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.
# S3 method for nSurv
print(x, digits = 4, ...)nSurv(
lambdaC = log(2)/6,
hr = 0.6,
hr0 = 1,
eta = 0,
etaE = NULL,
gamma = 1,
R = 12,
S = NULL,
T = NULL,
minfup = NULL,
ratio = 1,
alpha = 0.025,
beta = 0.1,
sided = 1,
tol = .Machine$double.eps^0.25
)
tEventsIA(x, timing = 0.25, tol = .Machine$double.eps^0.25)
nEventsIA(tIA = 5, x = NULL, target = 0, simple = TRUE)
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 = NULL,
minfup = NULL,
ratio = 1,
tol = .Machine$double.eps^0.25,
usTime = NULL,
lsTime = NULL
)
# S3 method for gsSurv
print(x, digits = 2, ...)
# S3 method for gsSurv
xtable(
x,
caption = NULL,
label = NULL,
align = NULL,
digits = NULL,
display = NULL,
auto = FALSE,
footnote = NULL,
fnwid = "9cm",
timename = "months",
...
)
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.
Number of digits past the decimal place to print
(print.gsSurv.); also a pass through to generic xtable() from
xtable.gsSurv().
other arguments that may be passed to generic functions underlying the methods here.
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.
hazard ratio (experimental/control) under the alternate hypothesis (scalar).
hazard ratio (experimental/control) under the null hypothesis (scalar).
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.
matrix dropout hazard rates for the experimental group specified
in like form as eta; if NULL, this is set equal to eta.
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.
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.
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.
study duration; if T is input as NULL, this will be
computed on output; see details.
follow-up of last patient enrolled; if minfup is input
as NULL, this will be computed on output; see details.
randomization ratio of experimental treatment divided by control; normally a scalar, but may be a vector with length equal to number of strata.
type I error rate. Default is 0.025 since 1-sided testing is default.
type II error rate. Default is 0.10 (90% power); NULL if power is to be computed based on other input values.
1 for 1-sided testing, 2 for 2-sided testing.
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.
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.
Timing of an interim analysis; should be between 0 and
y$T.
The targeted proportion of events at an interim analysis. This is used for root-finding will be 0 for normal use.
See output specification for nEventsIA().
Number of analyses planned, including interim and final.
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.
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.
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,
Spending_Function_Overview, manual and examples.
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.
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.
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.
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 user.
Default is NULL in which case upper bound spending time is
determined by timing. Otherwise, this should be a vector of length
codek with the spending time at each analysis (see Details in help for gsDesign).
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).
passed through to generic xtable().
passed through to generic xtable().
passed through to generic xtable().
passed through to generic xtable().
passed through to generic xtable().
footnote for xtable output; may be useful for describing some of the design parameters.
a text string controlling the width of footnote text at the bottom of the xtable output.
character string with plural of time units (e.g., "months")
nSurv() returns an object of type nSurv with the
following components:
As input.
As input.
Type II error; if missing, this is computed.
Power
corresponding to input beta or computed if output beta is
computed.
As input.
As input.
As input.
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.
As input.
As input unless T was
NULL on input.
As input.
As input.
As input.
As input.
As input.
Total expected sample size corresponding to output accrual rates and durations.
Total expected number of events under the alternate hypothesis.
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.
A vector of expected number of events by stratum in the control group under the alternate hypothesis.
A vector of expected number of events by stratum in the experimental group under the alternate hypothesis.
A vector of expected number of events by stratum in the control group under the null hypothesis.
A vector of expected number of events by stratum in the experimental group under the null hypothesis.
A vector of the expected accrual in each stratum in the control group.
A vector of the expected accrual in each stratum in the experimental group.
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:
A group sequential design (gsDesign) object.
As input.
As input.
As input.
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.
As input.
As input unless T was
NULL on input.
As input.
As input.
As input.
As input.
As input.
Total expected sample size corresponding to output accrual rates and durations.
Total expected sample size corresponding to output accrual rates and durations.
Total expected number of events under the alternate hypothesis.
Total expected number of events under the alternate hypothesis.
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.
A vector of expected number of events by stratum in the control group under the alternate hypothesis.
A vector of expected number of events by stratum in the experimental group under the alternate hypothesis.
A vector of expected number of events by stratum in the control group under the null hypothesis.
A vector of expected number of events by stratum in the experimental group under the null hypothesis.
A vector of the expected accrual in each stratum in the control group.
A vector of the expected accrual in each stratum in the experimental group.
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:
The input value tIA.
The expected number of events in the control group at time the
output time T.
The expected number of events in the
experimental group at the output time T.
The expected
enrollment in the control group at the output time T.
The
expected enrollment in the experimental group at the output time T.
tEventsIA() returns
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.
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.
gsBoundSummary, xprint,
gsDesign package overview, plot.gsDesign,
gsDesign, gsHR, nSurvival
# NOT RUN {
# 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
)
# example where only 1 month of follow-up is desired
# set failure rate to 0 after 1 month using lambdaC and S
nSurv(lambdaC = c(.4, 0), hr = 2 / 3, S = 1, minfup = 1)
# 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::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)
# }
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