bwsKSampleTest {PMCMRplus}

Murakami's k-Sample BWS Test

Description

Performs Murakami's k-Sample BWS Test.

Usage

bwsKSampleTest(x, ...)

## Default S3 method:
bwsKSampleTest(x, g, nperm = 1000, ...)

## S3 method for class 'formula'
bwsKSampleTest(formula, data, subset, na.action, nperm = 1000, ...)

Arguments

x

a numeric vector of data values, or a list of numeric data vectors.

...

further arguments to be passed to or from methods.

g

a vector or factor object giving the group for the corresponding elements of "x". Ignored with a warning if "x" is a list.

nperm

number of permutations for the assymptotic permutation test. Defaults to 1000.

formula

a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.

data

an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).

subset

an optional vector specifying a subset of observations to be used.

na.action

a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

Let X_{ij} ~ (1 \le i \le k,~ 1 \le 1 \le n_i) denote an identically and independently distributed variable that is obtained from an unknown continuous distribution F_i(x). Let R_{ij} be the rank of X_{ij}, where X_{ij} is jointly ranked from 1 to N, ~ N = \sum_{i=1}^k n_i. In the k-sample test the null hypothesis, H: F_i = F_j is tested against the alternative, A: F_i \ne F_j ~~(i \ne j) with at least one inequality beeing strict. Murakami (2006) has generalized the two-sample Baumgartner-Weiß-Schindler test (Baumgartner et al. 1998) and proposed a modified statistic B_k^* defined by

B_{k}^* = \frac{1}{k}\sum_{i=1}^k \left\{\frac{1}{n_i} \sum_{j=1}^{n_i} \frac{(R_{ij} - \mathsf{E}[R_{ij}])^2} {\mathsf{Var}[R_{ij}]}\right\},

where

\mathsf{E}[R_{ij}] = \frac{N + 1}{n_i + 1} j

and

\mathsf{Var}[R_{ij}] = \frac{j}{n_i + 1} \left(1 - \frac{j}{n_i + 1}\right) \frac{\left(N-n_i\right)\left(N+1\right)}{n_i + 2}.

The p-values are estimated via an assymptotic boot-strap method. It should be noted that the B_k^* detects both differences in the unknown location parameters and / or differences in the unknown scale parameters of the k-samples.

Value

A list with class "htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

One may increase the number of permutations to e.g. nperm = 10000 in order to get more precise p-values. However, this will be on the expense of computational time.

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for the general two-sample problem, Biometrics 54, 1129–1135.

Murakami, H. (2006) K-sample rank test based on modified Baumgartner statistic and its power comparison, J. Jpn. Comp. Statist. 19, 1–13.

See Also

sample, bwsAllPairsTest, bwsManyOneTest.

Examples

## Hollander & Wolfe (1973), 116.
## Mucociliary efficiency from the rate of removal of dust in normal
## subjects, subjects with obstructive airway disease, and subjects
## with asbestosis.
x <- c(2.9, 3.0, 2.5, 2.6, 3.2) # normal subjects
y <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway disease
z <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosis
g <- factor(x = c(rep(1, length(x)),
                   rep(2, length(y)),
                   rep(3, length(z))),
             labels = c("ns", "oad", "a"))
dat <- data.frame(
   g = g,
   x = c(x, y, z))

## AD-Test
adKSampleTest(x ~ g, data = dat)

## BWS-Test
bwsKSampleTest(x ~ g, data = dat)

## Kruskal-Test
## Using incomplete beta approximation
kruskalTest(x ~ g, dat, dist="KruskalWallis")
## Using chisquare distribution
kruskalTest(x ~ g, dat, dist="Chisquare")

## Not run: 
## Check with kruskal.test from R stats
kruskal.test(x ~ g, dat)

## End(Not run)
## Using Conover's F
kruskalTest(x ~ g, dat, dist="FDist")

## Not run: 
## Check with aov on ranks
anova(aov(rank(x) ~ g, dat))
## Check with oneway.test
oneway.test(rank(x) ~ g, dat, var.equal = TRUE)

## End(Not run)