Simulate Power of 2-Sample Test for Survival under Complex Conditions

Description

Given functions to generate random variables for survival times and censoring times, spower simulates the power of a user-given 2-sample test for censored data. By default, the logrank (Cox 2-sample) test is used, and a logrank function for comparing 2 groups is provided. For composing S-Plus functions to generate random survival times under complex conditions, the Quantile2 function allows the user to specify the intervention:control hazard ratio as a function of time, the probability of a control subject actually receiving the intervention (dropin) as a function of time, and the probability that an intervention subject receives only the control agent as a function of time (non-compliance, dropout). Quantile2 returns a function that generates either control or intervention uncensored survival times subject to non-constant treatment effect, dropin, and dropout. There is a plot method for plotting the results of Quantile2, which will aid in understanding the effects of the two types of non-compliance and non-constant treatment effects. Quantile2 assumes that the hazard function for either treatment group is a mixture of the control and intervention hazard functions, with mixing proportions defined by the dropin and dropout probabilities. It computes hazards and survival distributions by numerical differentiation and integration using a grid of (by default) 7500 equally-spaced time points.

The logrank function is intended to be used with spower but it can be used by itself as long as the group variable has only the values 1 and 2 and there are no missing data. It returns the 1 degree of freedom chi-square statistic.

The Weibull2 function accepts as input two vectors, one containing two times and one containing two survival probabilities, and it solves for the scale and shape parameters of the Weibull distribution (S(t)=exp(-alpha*t^ gamma)) which will yield those estimates. It creates an S-Plus function to evaluate survival probabilities from this Weibull distribution. Weibull2 is useful in creating functions to pass as the first argument to Quantile2.

The Lognorm2 and Gompertz2 functions are similar to Weibull2 except that they produce survival functions for the log-normal and Gompertz distributions.

Usage

spower(rcontrol, rinterv, rcens, nc, ni, 
       test=logrank, nsim=500, alpha=0.05, pr=TRUE)

Quantile2(scontrol, hratio, 
          dropin=function(times)0, dropout=function(times)0,
          m=7500, tmax, qtmax=.001, mplot=200, pr=TRUE, ...)

## S3 method for class 'Quantile2':
print(x, ...)

## S3 method for class 'Quantile2':
plot(x, 
     what=c('survival','hazard','both','drop','hratio','all'),
     dropsep=FALSE, lty=1:4, col=1, xlim, ylim=NULL,
     label.curves=NULL, ...)

logrank(S, group)

Gompertz2(times, surv)
Lognorm2(times, surv)
Weibull2(times, surv)

Arguments

Value

spower returns the power estimate (fraction of simulated chi-squares greater than the alpha-critical value). Quantile2 returns an S-Plus function of class "Quantile2" with attributes that drive the plot method. The major attribute is a list containing several lists. Each of these sub-lists contains a Time vector along with one of the following: survival probabilities for either treatment group and with or without contamination caused by non-compliance, hazard rates in a similar way, intervention:control hazard ratio function with and without contamination, and dropin and dropout functions. logrank returns a single chi-square statistic, and Weibull2, Lognorm2 and Gompertz2 return an S function with three arguments, only the first of which (the vector of times) is intended to be specified by the user.

Side Effects

spower prints the interation number every 10 iterations if pr=TRUE.

Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University School of Medicine
f.harrell@vanderbilt.edu

References

Lakatos E (1988): Sample sizes based on the log-rank statistic in complex clinical trials. Biometrics 44:229–241 (Correction 44:923).

Cuzick J, Edwards R, Segnan N (1997): Adjusting for non-compliance and contamination in randomized clinical trials. Stat in Med 16:1017–1029.

Cook, T (2003): Methods for mid-course corrections in clinical trials with survival outcomes. Stat in Med 22:3431–3447.

See Also

cpower, ciapower, bpower, cph, coxph, labcurve

Examples

# Simulate a simple 2-arm clinical trial with exponential survival so
# we can compare power simulations of logrank-Cox test with cpower()
# Hazard ratio is constant and patients enter the study uniformly
# with follow-up ranging from 1 to 3 years
# Drop-in probability is constant at .1 and drop-out probability is
# constant at .175.  Two-year survival of control patients in absence
# of drop-in is .8 (mortality=.2).  Note that hazard rate is -log(.8)/2
# Total sample size (both groups combined) is 1000
# % mortality reduction by intervention (if no dropin or dropout) is 25
# This corresponds to a hazard ratio of 0.7283 (computed by cpower)

cpower(2, 1000, .2, 25, accrual=2, tmin=1, 
       noncomp.c=10, noncomp.i=17.5)

ranfun <- Quantile2(function(x)exp(log(.8)/2*x),
                    hratio=function(x)0.7283156,
                    dropin=function(x).1,
                    dropout=function(x).175)

rcontrol <- function(n) ranfun(n, what='control')
rinterv  <- function(n) ranfun(n, what='int')
rcens    <- function(n) runif(n, 1, 3)

set.seed(11)   # So can reproduce results
spower(rcontrol, rinterv, rcens, nc=500, ni=500, 
       test=logrank, nsim=50)  # normally use nsim=500 or 1000

# Simulate a 2-arm 5-year follow-up study for which the control group's
# survival distribution is Weibull with 1-year survival of .95 and
# 3-year survival of .7.  All subjects are followed at least one year,
# and patients enter the study with linearly increasing probability  after that
# Assume there is no chance of dropin for the first 6 months, then the
# probability increases linearly up to .15 at 5 years
# Assume there is a linearly increasing chance of dropout up to .3 at 5 years
# Assume that the treatment has no effect for the first 9 months, then
# it has a constant effect (hazard ratio of .75)

# First find the right Weibull distribution for compliant control patients
sc <- Weibull2(c(1,3), c(.95,.7))
sc

# Inverse cumulative distribution for case where all subjects are followed
# at least a years and then between a and b years the density rises
# as (time - a) ^ d is a + (b-a) * u ^ (1/(d+1))

rcens <- function(n) 1 + (5-1) * (runif(n) ^ .5)
# To check this, type hist(rcens(10000), nclass=50)

# Put it all together

f <- Quantile2(sc, 
      hratio=function(x)ifelse(x<=.75, 1, .75),
      dropin=function(x)ifelse(x<=.5, 0, .15*(x-.5)/(5-.5)),
      dropout=function(x).3*x/5)

par(mfrow=c(2,2))
# par(mfrow=c(1,1)) to make legends fit
plot(f, 'all', label.curves=list(keys='lines'))

rcontrol <- function(n) f(n, 'control')
rinterv  <- function(n) f(n, 'intervention')

set.seed(211)
spower(rcontrol, rinterv, rcens, nc=350, ni=350, 
       test=logrank, nsim=50)  # normally nsim=500 or more
par(mfrow=c(1,1))