Power of Cox/log-rank Two-Sample Test

Assumes exponential distributions for both treatment groups. Uses the George-Desu method along with formulas of Schoenfeld that allow estimation of the expected number of events in the two groups. To allow for drop-ins (noncompliance to control therapy, crossover to intervention) and noncompliance of the intervention, the method of Lachin and Foulkes is used.

Usage

cpower(tref, n, mc, r, accrual, tmin, noncomp.c=0, noncomp.i=0, 
       alpha=0.05, nc, ni, pr=TRUE)

Arguments

tref: time at which mortalities estimated
n: total sample size (both groups combined). If allocation is unequal so that there are not n/2 observations in each group, you may specify the sample sizes in nc and ni.
mc: tref-year mortality, control
r: % reduction in mc by intervention
accrual: duration of accrual period
tmin: minimum follow-up time
noncomp.c: % non-compliant in control group (drop-ins)
noncomp.i: % non-compliant in intervention group (non-adherers)
alpha: type I error probability. A 2-tailed test is assumed.
nc: number of subjects in control group
ni: number of subjects in intervention group. nc and ni are specified exclusive of n.
pr: set to FALSE to suppress printing of details

Value

power

Side Effects

prints

Details

For handling noncompliance, uses a modification of formula (5.4) of Lachin and Foulkes. Their method is based on a test for the difference in two hazard rates, whereas cpower is based on testing the difference in two log hazards. It is assumed here that the same correction factor can be approximately applied to the log hazard ratio as Lachin and Foulkes applied to the hazard difference.

Note that Schoenfeld approximates the variance of the log hazard ratio by 4/m, where m is the total number of events, whereas the George-Desu method uses the slightly better 1/m1 + 1/m2. Power from this function will thus differ slightly from that obtained with the SAS samsizc program.

Author

Frank Harrell
Department of Biostatistics
Vanderbilt University
fh@fharrell.com

References

Peterson B, George SL: Controlled Clinical Trials 14:511–522; 1993.

Lachin JM, Foulkes MA: Biometrics 42:507–519; 1986.

Schoenfeld D: Biometrics 39:499–503; 1983.

Examples

#In this example, 4 plots are drawn on one page, one plot for each
#combination of noncompliance percentage.  Within a plot, the
#5-year mortality % in the control group is on the x-axis, and
#separate curves are drawn for several % reductions in mortality
#with the intervention.  The accrual period is 1.5y, with all
#patients followed at least 5y and some 6.5y.


par(mfrow=c(2,2),oma=c(3,0,3,0))


morts <- seq(10,25,length=50)
red <- c(10,15,20,25)


for(noncomp in c(0,10,15,-1)) {
  if(noncomp>=0) nc.i <- nc.c <- noncomp else {nc.i <- 25; nc.c <- 15}
  z <- paste("Drop-in ",nc.c,"%, Non-adherence ",nc.i,"%",sep="")
  plot(0,0,xlim=range(morts),ylim=c(0,1),
           xlab="5-year Mortality in Control Patients (%)",
           ylab="Power",type="n")
  title(z)
  cat(z,"\n")
  lty <- 0
  for(r in red) {
        lty <- lty+1
        power <- morts
        i <- 0
        for(m in morts) {
          i <- i+1
          power[i] <- cpower(5, 14000, m/100, r, 1.5, 5, nc.c, nc.i, pr=FALSE)
        }
        lines(morts, power, lty=lty)
  }
  if(noncomp==0)legend(18,.55,rev(paste(red,"% reduction",sep="")),
           lty=4:1,bty="n")
}
#> Drop-in 0%, Non-adherence 0% 
#> Drop-in 10%, Non-adherence 10% 
#> Drop-in 15%, Non-adherence 15% 
#> Drop-in 15%, Non-adherence 25% 
mtitle("Power vs Non-Adherence for Main Comparison",
           ll="alpha=.05, 2-tailed, Total N=14000",cex.l=.8)

#
# Point sample size requirement vs. mortality reduction
# Root finder (uniroot()) assumes needed sample size is between
# 1000 and 40000
#
nc.i <- 25; nc.c <- 15; mort <- .18
red <- seq(10,25,by=.25)
samsiz <- red


i <- 0
for(r in red) {
  i <- i+1
  samsiz[i] <- uniroot(function(x) cpower(5, x, mort, r, 1.5, 5,
                                          nc.c, nc.i, pr=FALSE) - .8,
                       c(1000,40000))$root
}


samsiz <- samsiz/1000
par(mfrow=c(1,1))
plot(red, samsiz, xlab='% Reduction in 5-Year Mortality',
   ylab='Total Sample Size (Thousands)', type='n')
lines(red, samsiz, lwd=2)
title('Sample Size for Power=0.80\nDrop-in 15%, Non-adherence 25%')
title(sub='alpha=0.05, 2-tailed', adj=0)