Two-parameter Spending Function Families

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.

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.

t

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 vignette("SpendingFunctionOverview") for further details.

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.

Author

Keaven Anderson keaven_anderson@merck.com