Sample size and power calculations using the APPLE, SEPPLE, APPLE+ and SEPPLE+ methods

Description

An R package for sample size and power calculation when the treatment time-lag effect is present. The package incorporates two specific lag assumptions:
1. the lag duration is homogeneous across the individual subject;
2. the lag duration varies heterogeneously from individual to individual within a certain domain and following a specific pattern.

Details

The four new methods in this package for performing the sample size and power calculations are:
1. Analytic Power calculation method based on Piecewise weighted Log-rank tEst (APPLE),
2. Simulation-based Empirical Power calculation method based on Piecewise weighted Log-rank tEst (SEPPLE),
3. Analytic Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect (APPLE+),
4. Simulation-based Empirical Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect (SEPPLE+).
See the reference for details of these methods. Specifically, APPLE and SEPPLE assume that the lag duration is homogeneous across the individual subject, whereas APPLE and SEPPLE assume that the lag duration varies heterogeneously from individual to individual or from study to study within a certain domain and following a specific pattern. The functions for computing power corresponding to the above methods are pow.APPLE, pow.SEPPLE, pow.APPLE.plus, pow.SEPPLE.plus and pow.SEPPLE.random.DE. These can be compared to pow.sim.logrk and pow.sim.logrk.rankdom.DE which compute the power from a simulation-based algorithm using the regular log-rank test which ignores the existence of lag effects. The package also includes the function N.APPLE, N.APPLE.plus to back calculate the sample size given the power and hazard ratio, and the functions HR.APPLE and HR.APPLE.plus to back calculate the hazard ratio given the power and sample size, respectively, using the close-from APPLE and APPLE+ methods.

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov> , Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

Generalized Piecewise Weighted Logrank Test

Description

Compute the p-value based on the generalized piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a uniform pattern.

Usage

GPW.logrank(data, obs.time, time.to.event, event.status, trt.group, tl, tu) 

Arguments

Value

The p-value of the test.

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov> , Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.SEPPLE.plus

Examples

  data(data, package="DelayedEffect.Design")
  GPW.logrank(data, "X", "time_to_event", "event_status", "Z", 30, 30*11) 

APPLE hazard ratio computation

Description

Perform the post-delay hazard ratio calculation given power and sample size using the close-form APPLE method based on the piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration is homogeneous across the individual subject

Usage

 HR.APPLE(lambda1, t1, p, N, tao, A, beta, ap=0.5, alpha=0.05) 

Arguments

Details

APPLE is an acronym for:
Analytic Power calculation method based on Piecewise weighted Log-rank tEst. See the reference for details of this method.

Out of the three input parameters lambda1, t1 and p, only two need to be specified, the remaining one will be computed internally from the formula lambda1 = -log(p)/t1. If all three are not NULL, then lambda1 will be set to -log(p)/t1 regardless of the user input value.

Value

The hazard ratio

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.APPLE, N.APPLE

Examples

  lambda1 <- NULL
  t1      <- 183
  p       <- 0.7
  N       <- 200
  tao     <- 365*3
  A       <- 365
  beta    <- 0.2
  HR.APPLE(lambda1, t1, p, N, tao, A, beta)

APPLE+ hazard ratio computation

Description

Perform the post-delay hazard ratio calculation given power and sample size using the close-form APPLE+ method based on the generalized piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a specific pattern.

Usage

 HR.APPLE.plus(lambda1, tl, tu, N, tao, A, beta, ap=0.5, alpha=0.05) 

Arguments

Details

APPLE+ is an acronym for:
Analytic Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect. See the reference for details of this method.

Value

The hazard ratio

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.APPLE.plus, N.APPLE.plus

Examples

  lambda1 <- 0.001982
  tl      <- 30
  tu      <- 30*11
  N       <- 200
  tao     <- 365*3
  A       <- 365
  beta    <- 0.2
  HR.APPLE.plus(lambda1, tl, tu, N, tao, A, beta)

APPLE sample size computation

Description

Perform the sample size calculation given the power and post-delay hazard ratio using the closeform APPLE method based on the piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration is homogeneous across the individual subject

Usage

 N.APPLE(lambda1, t1, p, HR, tao, A, beta, ap=0.5, alpha=0.05)

Arguments

Details

APPLE is an acronym for:
Analytic Power calculation method based on Piecewise weighted Log-rank tEst. See the reference for details of this method.

Out of the three input parameters lambda1, t1 and p, only two need to be specified, the remaining one will be computed internally from the formula lambda1 = -log(p)/t1. If all three are not NULL, then lambda1 will be set to -log(p)/t1 regardless of the user input value.

Value

The sample size

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.APPLE, HR.APPLE

Examples

  lambda1 <- NULL
  t1      <- 183
  p       <- 0.7
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  beta    <- 0.2
  N.APPLE(lambda1, t1, p, HR, tao, A, beta)

APPLE+ sample size computation

Description

Perform the sample size calculation given the power and post-delay hazard ratio using the close-form APPLE+ method based on the generalized piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a specific pattern.

Usage

 N.APPLE.plus(lambda1, tl, tu, HR, tao, A, beta, ap=0.5, alpha=0.05)

Arguments

Details

APPLE+ is an acronym for:
Analytic Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect. See the reference for details of this method.

Value

The sample size

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.APPLE.plus, HR.APPLE.plus

Examples

  lambda1 <- 0.001982
  tl      <- 30
  tu      <- 30*11
  HR      <- 1.3
  tao     <- 365*3
  A       <- 365
  beta    <- 0.2
  N.APPLE.plus(lambda1, tl, tu, HR, tao, A, beta)

Data for examples

Description

Data for examples.

Details

A data frame used in the examples.

Examples

 data(data, package="DelayedEffect.Design")

 # Display some of the data
 data[1:5, ]

APPLE power computation

Description

Perform the power calculation using the close-form APPLE method based on the piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration is homogeneous across the individual subject

Usage

 pow.APPLE(lambda1, t1, p, N, HR, tao, A, ap=0.5, alpha=0.05)

Arguments

Details

APPLE is an acronym for:
Analytic Power calculation method based on Piecewise weighted Log-rank tEst. See the reference for details of this method.

Out of the three input parameters lambda1, t1 and p, only two need to be specified, the remaining one will be computed internally from the formula lambda1 = -log(p)/t1. If all three are not NULL, then lambda1 will be set to -log(p)/t1 regardless of the user input value.

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

N.APPLE, HR.APPLE, pow.SEPPLE, pow.sim.logrk

Examples

  lambda1 <- NULL
  t1      <- 183
  p       <- 0.7
  N       <- 200
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  pow.APPLE(lambda1, t1, p, N, HR, tao, A)

APPLE+ power computation

Description

Perform the power calculation using the close-form APPLE+ method based on the generalized piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a specific pattern.

Usage

 pow.APPLE.plus(lambda1, tl, tu, N, HR, tao, A, ap=0.5, alpha=0.05)

Arguments

Details

APPLE+ is an acronym for:
Analytic Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect. See the reference for details of this method.

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

N.APPLE.plus, HR.APPLE.plus

Examples

  lambda1 <- 0.001982
  tl      <- 30
  tu      <- 30*11
  N       <- 200
  HR      <- 1.3
  tao     <- 365*3
  A       <- 365
  pow.APPLE.plus(lambda1, tl, tu, N, HR, tao, A)

SEPPLE power computation

Description

Perform the power calculation using the numeric SEPPLE method based on the piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration is homogeneous across the individual subject

Usage

pow.SEPPLE(lambda1, t1, p, N, HR, tao, A, ap=0.5, alpha=0.05, nsim=10000) 

Arguments

Details

SEPPLE is an acronym for:
Simulation-based Empirical Power calculation method based on Piecewise weighted Log-rank tEst. See the reference for details of this method.

Out of the three input parameters lambda1, t1 and p, only two need to be specified, the remaining one will be computed internally from the formula lambda1 = -log(p)/t1. If all three are not NULL, then lambda1 will be set to -log(p)/t1 regardless of the user input value.

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov> , Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.APPLE, pow.sim.logrk

Examples

  lambda1 <- NULL
  t1      <- 183
  p       <- 0.7
  N       <- 200
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  pow.SEPPLE(lambda1, t1, p, N, HR, tao, A, nsim=1000)

SEPPLE+ power computation

Description

Perform the power calculation using the numeric SEPPLE+ method based on the generalized piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a specific pattern.

Usage

pow.SEPPLE.plus(lambda1, tl, tu, N, HR, tao, A, dist="uniform",
              shape1=NULL, shape2=NULL, ap=0.5, alpha=0.05, nsim=10000) 

Arguments

Details

SEPPLE+ is an acronym for:
Simulation-based Empirical Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect. See the reference for details of this method.

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov> , Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.SEPPLE.random.DE, pow.sim.logrk.random.DE

Examples

  lambda1 <- 0.001982
  tl      <- 30
  tu      <- 30*11
  N       <- 200
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  shape1  <- 5
  shape2  <- 5
  pow.SEPPLE.plus(lambda1, tl, tu, N, HR, tao, A, dist="beta", 
                  shape1=shape1, shape2=shape2, nsim=1000)

SEPPLE+ power computation

Description

Perform the power calculation using the numeric SEPPLE method based on the piecewise weighted log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a specific pattern. The purpose of this function is to evaluate the property of SEPPLE which assumes the lag duration is homogeneous across the individual subject, when applied under the random scenario where the lag duration, in fact, varies heterogeneously.

Usage

pow.SEPPLE.random.DE(lambda1, tl, tu, N, HR, tao, A, t.fixed, dist="uniform",
              shape1=NULL, shape2=NULL, ap=0.5, alpha=0.05, nsim=10000) 

Arguments

Details

SEPPLE+ is an acronym for:
Simulation-based Empirical Power calculation method based on generalized Piecewise weighted Log-rank tEst with random treatment time-lag effect. See the reference for details of this method.

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov> , Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.SEPPLE.plus, pow.sim.logrk.random.DE

Examples

  lambda1 <- 0.001982
  tl      <- 30
  tu      <- 30*11
  N       <- 200
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  t.fixed <- (tl+tu)/2
  shape1  <- 5
  shape2  <- 5
  pow.SEPPLE.random.DE(lambda1, tl, tu, N, HR, tao, A, t.fixed, dist="beta", 
                       shape1=shape1, shape2=shape2, nsim=1000)

Simulated log-rank power computation

Description

Perform the power calculation using a simulation-based method based on the regular log-rank test when the treatment time-lag effect is present and the lag duration is homogeneous across the individual subject

Usage

pow.sim.logrk(lambda1, t1, p, N, HR, tao, A, ap=0.5, alpha=0.05, nsim=10000) 

Arguments

Details

Out of the three input parameters lambda1, t1 and p, only two need to be specified, the remaining one will be computed internally from the formula lambda1 = -log(p)/t1. If all three are not NULL, then lambda1 will be set to -log(p)/t1 regardless of the user input value.

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov>, Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.APPLE, pow.SEPPLE

Examples

  lambda1 <- NULL
  t1      <- 183
  p       <- 0.7
  N       <- 200
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  pow.sim.logrk(lambda1, t1, p, N, HR, tao, A, nsim=1000)

Simulated log-rank power computation

Description

Perform the power calculation using a simulation-based method based on the regular log-rank test when the treatment time-lag effect is present and the lag duration varies heterogeneously from individual to individual or from study to study, within a certain domain and following a specific pattern.

Usage

pow.sim.logrk.random.DE(lambda1, tl, tu, N, HR, tao, A, dist="uniform",
              shape1=NULL, shape2=NULL, ap=0.5, alpha=0.05, nsim=10000) 

Arguments

Details

The regular log-rank test is used here

Value

The power

Author(s)

Zhenzhen Xu <Zhenzhen.Xu@fda.hhs.gov> , Boguang Zhen<Boguang.Zhen@fda.hhs.gov>, Yongsoek Park <yongpark@pitt.edu> and Bin Zhu <bin.zhu@nih.gov>

References

Xu, Z., Park, Y., Zhen, B. & Zhu, B. (2017). Achieving optimal power of logrank test with random treatment time-lag effect. Biometrika. Under review.

Xu, Z., Zhen, B., Park, Y., & Zhu, B. (2017). Designing therapeutic cancer vaccine trials with delayed treatment effect. Statistics in medicine, 36(4), 592-605.

See Also

pow.SEPPLE.plus, pow.SEPPLE.random.DE

Examples

  lambda1 <- 0.001982
  tl      <- 30
  tu      <- 30*11
  N       <- 200
  HR      <- 0.55
  tao     <- 365*3
  A       <- 365
  shape1  <- 5
  shape2  <- 5
  pow.sim.logrk.random.DE(lambda1, tl, tu, N, HR, tao, A, dist="beta",
                          shape1=shape1, shape2=shape2, nsim=1000)