# spower

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

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
(\(S(t) = e^{-\alpha {t}^{\gamma}}\))
which will yield
those estimates. It creates an R 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.

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.

##### Usage

```
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)

##### Arguments

- rcontrol
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.- rinterv
similar to

`rcontrol`

but for the intervention group- rcens
a function of

`n`which returns`n`random censoring times. It is assumed that both treatment groups have the same censoring distribution.- nc
number of subjects in the control group

- ni
number in the intervention group

- scontrol
a function of a time vector which returns the survival probabilities for the control group at those times assuming that all patients are compliant.

- hratio
a function of time which specifies the intervention:control hazard ratio (treatment effect)

- x
an object of class “Quantile2” created by

`Quantile2`

, or of class “spower” created by`spower`

- conf.int
confidence level for determining fold-change margins of error in estimating the hazard ratio

- S
a

`Surv`

object or other two-column matrix for right-censored survival times- group
group indicators have length equal to the number of rows in

`S`

argument.- times
a vector of two times

- surv
a vector of two survival probabilities

- test
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`

.- cox
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.- nsim
number of simulations to perform (default=500)

- alpha
type I error (default=.05)

- pr
If

`FALSE`

prevents`spower`

from printing progress notes for simulations. If`FALSE`

prevents`Quantile2`

from printing`tmax`

when it calculates`tmax`

.- dropin
a function of time specifying the probability that a control subject actually is treated with the new intervention at the corresponding time

- dropout
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.- m
number of time points used for approximating functions (default is 7500)

- tmax
maximum time point to use in the grid of

`m`

times. Default is the time such that`scontrol(time)`

is`qtmax`

.- 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`

.- mplot
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`

.- what
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`

).- dropsep
If

`TRUE`

makes`plot.Quantile2`

separate pure and contaminated functions onto separate plots- lty
vector of line types

- col
vector of colors

- xlim
optional x-axis limits

- ylim
optional y-axis limits

- label.curves
optional list which is passed as the

`opts`

argument to`labcurve`

.

##### Value

`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.

##### Side Effects

`spower`

prints the interation number every 10 iterations if
`pr=TRUE`

.

##### 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.

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.

##### See Also

##### Examples

```
# 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)
# }
```

*Documentation reproduced from package Hmisc, version 4.3-0, License: GPL (>= 2)*