The functions `sfLogistic()`

, `sfNormal()`

,
`sfExtremeValue()`

, `sfExtremeValue2()`

, `sfCauchy()`

, and
`sfBetaDist()`

are all 2-parameter spending function families. These
provide increased flexibility in some situations where the flexibility of a
one-parameter spending function family is not sufficient. These functions
all allow fitting of two points on a cumulative spending function curve; in
this case, four parameters are specified indicating an x and a y coordinate
for each of 2 points. Normally each of these functions 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 in the examples; note, however, that an
automatic \(\alpha\)- and \(\beta\)-spending function plot
is also available.

`sfBetaDist(alpha,t,param)`

is simply `alpha`

times the incomplete
beta cumulative distribution function with parameters \(a\) and \(b\)
passed in `param`

evaluated at values passed in `t`

.

The other spending functions take the form $$f(t;\alpha,a,b)=\alpha
F(a+bF^{-1}(t))$$ where \(F()\) is a
cumulative distribution function with values \(> 0\) on the real line
(logistic for `sfLogistic()`

, normal for `sfNormal()`

, extreme
value for `sfExtremeValue()`

and Cauchy for `sfCauchy()`

) and
\(F^{-1}()\) is its inverse.

For the logistic spending function this simplifies to $$f(t;\alpha,a,b)=\alpha (1-(1+e^a(t/(1-t))^b)^{-1}).$$

For the extreme value distribution with $$F(x)=\exp(-\exp(-x))$$ this
simplifies to $$f(t;\alpha,a,b)=\alpha \exp(-e^a (-\ln t)^b).$$ Since the
extreme value distribution is not symmetric, there is also a version where
the standard distribution is flipped about 0. This is reflected in
`sfExtremeValue2()`

where $$F(x)=1-\exp(-\exp(x)).$$

## Usage

```
sfLogistic(alpha, t, param)
sfBetaDist(alpha, t, param)
sfCauchy(alpha, t, param)
sfExtremeValue(alpha, t, param)
sfExtremeValue2(alpha, t, param)
sfNormal(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
In the two-parameter specification,

`sfBetaDist()`

requires 2 positive values, while`sfLogistic()`

,`sfNormal()`

,`sfExtremeValue()`

,`sfExtremeValue2()`

and`sfCauchy()`

require the first parameter to be any real value and the second to be a positive value. The four parameter specification is`c(t1,t2,u1,u2)`

where the objective is that`sf(t1)=alpha*u1`

and`sf(t2)=alpha*u2`

. In this parameterization, all four values must be between 0 and 1 and`t1 < t2`

,`u1 < u2`

.

## 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

## Examples

```
library(ggplot2)
# 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
#> 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
# plot the alpha- and beta-spending functions
plot(x, plottype = 5)
# start by showing how to fit two points with sfLogistic
# plot the spending function using many points to obtain a smooth curve
# note that curve fits the points x=.1, y=.01 and x=.4, y=.1
# specified in the 3rd parameter of sfLogistic
t <- 0:100 / 100
plot(t, sfLogistic(1, t, c(.1, .4, .01, .1))$spend,
xlab = "Proportion of final sample size",
ylab = "Cumulative Type I error spending",
main = "Logistic Spending Function Examples",
type = "l", cex.main = .9
)
lines(t, sfLogistic(1, t, c(.01, .1, .1, .4))$spend, lty = 2)
# now just give a=0 and b=1 as 3rd parameters for sfLogistic
lines(t, sfLogistic(1, t, c(0, 1))$spend, lty = 3)
# try a couple with unconventional shapes again using
# the xy form in the 3rd parameter
lines(t, sfLogistic(1, t, c(.4, .6, .1, .7))$spend, lty = 4)
lines(t, sfLogistic(1, t, c(.1, .7, .4, .6))$spend, lty = 5)
legend(
x = c(.0, .475), y = c(.76, 1.03), lty = 1:5,
legend = c(
"Fit (.1, 01) and (.4, .1)", "Fit (.01, .1) and (.1, .4)",
"a=0, b=1", "Fit (.4, .1) and (.6, .7)",
"Fit (.1, .4) and (.7, .6)"
)
)
# set up a function to plot comparsons of all
# 2-parameter spending functions
plotsf <- function(alpha, t, param) {
plot(t, sfCauchy(alpha, t, param)$spend,
xlab = "Proportion of enrollment",
ylab = "Cumulative spending", type = "l", lty = 2
)
lines(t, sfExtremeValue(alpha, t, param)$spend, lty = 5)
lines(t, sfLogistic(alpha, t, param)$spend, lty = 1)
lines(t, sfNormal(alpha, t, param)$spend, lty = 3)
lines(t, sfExtremeValue2(alpha, t, param)$spend, lty = 6, col = 2)
lines(t, sfBetaDist(alpha, t, param)$spend, lty = 7, col = 3)
legend(
x = c(.05, .475), y = .025 * c(.55, .9),
lty = c(1, 2, 3, 5, 6, 7),
col = c(1, 1, 1, 1, 2, 3),
legend = c(
"Logistic", "Cauchy", "Normal", "Extreme value",
"Extreme value 2", "Beta distribution"
)
)
}
# do comparison for a design with conservative early spending
# note that Cauchy spending function is quite different
# from the others
param <- c(.25, .5, .05, .1)
plotsf(.025, t, param)
```