PMCMRplus (version 1.9.3)

bwsKSampleTest: Murakami's k-Sample BWS Test

Description

Performs Murakami's k-Sample BWS Test.

Usage

bwsKSampleTest(x, ...)
# S3 method for default
bwsKSampleTest(x, g, nperm = 1000, ...)
# S3 method for 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").

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.

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-Wei<U+00DF>-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.

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

# NOT RUN {
## 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)
# }
# 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)
# }