t-distribution Spending Function

The function sfTDist() provides perhaps the maximum flexibility among spending functions provided in the gsDesignCRT package. This function allows fitting of three points on a cumulative spending function curve; in this case, six parameters are specified indicating an x and a y coordinate for each of 3 points.

The t-distribution spending function takes the form $$f(t;\alpha)=\alpha F(a+bF^{-1}(t))$$ where $F()$ is a cumulative t-distribution function with df degrees of freedom and $F^{-1}()$ is its inverse.

Usage

sfTDist(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: In the three-parameter specification, the first paramater (a) may be any real value, the second (b) any positive value, and the third parameter (df=degrees of freedom) any real value 1 or greater. When gsDesignCRT() is called with a t-distribution spending function, this is the parameterization printed. The five parameter specification is c(t1,t2,u1,u2,df) where the objective is that the resulting cumulative proportion of spending at t represented by sf(t) satisfies sf(t1)=alpha*u1, sf(t2)=alpha*u2. The t-distribution used has df degrees of freedom. In this parameterization, all the first four values must be between 0 and 1 and t1 < t2, u1 < u2. The final parameter is any real value of 1 or more. This parameterization can fit any two points satisfying these requirements. The six parameter specification attempts to fit 3 points, but does not have flexibility to fit any three points. In this case, the specification for param is c(t1,t2,t3,u1,u2,u3) where the objective is that sf(t1)=alpha*u1, sf(t2)=alpha*u2, and sf(t3)=alpha*u3. See examples to see what happens when points are specified that cannot be fit.

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