kruskalTest: Kruskal-Wallis Rank Sum Test

Description

Performs a Kruskal-Wallis rank sum test.

Usage

kruskalTest(x, ...)
# S3 method for default
kruskalTest(x, g, dist = c("Chisquare", "KruskalWallis", "FDist"), ...)
# S3 method for formula
kruskalTest(
  formula,
  data,
  subset,
  na.action,
  dist = c("Chisquare", "KruskalWallis", "FDist"),
  ...
)

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.

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.
dist: the test distribution. Defaults's to "Chisquare".
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

For one-factorial designs with non-normally distributed residuals the Kruskal-Wallis rank sum test can be performed to test the H $_{0} : F_{1} (x) = F_{2} (x) = \dots = F_{k} (x)$ against the H $_{A} : F_{i} (x) \neq F_{j} (x) (i \neq j)$ with at least one strict inequality.

Let $R_{i j}$ be the joint rank of $X_{i j}$ , with $R_{(1) (1)} = 1, \dots, R_{(n) (n)} = N, N = \sum_{i = 1}^{k} n_{i}$ , The test statistic is calculated as $H = \sum_{i = 1}^{k} n_{i} ({\bar{R}}_{i} - \bar{R}) / σ_{R},$

with the mean rank of the $i$ -th group ${\bar{R}}_{i} = \sum_{j = 1}^{n_{i}} R_{i j} / n_{i},$

the expected value $\bar{R} = (N + 1) / 2$

and the expected variance as $σ_{R}^{2} = N (N + 1) / 12.$

In case of ties the statistic $H$ is divided by $(1 - \sum_{i = 1}^{r} t_{i}^{3} - t_{i}) / (N^{3} - N)$

According to Conover and Imam (1981), the statistic $H$ is related to the $F$ -quantile as $F = \frac{H / (k - 1)}{(N - 1 - H) / (N - k)}$ which is equivalent to a one-way ANOVA F-test using rank transformed data (see examples).

The function provides three different dist for $p$ -value estimation:

Chisquare: $p$ -values are computed from the Chisquare distribution with $v = k - 1$ degree of freedom.
KruskalWallis: $p$ -values are computed from the pKruskalWallis of the package SuppDists.
FDist: $p$ -values are computed from the FDist distribution with $v_{1} = k - 1, v_{2} = N - k$ degree of freedom.

References

Conover, W.J., Iman, R.L. (1981) Rank Transformations as a Bridge Between Parametric and Nonparametric Statistics. Am Stat 35, 124--129.

Kruskal, W.H., Wallis, W.A. (1952) Use of Ranks in One-Criterion Variance Analysis. J Am Stat Assoc 47, 583--621.

Sachs, L. (1997) Angewandte Statistik. Berlin: Springer.

Examples

Run this code

## 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")

if (FALSE) {
## Check with kruskal.test from R stats
kruskal.test(x ~ g, dat)
}
## Using Conover's F
kruskalTest(x ~ g, dat, dist="FDist")

if (FALSE) {
## Check with aov on ranks
anova(aov(rank(x) ~ g, dat))
## Check with oneway.test
oneway.test(rank(x) ~ g, dat, var.equal = TRUE)
}

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