adaptTest (version 1.1)

ovP: Function to compute and visualize overall p-values

Description

This function computes and plots overall p-values for adaptive two-stage tests.

Usage

ovP(typ = NA, fun = NA, dis = NA, p1 = 1:49/50, p2 = p1,
    a1 = 0, a0 = 1, grid = FALSE, plt = FALSE,
    invisible = FALSE, wire = FALSE, round = FALSE)

Value

A p-value, a vector of p-values or a matrix of p-values.

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

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

p1

the p-value \(p_1\) of the test after the first stage, or a vector of such p-values

p2

the p-value \(p_2\) of the test after the second stage, or a vector of such p-values; defaults to p1

a1

\(\alpha_1\), the efficacy stopping bound and local level of the test after the first stage (default: no stopping for efficacy)

a0

\(\alpha_0\), the futility stopping bound (default: no stopping for futility)

grid

logical determining whether a grid should be spanned by p1 and p2 (default: no grid is spanned)

plt

logical determining whether the overall p-values should be plotted or not (default: not)

invisible

logical determining whether the printing of the overall p-values should be suppressed or not (default: not)

wire

logical determining whether the overall p-values should be plotted in wireframe-style or in cloud-style (default: cloud-style)

round

rounding specification, logical or integer (see details; default: no rounding)

Author

Marc Vandemeulebroecke

Details

The overall p-value for an adaptive two-stage test is computed as \(p_1\) if \(p_1 <= \alpha_1\) or \(p_1 > \alpha_0\), and as $$\alpha_1 + \int_{\alpha_1}^{\alpha_0} cef_{(p_1,p_2)}(x) d x$$ otherwise, where \(cef_{(p_1,p_2)}\) is the conditional error function (of a specified family) running through the observed pair of p-values (p1,p2).

There are two alternative ways of specifying the family of conditional error functions (i.e., the test): through a type typ, or through an initial conditional error function fun and a distortion method dis; see CEF for details.

If p1 and p2 are of length 1, a single overall p-value is computed (and not plotted). Otherwise, the behavior of ovP depends on grid:

• If grid = FALSE, overall p-values are computed (and not plotted) for the elementwise pairs of p1 and p2. Here, p1 and p2 must be of the same length.

• If grid = TRUE, a grid is spanned by p1 and p2, and p-values are computed (and possibly plotted) over this grid. Here, p1 and p2 may be of different length. Plotting is triggered by plt = TRUE, and the style of the plot (wireframe or cloud) is determined by wire. invisible = TRUE suppresses the printing of the p-values.

The p-values are rounded to round digits after the comma (round = TRUE rounds to 1 digit; round = FALSE and round = 0 prevent rounding). The plot always shows unrounded values.

References

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.

See Also

adaptTest package description, CEF

Examples

## Visualize a Lehmacher Wassmer (1999) test to the overall level 0.1
## and compute and visualize the overall p-value for an observed (p1,p2)=(0.3,0.7)
alpha  <- .1
alpha0 <- .5
alpha1 <- .05
plotBounds(a1=alpha1, a0=alpha0, add=FALSE)
plotCEF(typ="l", a2=tsT(typ="l", a=alpha, a0=alpha0, a1=alpha1))
plotCEF(typ="l", p1=.3, p2=.7)
ovP(typ="l", p1=.3, p2=.7, a1=alpha1, a0=alpha0)
# The overall p-value is the area left of alpha1, plus the area below the 
# conditional error function running though (0.3,0.7) between alpha1 and alpha0.

## Investigate the p-values of the Lehmacher Wassmer (1999) test from above
ovP(typ="l", a1=alpha1, a0=alpha0, grid=TRUE, p1=1:9/10, round=3)
ovP(typ="l", a1=alpha1, a0=alpha0, grid=TRUE, plt=TRUE, invisible=TRUE, wire=TRUE)