4.1 Controlled Design
Standard parallel-group RCT with observed data from both treatment
(\(n_t\), \(\bar{y}_t\), \(s_t\)) and control (\(n_c\), \(\bar{y}_c\), \(s_c\)) groups. Both posteriors are updated
from the PoC data.
Vague prior — posterior probability:
# P(mu_t - mu_c > theta_TV | data) and P(mu_t - mu_c <= theta_MAV | data)
p_tv <- pbayespostpred1cont(
prob = 'posterior', design = 'controlled', prior = 'vague',
CalcMethod = 'NI', theta0 = 1.5, nMC = NULL,
n_t = 15, n_c = 15,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = 1.1, s_c = 1.8,
m_t = NULL, m_c = NULL,
kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
mu0_t = NULL, mu0_c = NULL, sigma0_t = NULL, sigma0_c = NULL,
r = NULL,
ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
bar_ye_t = NULL, se_t = NULL, bar_ye_c = NULL, se_c = NULL,
lower.tail = FALSE
)
p_mav <- pbayespostpred1cont(
prob = 'posterior', design = 'controlled', prior = 'vague',
CalcMethod = 'NI', theta0 = 0.5, nMC = NULL,
n_t = 15, n_c = 15,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = 1.1, s_c = 1.8,
m_t = NULL, m_c = NULL,
kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
mu0_t = NULL, mu0_c = NULL, sigma0_t = NULL, sigma0_c = NULL,
r = NULL,
ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
bar_ye_t = NULL, se_t = NULL, bar_ye_c = NULL, se_c = NULL,
lower.tail = FALSE
)
cat(sprintf("P(theta > theta_TV | data) = %.4f -> Go criterion: >= 0.80? %s\n",
p_tv, ifelse(p_tv >= 0.80, "YES", "NO")))
#> P(theta > theta_TV | data) = 0.7940 -> Go criterion: >= 0.80? NO
cat(sprintf("P(theta <= theta_MAV | data) = %.4f -> NoGo criterion: >= 0.20? %s\n",
1 - p_mav, ifelse((1 - p_mav) >= 0.20, "YES", "NO")))
#> P(theta <= theta_MAV | data) = 0.0178 -> NoGo criterion: >= 0.20? NO
cat(sprintf("Decision: %s\n",
ifelse(p_tv >= 0.80 & (1 - p_mav) < 0.20, "Go",
ifelse(p_tv < 0.80 & (1 - p_mav) >= 0.20, "NoGo",
ifelse(p_tv >= 0.80 & (1 - p_mav) >= 0.20, "Miss", "Gray")))))
#> Decision: Gray
N-Inv-\(\chi^2\) informative
prior:
Historical knowledge suggests treatment mean \(\sim 3.0\) and control mean \(\sim 1.0\) with prior pseudo-sample size
\(\kappa_{0t} = \kappa_{0c} = 5\)
(kappa0_t, kappa0_c) and degrees of freedom
\(\nu_{0t} = \nu_{0c} = 5\)
(nu0_t, nu0_c).
p_tv_inf <- pbayespostpred1cont(
prob = 'posterior', design = 'controlled', prior = 'N-Inv-Chisq',
CalcMethod = 'NI', theta0 = 1.5, nMC = NULL,
n_t = 15, n_c = 15,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = 1.1, s_c = 1.8,
m_t = NULL, m_c = NULL,
kappa0_t = 5, kappa0_c = 5,
nu0_t = 5, nu0_c = 5,
mu0_t = 3.0, mu0_c = 1.0,
sigma0_t = 2.0, sigma0_c = 1.8,
r = NULL,
ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
bar_ye_t = NULL, se_t = NULL, bar_ye_c = NULL, se_c = NULL,
lower.tail = FALSE
)
cat(sprintf("P(theta > theta_TV | data, N-Inv-Chisq prior) = %.4f\n", p_tv_inf))
#> P(theta > theta_TV | data, N-Inv-Chisq prior) = 0.8274
Posterior predictive probability (future Phase III: \(m_t = m_c = 60\)):
p_pred <- pbayespostpred1cont(
prob = 'predictive', design = 'controlled', prior = 'vague',
CalcMethod = 'NI', theta0 = 1.0, nMC = NULL,
n_t = 15, n_c = 15, m_t = 60, m_c = 60,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = 1.1, s_c = 1.8,
kappa0_t = NULL, kappa0_c = NULL, nu0_t = NULL, nu0_c = NULL,
mu0_t = NULL, mu0_c = NULL, sigma0_t = NULL, sigma0_c = NULL,
r = NULL,
ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
bar_ye_t = NULL, se_t = NULL, bar_ye_c = NULL, se_c = NULL,
lower.tail = FALSE
)
cat(sprintf("Predictive probability (m_t = m_c = 60) = %.4f\n", p_pred))
#> Predictive probability (m_t = m_c = 60) = 0.9966
4.2 Uncontrolled Design (Single-Arm)
When no concurrent control group is enrolled, the control
distribution is specified from external knowledge:
- Hypothetical control mean: \(\mu_{0c}\) (
mu0_c, fixed
scalar)
- Variance scaling: \(\sigma_c = \sqrt{r}
\cdot \sigma_t\), so \(r = 1\)
assumes equal variances
The posterior of the control mean is not updated from observed data;
only the treatment posterior is updated.
p_unctrl <- pbayespostpred1cont(
prob = 'posterior', design = 'uncontrolled', prior = 'vague',
CalcMethod = 'MM', theta0 = 1.5, nMC = NULL,
n_t = 15, n_c = NULL,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = NULL, s_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 = 1.0, # hypothetical control mean
sigma0_t = NULL, sigma0_c = NULL,
r = 1.0, # equal variance assumption
ne_t = NULL, ne_c = NULL, alpha0e_t = NULL, alpha0e_c = NULL,
bar_ye_t = NULL, se_t = NULL, bar_ye_c = NULL, se_c = NULL,
lower.tail = FALSE
)
cat(sprintf("P(theta > theta_TV | data, uncontrolled) = %.4f\n", p_unctrl))
#> P(theta > theta_TV | data, uncontrolled) = 0.8184
4.3 External Design (Power Prior)
In the external design, historical data from a prior study are
incorporated via a power prior with borrowing weight \(\alpha_{0e} \in (0, 1]\). The two prior
specifications yield different posterior update formulae.
Vague prior (prior = 'vague')
For group \(j\) with \(n_{e,j}\) external patients, external mean
\(\bar{y}_{e,j}\), and external SD
\(s_{e,j}\), the posterior parameters
after combining the power prior with the PoC data are given by Theorem 4
of Huang et al. (2025):
\[\mu_{nj}^* = \frac{\alpha_{0e,j}\,
n_{e,j}\, \bar{y}_{e,j} + n_j\, \bar{y}_j}
{\alpha_{0e,j}\, n_{e,j} + n_j},\]
\[\kappa_{nj}^* = \alpha_{0e,j}\, n_{e,j}
+ n_j,\]
\[\nu_{nj}^* = \alpha_{0e,j}\, n_{e,j} +
n_j - 1,\]
\[(\sigma_{nj}^*)^2 =
\frac{\alpha_{0e,j}(n_{e,j}-1)\, s_{e,j}^2 +
(n_j-1)\, s_j^2 +
\dfrac{\alpha_{0e,j}\, n_{e,j}\, n_j\, (\bar{y}_{e,j} - \bar{y}_j)^2}
{\alpha_{0e,j}\, n_{e,j} + n_j}}
{\alpha_{0e,j}\, n_{e,j} + n_j}.\]
The marginal posterior of \(\mu_j\)
is then
\[\mu_j \mid \mathbf{y}_j \;\sim\;
t_{\nu_{nj}^*}\!\!\left(\mu_{nj}^*,\;
\frac{\sigma_{nj}^*}{\sqrt{\kappa_{nj}^*}}\right).\]
Setting \(\alpha_{0e,j} = 1\)
corresponds to full borrowing; as \(\alpha_{0e,j} \to 0\) the result recovers
the vague-prior posterior from the current data alone.
N-Inv-\(\chi^2\) prior
(prior = 'N-Inv-Chisq')
When an informative N-Inv-\(\chi^2(\mu_{0j}, \kappa_{0j}, \nu_{0j},
\sigma_{0j}^2)\) initial prior is specified, the power prior is
first formed by combining the initial prior with the discounted external
data (Theorem 1 of Huang et al., 2025):
\[\mu_{e,j} = \frac{\alpha_{0e,j}\,
n_{e,j}\, \bar{y}_{e,j} + \kappa_{0j}\, \mu_{0j}}
{\alpha_{0e,j}\, n_{e,j} + \kappa_{0j}},\]
\[\kappa_{e,j} = \alpha_{0e,j}\, n_{e,j} +
\kappa_{0j},\]
\[\nu_{e,j} = \alpha_{0e,j}\, n_{e,j} +
\nu_{0j},\]
\[\sigma_{e,j}^2 =
\frac{\alpha_{0e,j}(n_{e,j}-1)\, s_{e,j}^2 +
\nu_{0j}\, \sigma_{0j}^2 +
\dfrac{\alpha_{0e,j}\, n_{e,j}\, \kappa_{0j}\, (\bar{y}_{e,j} -
\mu_{0j})^2}
{\alpha_{0e,j}\, n_{e,j} + \kappa_{0j}}}
{\nu_{e,j}}.\]
This intermediate result is then updated with the PoC data by
standard conjugate updating (Theorem 2 of Huang et al., 2025):
\[\mu_{nj}^* = \frac{\kappa_{e,j}\,
\mu_{e,j} + n_j\, \bar{y}_j}
{\kappa_{e,j} + n_j},\]
\[\kappa_{nj}^* = \kappa_{e,j} +
n_j,\]
\[\nu_{nj}^* = \nu_{e,j} +
n_j,\]
\[(\sigma_{nj}^*)^2 = \frac{\nu_{e,j}\,
\sigma_{e,j}^2 + (n_j-1)\, s_j^2 +
\dfrac{n_j\, \kappa_{e,j}\, (\mu_{e,j} - \bar{y}_j)^2}
{\kappa_{e,j} + n_j}}
{\nu_{nj}^*}.\]
The marginal posterior of \(\mu_j\)
is then
\[\mu_j \mid \mathbf{y}_j \;\sim\;
t_{\nu_{nj}^*}\!\!\left(\mu_{nj}^*,\;
\frac{\sigma_{nj}^*}{\sqrt{\kappa_{nj}^*}}\right).\]
Example: external control design, vague prior
Here, \(n_{e,c} = 20\)
(ne_c), \(\bar{y}_{e,c} =
0.9\) (bar_ye_c), \(s_{e,c} = 1.8\) (se_c), \(\alpha_{0e,c} = 0.5\)
(alpha0e_c) (50% borrowing from external control):
p_ext <- pbayespostpred1cont(
prob = 'posterior', design = 'external', prior = 'vague',
CalcMethod = 'MM', theta0 = 1.5, nMC = NULL,
n_t = 15, n_c = 15,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = 1.1, s_c = 1.8,
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, se_t = NULL, bar_ye_c = 0.9, se_c = 1.8,
lower.tail = FALSE
)
cat(sprintf("P(theta > theta_TV | data, external control design, alpha=0.5) = %.4f\n",
p_ext))
#> P(theta > theta_TV | data, external control design, alpha=0.5) = 0.8517
Effect of the borrowing weight: varying \(\alpha_{0e,c}\) from 0 to 1 shows how
external data influence the posterior.
# alpha0e_c must be in (0, 1]; start from 0.01 to avoid validation error
alpha_seq <- c(0.01, seq(0.1, 1.0, by = 0.1))
p_alpha <- sapply(alpha_seq, function(a) {
pbayespostpred1cont(
prob = 'posterior', design = 'external', prior = 'vague',
CalcMethod = 'MM', theta0 = 1.5, nMC = NULL,
n_t = 15, n_c = 15,
bar_y_t = 3.2, s_t = 2.0,
bar_y_c = 1.1, s_c = 1.8,
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 = a,
bar_ye_t = NULL, se_t = NULL, bar_ye_c = 0.9, se_c = 1.8,
lower.tail = FALSE
)
})
data.frame(alpha_ec = alpha_seq, P_gt_TV = round(p_alpha, 4))
#> alpha_ec P_gt_TV
#> 1 0.01 0.7994
#> 2 0.10 0.8133
#> 3 0.20 0.8259
#> 4 0.30 0.8361
#> 5 0.40 0.8446
#> 6 0.50 0.8517
#> 7 0.60 0.8577
#> 8 0.70 0.8629
#> 9 0.80 0.8674
#> 10 0.90 0.8713
#> 11 1.00 0.8748
