Exponential Spending Function

The function sfExponential implements the exponential spending function (Anderson and Clark, 2009).

An exponential spending function is defined for any positive nu and $0\le t\le 1$ as $$f(t;\alpha,\nu)=\alpha(t)=\alpha^{t^{-\nu}}.$$ A value of nu=0.8 approximates an O'Brien-Fleming spending function well.

The general class of spending functions this family is derived from requires a continuously increasing cumulative distribution function defined for $x>0$ and is defined as $$f(t;\alpha, \nu)=1-F\left(F^{-1}(1-\alpha)/ t^\nu\right).$$ The exponential spending function can be derived by letting $F(x)=1-\exp(-x)$, the exponential cumulative distribution function. This function was derived as a generalization of the Lan-DeMets (1983) spending function used to approximate an O'Brien-Fleming spending function (sfLDOF()), $$f(t; \alpha)=2-2\Phi \left( \Phi^{-1}(1-\alpha/2)/ t^{1/2} \right).$$

Usage

sfExponential(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 positive value specifying the nu parameter for which the exponential spending is to be computed; allowable range is (0, 1.5].

Value

An object of type spendfn. See vignette("SpendingFunctionOverview") for further details.

Note

The gsDesign technical manual shows how to use sfExponential() to closely approximate an O'Brien-Fleming design. The manual is available at <https://keaven.github.io/gsd-tech-manual/>.

References

Anderson KM and Clark JB (2009), Fitting spending functions. Statistics in Medicine; 29:321-327.

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

Lan, KKG and DeMets, DL (1983), Discrete sequential boundaries for clinical trials. Biometrika; 70:659-663.

Author

Keaven Anderson keaven_anderson@merck.com