gsDesign (version 2.8-7)

nSurv: Advanced time-to-event sample size calculation

Description

nSurv() is used to calculate the sample size for a clinical trial with a time-to-event endpoint and an assumption of proportional hazards. This set of routines is new with version 2.7 and will continue to be modified and refined to improve input error checking and output format with subsequent versions. It allows both the Lachin and Foulkes (1986) method (fixed trial duration) as well as the Kim and Tsiatis(1990) method (fixed enrollment rates and either fixed enrollment duration or fixed minimum follow-up). Piecewise exponential survival is supported as well as piecewise constant enrollment and dropout rates. The methods are for a 2-arm trial with treatment groups referred to as experimental and control. A stratified population is allowed as in Lachin and Foulkes (1986); this method has been extended to derive non-inferiority as well as superiority trials. Stratification also allows power calculation for meta-analyses. gsSurv() combines nSurv() with gsDesign() to derive a group sequential design for a study with a time-to-event endpoint. print(), xtable() and summary() methods are provided to operate on the returned value from gsSurv(), an object of class gsSurv. print() is also extended to nSurv objects. The functions gsBoundSummary (data frame for tabular output), xprint (application of xtable for tabular output) and summary.gsSurv (textual summary of gsDesign or gsSurv object) may be preferred summary functions; see example in vignettes. See also gsDesign print, summary and table summary functions for output of tabular summaries of bounds for designs produced by gsSurv(). Both nEventsIA and tEventsIA require a group sequential design for a time-to-event endpoing of class gsSurv as input. nEventsIA calculates the expected number of events under the alternate hypothesis at a given interim time. tEventsIA calculates the time that the expected number of events under the alternate hypothesis is a given proportion of the total events planned for the final analysis.

Usage

nSurv(lambdaC=log(2)/6, hr=.6, hr0=1, eta = 0, etaE=NULL, 
  gamma=1, R=12, S=NULL, T=NULL,  minfup = NULL, ratio = 1,
  alpha = 0.025, beta = 0.10,  sided = 1, tol = .Machine$double.eps^0.25)
## S3 method for class 'nSurv':
print(x,digits=4,...)
gsSurv(k=3, test.type=4, alpha=0.025, sided=1,  
  beta=0.1, astar=0, timing=1, sfu=sfHSD, sfupar=-4,
  sfl=sfHSD, sflpar=-2, r=18,
  lambdaC=log(2)/6, hr=.6, hr0=1, eta=0, etaE=NULL,
  gamma=1, R=12, S=NULL, T=NULL, minfup=NULL, ratio=1,
  tol = .Machine$double.eps^0.25)
## S3 method for class 'gsSurv':
print(x,digits=2,...)
## S3 method for class 'gsSurv':
xtable(x, caption=NULL, label=NULL, align=NULL, digits=NULL,
       display=NULL, footnote=NULL, fnwid="9cm", timename="months",...)
tEventsIA(x, timing=.25, tol = .Machine$double.eps^0.25)
nEventsIA(tIA=5, x=NULL, target=0, simple=TRUE)

Arguments

x

An object of class nSurv or gsSurv. print.nSurv() is used for an object of class nSurv which will generally be output from nSurv(). For print.gsSurv() is used for an object of c

digits

Number of digits past the decimal place to print (print.gsSurv.); also a pass through to generic xtable() from xtable.gsSurv().

lambdaC

scalar, vector or matrix of event hazard rates for the control group; rows represent time periods while columns represent strata; a vector implies a single stratum.

hr

hazard ratio (experimental/control) under the alternate hypothesis (scalar).

hr0

hazard ratio (experimental/control) under the null hypothesis (scalar).

eta

scalar, vector or matrix of dropout hazard rates for the control group; rows represent time periods while columns represent strata; if entered as a scalar, rate is constant accross strata and time periods; if entered as a vector, rates are constant accros

etaE

matrix dropout hazard rates for the experimental group specified in like form as eta; if NULL, this is set equal to eta.

gamma

a scalar, vector or matrix of rates of entry by time period (rows) and strata (columns); if entered as a scalar, rate is constant accross strata and time periods; if entered as a vector, rates are constant accross strata.

R

a scalar or vector of durations of time periods for recruitment rates specified in rows of gamma. Length is the same as number of rows in gamma. Note that when variable enrollment duration is specified (input T=NULL)

S

a scalar or vector of durations of piecewise constant event rates specified in rows of lambda, eta and etaE; this is NULL if there is a single event rate per stratum (exponential failure) or length of the number of r

T

study duration; if T is input as NULL, this will be computed on output; see details.

minfup

follow-up of last patient enrolled; if minfup is input as NULL, this will be computed on output; see details.

ratio

randomization ratio of experimental treatment divided by control; normally a scalar, but may be a vector with length equal to number of strata.

sided

1 for 1-sided testing, 2 for 2-sided testing.

alpha

type I error rate. Default is 0.025 since 1-sided testing is default.

beta

type II error rate. Default is 0.10 (90% power); NULL if power is to be computed based on other input values.

tol

for cases when T or minfup values are derived through root finding (T or minfup input as NULL), tol provides the level of error input to the uniroot() root-finding

k

Number of analyses planned, including interim and final.

test.type

1=one-sided 2=two-sided symmetric 3=two-sided, asymmetric, beta-spending with binding lower bound 4=two-sided, asymmetric, beta-spending with non-binding lower bound 5=two-sided, asymmet

astar

Normally not specified. If test.type=5 or 6, astar specifies the total probability of crossing a lower bound at all analyses combined. This will be changed to $1 -$alpha when default value of 0 is u

timing

Sets relative timing of interim analyses in gsSurv. Default of 1 produces equally spaced analyses. Otherwise, this is a vector of length k or k-1. The values should satisfy 0 < timing[1] < timing[2] < ...

sfu

A spending function or a character string indicating a boundary type (that is, WT for Wang-Tsiatis bounds, OF for O'Brien-Fleming bounds and Pocock for Pocock bounds). For one-sided and symmetric two-s

sfupar

Real value, default is $-4$ which is an O'Brien-Fleming-like conservative bound when used with the default Hwang-Shih-DeCani spending function. This is a real-vector for many spending functions. The parameter sfupar specifies any parameters

sfl

Specifies the spending function for lower boundary crossing probabilities when asymmetric, two-sided testing is performed (test.type = 3, 4, 5, or 6). Unlike the upper bound, only spending functions

sflpar

Real value, default is $-2$, which, with the default Hwang-Shih-DeCani spending function, specifies a less conservative spending rate than the default for the upper bound.

r

Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally r will not be changed by the

tIA

Timing of an interim analysis; should be between 0 and y$T.

target

The targeted proportion of events at an interim analysis. This is used for root-finding will be 0 for normal use.

simple

See output specification for nEventsIA().

footnote

footnote for xtable output; may be useful for describing some of the design parameters.

fnwid

a text string controlling the width of footnote text at the bottom of the xtable output.

timename

character string with plural of time units (e.g., "months")

caption

passed through to generic xtable().

label

passed through to generic xtable().

align

passed through to generic xtable().

display

passed through to generic xtable().

...

other arguments that may be passed to generic functions underlying the methods here.

Value

• nSurv() returns an object of type nSurv with the following components:
• alpha
  As input.
• sided
  As input.
• beta
  Type II error; if missing, this is computed.
• power
  Power corresponding to input beta or computed if output beta is computed.
• lambdaC
  As input.
• etaC
  As input.
• etaE
  As input.
• gamma
  As input unless none of the following are NULL: T, minfup, beta; otherwise, this is a constant times the input value required to power the trial given the other input variables.
• ratio
  As input.
• R
  As input unless T was NULL on input.
• S
  As input.
• T
  As input.
• minfup
  As input.
• hr
  As input.
• hr0
  As input.
• n
  Total expected sample size corresponding to output accrual rates and durations.
• d
  Total expected number of events under the alternate hypothesis.
• tol
  As input, except when not used in computations in which case this is returned as NULL. This and the remaining output below are not printed by the print() extension for the nSurv class.
• eDC
  A vector of expected number of events by stratum in the control group under the alternate hypothesis.
• eDE
  A vector of expected number of events by stratum in the experimental group under the alternate hypothesis.
• eDC0
  A vector of expected number of events by stratum in the control group under the null hypothesis.
• eDE0
  A vector of expected number of events by stratum in the experimental group under the null hypothesis.
• eNC
  A vector of the expected accrual in each stratum in the control group.
• eNE
  A vector of the expected accrual in each stratum in the experimental group.
• variable
  A text string equal to "Accrual rate" if a design was derived by varying the accrual rate, "Accrual duration" if a design was derived by varying the accrual duration, "Follow-up duration" if a design was derived by varying follow-up duration, or "Power" if accrual rates and duration as well as follow-up duration was specified and beta=NULL was input.
• gsSurv() returns much of the above plus an object of class gsDesign in a variable named gs; see gsDesign for general documentation on what is returned in gs. The value of gs$n.I represents the number of endpoints required at each analysis to adequately power the trial. Other items returned by gsSurv() are:
• gs
  A group sequential design (gsDesign) object.
• lambdaC
  As input.
• etaC
  As input.
• etaE
  As input.
• gamma
  As input unless none of the following are NULL: T, minfup, beta; otherwise, this is a constant times the input value required to power the trial given the other input variables.
• ratio
  As input.
• R
  As input unless T was NULL on input.
• S
  As input.
• T
  As input.
• minfup
  As input.
• hr
  As input.
• hr0
  As input.
• eNC
  Total expected sample size corresponding to output accrual rates and durations.
• eNE
  Total expected sample size corresponding to output accrual rates and durations.
• eDC
  Total expected number of events under the alternate hypothesis.
• eDE
  Total expected number of events under the alternate hypothesis.
• tol
  As input, except when not used in computations in which case this is returned as NULL. This and the remaining output below are not printed by the print() extension for the nSurv class.
• eDC
  A vector of expected number of events by stratum in the control group under the alternate hypothesis.
• eDE
  A vector of expected number of events by stratum in the experimental group under the alternate hypothesis.
• eDC0
  A vector of expected number of events by stratum in the control group under the null hypothesis.
• eDE0
  A vector of expected number of events by stratum in the experimental group under the null hypothesis.
• eNC
  A vector of the expected accrual in each stratum in the control group.
• eNE
  A vector of the expected accrual in each stratum in the experimental group.
• variable
  A text string equal to "Accrual rate" if a design was derived by varying the accrual rate, "Accrual duration" if a design was derived by varying the accrual duration, "Follow-up duration" if a design was derived by varying follow-up duration, or "Power" if accrual rates and duration as well as follow-up duration was specified and beta=NULL was input.
• nEventsIA() returns the expected proportion of the final planned events observed at the input analysis time minus target when simple=TRUE. When simple=FALSE, nEventsIA returns a list with following components:
• T
  The input value tIA.
• eDC
  The expected number of events in the control group at time the output time T.
• eDE
  The expected number of events in the experimental group at the output time T.
• eNC
  The expected enrollment in the control group at the output time T.
• eNE
  The expected enrollment in the experimental group at the output time T.
• tEventsIA() returns

Details

nSurv() produces an object of class nSurv with the number of subjects and events for a set of pre-specified trial parameters, such as accrual duration and follow-up period. The underlying power calculation is based on Lachin and Foulkes (1986) method for proportional hazards assuming a fixed underlying hazard ratio between 2 treatment groups. The method has been extended here to enable designs to test non-inferiority. Piecewise constant enrollment and failure rates are assumed and a stratified population is allowed. See also nSurvival for other Lachin and Foulkes (1986) methods assuming a constant hazard difference or exponential enrollment rate. When study duration (T) and follow-up duration (minfup) are fixed, nSurv applies exactly the Lachin and Foulkes (1986) method of computing sample size under the proportional hazards assumption when For this computation, enrollment rates are altered proportionately to those input in gamma to achieve the power of interest. Given the specified enrollment rate(s) input in gamma, nSurv may also be used to derive enrollment duration required for a trial to have defined power if T is input as NULL; in this case, both R (enrollment duration for each specified enrollment rate) and T (study duration) will be computed on output. Alternatively and also using the fixed enrollment rate(s) in gamma, if minimum follow-up minfup is specified as NULL, then the enrollment duration(s) specified in R are considered fixed and minfup and T are computed to derive the desired power. This method will fail if the specified enrollment rates and durations either over-powers the trial with no additional follow-up or underpowers the trial with infinite follow-up. This method produces a corresponding error message in such cases. The input to gsSurv is a combination of the input to nSurv() and gsDesign(). While this routine may change in the future, it is likely to be backwards compatible with the current version. nEventsIA() is provided to compute the expected number of events at a given point in time given enrollment, event and censoring rates. The routine is used with a root finding routine to approximate the approximate timing of an interim analysis. It is also used to extend enrollment or follow-up of a fixed design to obtain a sufficient number of events to power a group sequential design.

References

Kim KM and Tsiatis AA (1990), Study duration for clinical trials with survival response and early stopping rule. Biometrics, 46, 81-92 Lachin JM and Foulkes MA (1986), Evaluation of Sample Size and Power for Analyses of Survival with Allowance for Nonuniform Patient Entry, Losses to Follow-Up, Noncompliance, and Stratification. Biometrics, 42, 507-519. Schoenfeld D (1981), The Asymptotic Properties of Nonparametric Tests for Comparing Survival Distributions. Biometrika, 68, 316-319.

See Also

gsBoundSummary, xprint,gsDesign package overview, Plots for group sequential designs, gsDesign, gsHR, nSurvival

Examples

# vary accrual rate to obtain power
nSurv(lambdaC=log(2)/6, hr=.5, eta=log(2)/40,gamma=1, T=36,  minfup = 12)

# vary accrual duration to obtain power
nSurv(lambdaC=log(2)/6, hr=.5, eta=log(2)/40, gamma=6, minfup = 12)

# vary follow-up duration to obtain power
nSurv(lambdaC=log(2)/6, hr=.5, eta=log(2)/40, gamma=6, R=25)

# piecewise constant enrollment rates (vary accrual duration)
nSurv(lambdaC=log(2)/6, hr=.5, eta=log(2)/40, gamma=c(1,3,6), 
      R=c(3,6,9), minfup=12)

# stratified population (vary accrual duration)
nSurv(lambdaC=matrix(log(2)/c(6,12),ncol=2), hr=.5, eta=log(2)/40, 
      gamma=matrix(c(2,4),ncol=2), minfup=12)

# piecewise exponential failure rates (vary accrual duration)
nSurv(lambdaC=log(2)/c(6,12), hr=.5, eta=log(2)/40, S=3, gamma=6, minfup = 12)

# combine it all: 2 strata, 2 failure rate periods
nSurv(lambdaC=matrix(log(2)/c(6,12,18,24),ncol=2), hr=.5, 
      eta=matrix(log(2)/c(40,50,45,55),ncol=2), S=3, 
      gamma=matrix(c(3,6,5,7),ncol=2), R=c(5,10), minfup = 12)

# group sequential design (vary accrual rate to obtain power)
x<-gsSurv(k=4,sfl=sfPower,sflpar=.5,lambdaC=log(2)/6, hr=.5, 
          eta=log(2)/40,gamma=1, T=36,  minfup = 12)
x
print(xtable(x,footnote="This is a footnote; note that it can be wide.", 
              caption="Caption example."))
# find expected number of events at time 12 in the above trial
nEventsIA(x=x,tIA=10)

# find time at which 1/4 of events are expected
tEventsIA(x=x,timing=.25)