Piecewise Linear and Step Function Spending Functions

The function sfLinear() allows specification of a piecewise linear spending function. The function sfStep() specifies a step function spending function. Both functions provide complete flexibility in setting spending at desired timepoints in a group sequential design. Normally these function will be passed to gsDesignCRT() in the parameter sfu for the upper bound or sfl for the lower bound to specify a spending function family for a design. When passed to gsDesignCRT(), the value of param would be passed to sfLinear() or sfStep() through the gsDesignCRT() arguments sfupar for the upper bound and sflpar for the lower bound.

Note that sfStep() allows setting a particular level of spending when the timing is not strictly known; an example shows how this can inflate Type I error when timing of analyses are changed based on knowing the treatment effect at an interim.

Usage

sfLinear(alpha, t, param)

sfStep(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: A vector with a positive, even length. Values must range from 0 to 1, inclusive. Letting m <- length(param/2), the first m points in param specify increasing values strictly between 0 and 1 corresponding to interim timing (proportion of final total statistical information). The last m points in param specify non-decreasing values from 0 to 1, inclusive, with the cumulative proportion of spending at the specified timepoints.

Value

An object of type spendfn. The cumulative spending returned in sfLinear$spend is 0 for t <= 0 and alpha for t>=1. For t between specified points, linear interpolation is used to determine sfLinear$spend.

The cumulative spending returned in sfStep$spend is 0 for t<param[1] and alpha for t>=1. Letting m <- length(param/2), for i=1,2,...m-1 and param[i]<= t < param[i+1], the cumulative spending is set at alpha * param[i+m] (also for param[m]<=t<1).

Note that if param[2m] is 1, then the first time an analysis is performed after the last proportion of final planned information (param[m]) will be the final analysis, using any remaining error that was not previously spent.

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