wlr.Zbd: Rejection Boundary (z) for Weighted Logrank Test in Group...
In phe9480/rgs: What the Package Does (One Line, Title Case)

Description Usage Arguments Value Examples

Consider the general testing framework: At analysis i, the test is max(WLR_1, ..., WLR_ni). This function calculates the rejection boundary in z value for each analysis with the alpha allocation is given. The calculation is based on the asymptotic multivariate normal distribution. So it can be a challenge if the number of dimensions is large, in which case one can use simulation approach with a large number of draws like 100 million in order to achieve high precision.

wlr.Zbd(
  DCO = c(24, 36, 48),
  r = 1,
  alpha = c(0.01, 0.02, 0.02)/2,
  h0 = function(t) {     log(2)/12 },
  S0 = function(t) {     exp(-log(2)/12 * t) },
  h1 = function(t) {     log(2)/12 * 0.7 },
  S1 = function(t) {     exp(-log(2)/12 * 0.7 * t) },
  f.ws = list(IA1 = list(lr), IA2 = list(lr, fh11), FA = list(lr, fh01, fh11)),
  Lambda = function(t) {     (t/18)^1.5 * as.numeric(t <= 18) + as.numeric(t > 18) },
  G0 = function(t) {     0 },
  G1 = function(t) {     0 }
)

`DCO`	Analysis time.
`r`	Randomization ratio of experimental arm : control arm as r:1. When r = 1, it is equal allocation. Default r = 1.
`alpha`	A vector of allocated alpha levels for interim and final analyses. sum(alpha) = 0.025 (1-sided).
`h0`	Hazard function of control arm.
`S0`	Survival function of control arm.
`h1`	Hazard function of experimental arm.
`S1`	Survival function of experimental arm.
`f.ws`	Weight functions used for each analysis. For example (1), if there are 3 analyses planned including IA1, IA2, and FA. IA1 uses log-rank test, IA2 uses a maxcombo test of (logrank, FH01), and FA uses a maxcombo test of (logrank, FH01, FH11). Then specify as f.ws = list(IA1=list(lr), IA2=list(lr, fh01), FA=list(lr, fh01, fh11)), where define lr = function(s)1; fh01=function(s)1-s; fh11 = function(s)s*(1-s); For example (2), if only fh01 is used for all three analyses IA1, IA2 and FA. f.ws = list(IA1=list(fh01), IA2=list(fh01), FA1=list(fh01)). For example (3), if logrank is used in a study design with only one analysis, then f.ws = list(FA=list(lr)).
`Lambda`	Distribution function of enrollment. For example, Lambda(t) = (t/A)^psi*I(0<=t<=A) + I(t>A) for enrollment in (0, A) with weight psi.
`G0`	Cumulative distribution function of drop-off for control arm, eg, G.ltfu=function(t)1-exp(-0.03/12*t) is the distribution function for 3 percent drop-off in 12 months of followup.
`G1`	Cumulative distribution function of drop-off for experimental arm.

cov: Asymptotic covariance matrix, also correlation matrix under H0.
bd: A vector of rejection boundaries for all analyses.

m0 = 12; #median RFS for control arm
lam0 = log(2) / m0
h0 = function(t){lam0}; 
S0 = function(t){exp(-lam0 * t)}
HRd = 0.60 #hazard ratio after delay

h.D6=function(t){lam0*as.numeric(t<6)+HRd*lam0*as.numeric(t>=6)}
c6 = exp(-6*lam0*(1-HRd)); 
S.D6 = function(t){exp(-lam0*t)*as.numeric(t<6)+c6*exp(-HRd*lam0*t)*as.numeric(t>=6)}
f.logHR.D6 = function(t){log(as.numeric(t<6) + as.numeric(t>= 6)*HRd)}

#Define weight functions for weighted log-rank tests
lr = function(s){1}
fh01 = function(s){(1-s)}
fh11 = function(s){s*(1-s)}
#modestly log-rank
mfh01 = function(s){s1 = apply(cbind(s, 0.5), MARGIN=1,FUN=max); return(1/s1)}

#Define the testing strategy
test.IA1 = list(lr)
test.IA2 = list(lr, fh11)
test.IA3 = list(lr, fh01, fh11)

Lambda = function(t){(t/18)^1.5*as.numeric(t <= 18) + as.numeric(t > 18)}
G0 = G1 = function(t){0} 
  
wlr.Zbd(DCO = c(24, 36, 48), r = 1, alpha=c(0.01, 0.02, 0.02)/2,
h0 = h0, S0 = S0, h1 = h.D6, S1 = S.D6, 
f.ws = list(IA1=test.IA1, IA2=test.IA2, FA=test.IA3),
Lambda = Lambda, G0 = G0, G1 = G1)