sfpower: 4.2: Kim-DeMets (power) Spending Function

In gsDesign: Group Sequential Design

Description

The function sfPower() implements a Kim-DeMets (power) spending function. This is a flexible, one-parameter spending function recommended by Jennison and Turnbull (2000). Normally it 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 below in examples.

Usage

sfPower(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 rho parameter for which Kim-DeMets spending is to be computed; allowable range is (0,15]

Details

A Kim-DeMets spending function takes the form

f(t; alpha, rho) = alpha t^rho

where rho is the value passed in param. See examples below for a range of values of rho that may be of interest (param=0.75 to 3 are documented there).

Value

An object of type spendfn. See Spending function overview for further details.

Note

The manual is not linked to this help file, but is available in library/gsdesign/doc/gsDesignManual.pdf in the directory where R is installed.

Author(s)

Keaven Anderson keaven\_anderson@merck.

References

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

See Also

Spending function overview, gsDesign, gsDesign package overview

Examples

# 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 spending function using many points to obtain a smooth curve
# show rho=3 for approximation to O'Brien-Fleming and rho=.75 for 
# approximation to Pocock design.
# Also show rho=2 for an intermediate spending.
# Compare these to Hwang-Shih-DeCani spending with gamma=-4,  -2,  1
t <- 0:100/100
plot(t,  sfPower(0.025, t, 3)$spend, xlab="Proportion of sample size", 
    ylab="Cumulative Type I error spending", 
    main="Kim-DeMets (rho) versus Hwang-Shih-DeCani (gamma) Spending", 
    type="l", cex.main=.9)
lines(t, sfPower(0.025, t, 2)$spend, lty=2)
lines(t, sfPower(0.025, t, 0.75)$spend, lty=3)
lines(t, sfHSD(0.025, t, 1)$spend, lty=3, col=2)
lines(t, sfHSD(0.025, t, -2)$spend, lty=2, col=2)
lines(t, sfHSD(0.025, t, -4)$spend, lty=1, col=2)
legend(x=c(.0, .375), y=.025*c(.65, 1), lty=1:3, 
       legend=c("rho= 3", "rho= 2", "rho= 0.75"))
legend(x=c(.0, .357), y=.025*c(.65, .85), lty=1:3, bty="n", col=2, 
       legend=c("gamma= -4", "gamma= -2", "gamma=1"))

Example output

Loading required package: xtable
Loading required package: ggplot2
Asymmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Upper bound spending computations assume
trial continues if lower bound is crossed.

           Sample
            Size    ----Lower bounds----  ----Upper bounds-----
  Analysis Ratio*   Z   Nominal p Spend+  Z   Nominal p Spend++
         1  0.282 -0.52    0.3015 0.0125 3.36    0.0004  0.0004
         2  0.564  0.53    0.7028 0.0229 2.76    0.0029  0.0027
         3  0.846  1.32    0.9072 0.0296 2.36    0.0092  0.0074
         4  1.128  2.03    0.9788 0.0350 2.03    0.0212  0.0145
     Total                        0.1000                 0.0250 
+ lower bound beta spending (under H1):
 Kim-DeMets (power) spending function with rho = 1.5.
++ alpha spending:
 Kim-DeMets (power) spending function with rho = 3.
* Sample size ratio compared to fixed design with no interim

Boundary crossing probabilities and expected sample size
assume any cross stops the trial

Upper boundary (power or Type I Error)
          Analysis
   Theta      1      2      3      4  Total  E{N}
  0.0000 0.0004 0.0027 0.0073 0.0116 0.0221 0.579
  3.2415 0.0507 0.3248 0.3619 0.1626 0.9000 0.768

Lower boundary (futility or Type II Error)
          Analysis
   Theta      1      2      3      4  Total
  0.0000 0.3015 0.4138 0.2008 0.0619 0.9779
  3.2415 0.0125 0.0229 0.0296 0.0350 0.1000

gsDesign documentation built on May 2, 2019, 4:49 p.m.