adaptTest-package: Adaptive two-stage tests
In adaptTest: Adaptive two-stage tests

Description Details Note Author(s) References See Also Examples

The functions defined in this program serve for implementing adaptive two-stage tests.

Package:	adaptTest
Type:	Package
Version:	1.0
Date:	2009-10-14
License:	GPL (version 2 or later)
LazyLoad:	yes

An adaptive two-stage test can be considered as a family of decreasing functions f[c](p1) in the unit square. Each of these functions is a conditional error function, specifying the type I error conditional on the p-value p1 of the first stage. For example, f[c](p1) = min(1, c/p1) corresponds to Fisher's combination test (Bauer and Koehne, 1994). Based on this function family, the test can be put into practice by specifying the desired overall level alpha, stopping bounds alpha1 <= alpha0 and a parameter alpha2. After computing p1, the test stops with or without rejection of the null hypothesis if p1 <= alpha1 or p1 > alpha0, respectively. Otherwise, the null hypothesis is rejected if and only if p2 <= f[c](p1) holds for the p-value p2 of the second stage, where c is such that the local level of this latter test is alpha2 (e.g., c = c(alpha2) = exp(-chi2_{4,alpha2}/2) for Fisher's combination test).

This package provides functions for handling conditional error functions, performing calculations among the different parameters (alpha, alpha0, alpha1, alpha2 and c) and computing overall p-values, in addition to graphical visualization routines. Currently, four predefined tests are included: Bauer and Koehne (1994), Lehmacher and Wassmer (1999), Vandemeulebroecke (2006), and the horizontal conditional error function. User-defined tests can also be implemented.

This package contains the following functions:

Key functions are CEF, plotCEF, tsT, ovP.
Further functions are a1Table, getpar, parconv, pathCEF, plotBounds, eq, ne, ge, gt, le, lt.

The functions a1Table, getpar, parconv and tsT can handle the four predefined tests mentioned above. The functions CEF, plotCEF, pathCEF and ovP can also handle these, and user-defined tests in addition. The functions plotBounds, eq, ne, ge, gt, le and lt do not directly handle tests.

Note that a family of conditional error functions can be parameterized in two alternative ways: more "traditionally" by some parameter c that, in turn, depends on the local level alpha2 of the test after the second stage, or - perhaps more conveniently - by alpha2 itself.

In this implementation, early stopping bounds are not part of the conditional error function. Rather, they are specified separately and "imposed" on it.

I want to thank Niklas Hack for technical support.

Marc Vandemeulebroecke

Maintainer: Marc Vandemeulebroecke <marc.vandemeulebroecke(at)novartis.com>

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

Brannath, W., Posch, M., Bauer, P. (2002). Recursive combination tests. J. Amer. Statist. Assoc. 97, 236-244.

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.

Vandemeulebroecke, M. (2006). A general approach to two-stage tests. Doctoral thesis, Otto-von-Guericke-Universitaet Magdeburg, http://www.dissertation.de.

Vandemeulebroecke, M. (2008). Group sequential and adaptive designs - a review of basic concepts and points of discussion. Biometrical Journal 50, 541-557.

CEF, tsT

## Example from Bauer and Koehne (1994)
alpha  <- 0.1
alpha2 <- 0.1
alpha0 <- 0.5
alpha1 <- tsT(typ="b", a=alpha, a0=alpha0, a2=alpha2)
plotCEF(typ="b", a2=alpha2, add=FALSE)
plotBounds(alpha1, alpha0)
CEF(typ="b", a2=alpha2)