# CEF: Function to specify a conditional error function In adaptTest: Adaptive two-stage tests

## Description

This function returns a conditional error function.

## Usage

 `1` ```CEF(typ = NA, fun = NA, dis = NA, a2 = NA, c = NA, p1 = NA, p2 = p1) ```

## Arguments

 `typ` type of test: `"b"` for Bauer and Koehne (1994), `"l"` for Lehmacher and Wassmer (1999), `"v"` for Vandemeulebroecke (2006) and `"h"` for the horizontal conditional error function `c` the parameter c `a2` alpha2, the local level of the test after the second stage `p1` the p-value p1 of the test after the first stage `p2` the p-value p2 of the test after the second stage, defaults to `p1` `fun` a conditional error function `dis` a distortion method for a supplied conditional error function (see details): `"pl"` for power lines, `"vt"` for vertical translation

## Details

There are two alternative ways of specifying the desired conditional error function:

• through a type `typ`, and either a parameter (either `a2` or `c`) or a point `(p1,p2)`, OR

• through an initial conditional error function `fun`, and possibly a distortion method `dis` together with either the parameter `a2` or a point `(p1,p2)`

Most people will only need the first of these two ways; the second leads to user-defined non-standard tests.

If `typ` is specified, a parameter `a2` or `c` or the point `(p1,p2)` must be provided. In this case, `CEF` returns the conditional error function of the chosen type with the given parameter or running through the given point.

If `typ` is not specified, a conditional error function (i.e., a nonincreasing function defined on [0,1] with values in [0,1]) `fun` must be provided. If no distortion method is selected (`dis = NA`), `fun` is returned unchanged. Otherwise, the function is distorted using the chosen distortion method, either to match a desired second stage level `a2` or to run through a specified point `(p1,p2)` (one of which must be provided). Currently, two distortion methods are implemented:

• `dis = "pl"`, Power lines: For an initial function f, define f[r](x) = (f(x^r))^(1/r), r>0. Note that if f is a conditional error function of type `"b"` (Bauer and Koehne, 1994), so is f[r].

• `dis = "pl"`, Vertical translation: The initial function is shifted vertically.

See `parconv` for more information on the two alternative parameterizations by alpha2 and c.

## Value

These functions return a conditional error function (see details).

## Note

Provide either `typ` or `fun`, not both! If `typ` is provided, then also specify `a2`, `c`, or `p1` (and possibly `p2`). If `fun` is provided, then also specify `dis` and `a2`, or `dis` and `p1` (and possibly `p2`), or none of these.

Warning: Values of `a2` close to 0 or 1 may not work for `dis = "pl"`.

Note that in this implementation of adaptive two-stage tests, early stopping bounds are not part of the conditional error function. Rather, they are specified separately (see also `tsT`).

## Author(s)

Marc Vandemeulebroecke

## References

Bauer, P., Koehne, K. (1994). Evaluation of experiments with adaptive interim analyses. Biometrics 50, 1029-1041.

Lehmacher, W., Wassmer, G. (1999). Adaptive sample size calculations in group sequential trials. Biometrics 55, 1286-1290.

Vandemeulebroecke, M. (2006). An investigation of two-stage tests. Statistica Sinica 16, 933-951.

`adaptTest` package description, `parconv`, `plotCEF`, `tsT`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```## Plot two conditional error functions of the Lehmacher-Wassmer (1999) type: ## one to the local level alpha2=0.1, and one that runs through (p1,p2)=(0.3,0.7) foo1 <- CEF(typ="l", a2=0.1) foo2 <- CEF(typ="l", p1=0.3, p2=0.7) plot(foo1, xlim=0:1) plot(foo2, add=TRUE) ## A different way of doing the same plotCEF(typ="l", a2=0.1, add=FALSE) plotCEF(typ="l", p1=0.3, p2=0.7, plt.pt=FALSE) ```