Treatment effects are \theta_1 = \pi_{t1} - \pi_{c1} and \theta_2 = \pi_{t2} - \pi_{c2}.

Uncontrolled design. When design = 'uncontrolled', the hypothetical control distribution is specified as a Dirichlet distribution directly via the prior hyperparameters and hypothetical control counts z00, z01, z10, z11.

External design. When design = 'external', the power prior augments the Dirichlet prior parameters.

Value

A named numeric vector of region probabilities. For prob = 'posterior': length 9, named R1, ..., R9, corresponding to regions defined by TV and MAV thresholds for both endpoints (row-major order: Endpoint 1 varies slowest). For prob = 'predictive': length 4, named R1, ..., R4. All elements are non-negative and sum to 1.

Examples

# Example 1: Controlled design - posterior probability
pbayespostpred2bin(
  prob = 'posterior', design = 'controlled',
  theta_TV1 = 0.15, theta_MAV1 = 0.05, theta_TV2 = 0.10, theta_MAV2 = 0.0,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  x_t_00 = 2, x_t_01 = 3, x_t_10 = 5, x_t_11 = 2,
  x_c_00 = 3, x_c_01 = 2, x_c_10 = 4, x_c_11 = 1,
  a_t_00 = 1, a_t_01 = 1, a_t_10 = 1, a_t_11 = 1,
  a_c_00 = 1, a_c_01 = 1, a_c_10 = 1, a_c_11 = 1,
  m_t = NULL, m_c = NULL,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 2: Uncontrolled design - posterior probability
pbayespostpred2bin(
  prob = 'posterior', design = 'uncontrolled',
  theta_TV1 = 0.15, theta_MAV1 = 0.05, theta_TV2 = 0.10, theta_MAV2 = 0.0,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  x_t_00 = 2, x_t_01 = 3, x_t_10 = 5, x_t_11 = 2,
  x_c_00 = NULL, x_c_01 = NULL, x_c_10 = NULL, x_c_11 = NULL,
  a_t_00 = 1, a_t_01 = 1, a_t_10 = 1, a_t_11 = 1,
  a_c_00 = 1, a_c_01 = 1, a_c_10 = 1, a_c_11 = 1,
  m_t = NULL, m_c = NULL,
  z00 = 1, z01 = 2, z10 = 2, z11 = 1,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 3: External design - posterior probability
pbayespostpred2bin(
  prob = 'posterior', design = 'external',
  theta_TV1 = 0.15, theta_MAV1 = 0.05, theta_TV2 = 0.10, theta_MAV2 = 0.0,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  x_t_00 = 2, x_t_01 = 3, x_t_10 = 5, x_t_11 = 2,
  x_c_00 = 3, x_c_01 = 2, x_c_10 = 4, x_c_11 = 1,
  a_t_00 = 1, a_t_01 = 1, a_t_10 = 1, a_t_11 = 1,
  a_c_00 = 1, a_c_01 = 1, a_c_10 = 1, a_c_11 = 1,
  m_t = NULL, m_c = NULL,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = 2, xe_t_01 = 2, xe_t_10 = 3, xe_t_11 = 1,
  xe_c_00 = 2, xe_c_01 = 1, xe_c_10 = 2, xe_c_11 = 1,
  alpha0e_t = 0.5, alpha0e_c = 0.5
)

# Example 4: Controlled design - posterior predictive probability
pbayespostpred2bin(
  prob = 'predictive', design = 'controlled',
  theta_TV1 = NULL, theta_MAV1 = NULL, theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.05, theta_NULL2 = 0.0,
  x_t_00 = 2, x_t_01 = 3, x_t_10 = 5, x_t_11 = 2,
  x_c_00 = 3, x_c_01 = 2, x_c_10 = 4, x_c_11 = 1,
  a_t_00 = 1, a_t_01 = 1, a_t_10 = 1, a_t_11 = 1,
  a_c_00 = 1, a_c_01 = 1, a_c_10 = 1, a_c_11 = 1,
  m_t = 20, m_c = 20,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 5: Uncontrolled design - posterior predictive probability
pbayespostpred2bin(
  prob = 'predictive', design = 'uncontrolled',
  theta_TV1 = NULL, theta_MAV1 = NULL, theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.05, theta_NULL2 = 0.0,
  x_t_00 = 2, x_t_01 = 3, x_t_10 = 5, x_t_11 = 2,
  x_c_00 = NULL, x_c_01 = NULL, x_c_10 = NULL, x_c_11 = NULL,
  a_t_00 = 1, a_t_01 = 1, a_t_10 = 1, a_t_11 = 1,
  a_c_00 = 1, a_c_01 = 1, a_c_10 = 1, a_c_11 = 1,
  m_t = 20, m_c = 20,
  z00 = 1, z01 = 2, z10 = 2, z11 = 1,
  xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
  xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
  alpha0e_t = NULL, alpha0e_c = NULL
)

# Example 6: External design - posterior predictive probability
pbayespostpred2bin(
  prob = 'predictive', design = 'external',
  theta_TV1 = NULL, theta_MAV1 = NULL, theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.05, theta_NULL2 = 0.0,
  x_t_00 = 2, x_t_01 = 3, x_t_10 = 5, x_t_11 = 2,
  x_c_00 = 3, x_c_01 = 2, x_c_10 = 4, x_c_11 = 1,
  a_t_00 = 1, a_t_01 = 1, a_t_10 = 1, a_t_11 = 1,
  a_c_00 = 1, a_c_01 = 1, a_c_10 = 1, a_c_11 = 1,
  m_t = 20, m_c = 20,
  z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
  xe_t_00 = 2, xe_t_01 = 2, xe_t_10 = 3, xe_t_11 = 1,
  xe_c_00 = 2, xe_c_01 = 1, xe_c_10 = 2, xe_c_11 = 1,
  alpha0e_t = 0.5, alpha0e_c = 0.5
)

Region Probabilities for Two Continuous Endpoints

Description

Computes the posterior or posterior predictive probability that the bivariate treatment effect \theta = \mu_t - \mu_c falls in each of the 9 (posterior) or 4 (predictive) rectangular regions defined by the decision thresholds.

Usage

pbayespostpred2cont(
  prob,
  design,
  prior,
  theta_TV1 = NULL,
  theta_MAV1 = NULL,
  theta_TV2 = NULL,
  theta_MAV2 = NULL,
  theta_NULL1 = NULL,
  theta_NULL2 = NULL,
  n_t,
  n_c = NULL,
  ybar_t,
  S_t,
  ybar_c = NULL,
  S_c = NULL,
  m_t = NULL,
  m_c = NULL,
  kappa0_t = NULL,
  nu0_t = NULL,
  mu0_t = NULL,
  Lambda0_t = NULL,
  kappa0_c = NULL,
  nu0_c = NULL,
  mu0_c = NULL,
  Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL,
  ne_c = NULL,
  alpha0e_t = NULL,
  alpha0e_c = NULL,
  bar_ye_t = NULL,
  bar_ye_c = NULL,
  se_t = NULL,
  se_c = NULL,
  nMC = 10000L,
  CalcMethod = "MC"
)

Arguments

Details

Model. Both endpoints follow a bivariate Normal distribution y_{k,j} \sim N_2(\mu_k, \Sigma_k) for group k \in \{t, c\}. The treatment effect is \theta = \mu_t - \mu_c.

Posterior distribution (vague prior).

\mu_k | Y_k \sim t_{n_k - 2}\!\left(\bar{y}_k,\; \frac{S_k}{n_k(n_k - 2)}\right)

Posterior distribution (NIW prior).

\mu_k | Y_k \sim t_{\nu_{nk} - 1}\!\left(\mu_{nk},\; \frac{\Lambda_{nk}}{\kappa_{nk}(\nu_{nk} - 1)}\right)

with updated hyperparameters \kappa_{nk} = \kappa_{0k} + n_k, \nu_{nk} = \nu_{0k} + n_k, \mu_{nk} = (\kappa_{0k}\mu_{0k} + n_k\bar{y}_k)/\kappa_{nk}, and \Lambda_{nk} = \Lambda_{0k} + S_k + \kappa_{0k}n_k(\bar{y}_k - \mu_{0k})(\bar{y}_k - \mu_{0k})^T / \kappa_{nk}.

Predictive distribution. The scale matrix of a single future observation is inflated by (1 + n_k)/n_k (vague) or (1 + \kappa_{nk})/\kappa_{nk} (NIW) relative to the posterior. The mean of m_k future observations has scale divided by m_k.

Posterior probability regions (prob = 'posterior'). Row-major 3x3 grid; Endpoint 1 varies slowest:

• R1: \theta_1 > TV_1 AND \theta_2 > TV_2

• R2: \theta_1 > TV_1 AND TV_2 \ge \theta_2 > MAV_2

• R3: \theta_1 > TV_1 AND \theta_2 \le MAV_2

• R4: TV_1 \ge \theta_1 > MAV_1 AND \theta_2 > TV_2

• R5: TV_1 \ge \theta_1 > MAV_1 AND TV_2 \ge \theta_2 > MAV_2

• R6: TV_1 \ge \theta_1 > MAV_1 AND \theta_2 \le MAV_2

• R7: \theta_1 \le MAV_1 AND \theta_2 > TV_2

• R8: \theta_1 \le MAV_1 AND TV_2 \ge \theta_2 > MAV_2

• R9: \theta_1 \le MAV_1 AND \theta_2 \le MAV_2

Predictive probability regions (prob = 'predictive'). Row-major 2x2 grid; Endpoint 1 varies slowest:

• R1: \tilde\theta_1 > \theta_{\rm NULL1} AND \tilde\theta_2 > \theta_{\rm NULL2}

• R2: \tilde\theta_1 > \theta_{\rm NULL1} AND \tilde\theta_2 \le \theta_{\rm NULL2}

• R3: \tilde\theta_1 \le \theta_{\rm NULL1} AND \tilde\theta_2 > \theta_{\rm NULL2}

• R4: \tilde\theta_1 \le \theta_{\rm NULL1} AND \tilde\theta_2 \le \theta_{\rm NULL2}

Uncontrolled design. Under NIW prior: \mu_c \sim t_{\nu_{nt} - 1}(\mu_{0c},\; r\Lambda_{nt} / [\kappa_{nt}(\nu_{nt} - 1)]). The parameter r allows the variance scale of the hypothetical control to differ from the treatment group.

External design. Incorporated via the power prior with NIW conjugate representation. The effective posterior hyperparameters are obtained by constructing the power prior from external data with weight a_0, then updating with current PoC data (see Conjugate_power_prior.pdf, Theorem 5).

Vectorised usage. When ybar_t is supplied as an N \times 2 matrix and S_t as a list of N scatter matrices, the function computes region probabilities for all N observations in a single call, returning an N \times n_{\rm regions} matrix. For CalcMethod = 'MC', standard normal and chi-squared variates are pre-generated once (size nMC) and reused across all observations, with only the Cholesky factor of the replicate-specific scale matrix recomputed per observation. For CalcMethod = 'MM', the moment-matching parameters are computed per observation (since they depend on the replicate-specific V_k) and mvtnorm::pmvt is called once per region per observation.

Value

When ybar_t is a length-2 vector (single observation): a named numeric vector of length 9 (R1–R9) for prob = 'posterior' or length 4 (R1–R4) for prob = 'predictive'. All elements are non-negative and sum to 1.

    When \code{ybar_t} is an \eqn{N \times 2} matrix (vectorised call):
    a numeric matrix with \eqn{N} rows and 9 (or 4) columns, with
    column names \code{R1}--\code{R9} (or \code{R1}--\code{R4}). Each
    row sums to 1.

Examples

# Example 1: Controlled design - posterior probability, vague prior
S_t <- matrix(c(18.0, 3.6, 3.6, 9.0), 2, 2)
S_c <- matrix(c(16.0, 2.8, 2.8, 8.5), 2, 2)
pbayespostpred2cont(
  prob = 'posterior', design = 'controlled', prior = 'vague',
  theta_TV1 = 1.5, theta_MAV1 = 0.5,
  theta_TV2 = 1.0, theta_MAV2 = 0.3,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  n_t = 12L, n_c = 12L,
  ybar_t = c(3.5, 2.1), S_t = S_t,
  ybar_c = c(1.8, 1.0), S_c = S_c,
  m_t = NULL, m_c = NULL,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = NULL, Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 1000L
)

# Example 2: Uncontrolled design - posterior probability, NIW prior
S_t <- matrix(c(18.0, 3.6, 3.6, 9.0), 2, 2)
L0  <- matrix(c(20.0, 0.0, 0.0, 10.0), 2, 2)
pbayespostpred2cont(
  prob = 'posterior', design = 'uncontrolled', prior = 'N-Inv-Wishart',
  theta_TV1 = 1.5, theta_MAV1 = 0.5,
  theta_TV2 = 1.0, theta_MAV2 = 0.3,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  n_t = 12L, n_c = NULL,
  ybar_t = c(3.5, 2.1), S_t = S_t,
  ybar_c = NULL, S_c = NULL,
  m_t = NULL, m_c = NULL,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = c(1.0, 0.5), Lambda0_c = NULL,
  r = 1.0,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 1000L
)

# Example 3: External design - posterior probability, NIW prior
S_t  <- matrix(c(18.0, 3.6, 3.6, 9.0), 2, 2)
S_c  <- matrix(c(16.0, 2.8, 2.8, 8.5), 2, 2)
L0   <- matrix(c(20.0, 0.0, 0.0, 10.0), 2, 2)
se_c <- matrix(c(15.0, 2.5, 2.5, 7.5), 2, 2)
pbayespostpred2cont(
  prob = 'posterior', design = 'external', prior = 'N-Inv-Wishart',
  theta_TV1 = 1.5, theta_MAV1 = 0.5,
  theta_TV2 = 1.0, theta_MAV2 = 0.3,
  theta_NULL1 = NULL, theta_NULL2 = NULL,
  n_t = 12L, n_c = 12L,
  ybar_t = c(3.5, 2.1), S_t = S_t,
  ybar_c = c(1.8, 1.0), S_c = S_c,
  m_t = NULL, m_c = NULL,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = 2.0, nu0_c = 5.0, mu0_c = c(1.0, 0.5), Lambda0_c = L0,
  r = NULL,
  ne_t = NULL, ne_c = 10L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = c(1.5, 0.8), se_t = NULL, se_c = se_c,
  nMC = 1000L
)

# Example 4: Controlled design - posterior predictive probability, vague prior
S_t <- matrix(c(18.0, 3.6, 3.6, 9.0), 2, 2)
S_c <- matrix(c(16.0, 2.8, 2.8, 8.5), 2, 2)
pbayespostpred2cont(
  prob = 'predictive', design = 'controlled', prior = 'vague',
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.5, theta_NULL2 = 0.3,
  n_t = 12L, n_c = 12L,
  ybar_t = c(3.5, 2.1), S_t = S_t,
  ybar_c = c(1.8, 1.0), S_c = S_c,
  m_t = 30L, m_c = 30L,
  kappa0_t = NULL, nu0_t = NULL, mu0_t = NULL, Lambda0_t = NULL,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = NULL, Lambda0_c = NULL,
  r = NULL,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 1000L
)

# Example 5: Uncontrolled design - posterior predictive probability, NIW prior
S_t <- matrix(c(18.0, 3.6, 3.6, 9.0), 2, 2)
L0  <- matrix(c(20.0, 0.0, 0.0, 10.0), 2, 2)
pbayespostpred2cont(
  prob = 'predictive', design = 'uncontrolled', prior = 'N-Inv-Wishart',
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.5, theta_NULL2 = 0.3,
  n_t = 12L, n_c = NULL,
  ybar_t = c(3.5, 2.1), S_t = S_t,
  ybar_c = NULL, S_c = NULL,
  m_t = 30L, m_c = 30L,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = NULL, nu0_c = NULL, mu0_c = c(1.0, 0.5), Lambda0_c = NULL,
  r = 1.0,
  ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
  bar_ye_t = NULL, bar_ye_c = NULL, se_t = NULL, se_c = NULL,
  nMC = 1000L
)

# Example 6: External design - posterior predictive probability, NIW prior
S_t  <- matrix(c(18.0, 3.6, 3.6, 9.0), 2, 2)
S_c  <- matrix(c(16.0, 2.8, 2.8, 8.5), 2, 2)
L0   <- matrix(c(20.0, 0.0, 0.0, 10.0), 2, 2)
se_c <- matrix(c(15.0, 2.5, 2.5, 7.5), 2, 2)
pbayespostpred2cont(
  prob = 'predictive', design = 'external', prior = 'N-Inv-Wishart',
  theta_TV1 = NULL, theta_MAV1 = NULL,
  theta_TV2 = NULL, theta_MAV2 = NULL,
  theta_NULL1 = 0.5, theta_NULL2 = 0.3,
  n_t = 12L, n_c = 12L,
  ybar_t = c(3.5, 2.1), S_t = S_t,
  ybar_c = c(1.8, 1.0), S_c = S_c,
  m_t = 30L, m_c = 30L,
  kappa0_t = 2.0, nu0_t = 5.0, mu0_t = c(2.0, 1.0), Lambda0_t = L0,
  kappa0_c = 2.0, nu0_c = 5.0, mu0_c = c(1.0, 0.5), Lambda0_c = L0,
  r = NULL,
  ne_t = NULL, ne_c = 10L, alpha0e_t = NULL, alpha0e_c = 0.5,
  bar_ye_t = NULL, bar_ye_c = c(1.5, 0.8), se_t = NULL, se_c = se_c,
  nMC = 1000L
)

Cumulative Distribution Function of the Difference Between Two Independent Beta-Binomial Proportions

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent Beta-Binomial proportions by exact enumeration. Specifically, computes P((Y_t / m_t) - (Y_c / m_c) \le q) or P((Y_t / m_t) - (Y_c / m_c) > q), where Y_j \sim \mathrm{BetaBinomial}(m_j, \alpha_j, \beta_j) for j \in \{t, c\}.

Usage

pbetabinomdiff(
  q,
  m_t,
  m_c,
  alpha_t,
  alpha_c,
  beta_t,
  beta_c,
  lower.tail = TRUE
)

Arguments

Details

The probability mass function of Y_j \sim \mathrm{BetaBinomial}(m_j, \alpha_j, \beta_j) is:

P(Y_j = k) = \binom{m_j}{k} \frac{B(k + \alpha_j,\; m_j - k + \beta_j)}{B(\alpha_j, \beta_j)}, \quad k = 0, \ldots, m_j

where B(\cdot, \cdot) is the Beta function.

The exact CDF is obtained by enumerating all (m_t + 1)(m_c + 1) outcome combinations and summing the joint probabilities for which the proportion difference satisfies the specified condition. Computation time therefore grows quadratically in m_t and m_c; for large future sample sizes consider a normal approximation.

The Beta-Binomial distribution arises when the success probability in a Binomial model follows a Beta prior, making it appropriate for posterior predictive calculations in Bayesian binary-endpoint trials.

Value

A numeric scalar in [0, 1].

Examples

# P((Y_t/12) - (Y_c/12) > 0.2) with symmetric Beta(0.5, 0.5) priors
pbetabinomdiff(0.2, 12, 12, 0.5, 0.5, 0.5, 0.5, lower.tail = FALSE)

# P((Y_t/20) - (Y_c/15) > 0.1) with different future sample sizes
pbetabinomdiff(0.1, 20, 15, 1, 1, 1, 1, lower.tail = FALSE)

# P((Y_t/10) - (Y_c/10) > 0) with informative priors
pbetabinomdiff(0, 10, 10, 2, 3, 3, 2, lower.tail = FALSE)

# Lower tail: P((Y_t/15) - (Y_c/15) <= 0.05) with vague priors
pbetabinomdiff(0.05, 15, 15, 1, 1, 1, 1, lower.tail = TRUE)

Cumulative Distribution Function of the Difference Between Two Independent Beta Variables

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent Beta-distributed random variables. Specifically, computes P(\pi_t - \pi_c \le q) or P(\pi_t - \pi_c > q), where \pi_j \sim \mathrm{Beta}(\alpha_j, \beta_j) for j \in \{t, c\}.

Usage

pbetadiff(q, alpha_t, alpha_c, beta_t, beta_c, lower.tail = TRUE)

Arguments

Details

The upper-tail probability is obtained via the convolution formula:

P(\pi_t - \pi_c > q) = \int_0^1 F_{\mathrm{Beta}(\alpha_c, \beta_c)}(x - q)\, f_{\mathrm{Beta}(\alpha_t, \beta_t)}(x)\, dx

where f_{\mathrm{Beta}(\alpha_t, \beta_t)} is the density of \pi_t and F_{\mathrm{Beta}(\alpha_c, \beta_c)} is the CDF of \pi_c. Boundary cases are handled automatically by pbeta (0 for x - q \le 0, 1 for x - q \ge 1), so integrating over [0, 1] is safe.

This single-integral convolution replaces an equivalent double-integral formulation based on Appell's F1 hypergeometric function, yielding the same result with lower computational cost because both pbeta and dbeta are implemented in compiled C code.

Value

A numeric scalar in [0, 1].

Examples

# P(pi_t - pi_c > 0.2) with symmetric Beta(0.5, 0.5) priors
pbetadiff(0.2, 0.5, 0.5, 0.5, 0.5, lower.tail = FALSE)

# P(pi_t - pi_c > -0.1) with informative priors
pbetadiff(-0.1, 2, 1, 3, 4, lower.tail = FALSE)

# P(pi_t - pi_c > 0) with equal priors -- should be approximately 0.5
pbetadiff(0, 1, 1, 1, 1, lower.tail = FALSE)

# Lower tail: P(pi_t - pi_c <= 0.1) with symmetric priors
pbetadiff(0.1, 2, 2, 2, 2, lower.tail = TRUE)

Plot Method for getgamma1bin Objects

Description

Displays calibration curves of marginal Go and NoGo probabilities against the threshold grid \gamma for results returned by getgamma1bin.

Usage

## S3 method for class 'getgamma1bin'
plot(
  x,
  title = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  base_size = 28,
  ...
)

Arguments

Details

The x-axis represents candidate threshold values \gamma \in (0, 1). The y-axis represents the marginal probability:

• Go curve: \Pr(g_{\mathrm{Go}} \ge \gamma) evaluated under the Go-calibration scenario (typically the Null scenario).

• NoGo curve: \Pr(g_{\mathrm{NoGo}} \ge \gamma) evaluated under the NoGo-calibration scenario (typically the Alternative scenario).

Horizontal reference lines are drawn at target_go and target_nogo. Filled circles (geom_point) mark the optimal thresholds (\gamma_{\mathrm{go}},\, \Pr(\mathrm{Go})_{\mathrm{opt}}) and (\gamma_{\mathrm{nogo}},\, \Pr(\mathrm{NoGo})_{\mathrm{opt}}), with their values shown in the legend. If either optimal threshold is NA, the corresponding point is omitted.

Value

Invisibly returns a ggplot object.

Plot Method for getgamma1cont Objects

Description

Displays calibration curves of marginal Go and NoGo probabilities against the threshold grid \gamma for results returned by getgamma1cont.

Usage

## S3 method for class 'getgamma1cont'
plot(
  x,
  title = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  base_size = 28,
  ...
)

Arguments

Details

The x-axis represents candidate threshold values \gamma \in (0, 1). The y-axis represents the marginal probability:

• Go curve: \Pr(g_{\mathrm{Go}} \ge \gamma) evaluated under the Go-calibration scenario (typically the Null scenario).

• NoGo curve: \Pr(g_{\mathrm{NoGo}} \ge \gamma) evaluated under the NoGo-calibration scenario (typically the Alternative scenario).

Horizontal reference lines are drawn at target_go and target_nogo. Filled circles (geom_point) mark the optimal thresholds (\gamma_{\mathrm{go}},\, \Pr(\mathrm{Go})_{\mathrm{opt}}) and (\gamma_{\mathrm{nogo}},\, \Pr(\mathrm{NoGo})_{\mathrm{opt}}), with their values shown in the legend. If either optimal threshold is NA, the corresponding point is omitted.

Value

Invisibly returns a ggplot object.

Plot Method for getgamma2bin Objects

Description

Displays calibration curves of marginal Go and NoGo probabilities against the threshold grid \gamma for results returned by getgamma2bin.

Usage

## S3 method for class 'getgamma2bin'
plot(
  x,
  title = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  base_size = 28,
  ...
)

Arguments

Details

The x-axis represents candidate threshold values \gamma \in (0, 1). The y-axis represents the marginal probability:

• Go curve: \Pr(g_{\mathrm{Go}} \ge \gamma) evaluated under the Go-calibration scenario (typically the Null scenario).

• NoGo curve: \Pr(g_{\mathrm{NoGo}} \ge \gamma) evaluated under the NoGo-calibration scenario (typically the Alternative scenario).

Horizontal reference lines are drawn at target_go and target_nogo. Filled circles (geom_point) mark the optimal thresholds (\gamma_{\mathrm{go}},\, \Pr(\mathrm{Go})_{\mathrm{opt}}) and (\gamma_{\mathrm{nogo}},\, \Pr(\mathrm{NoGo})_{\mathrm{opt}}), with their values shown in the legend. If either optimal threshold is NA, the corresponding point is omitted.

Value

Invisibly returns a ggplot object.

Plot Method for getgamma2cont Objects

Description

Displays calibration curves of marginal Go and NoGo probabilities against the threshold grid \gamma for results returned by getgamma2cont.

Usage

## S3 method for class 'getgamma2cont'
plot(
  x,
  title = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  base_size = 28,
  ...
)

Arguments

Details

The x-axis represents candidate threshold values \gamma \in (0, 1). The y-axis represents the marginal probability:

• Go curve: \Pr(g_{\mathrm{Go}} \ge \gamma) evaluated under the Go-calibration scenario (typically the Null scenario).

• NoGo curve: \Pr(g_{\mathrm{NoGo}} \ge \gamma) evaluated under the NoGo-calibration scenario (typically the Alternative scenario).

Horizontal reference lines are drawn at target_go and target_nogo. Filled circles (geom_point) mark the optimal thresholds (\gamma_{\mathrm{go}},\, \Pr(\mathrm{Go})_{\mathrm{opt}}) and (\gamma_{\mathrm{nogo}},\, \Pr(\mathrm{NoGo})_{\mathrm{opt}}), with their values shown in the legend. If either optimal threshold is NA, the corresponding point is omitted.

Value

Invisibly returns a ggplot object.

Plot Method for pbayesdecisionprob1bin Objects

Description

Displays an operating characteristics curve of Go/NoGo/Gray decision probabilities against the true treatment effect for binary endpoint results returned by pbayesdecisionprob1bin.

Usage

## S3 method for class 'pbayesdecisionprob1bin'
plot(
  x,
  title = NULL,
  xlab = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  col_gray = "#939597",
  base_size = 28,
  ...
)

Arguments

Details

For design = 'controlled' or design = 'external', the x-axis represents the treatment-minus-control difference \theta = \pi_t - \bar{\pi}_c, where \bar{\pi}_c is the mean of the supplied pi_c values. For design = 'uncontrolled', the x-axis represents \pi_t directly.

Vertical reference lines are drawn at the decision thresholds:

• When prob = 'posterior': lines at \theta_{TV} and \theta_{MAV} (converted to the x-axis scale).

• When prob = 'predictive': a single line at \theta_{NULL}.

Value

Invisibly returns a ggplot object.

Plot Method for pbayesdecisionprob1cont Objects

Description

Displays an operating characteristics curve of Go/NoGo/Gray decision probabilities against the true treatment effect for continuous endpoint results returned by pbayesdecisionprob1cont.

Usage

## S3 method for class 'pbayesdecisionprob1cont'
plot(
  x,
  title = NULL,
  xlab = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  col_gray = "#939597",
  base_size = 28,
  ...
)

Arguments

Details

For design = 'controlled' or design = 'external', the x-axis represents the treatment-minus-control difference \theta = \mu_t - \bar{\mu}_c, where \bar{\mu}_c is the mean of the supplied mu_c values. For design = 'uncontrolled', the x-axis represents \mu_t directly.

Vertical reference lines are drawn at the decision thresholds:

• When prob = 'posterior': lines at \theta_{TV} and \theta_{MAV} (converted to the x-axis scale).

• When prob = 'predictive': a single line at \theta_{NULL}.

Value

Invisibly returns a ggplot object.

Plot Method for pbayesdecisionprob2bin Objects

Description

Displays operating characteristics for two-binary-endpoint results returned by pbayesdecisionprob2bin.

Usage

## S3 method for class 'pbayesdecisionprob2bin'
plot(
  x,
  which = "Go",
  title = NULL,
  xlab = NULL,
  ylab = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  col_gray = "#939597",
  base_size = 28,
  ...
)

Arguments

Details

When the input scenarios form a regular grid over (pi_t1, pi_t2) (i.e., every combination of the unique values of pi_t1 and pi_t2 is present) and rho_t is constant, the function produces a filled tile plot: each panel (Go, Gray, NoGo) is coloured by its own probability on a continuous gradient (white to the panel colour), so intensity directly reflects the probability magnitude. A solid white contour line is overlaid at the corresponding decision threshold (gamma_go for the Go panel, gamma_nogo for the NoGo panel, and their mean for the Gray panel) to mark the boundary where the probability equals the threshold. Otherwise the function falls back to a scatter plot in which point colour encodes the decision probability on a continuous scale.

When which = "all", the three panels are arranged side-by-side using gridExtra::grid.arrange, so each panel retains its own independent colour scale. This requires the gridExtra package.

For design = 'controlled' or design = 'external', both axes are expressed as treatment-minus-control differences: \theta_1 = \pi_{t1} - \bar{\pi}_{c1} and \theta_2 = \pi_{t2} - \bar{\pi}_{c2}, where \bar{\pi}_{c1} and \bar{\pi}_{c2} are the means of the supplied pi_c1 and pi_c2 vectors. For design = 'uncontrolled', the axes represent \pi_{t1} and \pi_{t2} directly.

Vertical and horizontal reference lines are drawn at the decision thresholds:

• When prob = 'posterior': vertical lines at \theta_{TV1} and \theta_{MAV1} (x-axis) and horizontal lines at \theta_{TV2} and \theta_{MAV2} (y-axis).

• When prob = 'predictive': a single vertical line at \theta_{NULL1} and a single horizontal line at \theta_{NULL2}.

Value

Invisibly returns a ggplot object (single panel) or a gtable object (which = "all").

Plot Method for pbayesdecisionprob2cont Objects

Description

Displays operating characteristics for two-continuous-endpoint results returned by pbayesdecisionprob2cont.

Usage

## S3 method for class 'pbayesdecisionprob2cont'
plot(
  x,
  which = "Go",
  title = NULL,
  xlab = NULL,
  ylab = NULL,
  col_go = "#658D1B",
  col_nogo = "#D91E49",
  col_gray = "#939597",
  base_size = 28,
  ...
)

Arguments

Details

When the input scenarios form a regular grid over (mu_t1, mu_t2) (i.e., every combination of the unique values of mu_t1 and mu_t2 is present) and rho_t is constant, the function produces a filled tile plot: each panel (Go, Gray, NoGo) is coloured by its own probability on a continuous gradient (white to the panel colour), so intensity directly reflects the probability magnitude. Otherwise the function falls back to a scatter plot in which point colour encodes the decision probability on a continuous scale.

When which = "all", the three panels are arranged side-by-side using gridExtra::grid.arrange, so each panel retains its own independent colour scale. This requires the gridExtra package.

For design = 'controlled' or design = 'external', both axes are expressed as treatment-minus-control differences: \theta_1 = \mu_{t1} - \bar{\mu}_{c1} and \theta_2 = \mu_{t2} - \bar{\mu}_{c2}, where \bar{\mu}_{c1} and \bar{\mu}_{c2} are the means of the supplied mu_c1 and mu_c2 vectors. For design = 'uncontrolled', the axes represent \mu_{t1} and \mu_{t2} directly.

Vertical and horizontal reference lines are drawn at the decision thresholds:

• When prob = 'posterior': vertical lines at \theta_{TV1} and \theta_{MAV1} (x-axis) and horizontal lines at \theta_{TV2} and \theta_{MAV2} (y-axis).

• When prob = 'predictive': a single vertical line at \theta_{NULL1} and a single horizontal line at \theta_{NULL2}.

Value

Invisibly returns a ggplot object (single panel) or a gtable object (which = "all").

Print Method for pbayesdecisionprob1bin Objects

Description

Displays a formatted summary of Go/NoGo/Gray decision probabilities for binary endpoint results returned by pbayesdecisionprob1bin.

Usage

## S3 method for class 'pbayesdecisionprob1bin'
print(x, digits = 4, ...)

Arguments

Value

Invisibly returns x.

Print Method for pbayesdecisionprob1cont Objects

Description

Displays a formatted summary of Go/NoGo/Gray decision probabilities for continuous endpoint results returned by pbayesdecisionprob1cont.

Usage

## S3 method for class 'pbayesdecisionprob1cont'
print(x, digits = 4, ...)

Arguments

Value

Invisibly returns x.

Print Method for pbayesdecisionprob2bin Objects

Description

Displays a formatted summary of Go/NoGo/Gray decision probabilities for two-binary-endpoint results returned by pbayesdecisionprob2bin.

Usage

## S3 method for class 'pbayesdecisionprob2bin'
print(x, digits = 4, ...)

Arguments

Value

Invisibly returns x.

Print Method for pbayesdecisionprob2cont Objects

Description

Displays a formatted summary of Go/NoGo/Gray decision probabilities for two-continuous-endpoint results returned by pbayesdecisionprob2cont.

Displays a formatted summary of Go/NoGo/Gray decision probabilities for two-continuous-endpoint results returned by pbayesdecisionprob2cont.

Usage

## S3 method for class 'pbayesdecisionprob2cont'
print(x, digits = 4, ...)

## S3 method for class 'pbayesdecisionprob2cont'
print(x, digits = 4, ...)

Arguments

Value

Invisibly returns x.

Invisibly returns x.

Cumulative Distribution Function of the Difference of Two Independent t-Distributed Variables via Monte Carlo Simulation

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent non-standardised t-distributed random variables using Monte Carlo simulation. Specifically, computes P(T_t - T_c \le q) or P(T_t - T_c > q), where T_k \sim t(\mu_k, \sigma_k^2, \nu_k) for k \in \{t, c\}.

Usage

ptdiff_MC(nMC, q, mu_t, mu_c, sd_t, sd_c, nu_t, nu_c, lower.tail = TRUE)

Arguments

Details

The algorithm proceeds as follows:

1. Generate an nMC x n matrix of standard t draws for T_t (\nu_t degrees of freedom), then scale and shift each column by sd_t[i] and mu_t[i].

2. Repeat for T_c with \nu_c degrees of freedom.

3. Compute the nMC x n difference matrix D = T_t - T_c.

4. Return colMeans(D > q) as the estimated P(T_t - T_c > q) for each parameter set.

All operations are matrix-based (no R-level loop over parameter sets), so performance scales well with n. However, note that the matrix size is nMC x n, so memory usage grows linearly with both nMC and n. When n is large (e.g., nsim x n_scenarios in pbayesdecisionprob1cont), memory requirements can become prohibitive; in such cases prefer CalcMethod = 'MM'.

Monte Carlo error is approximately \sqrt{p(1 - p) / \mathrm{nMC}}; near p = 0.5 this is roughly 0.5 / \sqrt{\mathrm{nMC}}.

Value

A numeric scalar or vector in [0, 1]. When mu_t, mu_c, sd_t, or sd_c are vectors of length n, a vector of length n is returned.

Examples

# P(T_t - T_c > 3) with equal parameters
ptdiff_MC(nMC = 1e5, q = 3, mu_t = 2, mu_c = 0, sd_t = 1, sd_c = 1,
          nu_t = 17, nu_c = 17, lower.tail = FALSE)

# P(T_t - T_c > 1) with unequal scales
ptdiff_MC(nMC = 1e5, q = 1, mu_t = 5, mu_c = 3, sd_t = 2, sd_c = 1.5,
          nu_t = 10, nu_c = 15, lower.tail = FALSE)

# P(T_t - T_c > 0) with different degrees of freedom
ptdiff_MC(nMC = 1e5, q = 0, mu_t = 1, mu_c = 1, sd_t = 1, sd_c = 1,
          nu_t = 5, nu_c = 20, lower.tail = FALSE)

# Lower tail: P(T_t - T_c <= 2)
ptdiff_MC(nMC = 1e5, q = 2, mu_t = 3, mu_c = 0, sd_t = 1.5, sd_c = 1.2,
          nu_t = 12, nu_c = 15, lower.tail = TRUE)

# Vectorised usage
ptdiff_MC(nMC = 1e5, q = 1, mu_t = c(2, 3, 4), mu_c = c(0, 1, 2),
          sd_t = c(1, 1.2, 1.5), sd_c = c(1, 1.1, 1.3),
          nu_t = 10, nu_c = 10, lower.tail = FALSE)

Cumulative Distribution Function of the Difference of Two Independent t-Distributed Variables via Moment-Matching Approximation

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent non-standardised t-distributed random variables using a Moment-Matching approximation. Specifically, computes P(T_t - T_c \le q) or P(T_t - T_c > q), where T_k \sim t(\mu_k, \sigma_k^2, \nu_k) for k \in \{t, c\}.

Usage

ptdiff_MM(q, mu_t, mu_c, sd_t, sd_c, nu_t, nu_c, lower.tail = TRUE)

Arguments

Details

The difference D = T_t - T_c is approximated by a single non-standardised t-distribution t(\mu^*, {\sigma^*}^2, \nu^*) whose parameters are determined by matching the first two even moments of D:

• \mu^* = \mu_t - \mu_c.

• {\sigma^*}^2 is obtained from the second-moment equation.

• \nu^* is obtained from the fourth-moment equation.

The approximation requires \nu_t > 4 and \nu_c > 4 for finite fourth moments. It is exact in the normal limit (\nu \to \infty) and works well in practice when \nu > 10. Because it reduces to a single call to pt(), it is orders of magnitude faster than the numerical integration method (ptdiff_NI) and is fully vectorised.

Value

A numeric scalar or vector in [0, 1]. When mu_t, mu_c, sd_t, or sd_c are vectors of length n, a vector of length n is returned.

Examples

# P(T_t - T_c > 3) with equal parameters
ptdiff_MM(q = 3, mu_t = 2, mu_c = 0, sd_t = 1, sd_c = 1,
          nu_t = 17, nu_c = 17, lower.tail = FALSE)

# P(T_t - T_c > 1) with unequal scales
ptdiff_MM(q = 1, mu_t = 5, mu_c = 3, sd_t = 2, sd_c = 1.5,
          nu_t = 10, nu_c = 15, lower.tail = FALSE)

# P(T_t - T_c > 0) with different degrees of freedom
ptdiff_MM(q = 0, mu_t = 1, mu_c = 1, sd_t = 1, sd_c = 1,
          nu_t = 5, nu_c = 20, lower.tail = FALSE)

# Lower tail: P(T_t - T_c <= 2)
ptdiff_MM(q = 2, mu_t = 3, mu_c = 0, sd_t = 1.5, sd_c = 1.2,
          nu_t = 12, nu_c = 15, lower.tail = TRUE)

# Vectorised usage
ptdiff_MM(q = 1, mu_t = c(2, 3, 4), mu_c = c(0, 1, 2),
          sd_t = c(1, 1.2, 1.5), sd_c = c(1, 1.1, 1.3),
          nu_t = 10, nu_c = 10, lower.tail = FALSE)

Cumulative Distribution Function of the Difference of Two Independent t-Distributed Variables via Numerical Integration

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent non-standardised t-distributed random variables using numerical integration. Specifically, computes P(T_t - T_c \le q) or P(T_t - T_c > q), where T_k \sim t(\mu_k, \sigma_k^2, \nu_k) for k \in \{t, c\}.

Usage

ptdiff_NI(q, mu_t, mu_c, sd_t, sd_c, nu_t, nu_c, lower.tail = TRUE)

Arguments

Details

The upper-tail probability is obtained via the convolution formula:

P(T_t - T_c > q) = \int_{-\infty}^{\infty} F_{t(\mu_c, \sigma_c^2, \nu_c)}(x - q)\, f_{t(\mu_t, \sigma_t^2, \nu_t)}(x)\, dx

where f_{t(\mu_t, \sigma_t^2, \nu_t)} is the density of T_t and F_{t(\mu_c, \sigma_c^2, \nu_c)} is the CDF of T_c. The integral is evaluated by adaptive Gauss-Kronrod quadrature via stats::integrate.

When the input parameters are vectors, mapply applies the scalar integration function across all parameter sets.

Advantages:

• Exact within numerical precision.

• Handles arbitrary parameter combinations.

Computational note: this method is substantially slower than the Moment-Matching approximation (ptdiff_MM) because it calls integrate() once per parameter set. For large-scale simulation (many parameter sets), prefer CalcMethod = 'MM' in pbayesdecisionprob1cont.

Value

A numeric scalar or vector in [0, 1]. When mu_t, mu_c, sd_t, or sd_c are vectors of length n, a vector of length n is returned.

Examples

# P(T_t - T_c > 3) with equal parameters
ptdiff_NI(q = 3, mu_t = 2, mu_c = 0, sd_t = 1, sd_c = 1,
          nu_t = 17, nu_c = 17, lower.tail = FALSE)

# P(T_t - T_c > 1) with unequal scales
ptdiff_NI(q = 1, mu_t = 5, mu_c = 3, sd_t = 2, sd_c = 1.5,
          nu_t = 10, nu_c = 15, lower.tail = FALSE)

# P(T_t - T_c > 0) with different degrees of freedom
ptdiff_NI(q = 0, mu_t = 1, mu_c = 1, sd_t = 1, sd_c = 1,
          nu_t = 5, nu_c = 20, lower.tail = FALSE)

# Lower tail: P(T_t - T_c <= 2)
ptdiff_NI(q = 2, mu_t = 3, mu_c = 0, sd_t = 1.5, sd_c = 1.2,
          nu_t = 12, nu_c = 15, lower.tail = TRUE)

# Vectorised usage
ptdiff_NI(q = 1, mu_t = c(2, 3, 4), mu_c = c(0, 1, 2),
          sd_t = c(1, 1.2, 1.5), sd_c = c(1, 1.1, 1.3),
          nu_t = 10, nu_c = 10, lower.tail = FALSE)

Generate Random Samples from a Dirichlet Distribution

Description

Generates random samples from a Dirichlet distribution using the Gamma representation: if Y_i \sim \mathrm{Gamma}(\alpha_i, 1) independently for i = 1, \ldots, K, then (Y_1 / S, \ldots, Y_K / S) \sim \mathrm{Dirichlet}(\alpha_1, \ldots, \alpha_K), where S = \sum_{i=1}^{K} Y_i.

Usage

rdirichlet(n, alpha)

Arguments

Details

The Dirichlet distribution is a multivariate generalisation of the Beta distribution and is commonly used as a conjugate prior for multinomial proportions in Bayesian statistics.

The probability density function is:

f(x_1, \ldots, x_K) = \frac{\Gamma\!\left(\sum_{i=1}^{K} \alpha_i\right)} {\prod_{i=1}^{K} \Gamma(\alpha_i)} \prod_{i=1}^{K} x_i^{\alpha_i - 1}

where x_i > 0 and \sum_{i=1}^{K} x_i = 1.

Key properties:

• Each marginal follows a Beta distribution: X_i \sim \mathrm{Beta}\!\left(\alpha_i,\, \sum_{l \neq i} \alpha_l\right).

• E[X_i] = \alpha_i / \sum_{l=1}^{K} \alpha_l.

• Components are negatively correlated unless one component dominates.

Implementation steps:

1. Generate independent Y_i \sim \mathrm{Gamma}(\alpha_i, 1) for each i = 1, \ldots, K.

2. Normalise: X_i = Y_i / \sum_{l=1}^{K} Y_l.

Value

A numeric matrix of dimensions n x K where each row is one random draw from the Dirichlet distribution, with all elements in [0, 1] and each row summing to 1. When n = 1, a numeric vector of length K is returned.

Examples

# Example 1: Generate 5 samples from Dirichlet(1, 1, 1) - uniform on simplex
samples <- rdirichlet(5, c(1, 1, 1))
print(samples)
rowSums(samples)  # Each row should sum to 1

# Example 2: Generate samples with unequal concentrations
samples <- rdirichlet(1000, c(2, 5, 3))
colMeans(samples)  # Expected values: approximately c(0.2, 0.5, 0.3)

# Example 3: Sparse Dirichlet (small alpha values)
samples <- rdirichlet(100, c(0.1, 0.1, 0.1, 0.1))
head(samples)  # Most weight concentrated on one component

# Example 4: Concentrated Dirichlet (large alpha values)
samples <- rdirichlet(100, c(100, 100, 100))
colMeans(samples)  # Concentrated around c(1/3, 1/3, 1/3)

# Example 5: Bayesian update with Jeffreys prior for 4 categories
prior_alpha     <- c(0.5, 0.5, 0.5, 0.5)
observed_counts <- c(10, 5, 8, 7)
posterior_samples <- rdirichlet(1000, prior_alpha + observed_counts)
colMeans(posterior_samples)  # Posterior mean