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.

1 2 3 4 5 6 7 8 | ```
## 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)
``` |

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