4. Operating Characteristics
4.1 Definition
For given true parameters \((\mathbf{p}_t, \mathbf{p}_c)\), the operating characteristics are computed by exact enumeration over all possible multinomial count combinations via a two-stage strategy.
Stage 1 (precomputation): all count combinations \(\mathbf{x}_{t,i}\) (\(K_t\) combinations) and \(\mathbf{x}_{c,j}\) (\(K_c\) combinations) are enumerated by
allmultinom(). For each pair \((i, j)\), pbayespostpred2bin()
computes the region probability vector once, yielding \(\hat{g}_{\mathrm{go},ij}\) and \(\hat{g}_{\mathrm{nogo},ij}\) independent of
any decision threshold.
Stage 2 (weighting): for given thresholds \((\gamma_{\mathrm{go}}, \gamma_{\mathrm{nogo}})\):
\[\Pr(\mathrm{Go}) = \sum_{i=1}^{K_t} \sum_{j=1}^{K_c} w_{ij}\, \mathbf{1}\!\left[\hat{g}_{\mathrm{go},ij} \ge \gamma_{\mathrm{go}} \text{ and } \hat{g}_{\mathrm{nogo},ij} < \gamma_{\mathrm{nogo}}\right],\]
\[\Pr(\mathrm{NoGo}) = \sum_{i=1}^{K_t} \sum_{j=1}^{K_c} w_{ij}\, \mathbf{1}\!\left[\hat{g}_{\mathrm{nogo},ij} \ge \gamma_{\mathrm{nogo}} \text{ and } \hat{g}_{\mathrm{go},ij} < \gamma_{\mathrm{go}}\right],\]
where \(w_{ij} = P_\mathrm{Mult}(\mathbf{x}_{t,i};\, n_t, \mathbf{p}_t) \times P_\mathrm{Mult}(\mathbf{x}_{c,j};\, n_c, \mathbf{p}_c)\), and
\[\hat{g}_{\mathrm{go},ij} = \sum_{r \in \text{GoRegions}} \hat{P}(R_r \mid \mathbf{x}_{t,i}, \mathbf{x}_{c,j}),\] \[\hat{g}_{\mathrm{nogo},ij} = \sum_{r \in \text{NoGoRegions}} \hat{P}(R_r \mid \mathbf{x}_{t,i}, \mathbf{x}_{c,j}).\]
A Miss (\(\hat{g}_{\mathrm{go},ij} \ge
\gamma_{\mathrm{go}}\) AND \(\hat{g}_{\mathrm{nogo},ij} \ge
\gamma_{\mathrm{nogo}}\)) indicates an inconsistent threshold
configuration and triggers an error by default
(error_if_Miss = TRUE). Gray comprises all remaining
outcomes.
For large \(n_t\) or \(n_c\), the CalcMethod = 'MC'
option replaces full enumeration with \(n_\mathrm{sim}\) multinomial draws,
deduplicates them into \(K \ll
n_\mathrm{sim}\) unique pairs, and evaluates
pbayespostpred2bin() only for those unique pairs.
Note on computation time. The number of multinomial count combinations grows rapidly with \(n\): for \(n_j\) patients and 4 response categories the number of combinations is \(\binom{n_j + 3}{3}\), which equals 286 for \(n_j = 10\) and 1771 for \(n_j = 20\). For the Exact method with two groups this means up to \(286^2 \approx 82{,}000\) unique pairs each requiring
nMCDirichlet draws. When predictive probability is used (\(m_j > 0\)), an additional layer of Multinomial sampling is added inside each Dirichlet draw, further multiplying computation time. Use smallnMC(100–500) andn_t = n_c = 10for demonstration; switch toCalcMethod = 'MC'with moderatensimfor larger sample sizes.
4.2 Example: Controlled Design, Posterior Probability
pi_t_seq <- seq(0.20, 0.90, by = 0.10)
n_scen <- length(pi_t_seq)
oc_ctrl <- pbayesdecisionprob2bin(
prob = 'posterior', design = 'controlled',
GoRegions = 1L, NoGoRegions = 9L,
gamma_go = 0.80, gamma_nogo = 0.80,
pi_t1 = rep(pi_t_seq, each = n_scen),
pi_t2 = rep(pi_t_seq, times = n_scen),
rho_t = rep(0.0, n_scen * n_scen),
pi_c1 = rep(0.20, n_scen * n_scen),
pi_c2 = rep(0.20, n_scen * n_scen),
rho_c = rep(0.0, n_scen * n_scen),
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,
m_t = NULL, m_c = NULL,
theta_TV1 = 0.20, theta_MAV1 = 0.10,
theta_TV2 = 0.20, theta_MAV2 = 0.10,
theta_NULL1 = NULL, theta_NULL2 = NULL,
z00 = NULL, z01 = NULL, z10 = NULL, z11 = NULL,
xe_t_00 = NULL, xe_t_01 = NULL, xe_t_10 = NULL, xe_t_11 = NULL,
xe_c_00 = NULL, xe_c_01 = NULL, xe_c_10 = NULL, xe_c_11 = NULL,
alpha0e_t = NULL, alpha0e_c = NULL,
nMC = 200L, CalcMethod = 'Exact',
error_if_Miss = TRUE, Gray_inc_Miss = FALSE
)
print(oc_ctrl)
#> Go/NoGo/Gray Decision Probabilities (Two Binary Endpoints)
#> -----------------------------------------------------------------------------
#> Probability type : posterior
#> Design : controlled
#> Threshold(s) : TV1 = 0.2, MAV1 = 0.1
#> TV2 = 0.2, MAV2 = 0.1
#> Go threshold : gamma_go = 0.8
#> NoGo threshold : gamma_nogo = 0.8
#> Go regions : {1}
#> NoGo regions : {9}
#> Sample size : n_t = 7, n_c = 7
#> Prior (treatment): a_t = (0.25, 0.25, 0.25, 0.25) [a_00, a_01, a_10, a_11]
#> Prior (control) : a_c = (0.25, 0.25, 0.25, 0.25) [a_00, a_01, a_10, a_11]
#> Method : Exact
#> MC draws : nMC = 200
#> Miss handling : error_if_Miss = TRUE, Gray_inc_Miss = FALSE
#> -----------------------------------------------------------------------------
#> pi_t1 pi_t2 rho_t pi_c1 pi_c2 rho_c Go Gray NoGo
#> 0.2 0.2 0 0.2 0.2 0 0.0002 0.8886 0.1112
#> 0.2 0.3 0 0.2 0.2 0 0.0008 0.9381 0.0611
#> 0.2 0.4 0 0.2 0.2 0 0.0021 0.9666 0.0313
#> 0.2 0.5 0 0.2 0.2 0 0.0045 0.9810 0.0145
#> 0.2 0.6 0 0.2 0.2 0 0.0082 0.9859 0.0059
#> 0.2 0.7 0 0.2 0.2 0 0.0136 0.9845 0.0019
#> 0.2 0.8 0 0.2 0.2 0 0.0205 0.9791 0.0004
#> 0.2 0.9 0 0.2 0.2 0 0.0290 0.9710 0.0000
#> 0.3 0.2 0 0.2 0.2 0 0.0009 0.9380 0.0611
#> 0.3 0.3 0 0.2 0.2 0 0.0031 0.9637 0.0332
#> 0.3 0.4 0 0.2 0.2 0 0.0076 0.9756 0.0168
#> 0.3 0.5 0 0.2 0.2 0 0.0151 0.9772 0.0077
#> 0.3 0.6 0 0.2 0.2 0 0.0262 0.9707 0.0030
#> 0.3 0.7 0 0.2 0.2 0 0.0415 0.9575 0.0010
#> 0.3 0.8 0 0.2 0.2 0 0.0610 0.9388 0.0002
#> 0.3 0.9 0 0.2 0.2 0 0.0848 0.9152 0.0000
#> 0.4 0.2 0 0.2 0.2 0 0.0024 0.9670 0.0306
#> 0.4 0.3 0 0.2 0.2 0 0.0080 0.9755 0.0165
#> 0.4 0.4 0 0.2 0.2 0 0.0186 0.9732 0.0082
#> 0.4 0.5 0 0.2 0.2 0 0.0352 0.9611 0.0037
#> 0.4 0.6 0 0.2 0.2 0 0.0586 0.9400 0.0014
#> 0.4 0.7 0 0.2 0.2 0 0.0892 0.9104 0.0005
#> 0.4 0.8 0 0.2 0.2 0 0.1272 0.8728 0.0000
#> 0.4 0.9 0 0.2 0.2 0 0.1728 0.8272 0.0000
#> 0.5 0.2 0 0.2 0.2 0 0.0053 0.9811 0.0136
#> 0.5 0.3 0 0.2 0.2 0 0.0167 0.9761 0.0073
#> 0.5 0.4 0 0.2 0.2 0 0.0368 0.9596 0.0036
#> 0.5 0.5 0 0.2 0.2 0 0.0667 0.9317 0.0016
#> 0.5 0.6 0 0.2 0.2 0 0.1070 0.8924 0.0006
#> 0.5 0.7 0 0.2 0.2 0 0.1574 0.8424 0.0002
#> 0.5 0.8 0 0.2 0.2 0 0.2176 0.7824 0.0000
#> 0.5 0.9 0 0.2 0.2 0 0.2876 0.7124 0.0000
#> 0.6 0.2 0 0.2 0.2 0 0.0098 0.9850 0.0051
#> 0.6 0.3 0 0.2 0.2 0 0.0296 0.9677 0.0027
#> 0.6 0.4 0 0.2 0.2 0 0.0628 0.9359 0.0013
#> 0.6 0.5 0 0.2 0.2 0 0.1101 0.8894 0.0006
#> 0.6 0.6 0 0.2 0.2 0 0.1709 0.8289 0.0002
#> 0.6 0.7 0 0.2 0.2 0 0.2440 0.7560 0.0000
#> 0.6 0.8 0 0.2 0.2 0 0.3274 0.6726 0.0000
#> 0.6 0.9 0 0.2 0.2 0 0.4198 0.5802 0.0000
#> 0.7 0.2 0 0.2 0.2 0 0.0161 0.9824 0.0016
#> 0.7 0.3 0 0.2 0.2 0 0.0466 0.9526 0.0008
#> 0.7 0.4 0 0.2 0.2 0 0.0958 0.9038 0.0004
#> 0.7 0.5 0 0.2 0.2 0 0.1632 0.8366 0.0002
#> 0.7 0.6 0 0.2 0.2 0 0.2470 0.7530 0.0000
#> 0.7 0.7 0 0.2 0.2 0 0.3437 0.6563 0.0000
#> 0.7 0.8 0 0.2 0.2 0 0.4488 0.5512 0.0000
#> 0.7 0.9 0 0.2 0.2 0 0.5577 0.4423 0.0000
#> 0.8 0.2 0 0.2 0.2 0 0.0235 0.9762 0.0003
#> 0.8 0.3 0 0.2 0.2 0 0.0663 0.9335 0.0002
#> 0.8 0.4 0 0.2 0.2 0 0.1331 0.8669 0.0000
#> 0.8 0.5 0 0.2 0.2 0 0.2221 0.7779 0.0000
#> 0.8 0.6 0 0.2 0.2 0 0.3295 0.6705 0.0000
#> 0.8 0.7 0 0.2 0.2 0 0.4491 0.5509 0.0000
#> 0.8 0.8 0 0.2 0.2 0 0.5724 0.4276 0.0000
#> 0.8 0.9 0 0.2 0.2 0 0.6897 0.3103 0.0000
#> 0.9 0.2 0 0.2 0.2 0 0.0310 0.9690 0.0000
#> 0.9 0.3 0 0.2 0.2 0 0.0861 0.9139 0.0000
#> 0.9 0.4 0 0.2 0.2 0 0.1702 0.8298 0.0000
#> 0.9 0.5 0 0.2 0.2 0 0.2804 0.7196 0.0000
#> 0.9 0.6 0 0.2 0.2 0 0.4108 0.5892 0.0000
#> 0.9 0.7 0 0.2 0.2 0 0.5518 0.4482 0.0000
#> 0.9 0.8 0 0.2 0.2 0 0.6892 0.3108 0.0000
#> 0.9 0.9 0 0.2 0.2 0 0.8073 0.1927 0.0000
#> -----------------------------------------------------------------------------
plot(oc_ctrl, base_size = 20)
The same function supports design = 'uncontrolled' (with
hypothetical count vector arguments z00, z01,
z10, z11), design = 'external'
(with power prior arguments xe_c_00, …,
xe_c_11 and alpha0e_c), and
prob = 'predictive' (with future sample size arguments
m_t, m_c and theta_NULL1,
theta_NULL2). The within-group correlation
rho_t can also be varied to study its effect on the
operating characteristics. The function signature and output format are
identical across all combinations.
