sfLogistic: Two-parameter Spending Function Families

Description

The functions sfLogistic(), sfNormal(), sfExtremeValue(), sfExtremeValue2(), sfCauchy(), and sfBetaDist() are all 2-parameter spending function families. These provide increased flexibility in some situations where the flexibility of a one-parameter spending function family is not sufficient. These functions all allow fitting of two points on a cumulative spending function curve; in this case, four parameters are specified indicating an x and a y coordinate for each of 2 points. Normally each of these functions 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 in the examples; note, however, that an automatic $\alpha$- and $\beta$-spending function plot is also available.

sfBetaDist(alpha,t,param) is simply alpha times the incomplete beta cumulative distribution function with parameters $a$ and $b$ passed in param evaluated at values passed in t.

The other spending functions take the form $$f(t;\alpha,a,b)=\alpha F(a+bF^{-1}(t))$$ where $F()$ is a cumulative distribution function with values $> 0$ on the real line (logistic for sfLogistic(), normal for sfNormal(), extreme value for sfExtremeValue() and Cauchy for sfCauchy()) and $F^{-1}()$ is its inverse.

For the logistic spending function this simplifies to $$f(t;\alpha,a,b)=\alpha (1-(1+e^a(t/(1-t))^b)^{-1}).$$

For the extreme value distribution with $$F(x)=\exp(-\exp(-x))$$ this simplifies to $$f(t;\alpha,a,b)=\alpha \exp(-e^a (-\ln t)^b).$$ Since the extreme value distribution is not symmetric, there is also a version where the standard distribution is flipped about 0. This is reflected in sfExtremeValue2() where $$F(x)=1-\exp(-\exp(x)).$$

Usage

sfLogistic(alpha, t, param)
sfBetaDist(alpha, t, param)
sfCauchy(alpha, t, param)
sfExtremeValue(alpha, t, param)
sfExtremeValue2(alpha, t, param)
sfNormal(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 or information.

A vector of points with increasing values from 0 to 1, inclusive. Values of the proportion of sample size or information for which the spending function will be computed.

param

In the two-parameter specification, sfBetaDist() requires 2 positive values, while sfLogistic(), sfNormal(), sfExtremeValue(),

sfExtremeValue2() and sfCauchy() require the first parameter to be any real value and the second to be a positive value. The four parameter specification is c(t1,t2,u1,u2) where the objective is that sf(t1)=alpha*u1 and sf(t2)=alpha*u2. In this parameterization, all four values must be between 0 and 1 and t1 < t2, u1 < u2.

Value

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

References

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

Examples

Run this code

# NOT RUN {
library(ggplot2)
# design a 4-analysis trial using a Kim-DeMets spending function
# for both lower and upper bounds
x <- gsDesign(k = 4, sfu = sfPower, sfupar = 3, sfl = sfPower, sflpar = 1.5)

# print the design
x

# plot the alpha- and beta-spending functions
plot(x, plottype = 5)

# start by showing how to fit two points with sfLogistic
# plot the spending function using many points to obtain a smooth curve
# note that curve fits the points x=.1,  y=.01 and x=.4,  y=.1
# specified in the 3rd parameter of sfLogistic
t <- 0:100 / 100
plot(t, sfLogistic(1, t, c(.1, .4, .01, .1))$spend,
  xlab = "Proportion of final sample size",
  ylab = "Cumulative Type I error spending",
  main = "Logistic Spending Function Examples",
  type = "l", cex.main = .9
)
lines(t, sfLogistic(1, t, c(.01, .1, .1, .4))$spend, lty = 2)

# now just give a=0 and b=1 as 3rd parameters for sfLogistic
lines(t, sfLogistic(1, t, c(0, 1))$spend, lty = 3)

# try a couple with unconventional shapes again using
# the xy form in the 3rd parameter
lines(t, sfLogistic(1, t, c(.4, .6, .1, .7))$spend, lty = 4)
lines(t, sfLogistic(1, t, c(.1, .7, .4, .6))$spend, lty = 5)
legend(
  x = c(.0, .475), y = c(.76, 1.03), lty = 1:5,
  legend = c(
    "Fit (.1, 01) and (.4, .1)", "Fit (.01, .1) and (.1, .4)",
    "a=0,  b=1", "Fit (.4, .1) and (.6, .7)",
    "Fit (.1, .4) and (.7, .6)"
  )
)

# set up a function to plot comparsons of all
# 2-parameter spending functions
plotsf <- function(alpha, t, param) {
  plot(t, sfCauchy(alpha, t, param)$spend,
    xlab = "Proportion of enrollment",
    ylab = "Cumulative spending", type = "l", lty = 2
  )
  lines(t, sfExtremeValue(alpha, t, param)$spend, lty = 5)
  lines(t, sfLogistic(alpha, t, param)$spend, lty = 1)
  lines(t, sfNormal(alpha, t, param)$spend, lty = 3)
  lines(t, sfExtremeValue2(alpha, t, param)$spend, lty = 6, col = 2)
  lines(t, sfBetaDist(alpha, t, param)$spend, lty = 7, col = 3)
  legend(
    x = c(.05, .475), y = .025 * c(.55, .9),
    lty = c(1, 2, 3, 5, 6, 7),
    col = c(1, 1, 1, 1, 2, 3),
    legend = c(
      "Logistic", "Cauchy", "Normal", "Extreme value",
      "Extreme value 2", "Beta distribution"
    )
  )
}
# do comparison for a design with conservative early spending
# note that Cauchy spending function is quite different
# from the others
param <- c(.25, .5, .05, .1)
plotsf(.025, t, param)
# }