parconv {adaptTest}

Function to convert between two different parameterizations of a family of conditional error functions

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 \alpha_2 specifying the local level of the test after the second stage.

Usage

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

\alpha_2, 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 \alpha_2 of the test after the second stage. However, it can be convenient to parameterize the family directly by \alpha_2. 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(-\chi^2_{4,\alpha_2}/2) for Fisher's combination test (Bauer and Koehne, 1994)

• c = \Phi^{-1}(1-\alpha_2) for the inverse normal method (Lehmacher and Wassmer, 1999)

• \alpha_2 = {(\Gamma(1+1/r))^2}/{\Gamma(1+2/r)} for Vandemeulebroecke (2006)

• c = \alpha_2 for the family of horizontal conditional error functions

Value

parconv returns \alpha_2 corresponding to the supplied c, or c corresponding to the supplied \alpha_2.

Note

Provide either a2 or c, not both!

\alpha_2 is the local level of the test after the second stage, and it equals the integral under the corresponding conditional error function:

\alpha_2 = \int_0^1 cef_{\alpha_2}(p_1) d p_1,

where cef_{\alpha_2} is the conditional error function (of a specified family) with parameter \alpha_2.

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).

\alpha_2 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.

See Also

adaptTest package description, getpar, CEF

Examples

## 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)