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. Optionally a Cox model is fitted for each each
simulated dataset and the log hazard ratios are saved (this requires
the survival package). A print method prints various
measures from these. For composing R 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. It returns the 1 degree of freedom
chi-square statistic, with the hazard ratio estimate as an attribute.
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
(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.
When cox=TRUE is specified to spower, the analyst may wish
to extract the two margins of error by using the print method
for spower objects (see example below) and take the maximum of
the two.
spower(rcontrol, rinterv, rcens, nc, ni,
test=logrank, cox=FALSE, nsim=500, alpha=0.05, pr=TRUE)# S3 method for spower
print(x, conf.int=.95, …)
Quantile2(scontrol, hratio,
dropin=function(times)0, dropout=function(times)0,
m=7500, tmax, qtmax=.001, mplot=200, pr=TRUE, …)
# S3 method for Quantile2
print(x, …)
# S3 method for 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)
a function of n which returns n random uncensored
failure times for the control group. spower assumes that
non-compliance (dropin) has been taken into account by this
function.
similar to rcontrol but for the intervention group
a function of n which returns n random censoring times. It is assumed that both treatment groups have the same censoring distribution.
number of subjects in the control group
number in the intervention group
a function of a time vector which returns the survival probabilities for the control group at those times assuming that all patients are compliant.
a function of time which specifies the intervention:control hazard ratio (treatment effect)
an object of class “Quantile2” created by Quantile2,
or of class “spower” created by spower
confidence level for determining fold-change margins of error in estimating the hazard ratio
a Surv object or other two-column matrix for right-censored
survival times
group indicators have length equal to the number of rows in S
argument.
a vector of two times
a vector of two survival probabilities
any function of a Surv object and a grouping variable which
computes a chi-square for a two-sample censored data test. The
default is logrank.
If true TRUE the two margins of error are available by using
the print method for spower objects (see example
below) and taking the maximum of the two.
number of simulations to perform (default=500)
type I error (default=.05)
If FALSE prevents spower from printing progress notes for
simulations.
If FALSE prevents Quantile2 from printing tmax
when it calculates tmax.
a function of time specifying the probability that a control subject actually is treated with the new intervention at the corresponding time
a function of time specifying the probability of an intervention
subject dropping out to control conditions. As a function of time,
dropout specifies the probability that a patient is treated
with the control therapy at time t. dropin and
dropout form mixing proportions for control and intervention
hazard functions.
number of time points used for approximating functions (default is 7500)
maximum time point to use in the grid of m times. Default is
the time such that scontrol(time) is qtmax.
survival probability corresponding to the last time point used for
approximating survival and hazard functions. Default is 0.001. For
qtmax of the time for which a simulated time is needed which
corresponds to a survival probability of less than qtmax, the
simulated value will be tmax.
number of points used for approximating functions for use in plotting (default is 200 equally spaced points)
optional arguments passed to the scontrol function when it's
evaluated by Quantile2. Unused for print.spower.
a single character constant (may be abbreviated) specifying which
functions to plot. The default is "both" meaning both
survival and hazard functions. Specify what="drop" to just
plot the dropin and dropout functions, what="hratio" to plot
the hazard ratio functions, or "all" to make 4 separate plots
showing all functions (6 plots if dropsep=TRUE).
If TRUE makes plot.Quantile2 separate pure and
contaminated functions onto separate plots
vector of line types
vector of colors
optional x-axis limits
optional y-axis limits
optional list which is passed as the opts argument to
labcurve.
spower returns the power estimate (fraction of simulated
chi-squares greater than the alpha-critical value). If
cox=TRUE, spower returns an object of class
“spower” containing the power and various other quantities.
Quantile2 returns an R 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.
Weibull2, Lognorm2 and Gompertz2 return an R
function with three arguments, only the first of which (the vector of
times) is intended to be specified by the user.
spower prints the interation number every 10 iterations if
pr=TRUE.
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.
Barthel FMS, Babiker A et al (2006): Evaluation of sample size and power for multi-arm survival trials allowing for non-uniform accrual, non-proportional hazards, loss to follow-up and cross-over. Stat in Med 25:2521--2542.
# NOT RUN {
# 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
# }
# NOT RUN {
# Run the same simulation but fit the Cox model for each one to
# get log hazard ratios for the purpose of assessing the tightness
# confidence intervals that are likely to result
set.seed(11)
u <- spower(rcontrol, rinterv, rcens, nc=500, ni=500,
test=logrank, nsim=50, cox=TRUE)
u
v <- print(u)
v[c('MOElower','MOEupper','SE')]
# }
# NOT RUN {
# 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))
# Compose a censoring time generator function such that at 1 year
# 5% of subjects are accrued, at 3 years 70% are accured, and at 10
# years 100% are accrued. The trial proceeds two years past the last
# accrual for a total of 12 years of follow-up for the first subject.
# Use linear interporation between these 3 points
rcens <- function(n)
{
times <- c(0,1,3,10)
accrued <- c(0,.05,.7,1)
# Compute inverse of accrued function at U(0,1) random variables
accrual.times <- approx(accrued, times, xout=runif(n))$y
censor.times <- 12 - accrual.times
censor.times
}
censor.times <- rcens(500)
# hist(censor.times, nclass=20)
accrual.times <- 12 - censor.times
# Ecdf(accrual.times)
# lines(c(0,1,3,10), c(0,.05,.7,1), col='red')
# spower(..., rcens=rcens, ...)
# }
# NOT RUN {
# To define a control survival curve from a fitted survival curve
# with coordinates (tt, surv) with tt[1]=0, surv[1]=1:
Scontrol <- function(times, tt, surv) approx(tt, surv, xout=times)$y
tt <- 0:6
surv <- c(1, .9, .8, .75, .7, .65, .64)
formals(Scontrol) <- list(times=NULL, tt=tt, surv=surv)
# To use a mixture of two survival curves, with e.g. mixing proportions
# of .2 and .8, use the following as a guide:
#
# Scontrol <- function(times, t1, s1, t2, s2)
# .2*approx(t1, s1, xout=times)$y + .8*approx(t2, s2, xout=times)$y
# t1 <- ...; s1 <- ...; t2 <- ...; s2 <- ...;
# formals(Scontrol) <- list(times=NULL, t1=t1, s1=s1, t2=t2, s2=s2)
# Check that spower can detect a situation where generated censoring times
# are later than all failure times
rcens <- function(n) runif(n, 0, 7)
f <- Quantile2(scontrol=Scontrol, hratio=function(x).8, tmax=6)
cont <- function(n) f(n, what='control')
int <- function(n) f(n, what='intervention')
spower(rcontrol=cont, rinterv=int, rcens=rcens, nc=300, ni=300, nsim=20)
# Do an unstratified logrank test
library(survival)
# From SAS/STAT PROC LIFETEST manual, p. 1801
days <- c(179,256,262,256,255,224,225,287,319,264,237,156,270,257,242,
157,249,180,226,268,378,355,319,256,171,325,325,217,255,256,
291,323,253,206,206,237,211,229,234,209)
status <- c(1,1,1,1,1,0,1,1,1,1,0,1,1,1,1,1,1,1,1,0,
0,rep(1,19))
treatment <- c(rep(1,10), rep(2,10), rep(1,10), rep(2,10))
sex <- Cs(F,F,M,F,M,F,F,M,M,M,F,F,M,M,M,F,M,F,F,M,
M,M,M,M,F,M,M,F,F,F,M,M,M,F,F,M,F,F,F,F)
data.frame(days, status, treatment, sex)
table(treatment, status)
logrank(Surv(days, status), treatment) # agrees with p. 1807
# For stratified tests the picture is puzzling.
# survdiff(Surv(days,status) ~ treatment + strata(sex))$chisq
# is 7.246562, which does not agree with SAS (7.1609)
# But summary(coxph(Surv(days,status) ~ treatment + strata(sex)))
# yields 7.16 whereas summary(coxph(Surv(days,status) ~ treatment))
# yields 5.21 as the score test, not agreeing with SAS or logrank() (5.6485)
# }
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