BWStest (version 0.2.3)

bws_stat: Compute the test statistic of the Baumgartner-Weiss-Schindler test.

Description

Compute the Baumgartner-Weiss-Schindler test statistic.

Usage

bws_stat(x, y)

Value

The BWS test statistic, \(B\).

Arguments

x

a vector.

y

a vector.

Author

Steven E. Pav

Details

Given vectors \(X\) and \(Y\), computes \(B_X\) and \(B_Y\) as described by Baumgartner et al., returning their average, \(B\). The test statistic approximates the variance-weighted square norm of the difference in CDFs of the two distributions. For sufficiently large sample sizes (more than 20, say), under the null the test statistic approaches the asymptotic value computed in bws_cdf.

The test value is an approximation of $$\tilde{B} = \frac{mn}{m+n} \int_0^1 \frac{1}{z(1-z)} \left(F_X(z) - F_Y(z)\right)^2 \mathrm{dz},$$ where \(m\) (\(n\)) is the number of elements in \(X\) (\(Y\)), and \(F_X(z)\) (\(F_Y(z)\)) is the CDF of \(X\) (\(Y\)).

The test statistic is based only on the ranks of the input. If the same monotonic transform is applied to both vectors, the result should be unchanged. Moreover, the test is inherently two-sided, so swapping \(X\) and \(Y\) should also leave the test statistic unchanged.

References

W. Baumgartner, P. Weiss, H. Schindler, 'A nonparametric test for the general two-sample problem', Biometrics 54, no. 3 (Sep., 1998): pp. 1129-1135. tools:::Rd_expr_doi("10.2307/2533862")

See Also

bws_cdf, bws_test

Examples

set.seed(1234)
x <- runif(1000)
y <- runif(100)
bval <- bws_stat(x,y)
# check a monotonic transform:
ftrans <- function(x) { log(1 + x) }
bval2 <- bws_stat(ftrans(x),ftrans(y))
stopifnot(all.equal(bval,bval2))
# check commutivity
bval3 <- bws_stat(y,x)
stopifnot(all.equal(bval,bval3))