pbayespostpred2cont: Region Probabilities for Two Continuous Endpoints
In BayesianQDM: Bayesian Quantitative Decision-Making Framework for Binary and Continuous Endpoints
View source: R/pbayespostpred2cont.R
pbayespostpred2cont R Documentation
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
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'.
theta_TV1
A numeric scalar giving the target value (TV) threshold
for Endpoint 1. Required when prob = 'posterior'; must
satisfy theta_TV1 > theta_MAV1. Set to NULL when
prob = 'predictive'.
theta_MAV1
A numeric scalar giving the minimum acceptable value
(MAV) threshold for Endpoint 1. Required when
prob = 'posterior'; must satisfy
theta_TV1 > theta_MAV1. Set to NULL when
prob = 'predictive'.
theta_TV2
A numeric scalar giving the target value (TV) threshold
for Endpoint 2. Required when prob = 'posterior'; must
satisfy theta_TV2 > theta_MAV2. Set to NULL when
prob = 'predictive'.
theta_MAV2
A numeric scalar giving the minimum acceptable value
(MAV) threshold for Endpoint 2. Required when
prob = 'posterior'; must satisfy
theta_TV2 > theta_MAV2. Set to NULL when
prob = 'predictive'.
theta_NULL1
A numeric scalar giving the null hypothesis threshold
for Endpoint 1. Required when prob = 'predictive'; set to
NULL when prob = 'posterior'.
theta_NULL2
A numeric scalar giving the null hypothesis threshold
for Endpoint 2. Required when prob = 'predictive'; set to
NULL when prob = 'posterior'.
n_t
A positive integer giving the number of patients in the
treatment group in the proof-of-concept (PoC) trial.
n_c
A positive integer giving the number of patients in the
control group in the PoC trial. For design = 'uncontrolled',
this is the hypothetical control sample size (required for
consistency with other designs).
ybar_t
A numeric vector of length 2 or a numeric matrix
with 2 columns giving the sample mean(s) for the treatment group.
When a matrix with N rows is supplied, the function computes
probabilities for all N observations simultaneously and
returns an N \times n_{\rm regions} matrix.
S_t
A 2x2 numeric matrix giving the sum-of-squares matrix for the
treatment group (single observation), or a list of N
such matrices (vectorised call). Must be consistent with
ybar_t: if ybar_t is a matrix, S_t must be a
list of the same length.
ybar_c
A numeric vector of length 2, a numeric matrix with 2
columns, or NULL giving the sample mean(s) for the control
group. Not used when design = 'uncontrolled'.
S_c
A 2x2 numeric matrix, a list of 2x2 matrices, or NULL
giving the sum-of-squares matrix/matrices for the control group.
Not used when design = 'uncontrolled'.
m_t
A positive integer giving the number of patients in the
treatment group for the future trial. Required when
prob = 'predictive'; otherwise set to NULL.
m_c
A positive integer giving the number of patients in the
control group for the future trial. Required when
prob = 'predictive'; otherwise set to NULL.
kappa0_t
A positive numeric scalar giving the NIW prior
concentration parameter for the treatment group. Required when
prior = 'N-Inv-Wishart'; otherwise set to NULL.
nu0_t
A numeric scalar giving the NIW prior degrees of freedom for
the treatment group. Must be greater than 3. Required when
prior = 'N-Inv-Wishart'; otherwise set to NULL.
mu0_t
A numeric vector of length 2 giving the NIW prior mean for
the treatment group. Required when
prior = 'N-Inv-Wishart'; otherwise set to NULL.
Lambda0_t
A 2x2 positive-definite numeric matrix giving the NIW
prior scale matrix for the treatment group. Required when
prior = 'N-Inv-Wishart'; otherwise set to NULL.
kappa0_c
A positive numeric scalar giving the NIW prior
concentration parameter for the control group. Required when
prior = 'N-Inv-Wishart' and
design != 'uncontrolled'; otherwise set to NULL.
nu0_c
A numeric scalar giving the NIW prior degrees of freedom for
the control group. Must be greater than 3. Required when
prior = 'N-Inv-Wishart' and
design != 'uncontrolled'; otherwise set to NULL.
mu0_c
A numeric vector of length 2 giving the NIW prior mean for
the control group, or the hypothetical control location when
design = 'uncontrolled'. Required when
prior = 'N-Inv-Wishart'; otherwise set to NULL.
Lambda0_c
A 2x2 positive-definite numeric matrix giving the NIW
prior scale matrix for the control group. Required when
prior = 'N-Inv-Wishart' and
design != 'uncontrolled'; otherwise set to NULL.
r
A positive numeric scalar giving the variance scaling factor for
the hypothetical control distribution. Required when
design = 'uncontrolled'; otherwise set to NULL.
ne_t
A positive integer giving the external treatment group sample
size. Required when design = 'external' and external
treatment data are used; otherwise set to NULL.
ne_c
A positive integer giving the external control group sample
size. Required when design = 'external' and external
control data are used; otherwise set to NULL.
alpha0e_t
A numeric scalar in (0, 1] giving the power prior
weight for the external treatment data. Required when external
treatment data are used; otherwise set to NULL.
alpha0e_c
A numeric scalar in (0, 1] giving the power prior
weight for the external control data. Required when external
control data are used; otherwise set to NULL.
bar_ye_t
A numeric vector of length 2 giving the external treatment
group sample mean. Required when external treatment data are used;
otherwise set to NULL.
bar_ye_c
A numeric vector of length 2 giving the external control
group sample mean. Required when external control data are used;
otherwise set to NULL.
se_t
A 2x2 numeric matrix giving the external treatment group
sum-of-squares matrix. Required when external treatment data are
used; otherwise set to NULL.
se_c
A 2x2 numeric matrix giving the external control group
sum-of-squares matrix. Required when external control data are
used; otherwise set to NULL.
nMC
A positive integer giving the number of Monte Carlo draws used
to estimate region probabilities. Default is 10000. Required
when CalcMethod = 'MC'. May be set to NULL when
CalcMethod = 'MM' and \nu_k > 4 (the MM method uses
mvtnorm::pmvt analytically); if CalcMethod = 'MM' but
\nu_k \le 4 causes a fallback to MC, nMC must be a
positive integer.
CalcMethod
A character string specifying the computation method.
Must be 'MC' (Monte Carlo, default) or 'MM'
(Moment-Matching via mvtnorm::pmvt). When
CalcMethod = 'MM' and \nu_k \le 4, a warning is issued
and the function falls back to CalcMethod = 'MC'.
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
)
BayesianQDM documentation built on April 22, 2026, 1:09 a.m.
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View source: R/pbayespostpred2cont.R
| pbayespostpred2cont | R Documentation |
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
prob |
A character string specifying the probability type.
Must be |
design |
A character string specifying the trial design.
Must be |
prior |
A character string specifying the prior distribution.
Must be |
theta_TV1 |
A numeric scalar giving the target value (TV) threshold
for Endpoint 1. Required when |
theta_MAV1 |
A numeric scalar giving the minimum acceptable value
(MAV) threshold for Endpoint 1. Required when
|
theta_TV2 |
A numeric scalar giving the target value (TV) threshold
for Endpoint 2. Required when |
theta_MAV2 |
A numeric scalar giving the minimum acceptable value
(MAV) threshold for Endpoint 2. Required when
|
theta_NULL1 |
A numeric scalar giving the null hypothesis threshold
for Endpoint 1. Required when |
theta_NULL2 |
A numeric scalar giving the null hypothesis threshold
for Endpoint 2. Required when |
n_t |
A positive integer giving the number of patients in the treatment group in the proof-of-concept (PoC) trial. |
n_c |
A positive integer giving the number of patients in the
control group in the PoC trial. For |
ybar_t |
A numeric vector of length 2 or a numeric matrix
with 2 columns giving the sample mean(s) for the treatment group.
When a matrix with |
S_t |
A 2x2 numeric matrix giving the sum-of-squares matrix for the
treatment group (single observation), or a list of |
ybar_c |
A numeric vector of length 2, a numeric matrix with 2
columns, or |
S_c |
A 2x2 numeric matrix, a list of 2x2 matrices, or |
m_t |
A positive integer giving the number of patients in the
treatment group for the future trial. Required when
|
m_c |
A positive integer giving the number of patients in the
control group for the future trial. Required when
|
kappa0_t |
A positive numeric scalar giving the NIW prior
concentration parameter for the treatment group. Required when
|
nu0_t |
A numeric scalar giving the NIW prior degrees of freedom for
the treatment group. Must be greater than 3. Required when
|
mu0_t |
A numeric vector of length 2 giving the NIW prior mean for
the treatment group. Required when
|
Lambda0_t |
A 2x2 positive-definite numeric matrix giving the NIW
prior scale matrix for the treatment group. Required when
|
kappa0_c |
A positive numeric scalar giving the NIW prior
concentration parameter for the control group. Required when
|
nu0_c |
A numeric scalar giving the NIW prior degrees of freedom for
the control group. Must be greater than 3. Required when
|
mu0_c |
A numeric vector of length 2 giving the NIW prior mean for
the control group, or the hypothetical control location when
|
Lambda0_c |
A 2x2 positive-definite numeric matrix giving the NIW
prior scale matrix for the control group. Required when
|
r |
A positive numeric scalar giving the variance scaling factor for
the hypothetical control distribution. Required when
|
ne_t |
A positive integer giving the external treatment group sample
size. Required when |
ne_c |
A positive integer giving the external control group sample
size. Required when |
alpha0e_t |
A numeric scalar in |
alpha0e_c |
A numeric scalar in |
bar_ye_t |
A numeric vector of length 2 giving the external treatment
group sample mean. Required when external treatment data are used;
otherwise set to |
bar_ye_c |
A numeric vector of length 2 giving the external control
group sample mean. Required when external control data are used;
otherwise set to |
se_t |
A 2x2 numeric matrix giving the external treatment group
sum-of-squares matrix. Required when external treatment data are
used; otherwise set to |
se_c |
A 2x2 numeric matrix giving the external control group
sum-of-squares matrix. Required when external control data are
used; otherwise set to |
nMC |
A positive integer giving the number of Monte Carlo draws used
to estimate region probabilities. Default is |
CalcMethod |
A character string specifying the computation method.
Must be |
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_1AND\theta_2 > TV_2R2:
\theta_1 > TV_1ANDTV_2 \ge \theta_2 > MAV_2R3:
\theta_1 > TV_1AND\theta_2 \le MAV_2R4:
TV_1 \ge \theta_1 > MAV_1AND\theta_2 > TV_2R5:
TV_1 \ge \theta_1 > MAV_1ANDTV_2 \ge \theta_2 > MAV_2R6:
TV_1 \ge \theta_1 > MAV_1AND\theta_2 \le MAV_2R7:
\theta_1 \le MAV_1AND\theta_2 > TV_2R8:
\theta_1 \le MAV_1ANDTV_2 \ge \theta_2 > MAV_2R9:
\theta_1 \le MAV_1AND\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
)
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