gsDesign() is used to find boundaries and trial size required for a
group sequential design.
Many parameters normally take on default values and thus do not require
explicit specification. One- and two-sided designs are supported. Two-sided
designs may be symmetric or asymmetric. Wang-Tsiatis designs, including
O'Brien-Fleming and Pocock designs can be generated. Designs with common
spending functions as well as other built-in and user-specified functions
for Type I error and futility are supported. Type I error computations for
asymmetric designs may assume binding or non-binding lower bounds. The print
function has been extended using print.gsDesign() to print
gsDesign objects; see examples.
The user may ignore the structure of the value returned by gsDesign()
if the standard printing and plotting suffice; see examples.
delta and n.fix are used together to determine what sample
size output options the user seeks. The default, delta=0 and
n.fix=1, results in a ‘generic’ design that may be used with
any sampling situation. Sample size ratios are provided and the user
multiplies these times the sample size for a fixed design to obtain the
corresponding group sequential analysis times. If delta>0,
n.fix is ignored, and delta is taken as the standardized
effect size - the signal to noise ratio for a single observation; for
example, the mean divided by the standard deviation for a one-sample normal
problem. In this case, the sample size at each analysis is computed. When
delta=0 and n.fix>1, n.fix is assumed to be the sample
size for a fixed design with no interim analyses. See examples below.
Following are further comments on the input argument test.type which
is used to control what type of error measurements are used in trial design.
The manual may also be worth some review in order to see actual formulas for
boundary crossing probabilities for the various options. Options 3 and 5
assume the trial stops if the lower bound is crossed for Type I and Type II
error computation (binding lower bound). For the purpose of computing Type
I error, options 4 and 6 assume the trial continues if the lower bound is
crossed (non-binding lower bound); that is a Type I error can be made by
crossing an upper bound after crossing a previous lower bound.
Beta-spending refers to error spending for the lower bound crossing
probabilities under the alternative hypothesis (options 3 and 4). In this
case, the final analysis lower and upper boundaries are assumed to be the
same. The appropriate total beta spending (power) is determined by adjusting
the maximum sample size through an iterative process for all options. Since
options 3 and 4 must compute boundary crossing probabilities under both the
null and alternative hypotheses, deriving these designs can take longer than
other options. Options 5 and 6 compute lower bound spending under the null
hypothesis.
gsDesign(
k = 3,
test.type = 4,
alpha = 0.025,
beta = 0.1,
astar = 0,
delta = 0,
n.fix = 1,
timing = 1,
sfu = sfHSD,
sfupar = -4,
sfl = sfHSD,
sflpar = -2,
tol = 1e-06,
r = 18,
n.I = 0,
maxn.IPlan = 0,
nFixSurv = 0,
endpoint = NULL,
delta1 = 1,
delta0 = 0,
overrun = 0,
usTime = NULL,
lsTime = NULL
)# S3 method for gsDesign
xtable(
x,
caption = NULL,
label = NULL,
align = NULL,
digits = NULL,
display = NULL,
...
)
An object of the class gsDesign. This class has the following
elements and upon return from gsDesign() contains:
As input.
As input.
As input.
As input.
As input, except when test.type=5 or 6
and astar is input as 0; in this case astar is changed to
1-alpha.
The standardized effect size for which the
design is powered. Will be as input to gsDesign() unless it was input
as 0; in that case, value will be computed to give desired power for fixed
design with input sample size n.fix.
Sample size
required to obtain desired power when effect size is delta.
A vector of length k containing the portion of the
total planned information or sample size at each analysis.
As input.
As input.
Vector of length k. If values
are input, same values are output. Otherwise, n.I will contain the
sample size required at each analysis to achieve desired timing and
beta for the output value of delta. If delta=0 was
input, then this is the sample size required for the specified group
sequential design when a fixed design requires a sample size of
n.fix. If delta=0 and n.fix=1 then this is the relative
sample size compared to a fixed design; see details and examples.
As input.
As input.
Sample
size for Lachin and Foulkes method when nSurvival is used for fixed
design input. If nSurvival is used to compute n.fix, then
nFixSurv is inflated by the same amount as n.fix and stored in
nSurv. Note that if you use gsSurv for time-to-event sample
size, this is not needed and a more complete output summary is given.
As input.
As input.
As input.
As input.
As input.
As input.
Upper bound spending function, boundary and boundary crossing probabilities under the NULL and alternate hypotheses. See Spending_Function_Overview and manual for further details.
Lower bound spending function, boundary and boundary
crossing probabilities at each analysis. Lower spending is under alternative
hypothesis (beta spending) for test.type=3 or 4. For
test.type=2, 5 or 6, lower spending is under the null
hypothesis. For test.type=1, output value is NULL. See
Spending_Function_Overview and manual.
Standarized
effect size under null (0) and alternate hypothesis. If delta is
input, theta[1]=delta. If n.fix is input, theta[1] is
computed using a standard sample size formula (pseudocode):
((Zalpha+Zbeta)/theta[1])^2=n.fix.
For
test.type=4 or 6, this contains false positive probabilities
under the null hypothesis assuming that crossing a futility bound does not
stop the trial.
Expected sample size accounting for early
stopping. For time-to-event outcomes, this would be the expected number of
events (although gsSurv will give expected sample size). For
information-based-design, this would give the expected information when the
trial stops. If overrun is specified, the expected sample size
includes the overrun at each interim.
An object of class "xtable" with attributes specifying formatting options for a table
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.
Type I error, always one-sided. Default value is 0.025.
Type II error, default value is 0.1 (90% power).
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.
Effect size for theta under alternative hypothesis. This can be
set to the standardized effect size to generate a sample size if
n.fix=NULL. See details and examples.
Sample size for fixed design with no interim; used to find
maximum group sequential sample size. For a time-to-event outcome, input
number of events required for a fixed design rather than sample size and
enter fixed design sample size (optional) in nFixSurv. See details
and examples.
Sets relative timing of interim analyses. 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.
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.
Tolerance for error (default is 0.000001). Normally this will not be changed by the user. This does not translate directly to number of digits of accuracy, so use extra decimal places.
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.
Used for re-setting bounds when timing of analyses changes from initial design; see examples.
Used for re-setting bounds when timing of analyses changes from initial design; see examples.
If a time-to-event variable is used, nFixSurv
computed as the sample size from nSurvival may be entered to have
gsDesign compute the total sample size required as well as the number
of events at each analysis that will be returned in n.fix; this is
rounded up to an even number.
An optional character string that should represent the type
of endpoint used for the study. This may be used by output functions. Types
most likely to be recognized initially are "TTE" for time-to-event outcomes
with fixed design sample size generated by nSurvival() and "Binomial"
for 2-sample binomial outcomes with fixed design sample size generated by
nBinomial().
delta1 and delta0 may be used to store
information about the natural parameter scale compared to delta that
is a standardized effect size. delta1 is the alternative hypothesis
parameter value on the natural parameter scale (e.g., the difference in two
binomial rates).
delta0 is the null hypothesis parameter value on the
natural parameter scale.
Scalar or vector of length k-1 with patients enrolled
that are not included in each interim analysis.
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).
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).
An R object of class found among methods(xtable). See
below on how to write additional method functions for xtable.
Character vector of length 1 or 2 containing the
table's caption or title. If length is 2, the second item is the
"short caption" used when LaTeX generates a "List of Tables". Set to
NULL to suppress the caption. Default value is NULL.
Character vector of length 1 containing the LaTeX label
or HTML anchor. Set to NULL to suppress the label. Default
value is NULL.
Character vector of length equal to the number of columns
of the resulting table, indicating the alignment of the corresponding
columns. Also, "|" may be used to produce vertical lines
between columns in LaTeX tables, but these are effectively ignored
when considering the required length of the supplied vector. If a
character vector of length one is supplied, it is split as
strsplit(align, "")[[1]] before processing. Since the row
names are printed in the first column, the length of align is
one greater than ncol(x) if x is a
data.frame. Use "l", "r", and "c" to
denote left, right, and center alignment, respectively. Use
"p{3cm}" etc. for a LaTeX column of the specified width. For
HTML output the "p" alignment is interpreted as "l",
ignoring the width request. Default depends on the class of
x.
Numeric vector of length equal to one (in which case it will be
replicated as necessary) or to the number of columns of the
resulting table or matrix of the same size as the resulting
table, indicating the number of digits to display in the
corresponding columns. Since the row names are printed in the first
column, the length of the vector digits or the number of
columns of the matrix digits is one greater than
ncol(x) if x is a data.frame. Default depends
on the class of x. If values of digits are negative, the
corresponding values of x are displayed in scientific format
with abs(digits) digits.
Character vector of length equal to the number of columns of the
resulting table, indicating the format for the corresponding columns.
Since the row names are printed in the first column, the length of
display is one greater than ncol(x) if x is a
data.frame. These values are passed to the formatC
function. Use "d" (for integers), "f", "e",
"E", "g", "G", "fg" (for reals), or
"s" (for strings). "f" gives numbers in the usual
xxx.xxx format; "e" and "E" give
n.ddde+nn or n.dddE+nn (scientific format); "g"
and "G" put x[i] into scientific format only if it
saves space to do so. "fg" uses fixed format as "f",
but digits as number of significant digits. Note that
this can lead to quite long result strings. Default depends on the
class of x.
Additional arguments. (Currently ignored.)
Keaven Anderson keaven_anderson@merck.com
Jennison C and Turnbull BW (2000), Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman and Hall. Lan KK, DeMets DL (1989). Group sequential procedures: calendar versus information time. Statistics in medicine 8(10):1191-8. Liu, Q, Lim, P, Nuamah, I, and Li, Y (2012), On adaptive error spending approach for group sequential trials with random information levels. Journal of biopharmaceutical statistics; 22(4), 687-699.
library(ggplot2)
# symmetric, 2-sided design with O'Brien-Fleming-like boundaries
# lower bound is non-binding (ignored in Type I error computation)
# sample size is computed based on a fixed design requiring n=800
x <- gsDesign(k = 5, test.type = 2, n.fix = 800)
# note that "x" below is equivalent to print(x) and print.gsDesign(x)
x
plot(x)
plot(x, plottype = 2)
# Assuming after trial was designed actual analyses occurred after
# 300, 600, and 860 patients, reset bounds
y <- gsDesign(
k = 3, test.type = 2, n.fix = 800, n.I = c(300, 600, 860),
maxn.IPlan = x$n.I[x$k]
)
y
# asymmetric design with user-specified spending that is non-binding
# sample size is computed relative to a fixed design with n=1000
sfup <- c(.033333, .063367, .1)
sflp <- c(.25, .5, .75)
timing <- c(.1, .4, .7)
x <- gsDesign(
k = 4, timing = timing, sfu = sfPoints, sfupar = sfup, sfl = sfPoints,
sflpar = sflp, n.fix = 1000
)
x
plot(x)
plot(x, plottype = 2)
# same design, but with relative sample sizes
gsDesign(
k = 4, timing = timing, sfu = sfPoints, sfupar = sfup, sfl = sfPoints,
sflpar = sflp
)
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