Maxcombo.sz: Sample size calculation
In lilywang1988/GSMC: Group Sequential Design for Maxcombo tests

Description Usage Arguments Details Value Author(s) References See Also Examples

Sample size calculation to control the type II error or the power of an interim analysis with Maxcombo tests.

Maxcombo.sz(
  Sigma1,
  mu1,
  z_alpha_vec,
  beta,
  interim_vec,
  R,
  n_range,
  sum_D,
  n.rep = 5
)

`Sigma1`	the correlation matrix under the alternative hypothesis.
`mu1`	the unit mu under the alternative hypothesis (the mean of the expectation of each subject scaled weighted log-rank test statistic, which can be approximated using the formula for E^* in Hasegawa 2014 paper. ).
`z_alpha_vec`	same as the one exported from Maxcombo.bd, which is the boundaries for ordered test statistics, its order should be consistent to the rows and columns in `Sigma1`.
`beta`	type II error.
`interim_vec`	the vector of the interims in each stages, not that it should be a repeat vector with same iterim values for all the test statitics at same stages.
`R`	end of the enrollment time, which is identical to `R` defined in other functions like `I.1`.
`n_range`	the range ot the expected patient numbers.
`sum_D`	same as the exported value from `sample.size_FH`, the summed D^* in Hasegawa (2014).
`n.rep`	number of repeats to take the median for output

Assume that there are 2 stages (1 interm, 1 final), and two tests for a max-combo in each stage, then we have 4 test statistics, and the two cutoff values for the two stages have been determined by Maxcombo.bd in advance using their correlation matrix and the error spending function α_1, α. The goal of this function is to control the sample size n (number of patients for both arms) or d (observed events) to achieve the ideal type II error β or the power (1-β), i.e. \P(Z_{11}<z_1,Z_{12}<z_1,Z_{21}<z_2,Z_{22}<z_2)=β.

`n`	the number of patients needed for the trial to achieve the predefined power.
`d`	the number of events needed for the trial to achieve the predefined power.
`sum_D`	the input `sum_D` value.

Lili Wang

Hasegawa, T. (2014). Sample size determination for the weighted log‐rank test with the Fleming–Harrington class of weights in cancer vaccine studies. Pharmaceutical statistics, 13(2), 128-135.

Maxcombo.beta.n

## Not run: 
# install.packages("mvtnorm")
library(mvtnorm)
# install.packages("gsDesign")
library(gsDesign)
alpha <- 0.025
beta <- 0.1
# If there are two stages (K = 2), with on interim stage and a final stage
# First we obtain the errors spent at each stage to be identical to the ones
 from regular interim analysis assuming that the interim stage happened at
  60% of events have been observed. The error spending function used below
   is O\'Brien-Fleming.
x <- gsDesign::gsDesign(
k = 2, 
test.type = 1, 
timing = 0.6, 
sfu = "OF", 
alpha = alpha, 
beta = beta,
delta = -log(0.7)
)
(z <- x$upper$bound)
x
Sigma0_v <- rep(0.5,6)
Sigma0 <- matrix(1, ncol = 4, nrow = 4)
Sigma0[upper.tri(Sigma0)]<- Sigma0_v
Sigma0[lower.tri(Sigma0)]<- t(Sigma0)[lower.tri(t(Sigma0))]
Sigma0
alpha_interim <- pnorm(z[1],lower.tail = F) # The error you would like to spend at the interim stage
zz <- Maxcombo.bd(
Sigma0 = Sigma0,
index = c(1, 1, 2, 2),
alpha_sp = c(alpha_interim,alpha)
)
zz$z_alpha # boundary value for each stage
zz$z_alpha_vec # boundary value for each test statistic correponding to index
# Correlation matrix under the alternative hypothesis
Sigma1_v<-rep(0.5,6)
Sigma1<-matrix(1, ncol=4,nrow=4)
Sigma1[upper.tri(Sigma1)]<- Sigma1_v
Sigma1[lower.tri(Sigma1)]<- t(Sigma1)[lower.tri(t(Sigma1))]
Sigma1
# Define mu1
mu1=c(0.1,0.1,0.2,0.2)
# Obtain the sample size
Maxcombo.sz(
Sigma1 = Sigma1,
mu1 = mu1,
z_alpha_vec = zz$z_alpha_vec,
beta = 0.1,
interim_vec=c(10,10,18,18),
R = 14,
n_range = c(100,1000),
sum_D = 0.6)
# need 232 patients, 140 deaths

## End(Not run)