Package 'BWStest' reference manual

Title:	Baumgartner Weiss Schindler Test of Equal Distributions
Description:	Performs the 'Baumgartner-Weiss-Schindler' two-sample test of equal probability distributions, <doi:10.2307/2533862>. Also performs similar rank-based tests for equal probability distributions due to Neuhauser <doi:10.1080/10485250108832874> and Murakami <doi:10.1080/00949655.2010.551516>.
Authors:	Steven E. Pav [aut, cre]
Maintainer:	Steven E. Pav <[email protected]>
License:	LGPL-3
Version:	0.2.3
Built:	2024-09-09 04:55:55 UTC
Source:	https://github.com/shabbychef/BWStest

Baumgartner Weiss Schindler test of equal distributions.

Description

Baumgartner Weiss Schindler test.

Background

The Baumgartner Weiss Schindler test is a two sample test of the null that the samples come from the same probability distribution, similar to the Kolmogorv-Smirnov, Wilcoxon, and Cramer-Von Mises tests. It is similar to the Cramer-Von Mises test in that it estimates the square norm of the difference in CDFs of the two samples. However, the Baumgartner Weiss Schindler test weights the integral by the variance of the difference in CDFs, "[emphasizing] the tails of the distributions, which increases the power of the test for a lot of applications."

Legal Mumbo Jumbo

BWStest is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

Author(s)

Steven E. Pav [email protected]

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. doi:10.2307/2533862

M. Neuhauser, 'Exact tests based on the Baumgartner-Weiss-Schindler Statistic–a survey', Statistical Papers 46, no. 1 (2005): pp. 1-30. doi:10.1007/BF02762032

M. Neuhauser, 'One-sided two-sample and trend tests based on a modified Baumgartner-Weiss-Schindler statistic', J. Nonparametric Statistics 13, no. 5 (2001): pp 729-739. doi:10.1080/10485250108832874

H. Murakami, 'K-sample rank test based on modified Baumgartner statistic and its power comparison', J. Jpn. Comp. Statist. 19, no. 1 (2006): pp. 1-13. doi:10.5183/jjscs1988.19.1

H. Murakami, 'Modified Baumgartner Statistics for the two-sample and multisample problems: a numerical comparison', J. Stat. Comp. and Sim. 82, no. 5 (2012): pp. 711-728. doi:10.1080/00949655.2010.551516

H. Murakami, 'Lepage type statistic based on the modified Baumgartner statistic', Comp. Stat. & Data Analysis 51 (2007): pp 5061-5067. doi:10.1016/j.csda.2006.04.026

CDF of the Baumgartner-Weiss-Schindler test under the null.

Description

Computes the CDF of the Baumgartner-Weiss-Schindler test statistic under the null hypothesis of equal distributions.

Usage

bws_cdf(b, maxj = 5L, lower_tail = TRUE)
bws_cdf(b, maxj = 5L, lower_tail = TRUE)

Arguments

`b`	a vector of BWS test statistics.
`maxj`	the maximum value of j to take in the approximate computation of the CDF via equation (2.5). Baumgartner et. al. claim that a value of 3 is sufficient.
`lower_tail`	boolean, when `TRUE` returns $\Psi$ , otherwise compute the upper tail, $1-\Psi$ , which is more useful for hypothesis tests.

Details

Given value $b$ , computes the CDF of the BWS statistic under the null, denoted as $\Psi(b)$ by Baumgartner et al. The CDF is computed from equation (2.5) via numerical quadrature.

The expression for the CDF contains the integral

$\int_0^1 \frac{1}{\sqrt{r^3 (1-r)}} \mathrm{exp}\left(\frac{rb}{8} - \frac{\pi^2 (4j+1)^2}{8rb}\right) \mathrm{dr}$

By making the change of variables $x = 2r - 1$ , this can be re-expressed as an integral of the form

$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}} f(x) \mathrm{dx},$

for some function $f(x)$ involving $b$ and $j$ . This integral can be approximated via Gaussian quadrature using Chebyshev nodes (of the first kind), which is the approach we take here.

Value

A vector of the CDF of $b$ , $\Psi(b)$ .

Author(s)

Steven E. Pav [email protected]

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. doi:10.2307/2533862

Examples


# do it 500 times
set.seed(123)
bvals <- replicate(500, bws_stat(rnorm(50),rnorm(50)))
pvals <- bws_cdf(bvals)
# these should be uniform!
 
  plot(ecdf(pvals)) 


# compare to Table 1 of Baumgartner et al.
bvals <- c(1.933,2.493,3.076,3.880,4.500,5.990)
tab1v <- c(0.9,0.95,0.975,0.990,0.995,0.999)
pvals <- bws_cdf(bvals,lower_tail=TRUE)
show(data.frame(B=bvals,BWS_psi=tab1v,our_psi=pvals))

# do it 500 times
set.seed(123)
bvals <- replicate(500, bws_stat(rnorm(50),rnorm(50)))
pvals <- bws_cdf(bvals)
# these should be uniform!
 
  plot(ecdf(pvals)) 


# compare to Table 1 of Baumgartner et al.
bvals <- c(1.933,2.493,3.076,3.880,4.500,5.990)
tab1v <- c(0.9,0.95,0.975,0.990,0.995,0.999)
pvals <- bws_cdf(bvals,lower_tail=TRUE)
show(data.frame(B=bvals,BWS_psi=tab1v,our_psi=pvals))

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

Description

Compute the Baumgartner-Weiss-Schindler test statistic.

Usage

bws_stat(x, y)
bws_stat(x, y)

Arguments

`x`	a vector.
`y`	a vector.

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.

Value

The BWS test statistic, $B$ .

Author(s)

Steven E. Pav [email protected]

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. doi:10.2307/2533862

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))

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))

Perform the Baumgartner-Weiss-Schindler hypothesis test.

Description

Perform the Baumgartner-Weiss-Schindler hypothesis test.

Usage

bws_test(
  x,
  y,
  method = c("default", "BWS", "Neuhauser", "B1", "B2", "B3", "B4", "B5"),
  alternative = c("two.sided", "greater", "less")
)
bws_test(
  x,
  y,
  method = c("default", "BWS", "Neuhauser", "B1", "B2", "B3", "B4", "B5"),
  alternative = c("two.sided", "greater", "less")
)

Arguments

`x`	a vector of the first sample.
`y`	a vector of the second sample.
`method`	a character string specifying the test statistic to use. should be one of the following: default This is “Hobson's choice”, which uses the classical BWS test for two-sided alternative, but Neuhauser for one sided alternatives. BWS Use the classical BWS test. Neuhauser Use Neuhauser's test. B1 Use Murakami's $B_1$ test. B2 Use Murakami's $B_2$ test, which is exactly Neuhauser's test. B3 Use Murakami's $B_3$ test. B4 Use Murakami's $B_4$ test. B5 Use Murakami's $B_5$ test. Only Neuhauser's test supports one-sided alternatives.
`alternative`	a character string specifying the alternative hypothesis, must be one of “two.sided” (default), “greater” or “less”. You can specify just the initial letter. “greater” corresponds to testing whether the survival function of `x` is greater than that of `y`; equivalently one can think of this as `x` being ‘greater’ than `y` in the sense of first order stochastic dominance.

Value

Object of class htest, a list of the test statistic, the p-value, and the method noted.

Note

The code will happily compute Murakami's $B_3$ through $B_5$ for large sample sizes, even though nominal coverage is not achieved. A warning will be thrown. User assumes all risk relying on results from this function.

Author(s)

Steven E. Pav [email protected]

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. doi:10.2307/2533862

Examples


# under the null
set.seed(123)
x <- rnorm(100)
y <- rnorm(100)
hval <- bws_test(x,y)

# under the alternative
set.seed(123)
x <- rnorm(100)
y <- rnorm(100,mean=1.0)
hval <- bws_test(x,y)
show(hval)
stopifnot(hval$p.value < 0.05)

# under the alternative with a one sided test.
set.seed(123)
x <- rnorm(100)
y <- rnorm(100,mean=0.7)
hval <- bws_test(x,y,alternative='less')
show(hval)
stopifnot(hval$p.value < 0.01)

hval <- bws_test(x,y,alternative='greater')
stopifnot(hval$p.value > 0.99)

hval <- bws_test(x,y,alternative='two.sided')
stopifnot(hval$p.value < 0.05)

# under the null
set.seed(123)
x <- rnorm(100)
y <- rnorm(100)
hval <- bws_test(x,y)

# under the alternative
set.seed(123)
x <- rnorm(100)
y <- rnorm(100,mean=1.0)
hval <- bws_test(x,y)
show(hval)
stopifnot(hval$p.value < 0.05)

# under the alternative with a one sided test.
set.seed(123)
x <- rnorm(100)
y <- rnorm(100,mean=0.7)
hval <- bws_test(x,y,alternative='less')
show(hval)
stopifnot(hval$p.value < 0.01)

hval <- bws_test(x,y,alternative='greater')
stopifnot(hval$p.value > 0.99)

hval <- bws_test(x,y,alternative='two.sided')
stopifnot(hval$p.value < 0.05)

News for package 'BWStest':

Description

News for package 'BWStest'

Version 0.2.3 (2023-10-10)

Update doi links.

Version 0.2.2 (2018-10-17)

Package maintenance–no new features.

Version 0.2.1 (2017-03-20)

Package maintenance–no new features.
move github figures to location CRAN understands.
package initialization mumbo jumbo, see Rcpp issue 636.

Version 0.2.0 (2016-04-29)

Adding Murakami statistics.

Version 0.1.0 (2016-04-07)

First CRAN release.

Initial Version 0.0.0 (2016-04-06)

Start work

Murakami test statistic distribution.

Description

Estimates the CDF of the Murakami test statistics via permutations.

Usage

murakami_cdf(B, n1, n2, flavor = 0L, lower_tail = TRUE)
murakami_cdf(B, n1, n2, flavor = 0L, lower_tail = TRUE)

Arguments

`B`	the Murakami test statistic or a vector of the same.
`n1`	number of elements in the first sample.
`n2`	number of elements in the second sample.
`flavor`	the 'flavor' of the test statistic. See `murakami_stat`.
`lower_tail`	boolean, when `TRUE` returns the CDF, $\Psi$ , otherwise compute the upper tail, $1-\Psi$ , which is potentially more useful for hypothesis tests.

Details

Given the Murakami test statistic $B_j$ for $0 \le j \le 5$ , computes the CDF under the null that the two samples come from the same distribution. The CDF is computed by permutation test and memoization.

Value

a vector of the same size as B of the CDF under the null.

Note

the CDF is approximately computed by evaluating the permutations up to some reasonably small sample size (currently the cutoff is 9). When larger sample sizes are used, the distribution of the test statistic may not converge. This is apparently seen in flavors 3 through 5.

Author(s)

Steven E. Pav [email protected]

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. doi:10.2307/2533862

M. Neuhauser, 'Exact tests based on the Baumgartner-Weiss-Schindler Statistic–a survey', Statistical Papers 46, no. 1 (2005): pp. 1-30. doi:10.1007/BF02762032

M. Neuhauser, 'One-sided two-sample and trend tests based on a modified Baumgartner-Weiss-Schindler statistic', J. Nonparametric Statistics 13, no. 5 (2001): pp 729-739. doi:10.1080/10485250108832874

H. Murakami, 'K-sample rank test based on modified Baumgartner statistic and its power comparison', J. Jpn. Comp. Statist. 19, no. 1 (2006): pp. 1-13. doi:10.5183/jjscs1988.19.1

H. Murakami, 'Lepage type statistic based on the modified Baumgartner statistic', Comp. Stat. & Data Analysis 51 (2007): pp 5061-5067. doi:10.1016/j.csda.2006.04.026

Examples


# basic usage:
xv <- seq(0,4,length.out=101)
yv <- murakami_cdf(xv, n1=8, n2=6, flavor=1L)
plot(xv,yv)
zv <- bws_cdf(xv)
lines(xv,zv,col='red')

# check under the null:

flavor <- 1L
n1 <- 8
n2 <- 8
set.seed(1234)
Bvals <- replicate(2000,murakami_stat(rnorm(n1),rnorm(n2),flavor))
# should be uniform:
plot(ecdf(murakami_cdf(Bvals,n1,n2,flavor)))


# basic usage:
xv <- seq(0,4,length.out=101)
yv <- murakami_cdf(xv, n1=8, n2=6, flavor=1L)
plot(xv,yv)
zv <- bws_cdf(xv)
lines(xv,zv,col='red')

# check under the null:

flavor <- 1L
n1 <- 8
n2 <- 8
set.seed(1234)
Bvals <- replicate(2000,murakami_stat(rnorm(n1),rnorm(n2),flavor))
# should be uniform:
plot(ecdf(murakami_cdf(Bvals,n1,n2,flavor)))

Compute Murakami's test statistic.

Description

Compute one of the modified Baumgartner-Weiss-Schindler test statistics proposed by Murakami, or Neuhauser.

Usage

murakami_stat(x, y, flavor = 0L)

murakami_stat_perms(nx, ny, flavor = 0L)
murakami_stat(x, y, flavor = 0L)

murakami_stat_perms(nx, ny, flavor = 0L)

Arguments

`x`	a vector of the first sample.
`y`	a vector of the second sample.
`flavor`	which ‘flavor’ of test statistic.
`nx`	the length of `x`, the first sample.
`ny`	the length of `y`, the second sample.

Details

Given vectors $X$ and $Y$ , computes $B_{jX}$ and $B_{jY}$ for some $j$ as described by Murakami and by Neuhauser, returning either their their average or their average distance. The test statistics approximate the weighted square norm of the difference in CDFs of the two distributions.

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.

The various ‘flavor’s of test statistic are:

0: The statistic of Baumgartner-Weiss-Schindler.
1: Murakami's $B_1$ statistic, from his 2006 paper.
2: Neuhauser's difference statistic, denoted by Murakami as $B_2$ in his 2012 paper.
3: Murakami's $B_3$ statistic, from his 2012 paper.
4: Murakami's $B_4$ statistic, from his 2012 paper.
5: Murakami's $B_5$ statistic, from his 2012 paper, with a log weighting.

Value

The BWS test statistic, $B_j$ . For murakami_stat_perms, a vector of the test statistics for all permutations of the input.

Note

NA and NaN are not yet dealt with!

Author(s)

Steven E. Pav [email protected]

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. doi:10.2307/2533862

M. Neuhauser, 'Exact tests based on the Baumgartner-Weiss-Schindler Statistic–a survey', Statistical Papers 46, no. 1 (2005): pp. 1-30. doi:10.1007/BF02762032

M. Neuhauser, 'One-sided two-sample and trend tests based on a modified Baumgartner-Weiss-Schindler statistic', J. Nonparametric Statistics 13, no. 5 (2001): pp 729-739. doi:10.1080/10485250108832874

H. Murakami, 'K-sample rank test based on modified Baumgartner statistic and its power comparison', J. Jpn. Comp. Statist. 19, no. 1 (2006): pp. 1-13. doi:10.5183/jjscs1988.19.1

H. Murakami, 'Lepage type statistic based on the modified Baumgartner statistic', Comp. Stat. & Data Analysis 51 (2007): pp 5061-5067. doi:10.1016/j.csda.2006.04.026

Examples


set.seed(1234)
x <- runif(1000)
y <- runif(100)
bval <- murakami_stat(x,y,1)


nx <- 6
ny <- 5
# monte carlo
set.seed(1234)
repli <- replicate(3000,murakami_stat(rnorm(nx),rnorm(ny),0L))
# under the null, perform the permutation test:
allem <- murakami_stat_perms(nx,ny,0L)
plot(ecdf(allem)) 
lines(ecdf(repli),col='red') 


set.seed(1234)
x <- runif(1000)
y <- runif(100)
bval <- murakami_stat(x,y,1)


nx <- 6
ny <- 5
# monte carlo
set.seed(1234)
repli <- replicate(3000,murakami_stat(rnorm(nx),rnorm(ny),0L))
# under the null, perform the permutation test:
allem <- murakami_stat_perms(nx,ny,0L)
plot(ecdf(allem)) 
lines(ecdf(repli),col='red')

Package 'BWStest'

Help Index

Baumgartner Weiss Schindler test of equal distributions.

Description

Background

Legal Mumbo Jumbo

Author(s)

References

CDF of the Baumgartner-Weiss-Schindler test under the null.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

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

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Perform the Baumgartner-Weiss-Schindler hypothesis test.

Description

Usage

Arguments

Value

Note

Author(s)

References

See Also

Examples

News for package 'BWStest':

Description

Version 0.2.3 (2023-10-10)

Version 0.2.2 (2018-10-17)

Version 0.2.1 (2017-03-20)

Version 0.2.0 (2016-04-29)

Version 0.1.0 (2016-04-07)

Initial Version 0.0.0 (2016-04-06)

Murakami test statistic distribution.

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Compute Murakami's test statistic.

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples