Maxcombo.bd: Boundary calculation for GSMC
In lilywang1988/GSMC: Group Sequential Design for Maxcombo tests

Description Usage Arguments Details Value Author(s) References Examples

Boundary calculation for interim analysis with max-combo tests based on correlation matrix and the alpha spending function.

1	Maxcombo.bd(Sigma0, index, alpha_sp, n.rep = 5)

`Sigma0`	correlation matrix for all the test statistics.
`index`	vector of non-decreasing integer starting from 1 to indicate which stage each column or row of the correlation matrix `Sigma0` corresponds to.
`alpha_sp`	vector of cumulative errors to spend up to each stage.
`n.rep`	number of repeats to take the median for output since the called likelihood generator of a multivariate normal distribution `pmvnorm` is not determinant. The default `n.rep` value is 5.

Suppose there are 2 stages (1 interim, 1 final), and two tests for a max-combo in each stage, then we have totally 4 test statistics. Let the alpha spending function to be c(alpha1,alpha), and the first two (Z_{11},Z_{12}) share one cutoff value z1, the latter two share another two (Z_{21},Z_{22}) share another cutoff value z2. Controlling the type I error is equivalent to ensuring that P(Z_{11}<z_1,Z_{12}<z_1)=α_1 and P(Z_{11}<z_1,Z_{12}<z_1,Z_{21}<z_2,Z_{22}<z_2)=α are both satisfied. Note that the vector [Z_{11},Z_{12},Z_{21},Z_{22}]^T\sim MVN(0,Σ_0). Sigma0 corresponds to Σ_0, index records the ordered stages of each test statistics, whose order should be identical to the order of rows or columns in Sigma0. Specifically, in this example, index should be c(1,1,2,2). alpha_sp is the alpha spending function, which records how much type I error you would like to spend up to every stage.

`z_alpha`	boundary values for all the stages.
`z_alpha_vec`	boundary values for all the test statistics following the `index`.

Lili Wang

Wang, L., Luo, X., & Zheng, C. (2021). A Simulation-free Group Sequential Design with Max-combo Tests in the Presence of Non-proportional Hazards. Journal of Pharmaceutical Statistics.

 ## Not run: 
  #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
  # The error you would like to spend at the interim stage:
  alpha_interim <- pnorm(z[1],lower.tail = F) 
  
  zz <- Maxcombo.bd(
  Sigma0 = Sigma0,
  index = c(1, 1, 2, 2),
  alpha_sp = c(alpha_interim, alpha))
  
 
  # boundary value for each stage
  zz$z_alpha 
  # boundary value for each test statistic correponding to index
  zz$z_alpha_vec 
  mvtnorm::pmvnorm(
  upper = rep(zz$z_alpha[1], 2),
  corr = Sigma0[1:2,1:2]
  )[[1]]
  
  1-alpha_interim
  1-mvtnorm::pmvnorm(
  upper = zz$z_alpha_vec,
  corr = Sigma0
  )[[1]]
  
  alpha
  # What if we do not consider interim stage but with only a final stage? 
  zz1 <- Maxcombo.bd(
  Sigma0 = Sigma0[3:4,3:4],
  index = c(1,1),
  alpha_sp = c(alpha)
  )
  mvtnorm::pmvnorm(
  upper = rep(zz1$z_alpha, 2),
  corr = Sigma0[1:2, 1:2]
  )[[1]]
  
  1-alpha
  # This function will also fit 2 or any number of interims (K>=3)
  # Let there are 3 stages, Let us try controlling the error spent 
  at each stage.
  stage_p <- c(0.5,0.7,0.8,0.9)
  x <- gsDesign::gsDesign(k=5, test.type=1, timing=stage_p, sfu="OF", 
  alpha=alpha, beta=beta,delta=-log(0.7))
  (z <- x$upper$bound)
  alpha_sp<- cumsum(x$upper$prob[,1]) # the theoretical cumulative
   errors spent at each stage
# 2 tests per stage
Sigma0_v<-rep(0.5,choose(10,2))
Sigma0<-matrix(1, ncol=10,nrow=10)
Sigma0[upper.tri(Sigma0)]<- Sigma0_v
Sigma0[lower.tri(Sigma0)]<- t(Sigma0)[lower.tri(t(Sigma0))]
Sigma0

zz<-Maxcombo.bd(Sigma0 = Sigma0,index=c(1,1,2,2,3,3,4,4,5,5),alpha_sp=alpha_sp)

zz$z_alpha # boundary value for each stage
zz$z_alpha_vec # boundary value for each test statistic correponding to index
# interim 1
mvtnorm::pmvnorm(upper=rep(zz$z_alpha[1],2),corr=Sigma0[1:2,1:2])[[1]] # expected error spent at this stage
1-alpha_sp[1] #compare with the expected error spent at this stage
# above two rows are very close to each other, same for the following pairs.
# interim 2
mvtnorm::pmvnorm(upper=rep(zz$z_alpha[1:2],each=2),corr=Sigma0[1:4,1:4])[[1]]
1-alpha_sp[2]
# interim 3
mvtnorm::pmvnorm(upper=rep(zz$z_alpha[1:3],each=2),corr=Sigma0[1:6,1:6])[[1]]
1-alpha_sp[3]
# interim 4
mvtnorm::pmvnorm(upper=rep(zz$z_alpha[1:4],each=2),corr=Sigma0[1:8,1:8])[[1]]
1-alpha_sp[4]
# final stage
mvtnorm::pmvnorm(upper=rep(zz$z_alpha[1:5],each=2),corr=Sigma0[1:10,1:10])[[1]]
1-alpha_sp[5]
 
## End(Not run)