sfHSD: 4.1: Hwang-Shih-DeCani Spending Function

In gsDesign: Group Sequential Design

Description

The function sfHSD implements a Hwang-Shih-DeCani spending function. This is the default spending function for gsDesign(). Normally it will be passed to gsDesign in the parameter sfu for the upper bound or sfl for the lower bound to specify a spending function family for a design. In this case, the user does not need to know the calling sequence. The calling sequence is useful, however, when the user wishes to plot a spending function as demonstrated below in examples.

Usage

sfHSD(alpha, t, param)

Arguments

alpha	Real value > 0 and no more than 1. Normally, alpha=0.025 for one-sided Type I error specification or alpha=0.1 for Type II error specification. However, this could be set to 1 if for descriptive purposes you wish to see the proportion of spending as a function of the proportion of sample size/information.
t	A vector of points with increasing values from 0 to 1, inclusive. Values of the proportion of sample size/information for which the spending function will be computed.
param	A single real value specifying the gamma parameter for which Hwang-Shih-DeCani spending is to be computed; allowable range is [-40, 40]

Details

A Hwang-Shih-DeCani spending function takes the form

f(t; alpha, gamma) = alpha * (1-exp(-gamma * t))/(1 - exp(-gamma))

where gamma is the value passed in param. A value of gamma=-4 is used to approximate an O'Brien-Fleming design (see sfExponential for a better fit), while a value of gamma=1 approximates a Pocock design well.

Value

An object of type spendfn. See Spending function overview for further details.

Note

The manual is not linked to this help file, but is available in library/gsdesign/doc/gsDesignManual.pdf in the directory where R is installed.

Author(s)

Keaven Anderson keaven\[e-mail]

References

Jennison C and Turnbull BW (2000), Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman and Hall.

See Also

Spending function overview, gsDesign, gsDesign package overview

Examples

# design a 4-analysis trial using a Hwang-Shih-DeCani spending function 
# for both lower and upper bounds 
x <- gsDesign(k=4, sfu=sfHSD, sfupar=-2, sfl=sfHSD, sflpar=1)

# print the design
x

# since sfHSD is the default for both sfu and sfl,
# this could have been written as
x <- gsDesign(k=4, sfupar=-2, sflpar=1)

# print again
x

# plot the spending function using many points to obtain a smooth curve
# show default values of gamma to see how the spending function changes
# also show gamma=1 which is supposed to approximate a Pocock design
t <- 0:100/100
plot(t,  sfHSD(0.025, t, -4)$spend,
   xlab="Proportion of final sample size", 
   ylab="Cumulative Type I error spending", 
   main="Hwang-Shih-DeCani Spending Function Example", type="l")
lines(t, sfHSD(0.025, t, -2)$spend, lty=2)
lines(t, sfHSD(0.025, t, 1)$spend, lty=3)
legend(x=c(.0, .375), y=.025*c(.8, 1), lty=1:3, 
    legend=c("gamma= -4", "gamma= -2", "gamma= 1"))

Example output

Loading required package: xtable
Loading required package: ggplot2
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  0.324 0.03    0.5136 0.0350 2.80    0.0025  0.0025
         2  0.649 0.88    0.8096 0.0273 2.58    0.0049  0.0042
         3  0.973 1.51    0.9349 0.0212 2.34    0.0096  0.0069
         4  1.297 2.09    0.9817 0.0165 2.09    0.0183  0.0114
     Total                       0.1000                 0.0250 
+ lower bound beta spending (under H1):
 Hwang-Shih-DeCani spending function with gamma = 1.
++ alpha spending:
 Hwang-Shih-DeCani spending function with gamma = -2.
* 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 0.0025 0.0042 0.0065 0.0072 0.0203 0.5477
  3.2415 0.1695 0.3553 0.2774 0.0978 0.9000 0.7533

Lower boundary (futility or Type II Error)
          Analysis
   Theta      1      2      3      4  Total
  0.0000 0.5136 0.3156 0.1169 0.0336 0.9797
  3.2415 0.0350 0.0273 0.0212 0.0165 0.1000
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  0.324 0.03    0.5136 0.0350 2.80    0.0025  0.0025
         2  0.649 0.88    0.8096 0.0273 2.58    0.0049  0.0042
         3  0.973 1.51    0.9349 0.0212 2.34    0.0096  0.0069
         4  1.297 2.09    0.9817 0.0165 2.09    0.0183  0.0114
     Total                       0.1000                 0.0250 
+ lower bound beta spending (under H1):
 Hwang-Shih-DeCani spending function with gamma = 1.
++ alpha spending:
 Hwang-Shih-DeCani spending function with gamma = -2.
* 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 0.0025 0.0042 0.0065 0.0072 0.0203 0.5477
  3.2415 0.1695 0.3553 0.2774 0.0978 0.9000 0.7533

Lower boundary (futility or Type II Error)
          Analysis
   Theta      1      2      3      4  Total
  0.0000 0.5136 0.3156 0.1169 0.0336 0.9797
  3.2415 0.0350 0.0273 0.0212 0.0165 0.1000

gsDesign documentation built on May 2, 2019, 4:49 p.m.