nSurv: Advanced time-to-event sample size calculation
In gsDesign: Group Sequential Design
tEventsIA R Documentation
Advanced time-to-event sample size calculation
Description
nSurv() is used to calculate the sample size for a two-arm clinical trial
with a time-to-event endpoint under the assumption of proportional hazards.
The default method assumes a fixed enrollment duration and fixed trial duration;
in this case the required sample size is achieved by increasing enrollment rates.
nSurv() implements the Lachin and Foulkes (1986) method as default.
Schoenfeld (1981), Freedman (1982), and Bernstein and Lagakos (1989) methods
are also supported; see Details.
gsSurv() combines nSurv() with gsDesign() to derive a
group sequential design for a study with a time-to-event endpoint.
Usage
tEventsIA(x, timing = 0.25, tol = .Machine$double.eps^0.25)
nEventsIA(tIA = 5, x = NULL, target = 0, simple = TRUE)
nSurv(
lambdaC = log(2)/6,
hr = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = 18,
minfup = 6,
ratio = 1,
alpha = 0.025,
beta = 0.1,
sided = 1,
tol = .Machine$double.eps^0.25,
method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)
## S3 method for class 'nSurv'
print(x, digits = 3, show_strata = TRUE, ...)
## S3 method for class 'gsSurv'
xtable(
x,
caption = NULL,
label = NULL,
align = NULL,
digits = NULL,
display = NULL,
auto = FALSE,
footnote = NULL,
fnwid = "9cm",
timename = "months",
...
)
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 = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = 18,
minfup = 6,
ratio = 1,
tol = .Machine$double.eps^0.25,
usTime = NULL,
lsTime = NULL,
method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)
## S3 method for class 'gsSurv'
print(x, digits = 3, show_gsDesign = FALSE, show_strata = TRUE, ...)
Arguments
x
An object of class nSurv or gsSurv.
print.nSurv() is used for an object of class nSurv which will
generally be output from nSurv(). For print.gsSurv() is used
for an object of class gsSurv which will generally be output from
gsSurv(). nEventsIA and tEventsIA operate on both the
nSurv and gsSurv class.
timing
Sets relative timing of interim analyses in gsSurv.
Default of 1 produces equally spaced analyses. Otherwise, this is a vector
of length k or k-1. The values should satisfy 0 <
timing[1] < timing[2] < ... < timing[k-1] < timing[k]=1. For
tEventsIA, this is a scalar strictly between 0 and 1 that indicates
the targeted proportion of final planned events available at an interim
analysis.
tol
For cases when T or minfup values are derived
through root finding (T or minfup input as NULL),
tol provides the level of error input to the uniroot()
root-finding function. The default is the same as for uniroot.
tIA
Timing of an interim analysis; should be between 0 and
y$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().
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.
Note that rates corresponding the final time period are extended indefinitely.
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 across strata and time periods; if
entered as a vector, rates are constant across strata.
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
across strata and time periods; if entered as a vector, rates are constant
across 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), the final enrollment period is extended as
long as needed; otherwise enrollment after sum(R) is 0.
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 rows in lambda minus 1, otherwise.
The final time period is extended indefinitely for each stratum.
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.
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.
sided
1 for 1-sided testing, 2 for 2-sided testing.
method
One of "LachinFoulkes" (default), "Schoenfeld",
"Freedman", or "BernsteinLagakos".
Note: "Schoenfeld" and "Freedman" methods only support
superiority testing (hr0 = 1). "Freedman" does not support
stratified populations.
digits
Number of digits past the decimal place to print
(print.gsSurv.); also a pass through to generic xtable() from
xtable.gsSurv().
show_strata
Logical; for print.gsSurv(), include strata summaries.
...
Other arguments that may be passed to generic functions
underlying the methods here.
caption
Passed through to generic xtable().
label
Passed through to generic xtable().
align
Passed through to generic xtable().
display
Passed through to generic xtable().
auto
Passed through to generic xtable().
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")
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, asymmetric, lower bound spending under the null
hypothesis with binding lower bound
6=two-sided, asymmetric,
lower bound spending under the null hypothesis with non-binding lower bound.
See details, examples and manual.
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 used. Since this is the expected usage, normally
astar is not specified by the user.
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-sided testing is used to completely specify
spending (test.type=1, 2), sfu. The default value is
sfHSD which is a Hwang-Shih-DeCani spending function. See details,
vignette("SpendingFunctionOverview"), manual and examples.
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 needed for
the spending function specified by sfu; this will be ignored for
spending functions (sfLDOF, sfLDPocock) or bound types
(“OF”, “Pocock”) that do not require parameters.
Note that sfupar can be specified as a positive scalar for
sfLDOF for a generalized O'Brien-Fleming spending function.
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 are used to specify the lower bound. The
default value is sfHSD which is a Hwang-Shih-DeCani spending
function. The parameter sfl is ignored for one-sided testing
(test.type=1) or symmetric 2-sided testing (test.type=2). See
details, spending functions, manual and examples.
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 (>= 1 and <= 80) controlling the number of numerical
integration grid points. Default is 18, as recommended by Jennison and
Turnbull (2000). Grid points are spread out in the tails for accurate
probability calculations. Larger values provide more grid points and greater
accuracy but slow down computation. Jennison and Turnbull (p. 350) note an
accuracy of 10^{-6} with r = 16. This parameter is normally
not changed by users.
usTime
Default is NULL in which case upper bound spending time is
determined by timing. Otherwise, this should be a vector of length
k with the spending time at each analysis (see Details in help for gsDesign).
lsTime
Default is NULL in which case lower bound spending time is
determined by timing. Otherwise, this should be a vector of length
k with the spending time at each analysis (see Details in help for gsDesign).
show_gsDesign
Logical; for print.gsSurv(), include gsDesign details.
Details
The Lachin and Foulkes method uses both null and alternate hypothesis
variances to derive sample size and is extended here to support
non-inferiority, super-superiority, and stratified designs.
As an alternative, the Kim and Tsiatis (1990) method can be used with fixed
enrollment rates and either fixed enrollment duration or fixed minimum
follow-up.
The Schoenfeld approach uses the asymptotic distribution of the log-rank
statistic under the assumption of proportional hazards and local alternatives
(i.e., \log(HR) is small). The Freedman approach uses the same
asymptotic distribution and, like the Schoenfeld approach, uses just the
null hypothesis variance to derive sample size.
The Bernstein and Lagakos (1989) approach was derived to compute sample size
for a stratified model with a common proportional hazards assumption across
strata. Like the Lachin and Foulkes (1986) method, it uses both null and
alternate hypothesis variances to compute sample size; however, the null
hypothesis variance is computed differently. The Bernstein and Lagakos (1989)
approach uses the alternate hypothesis failure rate assumptions for both the
control and experimental groups, while the Lachin and Foulkes method uses
null hypothesis rates that average the alternate hypothesis failure rates to
get similar numbers of expected events under the null and alternate
hypotheses. Since the Lachin and Foulkes method has fewer events under the
null hypothesis (less statistical information), it calculates less power
than the Bernstein and Lagakos method.
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.
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 gsBoundSummary 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 endpoint 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.
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().
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.
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 the same structure as nEventsIA(..., simple=TRUE) when
Author(s)
Keaven Anderson keaven_anderson@merck.com
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
uniroot
gsBoundSummary, xprint,
gsSurvCalendar, gsDesign-package,
plot.gsDesign, gsDesign, gsHR,
nSurvival
Normal
xtable
Examples
# Vary accrual rate gamma to obtain power
# T, minfup and R all specified, although R will be adjusted on output
# gamma as input will be multiplied in output to achieve desired power
# Default method is Lachin and Foulkes
x_nsurv <- nSurv(
lambdaC = log(2) / 6, R = 10, hr = .5, eta = .001, gamma = 1,
alpha = 0.02, beta = .15, T = 36, minfup = 12, method = "LachinFoulkes"
)
# Demonstrate print method
print(x_nsurv)
# Same assumptions for group sequential design
x_gs <- gsSurv(
k = 4, sfl = gsDesign::sfPower, sflpar = .5, lambdaC = log(2) / 6, hr = .5,
eta = .001, gamma = 1, T = 36, minfup = 12, method = "LachinFoulkes"
)
print(x_gs)
# Demonstrate xtable method for gsSurv
print(xtable::xtable(x_gs,
footnote = "This is a footnote; note that it can be wide.",
caption = "Caption example for xtable output."
))
# Demonstrate nEventsIA method
# find expected number of events at time 12 in the above trial
nEventsIA(x = x_gs, tIA = 10)
# find time at which 1/4 of events are expected
tEventsIA(x = x_gs, timing = .25)
# Adjust accrual duration R to achieve desired power
# Trial duration T input as NULL and will be computed based on
# accrual duration R and minimum follow-up duration minfup
# Minimum follow-up duration minfup is specified
# We use the Schoenfeld method to compute accrual duration R
# Control median survival time is 6
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
alpha = .025, beta = .1, minfup = 12, T = NULL, method = "Schoenfeld"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
T = 36, minfup = 12, method = "Schoenfeld"
) |>
print()
# Vary minimum follow-up duration minfup to obtain power
# Accrual duration R rate gamma are fixed and will not change on output.
# Trial duration T and minimum follow-up minfup are input as NULL
# and will be computed on output.
# We will use the Freedman method to compute sample size
# Control median survival time is 6
# Often this method will fail as the accrual duration and rate provide too
# little or too much sample size.
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6, R = 25,
alpha = .025, beta = .1, minfup = NULL, T = NULL, method = "Freedman"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
T = 36, minfup = 12, method = "Freedman"
) |>
print()
# piecewise constant enrollment rates (vary accrual rate to achieve power)
# also piecewise constant failure rates
# will specify annualized enrollment and failure rates
nSurv(
lambdaC = -log(c(.95, .97, .98)), # 5%, 3% and 2% annual failure rates
S = c(1, 1), # 1 year duration for first 2 failure rates, 3rd continues indefinitely
R = c(.25, .25, 1.5), # 2-year enrollment with ramp-up over first 1/2 year
gamma = c(1, 3, 6), # 1, 3 and 6 annualized enrollment rates will be
# multiplied by ratio to achieve desired power
hr = .5, eta = -log(1 - .01), # 1% annual censoring rate
minfup = 3, T = 5, # 5-year trial duration with 3-year minimum follow-up
alpha = .025, beta = .1, method = "LachinFoulkes"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
lambdaC = -log(c(.95, .97, .98)), # 5%, 3% and 2% annual failure rates
S = c(1, 1), # 1 year duration for first 2 failure rates, 3rd continues indefinitely
R = c(.25, .25, 1.5), # 2-year enrollment with ramp-up over first 1/2 year
gamma = c(1, 3, 6), # 1, 3 and 6 annualized enrollment rates will be
# multiplied by ratio to achieve desired power
hr = .5, eta = -log(1 - .01), # 1% annual censoring rate
minfup = 3, T = 5, # 5-year trial duration with 3-year minimum follow-up
alpha = .025, beta = .1, method = "LachinFoulkes"
) |>
print()
# combine it all: 2 strata, 2 failure rate periods
# Note that method = "LachinFoulkes" may be preferred here
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,
alpha = .025, beta = .1, method = "BernsteinLagakos"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
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,
alpha = .025, beta = .1, method = "BernsteinLagakos"
) |>
print()
# Example to compute power for a fixed design.
# Trial duration T, minimum follow-up minfup and accrual duration R are all
# specified and will not change on output.
# beta=NULL will compute power and output will be the same as if beta were specified.
# This option is not available for group sequential designs.
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6, R = 25,
alpha = .025, beta = NULL, minfup = 12, T = 36, method = "LachinFoulkes"
) |>
print()
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| tEventsIA | R Documentation |
Advanced time-to-event sample size calculation
Description
nSurv() is used to calculate the sample size for a two-arm clinical trial
with a time-to-event endpoint under the assumption of proportional hazards.
The default method assumes a fixed enrollment duration and fixed trial duration;
in this case the required sample size is achieved by increasing enrollment rates.
nSurv() implements the Lachin and Foulkes (1986) method as default.
Schoenfeld (1981), Freedman (1982), and Bernstein and Lagakos (1989) methods
are also supported; see Details.
gsSurv() combines nSurv() with gsDesign() to derive a
group sequential design for a study with a time-to-event endpoint.
Usage
tEventsIA(x, timing = 0.25, tol = .Machine$double.eps^0.25)
nEventsIA(tIA = 5, x = NULL, target = 0, simple = TRUE)
nSurv(
lambdaC = log(2)/6,
hr = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = 18,
minfup = 6,
ratio = 1,
alpha = 0.025,
beta = 0.1,
sided = 1,
tol = .Machine$double.eps^0.25,
method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)
## S3 method for class 'nSurv'
print(x, digits = 3, show_strata = TRUE, ...)
## S3 method for class 'gsSurv'
xtable(
x,
caption = NULL,
label = NULL,
align = NULL,
digits = NULL,
display = NULL,
auto = FALSE,
footnote = NULL,
fnwid = "9cm",
timename = "months",
...
)
gsSurv(
k = 3,
test.type = 4,
alpha = 0.025,
sided = 1,
beta = 0.1,
astar = 0,
timing = 1,
sfu = sfHSD,
sfupar = -4,
sfl = sfHSD,
sflpar = -2,
r = 18,
lambdaC = log(2)/6,
hr = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = 18,
minfup = 6,
ratio = 1,
tol = .Machine$double.eps^0.25,
usTime = NULL,
lsTime = NULL,
method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)
## S3 method for class 'gsSurv'
print(x, digits = 3, show_gsDesign = FALSE, show_strata = TRUE, ...)
Arguments
x |
An object of class |
timing |
Sets relative timing of interim analyses in |
tol |
For cases when |
tIA |
Timing of an interim analysis; should be between 0 and
|
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 |
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. Note that rates corresponding the final time period are extended indefinitely. |
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 across strata and time periods; if entered as a vector, rates are constant across strata. |
etaE |
Matrix dropout hazard rates for the experimental group specified
in like form as |
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 across strata and time periods; if entered as a vector, rates are constant across strata. |
R |
A scalar or vector of durations of time periods for recruitment
rates specified in rows of |
S |
A scalar or vector of durations of piecewise constant event rates
specified in rows of |
T |
Study duration; if |
minfup |
Follow-up of last patient enrolled; if |
ratio |
Randomization ratio of experimental treatment divided by control; normally a scalar, but may be a vector with length equal to number of strata. |
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. |
sided |
1 for 1-sided testing, 2 for 2-sided testing. |
method |
One of |
digits |
Number of digits past the decimal place to print
( |
show_strata |
Logical; for |
... |
Other arguments that may be passed to generic functions underlying the methods here. |
caption |
Passed through to generic |
label |
Passed through to generic |
align |
Passed through to generic |
display |
Passed through to generic |
auto |
Passed through to generic |
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") |
k |
Number of analyses planned, including interim and final. |
test.type |
|
astar |
Normally not specified. If |
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-sided testing is used to completely specify
spending ( |
sfupar |
Real value, default is |
sfl |
Specifies the spending function for lower boundary crossing
probabilities when asymmetric, two-sided testing is performed
( |
sflpar |
Real value, default is |
r |
Integer value (>= 1 and <= 80) controlling the number of numerical
integration grid points. Default is 18, as recommended by Jennison and
Turnbull (2000). Grid points are spread out in the tails for accurate
probability calculations. Larger values provide more grid points and greater
accuracy but slow down computation. Jennison and Turnbull (p. 350) note an
accuracy of |
usTime |
Default is NULL in which case upper bound spending time is
determined by |
lsTime |
Default is NULL in which case lower bound spending time is
determined by |
show_gsDesign |
Logical; for |
Details
The Lachin and Foulkes method uses both null and alternate hypothesis
variances to derive sample size and is extended here to support
non-inferiority, super-superiority, and stratified designs.
As an alternative, the Kim and Tsiatis (1990) method can be used with fixed
enrollment rates and either fixed enrollment duration or fixed minimum
follow-up.
The Schoenfeld approach uses the asymptotic distribution of the log-rank
statistic under the assumption of proportional hazards and local alternatives
(i.e., \log(HR) is small). The Freedman approach uses the same
asymptotic distribution and, like the Schoenfeld approach, uses just the
null hypothesis variance to derive sample size.
The Bernstein and Lagakos (1989) approach was derived to compute sample size
for a stratified model with a common proportional hazards assumption across
strata. Like the Lachin and Foulkes (1986) method, it uses both null and
alternate hypothesis variances to compute sample size; however, the null
hypothesis variance is computed differently. The Bernstein and Lagakos (1989)
approach uses the alternate hypothesis failure rate assumptions for both the
control and experimental groups, while the Lachin and Foulkes method uses
null hypothesis rates that average the alternate hypothesis failure rates to
get similar numbers of expected events under the null and alternate
hypotheses. Since the Lachin and Foulkes method has fewer events under the
null hypothesis (less statistical information), it calculates less power
than the Bernstein and Lagakos method.
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.
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 gsBoundSummary 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 endpoint 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.
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().
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.
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 |
lambdaC |
As input. |
etaC |
As input. |
etaE |
As input. |
gamma |
As input unless none of the following are |
ratio |
As input. |
R |
As input unless |
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 |
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
|
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 ( |
lambdaC |
As input. |
etaC |
As input. |
etaE |
As input. |
gamma |
As input unless none of the following are |
ratio |
As input. |
R |
As input unless |
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 |
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 |
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 |
eDC |
The expected number of events in the control group at time the
output time |
eDE |
The expected number of events in the experimental group at the
output time |
eNC |
The expected enrollment in the control group at the
output time |
eNE |
The expected enrollment in the experimental group at the
output time |
tEventsIA() returns the same structure as nEventsIA(..., simple=TRUE) when
Author(s)
Keaven Anderson keaven_anderson@merck.com
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
uniroot
gsBoundSummary, xprint,
gsSurvCalendar, gsDesign-package,
plot.gsDesign, gsDesign, gsHR,
nSurvival
Normal
xtable
Examples
# Vary accrual rate gamma to obtain power
# T, minfup and R all specified, although R will be adjusted on output
# gamma as input will be multiplied in output to achieve desired power
# Default method is Lachin and Foulkes
x_nsurv <- nSurv(
lambdaC = log(2) / 6, R = 10, hr = .5, eta = .001, gamma = 1,
alpha = 0.02, beta = .15, T = 36, minfup = 12, method = "LachinFoulkes"
)
# Demonstrate print method
print(x_nsurv)
# Same assumptions for group sequential design
x_gs <- gsSurv(
k = 4, sfl = gsDesign::sfPower, sflpar = .5, lambdaC = log(2) / 6, hr = .5,
eta = .001, gamma = 1, T = 36, minfup = 12, method = "LachinFoulkes"
)
print(x_gs)
# Demonstrate xtable method for gsSurv
print(xtable::xtable(x_gs,
footnote = "This is a footnote; note that it can be wide.",
caption = "Caption example for xtable output."
))
# Demonstrate nEventsIA method
# find expected number of events at time 12 in the above trial
nEventsIA(x = x_gs, tIA = 10)
# find time at which 1/4 of events are expected
tEventsIA(x = x_gs, timing = .25)
# Adjust accrual duration R to achieve desired power
# Trial duration T input as NULL and will be computed based on
# accrual duration R and minimum follow-up duration minfup
# Minimum follow-up duration minfup is specified
# We use the Schoenfeld method to compute accrual duration R
# Control median survival time is 6
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
alpha = .025, beta = .1, minfup = 12, T = NULL, method = "Schoenfeld"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
T = 36, minfup = 12, method = "Schoenfeld"
) |>
print()
# Vary minimum follow-up duration minfup to obtain power
# Accrual duration R rate gamma are fixed and will not change on output.
# Trial duration T and minimum follow-up minfup are input as NULL
# and will be computed on output.
# We will use the Freedman method to compute sample size
# Control median survival time is 6
# Often this method will fail as the accrual duration and rate provide too
# little or too much sample size.
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6, R = 25,
alpha = .025, beta = .1, minfup = NULL, T = NULL, method = "Freedman"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6,
T = 36, minfup = 12, method = "Freedman"
) |>
print()
# piecewise constant enrollment rates (vary accrual rate to achieve power)
# also piecewise constant failure rates
# will specify annualized enrollment and failure rates
nSurv(
lambdaC = -log(c(.95, .97, .98)), # 5%, 3% and 2% annual failure rates
S = c(1, 1), # 1 year duration for first 2 failure rates, 3rd continues indefinitely
R = c(.25, .25, 1.5), # 2-year enrollment with ramp-up over first 1/2 year
gamma = c(1, 3, 6), # 1, 3 and 6 annualized enrollment rates will be
# multiplied by ratio to achieve desired power
hr = .5, eta = -log(1 - .01), # 1% annual censoring rate
minfup = 3, T = 5, # 5-year trial duration with 3-year minimum follow-up
alpha = .025, beta = .1, method = "LachinFoulkes"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
lambdaC = -log(c(.95, .97, .98)), # 5%, 3% and 2% annual failure rates
S = c(1, 1), # 1 year duration for first 2 failure rates, 3rd continues indefinitely
R = c(.25, .25, 1.5), # 2-year enrollment with ramp-up over first 1/2 year
gamma = c(1, 3, 6), # 1, 3 and 6 annualized enrollment rates will be
# multiplied by ratio to achieve desired power
hr = .5, eta = -log(1 - .01), # 1% annual censoring rate
minfup = 3, T = 5, # 5-year trial duration with 3-year minimum follow-up
alpha = .025, beta = .1, method = "LachinFoulkes"
) |>
print()
# combine it all: 2 strata, 2 failure rate periods
# Note that method = "LachinFoulkes" may be preferred here
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,
alpha = .025, beta = .1, method = "BernsteinLagakos"
)
# Same assumptions for group sequential design
gsSurv(
k = 4, sfu = gsDesign::sfHSD, sfupar = -4, sfl = gsDesign::sfPower, sflpar = .5,
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,
alpha = .025, beta = .1, method = "BernsteinLagakos"
) |>
print()
# Example to compute power for a fixed design.
# Trial duration T, minimum follow-up minfup and accrual duration R are all
# specified and will not change on output.
# beta=NULL will compute power and output will be the same as if beta were specified.
# This option is not available for group sequential designs.
nSurv(
lambdaC = log(2) / 6, hr = .5, eta = .001, gamma = 6, R = 25,
alpha = .025, beta = NULL, minfup = 12, T = 36, method = "LachinFoulkes"
) |>
print()
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