| Type: | Package |
| Title: | Optimal Two-Stage Designs for Ordered Categorical Outcomes |
| Version: | 1.0.2 |
| Date: | 2025-08-17 |
| Description: | Functions to design and simulate optimal two-stage randomized controlled trials (RCTs) with ordered categorical outcomes, supporting rank-based tests and group-sequential decision rules. Methods build on classical and modern rank tests and two-stage/Group-Sequential designs, e.g., Park (2025) <doi:10.1371/journal.pone.0318211>. Please see the package reference manual and vignettes for details. |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| Depends: | R (≥ 4.0) |
| Imports: | stats |
| RoxygenNote: | 7.3.2 |
| NeedsCompilation: | no |
| Packaged: | 2025-09-01 09:24:41 UTC; ypark56 |
| Author: | Yeonhee Park [aut, cre], Yudi Wang [aut], Zhanpeng Xu [aut] |
| Maintainer: | Yeonhee Park <yeonheepark@skku.edu> |
| Repository: | CRAN |
| Date/Publication: | 2025-09-05 20:40:02 UTC |
Optimal Two-Stage Designs for Ordered Categorical Outcomes
Description
Functions to design and simulate optimal two-stage randomized controlled trials (RCTs) with ordered categorical outcomes, supporting rank-based tests and group-sequential decision rules. Methods build on classical and modern rank tests and two-stage/Group-Sequential designs, e.g., Park (2025) <doi: 10.1371/journal.pone.0318211>. Please see the package reference manual and vignettes for details.
Details
There are several main functoins. Decision_rule_S_1stage, Decision_rule_M_1stage, Decision_rule_W_1stage, ruleF, and ruleFS determine the decision rule for clinical trials. op.1stage, op.F, and op.FS calculate the operating characteristics for clinical trial designs, including type I error, power, and expected sample size, to investigate the performance of the designs.
Author(s)
Yeonhee Park [aut, cre], Yudi Wang [aut], Zhanpeng Xu [aut]
Maintainer: Yeonhee Park <yeonheepark@skku.edu>
References
Park, Y. (2025). Optimal two-stage group sequential designs based on Mann-Whitney-Wilcoxon test. PloS one, 20(2), e0318211.
Decision fule for the F design based on the Mann-Whitney-Wilcoxon test with specified values of alpha1 and beta1
Description
This is to determine the decision rule for a two-stage design based on the Mann-Whitney-Wilcoxon test with the specified values of alpha1 and beta1.
Usage
Decision_rule_M.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha1 |
The parameter used to define futility monitoring. Under the null hypothesis, 1 - alpha1 corresponds to the probability of stopping for futility at the interim analysis. |
beta1 |
The probability of stopping for futility at the interim analysis when the alternative hypothesis is true. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1 |
The threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
References
Park, Y. (2025). Optimal two-stage group sequential designs based on Mann-Whitney-Wilcoxon test. PloS one, 20(2), e0318211.
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
alpha1 <- 0.2
beta1 <- 0.1
Decision_rule_M.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
Decision rule for the FS design based on the Mann-Whitney-Wilcoxon test with the specified values of alpha1, alpha2, and beta1
Description
This is the function to determine the decision rule for the FS design based on the Mann-Whitney-Wilcoxon test with the specified values of alpha1, alpha2, and beta1.
Usage
Decision_rule_M.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha1 |
The parameter used to define futility monitoring. Under the null hypothesis, 1 - alpha1 corresponds to the probability of stopping for futility at the interim analysis. |
alpha2 |
The probability of stopping for superiority at the interim analysis when the null hypothesis is true. |
beta1 |
The probability of stopping for futility at the interim analysis when the alternative hypothesis is true. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1l |
The lower threshold of the test statistic at the 1st analysis. |
t1u |
The upper threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
beta2 |
Under the null hypothesis, 1 - beta2 denotes the probability of stopping for superiority at the interim analysis. |
References
Park, Y. (2025). Optimal two-stage group sequential designs based on Mann-Whitney-Wilcoxon test. PloS one, 20(2), e0318211.
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
alpha1 <- 0.2
alpha2 <- 0.025
beta1 <- 0.1
Decision_rule_M.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
One-stage clinical trial design based on the Mann-Whitney-Wilcoxon test
Description
This is the function to determine the decision rule for a one-stage clinical trial designs based on the Mann-Whitney-Wilcoxon test.
Usage
Decision_rule_M_1stage(p1, p2, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n2 |
The total sample size at the final analysis including both the control and experimental groups. |
t2 |
The threshold of the test statistic at the analysis. |
References
Park, Y. (2025). Optimal two-stage group sequential designs based on Mann-Whitney-Wilcoxon test. PloS one, 20(2), e0318211.
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
Decision_rule_M_1stage(p1, p2, alpha, beta, lambda = 1)
Decision rule for the F design based on the score test with the specified values of alpha1 and beta1
Description
This is to determine the decision rule for a two-stage design based on the score test with the specified values of alpha1 and beta1.
Usage
Decision_rule_S.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha1 |
The parameter used to define futility monitoring. Under the null hypothesis, 1 - alpha1 corresponds to the probability of stopping for futility at the interim analysis. |
beta1 |
The probability of stopping for futility at the interim analysis when the alternative hypothesis is true. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1 |
The threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
Examples
alpha = 0.05; beta = 0.2; or = 3.06
p1 = c(0.075, 0.182, 0.319, 0.243, 0.015, 0.166) # control prob
p2 = p2_fun(p1, log(or)) # experimental prob
p2
alpha1 <- 0.2
beta1 <- 0.1
Decision_rule_S.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
Decision rule of the FS design based on the score test with the specified values of alpha1, alpha2, and beta1
Description
This is the function to determine the decision rule for the FS design based on the score test with the specified values of alpha1, alpha2, and beta1.
Usage
Decision_rule_S.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha1 |
The parameter used to define futility monitoring. Under the null hypothesis, 1 - alpha1 corresponds to the probability of stopping for futility at the interim analysis. |
alpha2 |
The probability of stopping for superiority at the interim analysis when the null hypothesis is true. |
beta1 |
The probability of stopping for futility at the interim analysis when the alternative hypothesis is true. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1l |
The lower threshold of the test statistic at the 1st analysis. |
t1u |
The upper threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
beta2 |
Under the null hypothesis, 1 - beta2 denotes the probability of stopping for superiority at the interim analysis. |
Examples
alpha = 0.05; beta = 0.2; or = 3.06
p1 = c(0.075, 0.182, 0.319, 0.243, 0.015, 0.166) # control prob
p2 = p2_fun(p1, log(or)) # experimental prob
p2
alpha1 <- 0.2
alpha2 <- 0.025
beta1 <- 0.1
Decision_rule_S.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
One-stage clinical trial design based on the score test
Description
This is the function to determine the decision rule for a one-stage clinical trial designs based on the score test.
Usage
Decision_rule_S_1stage(p1, p2, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n2 |
The total sample size at the final analysis including both the control and experimental groups. |
t2 |
The threshold of the test statistic at the analysis. |
References
Park, Y. (2025). Optimal two-stage group sequential designs based on Mann-Whitney-Wilcoxon test. PloS one, 20(2), e0318211.
Examples
alpha = 0.05; beta = 0.2; or = 3.06
p1 = c(0.075, 0.182, 0.319, 0.243, 0.015, 0.166) # control prob
p2 = p2_fun(p1, log(or)) # experimental prob
p2
Decision_rule_S_1stage(p1, p2, alpha, beta, lambda = 1)
Decision rule of the F design based on the Win Odds test with the specified values of alpha1 and beta1
Description
This is to determine the decision rule for a two-stage design based on the Win Odds test with the specified values of alpha1 and beta1.
Usage
Decision_rule_W.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha1 |
The parameter used to define futility monitoring. Under the null hypothesis, 1 - alpha1 corresponds to the probability of stopping for futility at the interim analysis. |
beta1 |
The probability of stopping for futility at the interim analysis when the alternative hypothesis is true. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1 |
The threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
alpha1 <- 0.2
beta1 <- 0.1
Decision_rule_W.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
Decision rule of the FS design based on the Win Odds test with the specified values of alpha1, alpha2, and beta1
Description
This is the function to determine the decision rule for the FS design based on the Win Odds test with the specified values of alpha1, alpha2, and beta1.
Usage
Decision_rule_W.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha1 |
The parameter used to define futility monitoring. Under the null hypothesis, 1 - alpha1 corresponds to the probability of stopping for futility at the interim analysis. |
alpha2 |
The probability of stopping for superiority at the interim analysis when the null hypothesis is true. |
beta1 |
The probability of stopping for futility at the interim analysis when the alternative hypothesis is true. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1l |
The lower threshold of the test statistic at the 1st analysis. |
t1u |
The upper threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
beta2 |
Under the null hypothesis, 1 - beta2 denotes the probability of stopping for superiority at the interim analysis. |
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
alpha1 <- 0.2
alpha2 <- 0.025
beta1 <- 0.1
Decision_rule_W.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
One-stage clinical trial design based on the Win Odds test
Description
This is the function to determine the decision rule for a one-stage clinical trial designs based on the Win Odds test.
Usage
Decision_rule_W_1stage(p1, p2, alpha, beta, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
n2 |
The total sample size at the final analysis including both the control and experimental groups. |
t2 |
The threshold of the test statistic at the analysis. |
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
Decision_rule_W_1stage(p1, p2, alpha, beta, lambda = 1)
Checking proportional odds assumption
Description
This is the function to check the proportional odds assumption for the score test.
Usage
Proportional_odds_assumption(p1, p2)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
Value
Indicates whether the proportional odds assumption holds. If the assumption holds, the function returns the log-odds ratio from the score test. If the assumption does not hold, the function returns NA.
Calculateion of Q or R
Description
This is the function to compute Q or R, which is required for the Mann-Whitney-Wilcoxon test.
Usage
QR_fun(p1, p2)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
Value
The value of Q or R.
Calcualtion of V
Description
This is the function to compute the value of V over n_{..k}, which is requited for the score test, where n_{..k} denotes the total sample size at the kth analysis.
Usage
V_S.over.nk(p1, p2, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
The value of V over n_{..k}.
Calculation of W
Description
This is the function to compute W, which is required for the Win Odds test.
Usage
W_W(p1, p2, lambda = 1)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
The value of W.
Performance evaluation of a one-stage design
Description
This is the function to calculate the operating characteristics for a one-stage design, including type I error, power, and expected sample size.
Usage
op.1stage(alpha, beta, p1, p2, method, n2, t2, nsim = 10000, lambda = 1)
Arguments
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
method |
"S", "M" or "W", denotes score test, Mann-Whitney-Wilcoxon test and wi n odds test respectively. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
nsim |
The number of simulations. nsim = 10000 by default |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
The probability of wrong decision and the expected total sample size under the true hypothesis.
Examples
set.seed(1234)
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
out <- Decision_rule_W_1stage(p1, p2, alpha, beta, lambda = 1)
# report the power and EN_a
op.1stage(alpha, beta, p1, p2, method="W", n2=out[1], t2=out[2], nsim = 1000, lambda = 1)
# report the overall type I error rate and EN_0
op.1stage(alpha, beta, p1, p1, method="W", n2=out[1], t2=out[2], nsim = 1000, lambda = 1)
Performance evaluation of the F design
Description
This is the function to calculate the operating characteristics for the F design, including type I error, power, and expected sample size.
Usage
op.F(alpha, beta, p1, p2, method, n1, t1, n2, t2, nsim = 10000, lambda = 1)
Arguments
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
method |
"S", "M" or "W", denotes score test, Mann-Whitney-Wilcoxon test and wi n odds test respectively. |
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1 |
The threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
nsim |
The number of simulations. nsim = 10000 by default |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
The probability of wrong decision and the expected total sample size under the true hypothesis.
Examples
set.seed(1234)
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
alpha1 <- 0.2
beta1 <- 0.1
out <- Decision_rule_W.F(p1, p2, alpha1, beta1, alpha, beta, lambda = 1)
# heavier example for illustration (skipped on CRAN timing checkes)
# report the power and EN_a
op.F(alpha, beta, p1, p2, method="W", n1=out[1], t1=out[2], n2=out[3],
t2=out[4], nsim = 10000, lambda = 1)
# report the overall type I error rate and EN_0
op.F(alpha, beta, p1, p1, method="W", n1=out[1], t1=out[2], n2=out[3],
t2=out[4], nsim = 10000, lambda = 1)
Performance evaluation of the FS design
Description
This is the function to calculate the operating characteristics for the FS design, including type I error, power, and expected sample size.
Usage
op.FS(alpha, beta, p1, p2, method, n1, t1l, t1u, n2, t2, nsim = 10000, lambda = 1)
Arguments
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
method |
"S", "M" or "W", denotes score test, Mann-Whitney-Wilcoxon test and win odds test respectively. |
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1l |
The lower threshold of the test statistic at the 1st analysis. |
t1u |
The upper threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
nsim |
The number of simulations. nsim = 10000 by default |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
The probability of wrong decision and the expected total sample size under the true hypothesis.
Examples
set.seed(1234)
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
alpha1 <- 0.2
alpha2 <- 0.025
beta1 <- 0.1
out <- Decision_rule_W.FS(p1, p2, alpha1, alpha2, beta1, alpha, beta, lambda = 1)
# heavier example for illustration (skipped on CRAN timing checkes)
# report the power and EN_a
op.FS(alpha, beta, p1, p2, method="W", n1=out[1], t1l=out[2],
t1u=out[3], n2=out[4], t2=out[5], nsim = 10000, lambda = 1)
# report the overall type I error rate and EN_0
op.FS(alpha, beta, p1, p1, method="W", n1=out[1], t1l=out[2],
t1u=out[3], n2=out[4], t2=out[5], nsim = 10000, lambda = 1)
p2
Description
This is the function to calculate the probability p2 when p1 and odds ratio are given.
Usage
p2_fun(p1, theta)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
theta |
The log odds ratio according to expected effect of the experimental treatment. |
Value
A numeric vector representing the expected probability distribution of outcomes across levels for the experimental group.
p_minus
Description
This is the function to comupute p-, which is required for the Mann-Whitney-Wilcoxon test.
Usage
p_minus(p1, p2)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
Value
The value of p-.
p_plus
Description
This is the function to comupute p+, which is required for the Mann-Whitney-Wilcoxon test.
Usage
p_plus(p1, p2)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
Value
The value of p+.
Calculation of pq
Description
This is the function to compute p_{jk} and q_{ik}, which are required for the Win Odds test.
Usage
pq_fun(p1, p2)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
Value
The value of p_{jk} and q_{ik}.
Decision fule of the F design
Description
This is the function to determine the decision rule for the F design.
Usage
ruleF(alpha, beta, p1, p2, method, criterion, lambda = 1)
Arguments
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
method |
"S", "M" or "W", denotes score test, Mann-Whitney-Wilcoxon test and win odds test respectively. |
criterion |
1: minimizing the expected total sample size under the null hypothesis, 2: minimizing the expected total sample size under the alternative hypothesis, 3: minimizing the expected total sample size assuming that Pr(H0) = Pr(Ha), 4: balancing sample sizes of the two stages prioritizing EN0, 5: balancing sample sizes of the two stages prioritizing maximum sample size n2. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
method |
Statistical test chosen. |
criterion |
Criterion chosen. |
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1 |
The threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
ruleF(alpha, beta, p1, p2, method="M", criterion="1", lambda = 1)
Decision fule of the FS design
Description
This is the function to determine the decision rule for the FS design.
Usage
ruleFS(alpha, beta, p1, p2, method, criterion, lambda = 1)
Arguments
alpha |
Target type I error rate. |
beta |
Target type II error rate. |
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
method |
"S", "M" or "W", denotes score test, Mann-Whitney-Wilcoxon test and win odds test respectively. |
criterion |
1: minimizing the expected total sample size under the null hypothesis, 2: minimizing the expected total sample size under the alternative hypothesis, 3: minimizing the expected total sample size assuming that Pr(H0) = Pr(Ha), 4: balancing sample sizes of the two stages prioritizing EN0, 5: balancing sample sizes of the two stages prioritizing maximum sample size n2. |
lambda |
The ratio of sample sizes between the experimental and control groups, defined as sample size (experimental): sample size (control) = lambda:1. The default value is 1. |
Value
method |
Statistical test chosen. |
criterion |
Criterion chosen. |
n1 |
The total sample size of the control and experimental groups required at the 1st analysis. |
t1l |
The lower threshold of the test statistic at the 1st analysis. |
t1u |
The upper threshold of the test statistic at the 1st analysis. |
n2 |
The cumulative total sample size of the control and experimental groups required at the 2nd analysis. |
t2 |
The threshold of the test statistic at the 2nd analysis. |
Examples
alpha = 0.05; beta = 0.2;
p1 = c(0.2, 0.5, 0.2, 0.1)
p2 = c(0.4, 0.3, 0.2, 0.1)
ruleFS(alpha, beta, p1, p2, method="M", criterion="1", lambda = 1)
Calculation of theta
Description
This is the function to compute theta (i.e., the expectation of T), which is required for the Win Odds test.
Usage
theta(p1, p2)
Arguments
p1 |
A vector containing the probabilities of the outcome falling into each level of the control arm. |
p2 |
A vector containging the probabilities of the outcome falling into each level of the control arm. |
Value
The value of theta