Design Derivation

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.

Usage

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 class 'gsDesign'
xtable(
  x,
  caption = NULL,
  label = NULL,
  align = NULL,
  digits = NULL,
  display = NULL,
  ...
)

Arguments

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.

alpha

Type I error, always one-sided. Default value is 0.025.

beta

Type II error, default value is 0.1 (90% power).

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.

delta

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.

n.fix

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.

timing

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.

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 is not needed 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.

tol

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.

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 user.

n.I

Used for re-setting bounds when timing of analyses changes from initial design; see examples.

maxn.IPlan

Used for re-setting bounds when timing of analyses changes from initial design; see examples.

nFixSurv

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.

endpoint

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

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

delta0 is the null hypothesis parameter value on the natural parameter scale.

overrun

Scalar or vector of length k-1 with patients enrolled that are not included in each interim analysis.

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).

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).

x

An R object of class found among methods(xtable). See below on how to write additional method functions for xtable.

caption

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.

label

Character vector of length 1 containing the LaTeX label or HTML anchor. Set to NULL to suppress the label. Default value is NULL.

align

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.

digits

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.

display

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.)

Value

An object of the class gsDesign. This class has the following elements and upon return from gsDesign() contains:

k

As input.

test.type

As input.

alpha

As input.

beta

As input.

astar

As input, except when test.type=5 or 6 and astar is input as 0; in this case astar is changed to 1-alpha.

delta

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.

n.fix

Sample size required to obtain desired power when effect size is delta.

timing

A vector of length k containing the portion of the total planned information or sample size at each analysis.

tol

As input.

r

As input.

n.I

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.

maxn.IPlan

As input.

nFixSurv

As input.

nSurv

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.

endpoint

As input.

delta1

As input.

delta0

As input.

overrun

As input.

usTime

As input.

lsTime

As input.

upper

Upper bound spending function, boundary and boundary crossing probabilities under the NULL and alternate hypotheses. See vignette("SpendingFunctionOverview") and manual for further details.

lower

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 vignette("SpendingFunctionOverview") and manual.

theta

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.

falseprobnb

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.

en

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

Note

The gsDesign technical manual is available at https://keaven.github.io/gsd-tech-manual/.

References

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.

See also

vignette("gsDesignPackageOverview"), gsBoundSummary, plot.gsDesign, gsProbability, vignette("SpendingFunctionOverview"),

Normal xtable

Author

Keaven Anderson keaven_anderson@merck.com

Examples

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
#> Symmetric two-sided group sequential design with
#> 90 % power and 2.5 % Type I Error.
#> Spending computations assume trial stops
#> if a bound is crossed.
#> 
#>               
#>   Analysis  N   Z   Nominal p  Spend
#>          1 164 3.25    0.0006 0.0006
#>          2 328 2.99    0.0014 0.0013
#>          3 492 2.69    0.0036 0.0028
#>          4 656 2.37    0.0088 0.0063
#>          5 819 2.03    0.0214 0.0140
#>      Total                    0.0250 
#> 
#> ++ alpha spending:
#>  Hwang-Shih-DeCani spending function with gamma = -4.
#> 
#> Boundary crossing probabilities and expected sample size
#> assume any cross stops the trial
#> 
#> Upper boundary (power or Type I Error)
#>           Analysis
#>    Theta      1      2      3      4      5 Total  E{N}
#>   0.0000 0.0006 0.0013 0.0028 0.0063 0.0140 0.025 812.8
#>   0.1146 0.0370 0.1512 0.2647 0.2699 0.1771 0.900 589.3
#> 
#> Lower boundary (futility or Type II Error)
#>           Analysis
#>    Theta     1      2      3      4     5 Total
#>   0.0000 6e-04 0.0013 0.0028 0.0063 0.014 0.025
#>   0.1146 0e+00 0.0000 0.0000 0.0000 0.000 0.000
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
#> Symmetric two-sided group sequential design with
#> 90 % power and 2.5 % Type I Error.
#> Spending computations assume trial stops
#> if a bound is crossed.
#> 
#>               
#>   Analysis  N   Z   Nominal p  Spend
#>          1 300 2.96    0.0016 0.0016
#>          2 600 2.44    0.0074 0.0067
#>          3 860 2.01    0.0220 0.0167
#>      Total                    0.0250 
#> 
#> ++ alpha spending:
#>  Hwang-Shih-DeCani spending function with gamma = -4.
#> 
#> Boundary crossing probabilities and expected sample size
#> assume any cross stops the trial
#> 
#> Upper boundary (power or Type I Error)
#>           Analysis
#>    Theta      1      2      3  Total  E{N}
#>   0.0000 0.0016 0.0067 0.0167 0.0250 854.8
#>   0.1146 0.1655 0.4833 0.2654 0.9142 641.6
#> 
#> Lower boundary (futility or Type II Error)
#>           Analysis
#>    Theta      1      2      3 Total
#>   0.0000 0.0016 0.0067 0.0167 0.025
#>   0.1146 0.0000 0.0000 0.0000 0.000

#  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
#> Asymmetric two-sided group sequential design with
#> 90 % power and 2.5 % Type I Error.
#> Upper bound spending computations assume
#> trial continues if lower bound is crossed.
#> 
#>                   ----Lower bounds----  ----Upper bounds-----
#>   Analysis   N    Z   Nominal p Spend+  Z   Nominal p Spend++
#>          1  123 -0.83    0.2041  0.025 3.14    0.0008  0.0008
#>          2  489  0.39    0.6513  0.025 3.16    0.0008  0.0008
#>          3  855  1.26    0.8966  0.025 3.06    0.0011  0.0009
#>          4 1222  1.98    0.9761  0.025 1.98    0.0239  0.0225
#>      Total                      0.1000                 0.0250 
#> + lower bound beta spending (under H1):
#>  User-specified spending function with Points = 0.25, Points = 0.5, Points = 0.75, Points = 1.
#> ++ alpha spending:
#>  User-specified spending function with Points = 0.03333, Points = 0.06337, Points = 0.1, Points = 1.
#> 
#> Boundary crossing probabilities and expected sample size
#> assume any cross stops the trial
#> 
#> Upper boundary (power or Type I Error)
#>           Analysis
#>    Theta      1      2      3      4  Total  E{N}
#>   0.0000 0.0008 0.0007 0.0009 0.0177 0.0202 564.0
#>   0.1025 0.0222 0.1716 0.2969 0.4094 0.9000 907.4
#> 
#> Lower boundary (futility or Type II Error)
#>           Analysis
#>    Theta      1      2      3      4  Total
#>   0.0000 0.2041 0.4703 0.2361 0.0693 0.9798
#>   0.1025 0.0250 0.0250 0.0250 0.0250 0.1000
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
)
#> Asymmetric two-sided group sequential design with
#> 90 % power and 2.5 % Type I Error.
#> Upper bound spending computations assume
#> trial continues if lower bound is crossed.
#> 
#>            Sample
#>             Size    ----Lower bounds----  ----Upper bounds-----
#>   Analysis Ratio*   Z   Nominal p Spend+  Z   Nominal p Spend++
#>          1  0.122 -0.83    0.2041  0.025 3.14    0.0008  0.0008
#>          2  0.488  0.39    0.6513  0.025 3.16    0.0008  0.0008
#>          3  0.855  1.26    0.8966  0.025 3.06    0.0011  0.0009
#>          4  1.221  1.98    0.9761  0.025 1.98    0.0239  0.0225
#>      Total                        0.1000                 0.0250 
#> + lower bound beta spending (under H1):
#>  User-specified spending function with Points = 0.25, Points = 0.5, Points = 0.75, Points = 1.
#> ++ alpha spending:
#>  User-specified spending function with Points = 0.03333, Points = 0.06337, Points = 0.1, Points = 1.
#> * Sample size ratio compared to fixed design with no interim
#> 
#> Boundary crossing probabilities and expected sample size
#> assume any cross stops the trial
#> 
#> Upper boundary (power or Type I Error)
#>           Analysis
#>    Theta      1      2      3      4  Total   E{N}
#>   0.0000 0.0008 0.0007 0.0009 0.0177 0.0202 0.5640
#>   3.2415 0.0222 0.1716 0.2969 0.4094 0.9000 0.9074
#> 
#> Lower boundary (futility or Type II Error)
#>           Analysis
#>    Theta      1      2      3      4  Total
#>   0.0000 0.2041 0.4703 0.2361 0.0693 0.9798
#>   3.2415 0.0250 0.0250 0.0250 0.0250 0.1000