# parconv: Function to convert between two different parameterizations... In adaptTest: Adaptive two-stage tests

## Description

This function converts between two different parameterizations of a family of conditional error functions: a (more ‘traditional’) parameter c, and a (more convenient) parameter alpha2 specifying the local level of the test after the second stage.

## Usage

 `1` ```parconv(typ, a2 = NA, c = NA) ```

## 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 `a2` alpha2, the local level of the test after the second stage (see details) `c` the parameter c (see details)

## Details

Traditionally, a family of conditional error functions is often parameterized by some parameter c that, in turn, depends on the local level alpha2 of the test after the second stage. However, it can be convenient to parameterize the family directly by alpha2. The function `parconv` converts one parameter into the other: provide one, and it returns the other.

Essentially, the relation between the two parameterizations is implemented as:

• c = exp(-chi2_{4,alpha2}/2) for Fisher's combination test (Bauer and Koehne, 1994)

• c = Phi^{-1}(1-alpha2) for the inverse normal method (Lehmacher and Wassmer, 1999)

• alpha2 = {(Gamma(1+1/r))^2}/{Gamma(1+2/r)} for Vandemeulebroecke (2006)

• c = alpha2 for the family of horizontal conditional error functions

## Value

`parconv` returns alpha2 corresponding to the supplied c, or c corresponding to the supplied alpha2.

## Note

Provide either `a2` or `c`, not both!

alpha2 is the local level of the test after the second stage, and it equals the integral under the corresponding conditional error function:

alpha2 = int_0^1 cef_{alpha2}(p1) d p1,

where cef_{alpha2} is the conditional error function (of a specified family) with parameter alpha2.

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`).

alpha2 can take any value in [0,1]; c can take values in

• [0,1] for Fisher's combination test (Bauer and Koehne, 1994)

• (-infty, infty) for the inverse normal method (Lehmacher and Wassmer, 1999)

• [0,infty) for Vandemeulebroecke (2006)

• [0,1] for the family of horizontal conditional error functions

## 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, `getpar`, `CEF`
 ```1 2``` ```## Obtain the parameter c for Fisher's combination test, using the local level 0.05 for the test after the second stage parconv(typ="b", a2=0.05) ```