Description Usage Arguments Details Value Note Author(s) References See Also Examples

The function `sfExponential`

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

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.

1 | ```
sfExponential(alpha, t, param)
``` |

`alpha` |
Real value |

`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]. |

An exponential spending function is defined for any positive `nu`

and *0≤ t≤ 1* as

*f(t;alpha,nu)=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(F^(-1)(1-alpha)/ t^nu).*

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(Phi^(-1)(1-alpha/2)/t^(1/2)).*

An object of type `spendfn`

.

The manual shows how to use `sfExponential()`

to closely approximate an O'Brien-Fleming design.
An example is given below.
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.

Keaven Anderson keaven\[email protected]

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.

Spending function overview, `gsDesign`

, gsDesign package overview

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
# use 'best' exponential approximation for k=6 to O'Brien-Fleming design
# (see manual for details)
gsDesign(k=6, sfu=sfExponential, sfupar=0.7849295,
test.type=2)$upper$bound
# show actual O'Brien-Fleming bound
gsDesign(k=6, sfu="OF", test.type=2)$upper$bound
# show Lan-DeMets approximation
# (not as close as sfExponential approximation)
gsDesign(k=6, sfu=sfLDOF, test.type=2)$upper$bound
# plot exponential spending function across a range of values of interest
t <- 0:100/100
plot(t, sfExponential(0.025, t, 0.8)$spend,
xlab="Proportion of final sample size",
ylab="Cumulative Type I error spending",
main="Exponential Spending Function Example", type="l")
lines(t, sfExponential(0.025, t, 0.5)$spend, lty=2)
lines(t, sfExponential(0.025, t, 0.3)$spend, lty=3)
lines(t, sfExponential(0.025, t, 0.2)$spend, lty=4)
lines(t, sfExponential(0.025, t, 0.15)$spend, lty=5)
legend(x=c(.0, .3), y=.025*c(.7, 1), lty=1:5,
legend=c("nu = 0.8", "nu = 0.5", "nu = 0.3", "nu = 0.2",
"nu = 0.15"))
text(x=.59, y=.95*.025, labels="<--approximates O'Brien-Fleming")
``` |

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