Package 'esDesign' reference manual

Title:	Adaptive Enrichment Designs with Sample Size Re-Estimation
Description:	Software of 'esDesign' is developed to implement the adaptive enrichment designs with sample size re-estimation presented in Lin et al. (2021) <doi: 10.1016/j.cct.2020.106216>. In details, three-proposed trial designs are provided, including the AED1-SSR (or ES1-SSR), AED2-SSR (or ES2-SSR) and AED3-SSR (or ES3-SSR). In addition, this package also contains several widely used adaptive designs, such as the Marker Sequential Test (MaST) design proposed Freidlin et al. (2014) <doi:10.1177/1740774513503739>, the adaptive enrichment designs without early stopping (AED or ES), the sample size re-estimation procedure (SSR) based on the conditional power proposed by Proschan and Hunsberger (1995), and some useful functions. In details, we can calculate the futility and/or efficacy stopping boundaries, the sample size required, calibrate the value of the threshold of the difference between subgroup-specific test statistics, conduct the simulation studies in AED, SSR, AED1-SSR, AED2-SSR and AED3-SSR.
Authors:	Zhao Yang, Ruitao Lin, Guosheng Yin and Ying Yuan
Maintainer:	Zhao Yang <[email protected]>
License:	GPL-2
Version:	1.0.3
Built:	2025-02-11 03:25:59 UTC
Source:	https://github.com/cran/esDesign

Calculate the critical value used at the final analysis in AED

Description

AED.boundary() is used to calculate the critical value used at the final analysis in AED design, meanwhile preserving the overall type I error rate at $\alpha$ level

Usage

AED.boundary(rho, alpha, Info, epsilon)
AED.boundary(rho, alpha, Info, epsilon)

Arguments

`rho`	The proportion of subgroup 1
`alpha`	The overall type I error rate
`Info`	The infromation fraction
`epsilon`	The threshold of difference between the subgroup-specific test statistics

Value

The critical value used at the final analysis

References

Lin, R., Yang, Z., Yuan, Y. and Yin, G., 2021. Sample size re-estimation in adaptive enrichment design. Contemporary Clinical Trials, 100, p.106216. <doi: 10.1016/j.cct.2020.106216>

Examples

AED.boundary(rho = 0.5, alpha = 0.05, Info = 0.5, epsilon = 0.5)
AED.boundary(rho = 0.5, alpha = 0.05, Info = 0.5, epsilon = 0.5)

Conduct the simulation studies of the Adaptive Enrichment Design without early stopping boundary

Description

The AED.sim() is used to conduct the simulation studies of the Adaptive Enrichment Design without early stopping boundary. The AED design is quite similar with the AED1_SSR design. But, in the AED design, the futility stopping boundary and the Sample Size Re-estimation Procedure are removed. On the contrary, a fixed sample size is used to replace the sample size re-estimated procedure. In addition, an $\epsilon$ -rule is also introduced to select the subgroup with larger subgroup-specific test statistic.

Usage

AED.sim(
  N1,
  N2,
  rho,
  alpha,
  beta,
  theta,
  theta0,
  K,
  Info,
  epsilon,
  sigma0,
  nSim,
  Seed
)
AED.sim(
  N1,
  N2,
  rho,
  alpha,
  beta,
  theta,
  theta0,
  K,
  Info,
  epsilon,
  sigma0,
  nSim,
  Seed
)

Arguments

`N1`	The sample size used at the first stage
`N2`	The sample size used at the second stage
`rho`	The proportion of the subgroup 1
`alpha`	The overall Type I error rate
`beta`	The (1 - Power)
`theta`	The sizes of treatment effects in subgroups 1 and 2 among the experimental arm
`theta0`	The size of treatment effect in standard arm
`K`	The number of subgroups
`Info`	The observed information
`epsilon`	The threshold of difference between the subgroup-specific test statistics
`sigma0`	The variance of the treatment effect
`nSim`	The number of simulated studies
`Seed`	The random Seed

Value

A list contains

nTotal The average expected sample size
H00 The probability of rejecting the null hypothesis of $H_{00}$
H01 The probability of rejecting the null hypothesis of $H_{01}$
H02 The probability of rejecting the null hypothesis of $H_{02}$
H0 The probabilities of rejecting at least one of the null hypothesis
Enrich01 The prevalence of adaptive enrichment of subgroup 1
Enrich02 The prevalence of adaptive enrichment of subgroup 2

References

Lin, R., Yang, Z., Yuan, Y. and Yin, G., 2021. Sample size re-estimation in adaptive enrichment design. Contemporary Clinical Trials, 100, p.106216. <doi: 10.1016/j.cct.2020.106216>

Examples

N1 <- 310
N2 <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.20
theta <- c(0,0)
theta0 <- 0
K <- 2
Info <- 0.5
epsilon <- 0.5
sigma0 <- 1
nSim <- 1000
Seed <- 6
AED.sim(N1 = N1, N2 = N2, rho = rho, alpha = alpha,
        beta = beta, theta = theta, theta0 = theta0,
        K  = K, Info = Info, epsilon = epsilon,
        sigma0 = sigma0, nSim = nSim, Seed = Seed)
N1 <- 310
N2 <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.20
theta <- c(0,0)
theta0 <- 0
K <- 2
Info <- 0.5
epsilon <- 0.5
sigma0 <- 1
nSim <- 1000
Seed <- 6
AED.sim(N1 = N1, N2 = N2, rho = rho, alpha = alpha,
        beta = beta, theta = theta, theta0 = theta0,
        K  = K, Info = Info, epsilon = epsilon,
        sigma0 = sigma0, nSim = nSim, Seed = Seed)

Calculate the critical value used at the final analysis of the Adaptive Enrichment Design (Strategy 1) with Sample Size Re-estimation Procedure

Description

The AED1_SSR.boundary() is used to calculate the critical value required at the final analysis of the Adaptive Enrichment Design (Strategy 1) with sample size re-estimation procedure. In the AED1-SSR design, the adaptive enrichment strategy is guided by a pre-specified futility stopping boundary and a threshold of the difference between the subgroup-specific test statistics.

Usage

AED1_SSR.boundary(rho, alpha, pstar, Info, epsilon)
AED1_SSR.boundary(rho, alpha, pstar, Info, epsilon)

Arguments

`rho`	The proportion of subgroup 1.
`alpha`	The overall Type I error rate.
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`Info`	The observation information, which is commonly calculated through the sample size used at each stage of the trial.
`epsilon`	The threshold of the difference between subgroup-specific test statistics.

References

Lin, R., Yang, Z., Yuan, Y. and Yin, G., 2021. Sample size re-estimation in adaptive enrichment design. Contemporary Clinical Trials, 100, p.106216. <doi: 10.1016/j.cct.2020.106216>

Examples

AED1_SSR.boundary(rho = 0.5, alpha = 0.05, pstar = 0.2, Info = 0.5, epsilon = 0.5)
AED1_SSR.boundary(rho = 0.5, alpha = 0.05, pstar = 0.2, Info = 0.5, epsilon = 0.5)

Calculate the conditional power of the Adaptive Enrichment Design with (Strategy 1) Sample Size Re-estimation Procedure

Description

The AED1_SSR.CP() is used to calculate the conditional power of the Adaptive Enrichment Design (Strategy 1) with sample size re-estimation procedure

Usage

AED1_SSR.CP(c, Z1, N1, N2)
AED1_SSR.CP(c, Z1, N1, N2)

Arguments

`c`	The critical value used at the final analysis
`Z1`	The test statistic obtained at the interim analysis
`N1`	The sample size used at the first stage
`N2`	The sample size used at the second stage

Value

A list contains

Critical.Value The critical value used at the final analysis
Conditional.Power The value of conditional power given the observed data

References

Lin, R., Yang, Z., Yuan, Y. and Yin, G., 2021. Sample size re-estimation in adaptive enrichment design. Contemporary Clinical Trials, 100, p.106216. <doi: 10.1016/j.cct.2020.106216>

Examples

c <- 2.258
Z1 <- 1.975
N1 <- 248
N2 <- 200
AED1_SSR.CP(c = 2.258, Z1 = 1.974, N1 = 248, N2 = 200)
c <- 2.258
Z1 <- 1.975
N1 <- 248
N2 <- 200
AED1_SSR.CP(c = 2.258, Z1 = 1.974, N1 = 248, N2 = 200)

Calculate the sample size required at the second stage of the adaptive enrichment design (Strategy1) with Sample Size Re-estimation Procedure

Description

The AED1_SSR.N2() is used to calculated the sample size required at the second stage of the Adaptive Enrichment Design (Strategy 1) with Sample Size Re-estimation Procedure.

Usage

AED1_SSR.N2(c, z1, N1, beta)
AED1_SSR.N2(c, z1, N1, beta)

Arguments

`c`	The critical value used at the final analysis
`z1`	The test statistic obtained at the interim analysis
`N1`	The sample size used at the first stage
`beta`	The (1 - power)

Value

The Value of the re-estimated sample size

References

Lin, R., Yang, Z., Yuan, Y. and Yin, G., 2021. Sample size re-estimation in adaptive enrichment design. Contemporary Clinical Trials, 100, p.106216. <doi: 10.1016/j.cct.2020.106216>

Examples

c <- 2.258
z1 <- 1.974
N1 <- 248
beta <- 0.2
AED1_SSR.N2(c = c, z1 = z1, N1 = N1, beta = beta)

c <- 2.258
z1 <- 1.974
N1 <- 248
beta <- 0.2
AED1_SSR.N2(c = c, z1 = z1, N1 = N1, beta = beta)

Conduct the simulation studies of the Adaptive Enrichment Design (Strategy 1) with Sample Size Re-estimation Procedure

Description

The AED1_SSR.sim() is used to conduct the simulation study of the Adaptive Enrichment Design (Strategy 1) with Sample Size Re-estimation procedure

Usage

AED1_SSR.sim(
  N1,
  rho,
  alpha,
  beta,
  pstar,
  theta,
  theta0,
  Info,
  K = 2,
  epsilon,
  sigma0,
  nSim,
  Seed
)
AED1_SSR.sim(
  N1,
  rho,
  alpha,
  beta,
  pstar,
  theta,
  theta0,
  Info,
  K = 2,
  epsilon,
  sigma0,
  nSim,
  Seed
)

Arguments

`N1`	The sample size used at the first stage
`rho`	The proportion of subgroup 1 among the overall patients
`alpha`	The overall Type I error rate
`beta`	The (1 - Power)
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`theta`	The sizes of the treatment effect in subgroups 1 and 2 with the experimental arm
`theta0`	The size of the treatment effect in standard arm
`Info`	The observation information
`K`	The number of subgroups. The default value is `K = 2`
`epsilon`	The threshold of the difference between the subgroup-specific test statistic
`sigma0`	The variance of the treatment effect
`nSim`	The number of simulated studies
`Seed`	The random seed

Value

A list contains

nTotal The average expected sample size
H00 The probability of rejecting the null hypothesis of $H_{00}$
H01 The probability of rejecting the null hypothesis of $H_{01}$
H02 The probability of rejecting the null hypothesis of $H_{02}$
H0 The probabilities of rejecting at least one of the null hypothesis
ESF The probability of early stopping for futility
ESE The probability of early stopping for efficacy
Enrich01 The prevalence of adaptive enrichment of subgroup 1
Enrich02 The prevalence of adaptive enrichment of subgroup 2

References

Lin, R., Yang, Z., Yuan, Y. and Yin, G., 2021. Sample size re-estimation in adaptive enrichment design. Contemporary Clinical Trials, 100, p.106216. <doi: 10.1016/j.cct.2020.106216>

Examples

res <- AED1_SSR.sim(
  N1 = 310, rho = 0.5,
  alpha = 0.05, beta = 0.2, pstar = 0.2,
  theta = c(0,0), theta0 = 0, Info = 0.5,
  epsilon = 0.5, sigma0 = 1, nSim = 1000, Seed = 6)
res <- AED1_SSR.sim(
  N1 = 310, rho = 0.5,
  alpha = 0.05, beta = 0.2, pstar = 0.2,
  theta = c(0,0), theta0 = 0, Info = 0.5,
  epsilon = 0.5, sigma0 = 1, nSim = 1000, Seed = 6)

Calculate the futility and efficacy stopping boundaries of the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Description

The AED2_SSR.boundary() is used to calculate the futility and efficacy stopping boundaries of the Adaptive Enrichment Design (strategy 2) with Sample Size Re-estimation Procedure. In the AED2-SSR design, an $\epsilon$ -rule is introduced to select the subgroup with larger test statistic. In practice, the value of $\epsilon$ should be calibrated to fit the requirement of the trial.

Usage

AED2_SSR.boundary(rho, alpha, pstar, epsilon)
AED2_SSR.boundary(rho, alpha, pstar, epsilon)

Arguments

`rho`	The proportion of subgroup 1
`alpha`	The overall Type I error rate
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`epsilon`	The threshold of difference between the subgroup-specific test statistics

Value

A list contains

upper.boundary The upper and efficacy stopping boundary
lower.boundary The lower and futility stopping boundary

Examples

rho <- 0.5
alpha <- 0.05
pstar <- 0.15
epsilon <- 0.5
AED2_SSR.boundary(rho = rho, alpha = alpha, pstar = pstar, epsilon = epsilon)
rho <- 0.5
alpha <- 0.05
pstar <- 0.15
epsilon <- 0.5
AED2_SSR.boundary(rho = rho, alpha = alpha, pstar = pstar, epsilon = epsilon)

Calculate the $N2$ and the critical value $C$ in the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Description

The AED2_SSR.CP() is used to calculate the sample size required at the second stage and the critical value used at the final analysis in the Adaptive Enrichment Design with Sample Size Re-estimation Procedure. In addition, this function can also used to conduct the conditional power analysis in terms of $N2$

Usage

AED2_SSR.CP(
  Z1 = NULL,
  delta = NULL,
  N1 = NULL,
  pstar,
  rho,
  epsilon,
  alpha,
  beta,
  N2 = NULL
)
AED2_SSR.CP(
  Z1 = NULL,
  delta = NULL,
  N1 = NULL,
  pstar,
  rho,
  epsilon,
  alpha,
  beta,
  N2 = NULL
)

Arguments

`Z1`	The test statistic obtained at the interim analysis
`delta`	The standardized size of treatment effect, which can be estimated by using $(\mu_{X} - \mu_{Y})/\sqrt{\sigma^2}$ .
`N1`	The sample size used at the first stage
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`rho`	The proportion of subgroup 1
`epsilon`	The threshold of the difference between subgroup-specific test statistics.
`alpha`	The overall Type I error rate
`beta`	The `(1 - Power)`
`N2`	The pre-specified sample size used at the second stage, which is used to conduct the conditional power analysis

Value

A list contains

upper.boundary The efficacy stopping boundary
lower.boundary The futility stopping boundary
N2 The pre-specified sample size used at the second stage, which is used to implement the conditional power analysis
Conditional.Power The value of conditional power given the value of N2 in the conditional power analysis
P.Value The corresponding P-Value used at the final analysis in the conditional power analysis
N2.CP The re-estimated sample size of N2 to ensure an adequate conditional power
c.CP The estimated the critical value used at the final analysis based the conditional power

Examples

Z1 <- 1.974
delta <- 0.355
N1 <- 248
pstar <- 0.15
alpha <- 0.05
rho <- 0.5
epsilon <- 0.5
beta <- 0.20
N2 <- 104
res <- AED2_SSR.CP(Z1 = Z1, delta = delta, N1 = N1, pstar = pstar,
           alpha = alpha, rho = rho, epsilon = epsilon,
           beta = beta, N2 = N2)

Z1 <- 1.974
delta <- 0.355
N1 <- 248
pstar <- 0.15
alpha <- 0.05
rho <- 0.5
epsilon <- 0.5
beta <- 0.20
N2 <- 104
res <- AED2_SSR.CP(Z1 = Z1, delta = delta, N1 = N1, pstar = pstar,
           alpha = alpha, rho = rho, epsilon = epsilon,
           beta = beta, N2 = N2)

Conduct the simulation studies of the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Description

The AED2_SSR.sim() is used to conduct the simulation studies of the Adaptive Enrichment Design (Strategy) with sample size re-estimation procedure. The AED2-SSR is different from the AED3-SSR, in which an $\epsilon$ -rule is introduced to select the subgroup with larger subgroup-specific test statistic.

Usage

AED2_SSR.sim(
  N1,
  rho,
  alpha,
  beta,
  pstar,
  theta,
  theta0,
  sigma0,
  epsilon,
  nSim,
  Seed
)
AED2_SSR.sim(
  N1,
  rho,
  alpha,
  beta,
  pstar,
  theta,
  theta0,
  sigma0,
  epsilon,
  nSim,
  Seed
)

Arguments

`N1`	The sample size used in the first stage
`rho`	The proportion of subgroup 1
`alpha`	The overall Type I error rate
`beta`	The (1 - power)
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`theta`	The sizes of treatment effect in subgroups 1 and 2 with the experimental treatment
`theta0`	The size of treatment effect with the standard treatment
`sigma0`	The variance of the treatment effect
`epsilon`	The threshold of the difference between subgroup-specific test statistics
`nSim`	The number of simulated studies
`Seed`	The random seed

Value

A list contains

nTotal The average expected sample size
H00 The probability of rejecting the null hypothesis of $H_{00}$
H01 The probability of rejecting the null hypothesis of $H_{01}$
H02 The probability of rejecting the null hypothesis of $H_{02}$
H0 The probabilities of rejecting at least one of the null hypothesis
ESF The probability of early stopping for futility
ESE The probability of early stopping for efficacy
Enrich01 The prevalence of adaptive enrichment of subgroup 1
Enrich02 The prevalence of adaptive enrichment of subgroup 2
Trigger03 The prevalence of no enrichment

Examples

N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.2
theta <- c(0,0)
theta0 <- 0
sigma0 <- 1
epsilon <- 0.5
pstar <- 0.20
nSim <- 1000
Seed <- 6
res <- AED2_SSR.sim(N1 = N, rho = rho, alpha = alpha,
             beta = beta, theta = theta, theta0 = theta0,
             sigma0 = sigma0, pstar = pstar, epsilon = epsilon,
             nSim = nSim, Seed = Seed)
N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.2
theta <- c(0,0)
theta0 <- 0
sigma0 <- 1
epsilon <- 0.5
pstar <- 0.20
nSim <- 1000
Seed <- 6
res <- AED2_SSR.sim(N1 = N, rho = rho, alpha = alpha,
             beta = beta, theta = theta, theta0 = theta0,
             sigma0 = sigma0, pstar = pstar, epsilon = epsilon,
             nSim = nSim, Seed = Seed)

Calculate the futility and efficacy stopping boundaries in Adaptive enrichment design (Strategy 3) with Sample Size Re-estimation Procedure for the continuous endpoint

Description

The AED3_SSR.boundary() is used to calculate the futility and efficacy stopping boundaries in the Adaptive Enrichment Design with Sample Size Re-estimation Procedure.

Usage

AED3_SSR.boundary(rho, alpha, pstar)
AED3_SSR.boundary(rho, alpha, pstar)

Arguments

`rho`	The proportion of subgroup 1
`alpha`	The overall Type I error rate
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.

Value

A list contains

upper.boundary The upper or the efficacy stopping boundary
lower.boundary The lower or the futility stopping boundary

Examples

rho <- 0.5
alpha <- 0.05
pstar <- 0.15
res <- AED3_SSR.boundary(rho = rho, alpha = alpha, pstar = pstar)
rho <- 0.5
alpha <- 0.05
pstar <- 0.15
res <- AED3_SSR.boundary(rho = rho, alpha = alpha, pstar = pstar)

Calculate the $N2$ and the critical value $C$ in the Adaptive Enrichment Design (Strategy 3) with Sample Size Re-estimation Procedure

Description

The AED3_SSR.CP() is used to calculate the sample size required at the second stage and the critical value used at the final analysis in the Adaptive Enrichment Design with Sample Size Re-estimation Procedure. In addition, this function can also used to conduct the conditional power analysis in terms of $N2$

Usage

AED3_SSR.CP(
  Z1 = NULL,
  delta = NULL,
  N1 = NULL,
  pstar,
  rho,
  alpha,
  beta,
  N2 = NULL
)
AED3_SSR.CP(
  Z1 = NULL,
  delta = NULL,
  N1 = NULL,
  pstar,
  rho,
  alpha,
  beta,
  N2 = NULL
)

Arguments

`Z1`	The test statistic obtained at the interim analysis
`delta`	The standardized size of treatment effect, which can be estimated by using $(\mu_{X} - \mu_{Y})/\sqrt{\sigma^2}$ .
`N1`	The sample size used at the first stage
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`rho`	The proportion of subgroup 1
`alpha`	The overall Type I error rate
`beta`	The `(1 - Power)`
`N2`	The pre-specified sample size used at the second stage, which is used to conduct the conditional power analysis

Value

A list contains

N2 The pre-specified sample size used at the second stage, which is used to implement the conditional power analysis
Conditional.Power The value of conditional power given the value of N2 in the conditional power analysis
P.Value The corresponding P-Value used at the final analysis in the conditional power analysis
N2.CP The re-estimated sample size of N2 to ensure an adequate conditional power
c.CP The estimated the critical value used at the final analysis based the conditional power

Examples

Z1 <- 1.974
delta <- 0.355
N1 <- 248
pstar <- 0.15
alpha <- 0.05
rho <- 0.5
beta <- 0.20
N2 <- 108
AED3_SSR.CP(Z1 = Z1, delta = delta, N1 = N1, pstar = pstar,
           alpha = alpha, rho = rho, beta = beta, N2 = N2)

Z1 <- 1.974
delta <- 0.355
N1 <- 248
pstar <- 0.15
alpha <- 0.05
rho <- 0.5
beta <- 0.20
N2 <- 108
AED3_SSR.CP(Z1 = Z1, delta = delta, N1 = N1, pstar = pstar,
           alpha = alpha, rho = rho, beta = beta, N2 = N2)

Conduct the simulation studies of the Adaptive Enrichment Design (Strategy 3) with Sample Size Re-estimation Procedure based on Futility and Efficacy Stopping Boundaries for the continuous endpoint

Description

The AED3_SSR.sim() is used to conduct the adaptive enrichment design with Sample Size Re-estimation, in which futility and efficacy stopping boundaries are used to guide the adaptive enrichment process. For the adaptively enriched subgroup, we re-estimate the sample size to maintain an adequate conditional power meanwhile protect the overall Type I error rate.

Usage

AED3_SSR.sim(N1, rho, alpha, beta, theta, theta0, sigma0, pstar, nSim, Seed)
AED3_SSR.sim(N1, rho, alpha, beta, theta, theta0, sigma0, pstar, nSim, Seed)

Arguments

`N1`	The sample size used at the first stage
`rho`	The proportion of subgroup 1 among the overall patients
`alpha`	The overall Type I error rate
`beta`	The `(1 - Power)`
`theta`	The sizes of treatment effect in subgroups 1 and 2 with experimental treatment
`theta0`	The size of treatment effect in standard treatment
`sigma0`	The known variance of the treatment effect
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`nSim`	The number of simulated studies.
`Seed`	The random seed

Value

A list contains

nTotal The average expected sample size
H00 The probability of rejecting the null hypothesis of $H_{00}$
H01 The probability of rejecting the null hypothesis of $H_{01}$
H02 The probability of rejecting the null hypothesis of $H_{02}$
H0 The probabilities of rejecting at least one of the null hypothesis
Enrich01 The prevalence of adaptive enrichment of subgroup 1
Enrich02 The prevalence of adaptive enrichment of subgroup 2
Trigger03 The prevalence of early stopping for the situation, in which the treatment effect in subgroup 1 is superiority, while the treatment effect in subgroup 2 is inconclusive
Trigger04 The prevalence of early stopping for the situation, in which the treatment effect in subgroup 2 is superiority, while the treatment effect in subgroup 2 is inconclusive
ESF The probability of early stopping for futility
ESE The probability of early stopping for efficacy

Examples

N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.2
theta <- c(0,0)
theta0 <- 0
sigma0 <- 1
pstar <- 0.20
nSim <- 100
Seed <- 6
res <- AED3_SSR.sim(N1 = N, rho = rho, alpha = alpha,
             beta = beta, theta = theta, theta0 = theta0,
             sigma0 = sigma0, pstar = pstar, nSim = nSim,
             Seed = Seed)
N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.2
theta <- c(0,0)
theta0 <- 0
sigma0 <- 1
pstar <- 0.20
nSim <- 100
Seed <- 6
res <- AED3_SSR.sim(N1 = N, rho = rho, alpha = alpha,
             beta = beta, theta = theta, theta0 = theta0,
             sigma0 = sigma0, pstar = pstar, nSim = nSim,
             Seed = Seed)

Conduct the simulation studies of the Marker Sequential Test design

Description

The MaST.sim() is used to conduct the simulation studies of the marker sequential test design (MaST).

Usage

MaST.sim(N, rho, alpha, beta, theta, theta0, sigma0, nSim, Seed)
MaST.sim(N, rho, alpha, beta, theta, theta0, sigma0, nSim, Seed)

Arguments

`N`	The total sample size used at the trial
`rho`	The proportion of subgroup 1 among the overall patients
`alpha`	The overall Type I error rate
`beta`	The (1 - Power)
`theta`	The sizes of treatment effect in subgroups 1 and 2 with the experimental arm
`theta0`	The size of treatment effect in the standard arm
`sigma0`	The variance of the treatment effect
`nSim`	The number of simulated studies
`Seed`	The random seed

Value

A list contains

nTotal The average expected sample size
H00 The probability of rejecting the null hypothesis of $H_{00}$
H01 The probability of rejecting the null hypothesis of $H_{01}$
H02 The probability of rejecting the null hypothesis of $H_{02}$
H0 The probabilities of rejecting at least one of the null hypothesis

References

Freidlin, B., Korn, E. L., and Gray, R. (2014). Marker sequential test (MaST) design. Clinical trials, 11(1), 19-27. <doi:10.1177/1740774513503739>

Examples

N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.20
theta <- c(0,0)
theta0 <- 0
sigma0 <- 1
nSim <- 1000
Seed <- 6
MaST.sim(N = N, rho = rho, alpha = alpha, beta = beta,
         theta = theta, theta0 = theta0, sigma0 = sigma0,
         nSim = nSim, Seed = Seed)
N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.20
theta <- c(0,0)
theta0 <- 0
sigma0 <- 1
nSim <- 1000
Seed <- 6
MaST.sim(N = N, rho = rho, alpha = alpha, beta = beta,
         theta = theta, theta0 = theta0, sigma0 = sigma0,
         nSim = nSim, Seed = Seed)

Conduct the simulation studies of the standard design

Description

The SD.sim() is used to implement the simulation studies of the standard design.

Usage

SD.sim(N, rho, alpha, beta, theta, theta0, sigma0, nSim, Seed)
SD.sim(N, rho, alpha, beta, theta, theta0, sigma0, nSim, Seed)

Arguments

`N`	The total sample size required
`rho`	The proportion of subgroup 1
`alpha`	The overall Type I error rate
`beta`	The `(1 -Power)`
`theta`	The sizes of treatment effects for subgroups 1 and 2 in experimental arm
`theta0`	The size of treatment effect for the control arm
`sigma0`	The variance of the treatment effect
`nSim`	The number of simulated studies
`Seed`	The random seed

Value

A list contains,

nTotal the total sample used
The power of the specified trial. Here, the power is defined as the probability of rejecting the null hypothesis.

Examples

N <- 620
rho <- 0.5
alpha <- 0.05
beta <- 0.2
theta <- c(0.2,0.0)
theta0 <- 0
sigma0 <- 1
nSim <- 1000
Seed <- 6
SD.sim(N = N, rho = rho,
       alpha = alpha, beta = beta, theta = theta, theta0 = theta0,
       sigma0 = sigma0, nSim = nSim, Seed = Seed)

N <- 620
rho <- 0.5
alpha <- 0.05
beta <- 0.2
theta <- c(0.2,0.0)
theta0 <- 0
sigma0 <- 1
nSim <- 1000
Seed <- 6
SD.sim(N = N, rho = rho,
       alpha = alpha, beta = beta, theta = theta, theta0 = theta0,
       sigma0 = sigma0, nSim = nSim, Seed = Seed)

Commonly used $\alpha$ -spending functions

Description

The SigP() is used to calculate the reduced significant level based on several widely used $\alpha$ -spending functions, such as the "Pocock", "Lan-DeMets", "O'Brein-Fleming" and "Power" functions.

Usage

SigP(alpha, Info, esFunction = "Pocock", gamma = 1)
SigP(alpha, Info, esFunction = "Pocock", gamma = 1)

Arguments

`alpha`	The overall Type I error rate
`Info`	The fraction of the observed information
`esFunction`	The specific $\alpha$ -spending function. For example, `esFunction = "Pocock"` for the Pocock method, `esFunction = "LD"` for the Lan-Demets method, `esFunction = "OF"` for the O'Brein-Fleming method, and `esFunction = "Power"` for the Power method.
`gamma`	The parameter used in the Power method. The default value is `gamma = 1`.

Value

The reduced significant level

Examples

alpha <- 0.05
Info <- 0.5
esFunction = "OF"
SigP(alpha = alpha, Info = Info, esFunction = esFunction)
alpha <- 0.05
Info <- 0.5
esFunction = "OF"
SigP(alpha = alpha, Info = Info, esFunction = esFunction)

Sample size calculation for the standard design with continuous endpoint

Description

The sSize.norm() is used to calculate the sample size used in the standard design with continuous endpoint.

Usage

sSize.norm(alpha, beta, theta, side, r, sigma2)
sSize.norm(alpha, beta, theta, side, r, sigma2)

Arguments

`alpha`	The Type I error rate or the significant level
`beta`	beta The `(1 -Power)`
`theta`	The size of treatment effect
`side`	One-sided or two-sided Test
`r`	The ratio of sample size between the experimental and control arms
`sigma2`	The variance of the treatment effect

Value

A list contains the total sample size, and the sample sizes required for the experimental and control arms.

Examples

alpha <- 0.05
beta <- 0.2
theta <- 0.2
side <- 1
r <- 1
sigma2 <- 0.8
sSize.norm(alpha = alpha, beta = beta, theta = theta,
side = side, r = r, sigma2 = sigma2)

alpha <- 0.05
beta <- 0.2
theta <- 0.2
side <- 1
r <- 1
sigma2 <- 0.8
sSize.norm(alpha = alpha, beta = beta, theta = theta,
side = side, r = r, sigma2 = sigma2)

Calculate the futility and efficacy stopping boundaries for Sample Size Re-estimation Procedure based on the conditional error function

Description

The SSD.boundary() is used to calculate the futility and efficacy stopping boundaries, meanwhile protect the overall Type I error rate at the pre-specified level.

Usage

SSR.boundary(alpha, pstar)
SSR.boundary(alpha, pstar)

Arguments

`alpha`	The overall Type I error rate
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.

Value

A list contain

upper.boundary The efficacy stopping boundary at the interim analysis
lower.boundary The futility stopping boundary at the interim analysis

References

Proschan MA, Hunsberger SA. Designed extension of studies based on conditional power. Biometrics 1995:1315-24. <doi:10.2307/2533262>

Examples

alpha <- 0.05
pstar <- 0.2
res <- SSR.boundary(alpha = alpha, pstar = pstar)

alpha <- 0.05
pstar <- 0.2
res <- SSR.boundary(alpha = alpha, pstar = pstar)

Calculate the $N2$ and the critical value $C$ in Sample Size Re-estimation Procedure

Description

The SSR.CP() is used to calculate the sample size required at the second stage and the critical value used at the final analysis. In addition, this function can also used to conduct the conditional power analysis in terms of $N2$

Usage

SSR.CP(Z1 = NULL, delta = NULL, N1 = NULL, pstar, alpha, beta, N2 = NULL)
SSR.CP(Z1 = NULL, delta = NULL, N1 = NULL, pstar, alpha, beta, N2 = NULL)

Arguments

`Z1`	The test statistic obtained at the interim analysis
`delta`	The standardized size of treatment effect, which can be estimated by using $(\mu_{X} - \mu_{Y})/\sqrt{\sigma^2}$ .
`N1`	The sample size used at the first stage
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`alpha`	The overall Type I error rate
`beta`	The `(1 - Power)`
`N2`	The pre-specified sample size used at the second stage, which is used to conduct the conditional power analysis

Value

A list contains

N2 The pre-specified sample size used at the second stage, which is used to implement the conditional power analysis
Conditional.Power The value of conditional power given the value of N2 in the conditional power analysis
P.Value The corresponding P-Value used at the final analysis in the conditional power analysis
N2.CP The re-estimated sample size of N2 to ensure an adequate conditional power
c.CP The estimated the critical value used at the final analysis based the conditional power

References

Proschan MA, Hunsberger SA. Designed extension of studies based on conditional power. Biometrics 1995:1315-1324. <doi:10.2307/2533262>

Examples

Z1 <- 1.527
delta <- 0.137
N1 <- 248
pstar <- 0.15
alpha <- 0.05
beta <- 0.2
res <- SSR.CP(Z1 = Z1, delta = delta, N1 = N1,
       pstar = pstar, alpha = alpha, beta = beta)

Z1 <- 1.527
delta <- 0.137
N1 <- 248
pstar <- 0.15
alpha <- 0.05
beta <- 0.2
res <- SSR.CP(Z1 = Z1, delta = delta, N1 = N1,
       pstar = pstar, alpha = alpha, beta = beta)

Conduct the simulation studies using SSR

Description

The SSR.sim() is used to implement the simulation studies based on the Sample Size Re-estimation Procedure.

Usage

SSR.sim(N, rho, alpha, beta, theta, theta0, sigma0, pstar, nSim, Seed)
SSR.sim(N, rho, alpha, beta, theta, theta0, sigma0, pstar, nSim, Seed)

Arguments

`N`	The sample size used at the first stage. Note that this `N` is not the initial total sample size calculated using the standard design
`rho`	The proportion of subgroup 1
`alpha`	The overall Type I error rate
`beta`	The `(1 - Power)`
`theta`	The sizes of treatment effects for subgroups 1 and 2 in the experimental arm
`theta0`	The size of treatment effect in the control arm
`sigma0`	The variance of the treatment effect
`pstar`	The `(1 - power)` of accepting the null hypothesis at the interim analysis.
`nSim`	The number of simulated studies
`Seed`	The random seed

Value

A list contains

nTotal The average total sample size used in SSR
H0 The power of SSR under the specific trial design. Here, the power is defined as the probability of rejecting the null hypothesis
ESF The percentage of early stopping for futility
ESE The percentage of early stopping for efficacy

References

Proschan MA, Hunsberger SA. Designed extension of studies based on conditional power. Biometrics 1995:1315-1324. <doi:10.2307/2533262>

Examples

N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.2
pstar <- 0.2
theta <- c(0.2,0)
theta0 <- 0
sigma0 <- 1.0
nSim <- 1000
Seed <- 6
res <- SSR.sim(N = N, rho = rho, alpha = alpha, beta = beta, theta = theta,
        theta0 = theta0, sigma0 = sigma0, pstar = pstar,
        nSim = nSim, Seed = Seed)
N <- 310
rho <- 0.5
alpha <- 0.05
beta <- 0.2
pstar <- 0.2
theta <- c(0.2,0)
theta0 <- 0
sigma0 <- 1.0
nSim <- 1000
Seed <- 6
res <- SSR.sim(N = N, rho = rho, alpha = alpha, beta = beta, theta = theta,
        theta0 = theta0, sigma0 = sigma0, pstar = pstar,
        nSim = nSim, Seed = Seed)

Package 'esDesign'

Help Index

Calculate the critical value used at the final analysis in AED

Description

Usage

Arguments

Value

References

Examples

Conduct the simulation studies of the Adaptive Enrichment Design without early stopping boundary

Description

Usage

Arguments

Value

References

Examples

Calculate the critical value used at the final analysis of the Adaptive Enrichment Design (Strategy 1) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

References

Examples

Calculate the conditional power of the Adaptive Enrichment Design with (Strategy 1) Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

References

Examples

Calculate the sample size required at the second stage of the adaptive enrichment design (Strategy1) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

References

Examples

Conduct the simulation studies of the Adaptive Enrichment Design (Strategy 1) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

References

Examples

Calculate the futility and efficacy stopping boundaries of the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

Examples

Calculate the N2N2N2 and the critical value CCC in the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

Examples

Conduct the simulation studies of the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

Examples

Calculate the futility and efficacy stopping boundaries in Adaptive enrichment design (Strategy 3) with Sample Size Re-estimation Procedure for the continuous endpoint

Description

Usage

Arguments

Value

Examples

Calculate the N2N2N2 and the critical value CCC in the Adaptive Enrichment Design (Strategy 3) with Sample Size Re-estimation Procedure

Description

Usage

Arguments

Value

Examples

Conduct the simulation studies of the Adaptive Enrichment Design (Strategy 3) with Sample Size Re-estimation Procedure based on Futility and Efficacy Stopping Boundaries for the continuous endpoint

Description

Usage

Arguments

Value

Examples

Conduct the simulation studies of the Marker Sequential Test design

Calculate the $N2$ and the critical value $C$ in the Adaptive Enrichment Design (Strategy 2) with Sample Size Re-estimation Procedure

Calculate the $N2$ and the critical value $C$ in the Adaptive Enrichment Design (Strategy 3) with Sample Size Re-estimation Procedure

Commonly used $\alpha$ -spending functions

Calculate the $N2$ and the critical value $C$ in Sample Size Re-estimation Procedure