| pwr2n.NPH {nphPower} | R Documentation |
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 <= entry)
},
entry_pdf1 = entry_pdf0,
ratio = 1,
alpha = 0.05,
beta = 0.1,
alternative = c("two.sided"),
criteria = 500,
k = 100,
m = 0,
nreps = 10,
varianceType = c("equal"),
weightBase = "C",
summary = TRUE
)
Arguments
method |
a text specifying the calculation method, either
|
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 |
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 " |
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: |
weightBase |
A character, either " |
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.
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 |
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
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
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)