Package 'BayesianQDM'

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

n	A single non-negative integer giving the sample size of the group ($n_j$).

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

prob	A character string specifying the probability type. Must be 'posterior' or 'predictive'.
design	A character string specifying the trial design. Must be 'controlled', 'uncontrolled', or 'external'.
theta_TV	A numeric scalar in (-1, 1) giving the Target Value (TV) threshold for the treatment effect. Required when prob = 'posterior'; set to NULL otherwise.
theta_MAV	A numeric scalar in (-1, 1) giving the Minimum Acceptable Value (MAV) threshold. Must satisfy theta_TV > theta_MAV. Required when prob = 'posterior'; set to NULL otherwise.
theta_NULL	A numeric scalar in (-1, 1) giving the null hypothesis threshold used for the predictive probability. Required when prob = 'predictive'; set to NULL otherwise.
pi_t_go	A numeric scalar in (0, 1) giving the true treatment response rate under the Go-calibration scenario (typically Null).
pi_c_go	A numeric scalar in (0, 1) giving the true control response rate under the Go-calibration scenario. Set to NULL for design = 'uncontrolled'.
pi_t_nogo	A numeric scalar in (0, 1) giving the true treatment response rate under the NoGo-calibration scenario (typically Alternative).
pi_c_nogo	A numeric scalar in (0, 1) giving the true control response rate under the NoGo-calibration scenario. Set to NULL for design = 'uncontrolled'.
target_go	A numeric scalar in (0, 1) giving the upper bound on the marginal Go probability under the Go-calibration scenario. The optimal $\gamma_{\mathrm{go}}$ is the smallest grid value satisfying $\Pr(\mathrm{Go}) < \code{target\_go}$.
target_nogo	A numeric scalar in (0, 1) giving the upper bound on the marginal NoGo probability under the NoGo-calibration scenario. The optimal $\gamma_{\mathrm{nogo}}$ is the largest grid value satisfying $\Pr(\mathrm{NoGo}) < \code{target\_nogo}$.
n_t	A positive integer giving the number of patients in the treatment group in the PoC trial.
n_c	A positive integer giving the number of patients in the control group in the PoC trial.
a_t	A positive numeric scalar giving the first shape parameter of the Beta prior for the treatment group.
a_c	A positive numeric scalar giving the first shape parameter of the Beta prior for the control group.
b_t	A positive numeric scalar giving the second shape parameter of the Beta prior for the treatment group.
b_c	A positive numeric scalar giving the second shape parameter of the Beta prior for the control group.
z	A non-negative integer giving the hypothetical number of responders in the control group. Required when design = 'uncontrolled'; set to NULL otherwise.
m_t	A positive integer giving the future sample size for the treatment group. Required when prob = 'predictive'; set to NULL otherwise.
m_c	A positive integer giving the future sample size for the control group. Required when prob = 'predictive'; set to NULL otherwise.
ne_t	A positive integer giving the number of patients in the treatment group of the external data set. Required when design = 'external'; set to NULL otherwise.
ne_c	A positive integer giving the number of patients in the control group of the external data set. Required when design = 'external'; set to NULL otherwise.
ye_t	A non-negative integer giving the number of responders in the treatment group of the external data set. Required when design = 'external'; set to NULL otherwise.
ye_c	A non-negative integer giving the number of responders in the control group of the external data set. Required when design = 'external'; set to NULL otherwise.
alpha0e_t	A numeric scalar in (0, 1] giving the power prior weight for the treatment group. Required when design = 'external'; set to NULL otherwise.
alpha0e_c	A numeric scalar in (0, 1] giving the power prior weight for the control group. Required when design = 'external'; set to NULL otherwise.
gamma_grid	A numeric vector of candidate threshold values in (0, 1) to search over. Defaults to seq(0.01, 0.99, by = 0.01).

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

nsim	A positive integer giving the number of Monte Carlo simulation replicates used to approximate the operating characteristics.
prob	A character string specifying the probability type. Must be 'posterior' or 'predictive'.
design	A character string specifying the trial design. Must be 'controlled', 'uncontrolled', or 'external'.
prior	A character string specifying the prior distribution. Must be 'vague' or 'N-Inv-Chisq'.
CalcMethod	A character string specifying the computation method for pbayespostpred1cont. Must be 'NI', 'MC', or 'MM'.
theta_TV	A numeric scalar giving the Target Value (TV) threshold for the treatment effect. Required when prob = 'posterior'; set to NULL otherwise.
theta_MAV	A numeric scalar giving the Minimum Acceptable Value (MAV) threshold. Must satisfy theta_TV > theta_MAV. Required when prob = 'posterior'; set to NULL otherwise.
theta_NULL	A numeric scalar giving the null hypothesis threshold for the predictive probability. Required when prob = 'predictive'; set to NULL otherwise.
nMC	A positive integer giving the number of inner Monte Carlo draws passed to pbayespostpred1cont. Required when CalcMethod = 'MC'; set to NULL otherwise.
mu_t_go	A numeric scalar giving the true mean for the treatment group under the Go-calibration scenario (typically Null).
mu_c_go	A numeric scalar giving the true mean for the control group under the Go-calibration scenario. Set to NULL for design = 'uncontrolled'.
sigma_t_go	A positive numeric scalar giving the true standard deviation for the treatment group under the Go-calibration scenario.
sigma_c_go	A positive numeric scalar giving the true standard deviation for the control group under the Go-calibration scenario. Set to NULL for design = 'uncontrolled'.
mu_t_nogo	A numeric scalar giving the true mean for the treatment group under the NoGo-calibration scenario (typically Alternative).
mu_c_nogo	A numeric scalar giving the true mean for the control group under the NoGo-calibration scenario. Set to NULL for design = 'uncontrolled'.
sigma_t_nogo	A positive numeric scalar giving the true standard deviation for the treatment group under the NoGo-calibration scenario.
sigma_c_nogo	A positive numeric scalar giving the true standard deviation for the control group under the NoGo-calibration scenario. Set to NULL for design = 'uncontrolled'.
target_go	A numeric scalar in (0, 1) giving the upper bound on the marginal Go probability under the Go-calibration scenario. The optimal $\gamma_{\mathrm{go}}$ is the smallest grid value satisfying $\Pr(\mathrm{Go}) < \code{target\_go}$.
target_nogo	A numeric scalar in (0, 1) giving the upper bound on the marginal NoGo probability under the NoGo-calibration scenario. The optimal $\gamma_{\mathrm{nogo}}$ is the smallest grid value satisfying $\Pr(\mathrm{NoGo}) < \code{target\_nogo}$.
n_t	A positive integer giving the number of patients in the treatment group in the PoC trial.
n_c	A positive integer giving the number of patients in the control group in the PoC trial. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
m_t	A positive integer giving the future sample size for the treatment group. Required when prob = 'predictive'; set to NULL otherwise.
m_c	A positive integer giving the future sample size for the control group. Required when prob = 'predictive'; set to NULL otherwise.
kappa0_t	A positive numeric scalar giving the prior precision parameter for the treatment group under the N-Inv-Chi-squared prior. Required when prior = 'N-Inv-Chisq'; set to NULL otherwise.
kappa0_c	A positive numeric scalar giving the prior precision parameter for the control group. Required when prior = 'N-Inv-Chisq' and design != 'uncontrolled'; set to NULL otherwise.
nu0_t	A positive numeric scalar giving the prior degrees of freedom for the treatment group under the N-Inv-Chi-squared prior. Required when prior = 'N-Inv-Chisq'; set to NULL otherwise.
nu0_c	A positive numeric scalar giving the prior degrees of freedom for the control group. Required when prior = 'N-Inv-Chisq' and design != 'uncontrolled'; set to NULL otherwise.
mu0_t	A numeric scalar giving the prior mean for the treatment group under the N-Inv-Chi-squared prior. Required when prior = 'N-Inv-Chisq'; set to NULL otherwise.
mu0_c	A numeric scalar giving the prior mean for the control group (or the fixed hypothetical control mean for uncontrolled design). Required when prior = 'N-Inv-Chisq'; set to NULL otherwise.
sigma0_t	A positive numeric scalar giving the prior scale parameter for the treatment group under the N-Inv-Chi-squared prior. Required when prior = 'N-Inv-Chisq'; set to NULL otherwise.
sigma0_c	A positive numeric scalar giving the prior scale parameter for the control group. Required when prior = 'N-Inv-Chisq' and design != 'uncontrolled'; set to NULL otherwise.
r	A positive numeric scalar giving the variance ratio $\sigma_c^2 / \sigma_t^2$ used for uncontrolled design. Required when design = 'uncontrolled'; set to NULL otherwise.
ne_t	A positive integer giving the number of patients in the treatment group of the external data set. Required when design = 'external' and external treatment data are available; otherwise set to NULL.
ne_c	A positive integer giving the number of patients in the control group of the external data set. Required when design = 'external' and external control data are available; otherwise set to NULL.
alpha0e_t	A numeric scalar in (0, 1] giving the power prior weight for external treatment group data. Required when design = 'external' and ne_t is non-NULL; set to NULL otherwise.
alpha0e_c	A numeric scalar in (0, 1] giving the power prior weight for external control group data. Required when design = 'external' and ne_c is non-NULL; set to NULL otherwise.
bar_ye_t	A numeric scalar giving the sample mean of the external treatment group data. Required when ne_t is non-NULL; set to NULL otherwise.
bar_ye_c	A numeric scalar giving the sample mean of the external control group data. Required when ne_c is non-NULL; set to NULL otherwise.
se_t	A positive numeric scalar giving the sample standard deviation of the external treatment group data. Required when ne_t is non-NULL; set to NULL otherwise.
se_c	A positive numeric scalar giving the sample standard deviation of the external control group data. Required when ne_c is non-NULL; set to NULL otherwise.
gamma_grid	A numeric vector of candidate threshold values in (0, 1) to search over. Defaults to seq(0.01, 0.99, by = 0.01).
seed	A numeric scalar for reproducible random number generation. The Go-calibration simulation uses seed and the NoGo-calibration simulation uses seed + 1 to ensure independence between the two scenarios.

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

prob	A character string specifying the probability type. Must be 'posterior' or 'predictive'.
design	A character string specifying the trial design. Must be 'controlled', 'uncontrolled', or 'external'.
GoRegions	An integer vector of region indices (subset of 1:9 for prob = 'posterior' or 1:4 for prob = 'predictive') that constitute the Go region.
NoGoRegions	An integer vector of region indices that constitute the NoGo region. Must be disjoint from GoRegions.
pi_t1_go	A numeric scalar in (0, 1) giving the true treatment response probability for Endpoint 1 under the Go-calibration scenario (typically Null).
pi_t2_go	A numeric scalar in (0, 1) giving the true treatment response probability for Endpoint 2 under the Go-calibration scenario.
rho_t_go	A numeric scalar giving the within-group correlation in the treatment group under the Go-calibration scenario.
pi_c1_go	A numeric scalar in (0, 1) giving the true control response probability for Endpoint 1 under the Go-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
pi_c2_go	A numeric scalar in (0, 1) giving the true control response probability for Endpoint 2 under the Go-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
rho_c_go	A numeric scalar giving the within-group correlation in the control group under the Go-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
pi_t1_nogo	A numeric scalar in (0, 1) giving the true treatment response probability for Endpoint 1 under the NoGo-calibration scenario (typically Alternative).
pi_t2_nogo	A numeric scalar in (0, 1) giving the true treatment response probability for Endpoint 2 under the NoGo-calibration scenario.
rho_t_nogo	A numeric scalar giving the within-group correlation in the treatment group under the NoGo-calibration scenario.
pi_c1_nogo	A numeric scalar in (0, 1) giving the true control response probability for Endpoint 1 under the NoGo-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
pi_c2_nogo	A numeric scalar in (0, 1) giving the true control response probability for Endpoint 2 under the NoGo-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
rho_c_nogo	A numeric scalar giving the within-group correlation in the control group under the NoGo-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
target_go	A numeric scalar in (0, 1) giving the upper bound on the worst-case marginal Go probability under the Go-calibration scenario. The optimal $\gamma_{\mathrm{go}}$ is the smallest grid value satisfying the constraint.
target_nogo	A numeric scalar in (0, 1) giving the upper bound on the worst-case marginal NoGo probability under the NoGo-calibration scenario. The optimal $\gamma_{\mathrm{nogo}}$ is the smallest grid value satisfying the constraint.
n_t	A positive integer giving the number of patients in the treatment group in the PoC trial.
n_c	A positive integer giving the number of patients in the control group in the PoC trial.
a_t_00	A positive numeric scalar giving the Dirichlet prior parameter for the (0,0) response pattern in the treatment group.
a_t_01	A positive numeric scalar; see a_t_00.
a_t_10	A positive numeric scalar; see a_t_00.
a_t_11	A positive numeric scalar; see a_t_00.
a_c_00	A positive numeric scalar giving the Dirichlet prior parameter for the (0,0) response pattern in the control group.
a_c_01	A positive numeric scalar; see a_c_00.
a_c_10	A positive numeric scalar; see a_c_00.
a_c_11	A positive numeric scalar; see a_c_00.
theta_TV1	A numeric scalar giving the TV threshold for Endpoint 1. Required when prob = 'posterior'; otherwise set to NULL.
theta_MAV1	A numeric scalar giving the MAV threshold for Endpoint 1. Required when prob = 'posterior'; otherwise set to NULL.
theta_TV2	A numeric scalar giving the TV threshold for Endpoint 2. Required when prob = 'posterior'; otherwise set to NULL.
theta_MAV2	A numeric scalar giving the MAV threshold for Endpoint 2. Required when prob = 'posterior'; otherwise set to NULL.
theta_NULL1	A numeric scalar giving the null hypothesis threshold for Endpoint 1. Required when prob = 'predictive'; otherwise set to NULL.
theta_NULL2	A numeric scalar giving the null hypothesis threshold for Endpoint 2. Required when prob = 'predictive'; otherwise set to NULL.
m_t	A positive integer giving the future sample size for the treatment group. Required when prob = 'predictive'; set to NULL otherwise.
m_c	A positive integer giving the future sample size for the control group. Required when prob = 'predictive'; set to NULL otherwise.
z00	A non-negative integer giving the hypothetical control count for pattern (0,0). Required when design = 'uncontrolled'; otherwise set to NULL.
z01	A non-negative integer; see z00.
z10	A non-negative integer; see z00.
z11	A non-negative integer; see z00.
xe_t_00	A non-negative integer giving the external treatment group count for pattern (0,0). Required when design = 'external' and external treatment data are used; otherwise NULL.
xe_t_01	A non-negative integer; see xe_t_00.
xe_t_10	A non-negative integer; see xe_t_00.
xe_t_11	A non-negative integer; see xe_t_00.
xe_c_00	A non-negative integer giving the external control group count for pattern (0,0). Required when design = 'external' and external control data are used; otherwise NULL.
xe_c_01	A non-negative integer; see xe_c_00.
xe_c_10	A non-negative integer; see xe_c_00.
xe_c_11	A non-negative integer; see xe_c_00.
alpha0e_t	A numeric scalar in (0, 1] giving the power prior weight for external treatment group data. Required when external treatment data are used; otherwise NULL.
alpha0e_c	A numeric scalar in (0, 1] giving the power prior weight for external control group data. Required when external control data are used; otherwise NULL.
nMC	A positive integer giving the number of Dirichlet draws used to evaluate region probabilities for each count combination in Stage 1. Default is 1000L.
gamma_go_grid	A numeric vector of candidate Go threshold values in (0, 1) to search over. Defaults to seq(0.01, 0.99, by = 0.01).
gamma_nogo_grid	A numeric vector of candidate NoGo threshold values in (0, 1) to search over. Defaults to seq(0.01, 0.99, by = 0.01).

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

nsim	A positive integer giving the number of Monte Carlo datasets to simulate per calibration scenario. Default is 10000L.
prob	A character string specifying the probability type. Must be 'posterior' or 'predictive'.
design	A character string specifying the trial design. Must be 'controlled', 'uncontrolled', or 'external'.
prior	A character string specifying the prior distribution. Must be 'vague' or 'N-Inv-Wishart'.
GoRegions	An integer vector of region indices (subset of 1:9) that constitute the Go region. The 9 regions are defined by the cross-classification of the two treatment effects $(\theta_1, \theta_2)$ relative to (TV1, MAV1) and (TV2, MAV2): region $k = (z_1 - 1) \times 3 + z_2$ where $z_i = 1$ if $\theta_i > TV_i$, $z_i = 2$ if $MAV_i < \theta_i \le TV_i$, $z_i = 3$ if $\theta_i \le MAV_i$.
NoGoRegions	An integer vector of region indices (subset of 1:9) that constitute the NoGo region. Must be disjoint from GoRegions.
mu_t_go	A length-2 numeric vector giving the true bivariate mean for the treatment group under the Go-calibration scenario (typically Null).
Sigma_t_go	A 2x2 positive-definite numeric matrix giving the true within-group covariance in the treatment group under the Go-calibration scenario.
mu_c_go	A length-2 numeric vector giving the true bivariate mean for the control group under the Go-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
Sigma_c_go	A 2x2 positive-definite numeric matrix giving the true within-group covariance in the control group under the Go-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
mu_t_nogo	A length-2 numeric vector giving the true bivariate mean for the treatment group under the NoGo-calibration scenario (typically Alternative).
Sigma_t_nogo	A 2x2 positive-definite numeric matrix giving the true within-group covariance in the treatment group under the NoGo-calibration scenario.
mu_c_nogo	A length-2 numeric vector giving the true bivariate mean for the control group under the NoGo-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
Sigma_c_nogo	A 2x2 positive-definite numeric matrix giving the true within-group covariance in the control group under the NoGo-calibration scenario. Required for design = 'controlled' or 'external'; set to NULL for design = 'uncontrolled'.
target_go	A numeric scalar in (0, 1) giving the upper bound on the marginal Go probability under the Go-calibration scenario. The optimal $\gamma_{\mathrm{go}}$ is the smallest grid value satisfying the constraint.
target_nogo	A numeric scalar in (0, 1) giving the upper bound on the marginal NoGo probability under the NoGo-calibration scenario. The optimal $\gamma_{\mathrm{nogo}}$ is the smallest grid value satisfying the constraint.
n_t	A positive integer giving the number of patients in the treatment group in the PoC trial.
n_c	A positive integer giving the number of patients in the control group in the PoC trial. Set to NULL for design = 'uncontrolled'.
theta_TV1	A numeric scalar giving the TV threshold for Endpoint 1. Required when prob = 'posterior'; otherwise set to NULL.
theta_MAV1	A numeric scalar giving the MAV threshold for Endpoint 1. Required when prob = 'posterior'; otherwise set to NULL.
theta_TV2	A numeric scalar giving the TV threshold for Endpoint 2. Required when prob = 'posterior'; otherwise set to NULL.
theta_MAV2	A numeric scalar giving the MAV threshold for Endpoint 2. Required when prob = 'posterior'; otherwise set to NULL.
theta_NULL1	A numeric scalar giving the null hypothesis threshold for Endpoint 1. Required when prob = 'predictive'; otherwise set to NULL.
theta_NULL2	A numeric scalar giving the null hypothesis threshold for Endpoint 2. Required when prob = 'predictive'; otherwise set to NULL.
m_t	A positive integer giving the future sample size for the treatment group. Required when prob = 'predictive'; set to NULL otherwise.
m_c	A positive integer giving the future sample size for the control group. Required when prob = 'predictive'; set to NULL otherwise.
kappa0_t	Positive numeric scalar. NIW prior hyperparameter $\kappa_{01}$ for the treatment group. Required when prior = 'N-Inv-Wishart'; otherwise NULL.
nu0_t	Positive numeric scalar. NIW prior degrees of freedom $\nu_{01}$ for the treatment group. Required when prior = 'N-Inv-Wishart'; otherwise NULL.
mu0_t	Length-2 numeric vector. NIW prior mean $\mu_{01}$ for the treatment group. Required when prior = 'N-Inv-Wishart'; otherwise NULL.
Lambda0_t	A 2x2 positive-definite numeric matrix. NIW prior scale matrix $\Lambda_{01}$ for the treatment group. Required when prior = 'N-Inv-Wishart'; otherwise NULL.
kappa0_c	Positive numeric scalar; see kappa0_t. For the control group.
nu0_c	Positive numeric scalar; see nu0_t. For the control group.
mu0_c	Length-2 numeric vector; see mu0_t. For the control group. May be required for the vague prior uncontrolled design; see pbayesdecisionprob2cont.
Lambda0_c	A 2x2 matrix; see Lambda0_t. For the control group.
r	A positive numeric scalar giving the power prior weight for the control group when design = 'uncontrolled' and prior = 'vague'. Otherwise NULL.
ne_t	A positive integer giving the external treatment sample size. Required when design = 'external' and external treatment data are used; otherwise NULL.
ne_c	A positive integer giving the external control sample size. Required when design = 'external' and external control data are used; otherwise NULL.
alpha0e_t	A numeric scalar in (0, 1] giving the power prior weight for external treatment data. Required when external treatment data are used; otherwise NULL.
alpha0e_c	A numeric scalar in (0, 1] giving the power prior weight for external control data. Required when external control data are used; otherwise NULL.
bar_ye_t	A length-2 numeric vector. External treatment sample mean. Required when external treatment data are used; otherwise NULL.
bar_ye_c	A length-2 numeric vector. External control sample mean. Required when external control data are used; otherwise NULL.
se_t	A 2x2 numeric matrix. External treatment sum-of-squares matrix. Required when external treatment data are used; otherwise NULL.
se_c	A 2x2 numeric matrix. External control sum-of-squares matrix. Required when external control data are used; otherwise NULL.
nMC	A positive integer giving the number of Monte Carlo draws passed to pbayespostpred2cont. Required when CalcMethod = 'MC'. May be set to NULL when CalcMethod = 'MM' and $\nu_k > 4$; if CalcMethod = 'MM' but $\nu_k \le 4$ causes a fallback to MC, nMC must be a positive integer. Default is NULL.
CalcMethod	A character string specifying the computation method passed to pbayespostpred2cont. Must be 'MC' (default) or 'MM'.
gamma_go_grid	A numeric vector of candidate Go threshold values in (0, 1) to search over. Defaults to seq(0.01, 0.99, by = 0.01).
gamma_nogo_grid	A numeric vector of candidate NoGo threshold values in (0, 1) to search over. Defaults to seq(0.01, 0.99, by = 0.01).
seed	A numeric scalar for reproducible random number generation. The Go-calibration simulation uses seed and the NoGo-calibration simulation uses seed + 1 to ensure independence between the two scenarios.

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
)

---