gsDesign (version 3.5.0)

sfHSD: Hwang-Shih-DeCani Spending Function

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.

A Hwang-Shih-DeCani spending function takes the form $$f(t;\alpha, \gamma)=\alpha(1-e^{-\gamma t})/(1-e^{-\gamma})$$ 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.

Usage

sfHSD(alpha, t, param)

Value

An object of type spendfn. See Spending_Function_Overview for further details.

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]

Author

Keaven Anderson

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

library(ggplot2)
# 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")
)