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.
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)
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.
rcontrol
but for the intervention group
n
which returns
n
random censoring times. It is
assumed that both treatment groups have the same censoring distribution.
"Quantile2"
created by
Quantile2
Surv
object or other two-column matrix for right-censored survival
times
S
. Only
values allowed are 1 and 2.
Surv
object and a grouping variable which computes
a chi-square for a two-sample censored data test. The default is
logrank
.
FALSE
to cause
spower
to suppress progress notes for
simulations.
Set to
FALSE
to prevent
Quantile2
from printing
tmax
when it
calculates
tmax
.
m
times. Default is the
time such that
scontrol(time)
is
qtmax
.
.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
.
scontrol
function when it's
evaluated by
Quantile2
"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
).
dropsep=TRUE
to make
plot.Quantile2
separate pure and
contaminated functions onto separate plots
opts
argument to
labcurve
.
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.
Frank Harrell
Department of Biostatistics
Vanderbilt University School of Medicine
f.harrell@vanderbilt.edu
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.
# 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))