nphPower (version 1.1.0)

pwr2n.NPH: Sample Size Calculation with Combination Test

Description

pwr2n.NPH calculates the number of events and subjects required to achieve pre-specified power in the setup of two groups. The method extends the calculation in the framework of the Markov model by Lakatos, allowing for using the maximum weighted logrank tests or projection test with an arbitrary number of weight functions. For maximum weighted logrank type test, if only one weight function is provided, the test is essentially the classic (weighted) logrank test.

Usage

pwr2n.NPH(
  method = "MaxLR",
  entry = 1,
  fup = 1,
  maxfup = entry + fup,
  CtrlHaz,
  hazR,
  transP1 = c(0, 0),
  transP0 = c(0, 0),
  Wlist = list(function(x) {
     x^0
 }),
  entry_pdf0 = function(x) {
     (1/entry) * (x >= 0 & x 

Value

An object of class "NPHpwr" with corresponding plot function. The object is a list containing the following components:

eventN

total number of events

totalN

total number of subjects

pwr

actual power given the number of events

prob_event

event probability at the end of trial

prob1

event probability for the treatment group

prob0

event probability for the control group

L_trans

a list, consisting of transition matrix at each interval

pdat

a dataframe including all the intermediate variables in the calculation. see Details.

studytime

a vector of length 2, including the entry and follow-up time as input

RandomizationRatio

as input

eventlist

a vector containing the number of events using each weight function alone

inputfun

a list containing all the input functions specified by users

Arguments

method

a text specifying the calculation method, either "MaxLR" or "Projection". Maximum weighted logrank test is used if "MaxLR" is specified; otherwise, projection test is used.

entry

a numeric value indicating the enrollment time, Default: 1

fup

a numeric value indicating the minimum follow-up time for subjects. , Default: 1

maxfup

maximum follow-up time

CtrlHaz

a function, specifying the hazard function for control group.

hazR

a function, specifying the hazard ratio function between treatment and control group

transP1

a numeric vector of length 2, consisting of the transition probability from receiving treatment to drop-out (drop-out rate) and from receiving treatment to receiving control (drop-in rate) per time unit.

transP0

a numeric vector of length 2, consisting of the transition probability from receiving control to drop-out (drop-out rate) and from receiving control to receiving treatment (drop-in rate) per time unit.

Wlist

a list, consisting of weight functions applied to the test. The element of the list must be functions. Default is a list of one constant function, corresponding to the logrank test.

entry_pdf0

a function, indicating the probability density function (pdf) of enrollment time for control group. The default assumes a uniform distribution corresponding to the constant enrollment rate.

entry_pdf1

a pdf of enrollment time for treatment group. See entry_pdf0, Default: assume same pdf as control group.

ratio

an integer, indicating the randomization ratio between treatment and control group, Default: 1

alpha

type I error rate, Default: 0.05

beta

type II error rate, Default: 0.1

alternative

a character string specifying the alternative hypothesis, must be one of "two.sided", "greater","less". See details. For "Projection" method, only "two-sided" alternative is supported. Default: c("two.sided")

criteria

an integer indicating the maximum iteration allowed in obtaining the number of events. See details , Default: 500

k

an integer, indicating number of sub-intervals per time unit, Default: 100

m

a value within 0 and 1.

nreps

an integer, indicating number of iterations in calculating the quantile of multivariate normal. See Details.

varianceType

Default: equal. Indicates different variance assumptions for the sample size calculation. It is not applicable for the maximum weighted logrank test. See details.

weightBase

A character, either "F" or "T". F indicates a CDF is the base for the weight function used in the weighted logrank or maximum weighted logrank test. T indicates time is the base for weight function. Default: F

summary

a logical value, controlling whether to print the summary of calculation, Default: TRUE

Details

The detailed methods can be found in the reference papers. The number of subjects is determined by several factors, including the control hazard function, hazard ratio function, entry time distribution, follow-up time, etc. Under proportional hazard assumption, the number of events is mainly determined by the hazard ratio besides type i/ii error rates. However, under nonproportional hazards, all the above design parameters may have an impact on the number of events. The study design assumes entry time units of enrollment and at least fup time units of follow-up. If enrollment time entry is set to zero, all subjects are enrolled simultaneously, so there is no staggered entry. Otherwise, if entry is greater than 0, administrative censoring is considered. The user-defined enrollment time function, hazard function for the control group and hazard ratio function can be either discrete or continuous. Various non-proportional hazards types are accommodated. See examples below. If multiple weight functions are provided in Wlist, a maximum weighted logrank test or combination test is implemented. An iterative procedure is used to obtain the event number based on the multivariate normal distribution. Package mvtnorm is used to calculate the quantiles. Because the algorithm is slightly seed dependent, the quantiles are mean values of ten replicates by default. The number of replicates is controlled by argument ninter.

The "alternative" option supports both two-sided and one-sided test. Let \(\Lambda_1\) and \(\Lambda_0\) denote the cumulative hazard of treatment and control group. The less option tests \(H_0: \Lambda_1 > \Lambda_0\) against \(H_a: \Lambda_1 <= \Lambda_0\). The greater option tests \(H_0: \Lambda_1 < \Lambda_0\) against \(H_a: \Lambda_1 >= \Lambda_0\).

When varianceType is equal, the sample size for a two sided test is \((z_{1-\alpha/2}+z_{1-\beta})^2\tilde{\sigma}^2/\mu_w^2\), where \(\tilde{\sigma}^2\) is the variance estimate under alternative. when varianceType is not equal. The formula is \((z_{1-\alpha/2}\sigma_w+z_{1-\beta}\tilde{\sigma})^2/\mu_w^2\). Please use equal variance type for the maximum weighted logrank test.

References

Brendel, M., Janssen, A., Mayer, C. D., & Pauly, M. (2014). Weighted logrank permutation tests for randomly right censored life science data. Scandinavian Journal of Statistics, 41(3), 742-761.

Cheng, H., & He, J. (2021). A Maximum Weighted Logrank Test in Detecting Crossing Hazards. arXiv preprint arXiv:2110.03833.

Cheng H, He J. Sample size calculation for the combination test under nonproportional hazards. Biom J. 2023 Apr;65(4):e2100403. doi: 10.1002/bimj.202100403. Epub 2023 Feb 15. PMID: 36789566

See Also

pwr2n.LR gen.wgt, evalfup

Examples

 #------------------------------------------------------------
 ## Delayed treatment effects using maxcombo test
 ## generate a list of weight functions for maxcombot test
 wmax <- gen.wgt(method = "Maxcombo" )
 t_enrl <- 12; t_fup <- 18; lmd0 <- log(2)/12
 ## delayed treatment effects
 f_hr_delay <- function(x){(x<=6)+(x>6)*0.75}
 f_haz0 <- function(x){lmd0*x^0}
 ##  The following code takes more than 5 seconds to run
 # \donttest{
 snph1 <- pwr2n.NPH(entry = t_enrl, fup = t_fup, Wlist = wmax,
                    k = 100, ratio = 2, CtrlHaz = f_haz0, hazR = f_hr_delay)
 # }
 #-------------------------------------------------------------
 # same setting using projection test
 snph2 <- pwr2n.NPH(method = "Projection", entry = t_enrl,
  fup = t_fup, Wlist = wmax, k = 10, ratio = 2, CtrlHaz = f_haz0,
  hazR = f_hr_delay)

 #-------------------------------------------------------------
 #proportional hazards with weibull survival for control group
 #logrank test
 wlr <- gen.wgt(method = "LR" )
 b0 <- 3
 th0 <- 10/(-log(0.2))^(1/b0)
 #Weibull hazard function
 f_hz_weibull <- function(x){b0/th0^b0*x^(b0-1)}
 #hazard ratio function
 f_hr <- function(x){0.5*x^0}
 # define entry and follow-up time
 t_enrl <- 5; t_fup <- 5
 exph1 <- pwr2n.NPH(entry = t_enrl, fup = t_fup, k = 100,
 Wlist = wlr,  CtrlHaz = f_hz_weibull, hazR = f_hr, summary = FALSE)
 summary(exph1)