kruskalTest function

Kruskal-Wallis Rank Sum Test

Performs a Kruskal-Wallis rank sum test.

kruskalTest(x, ...)

## Default S3 method:
kruskalTest(x, g, dist = c("Chisquare", "KruskalWallis", "FDist"), ...)

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

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").

Returns

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

For one-factorial designs with non-normally distributed residuals the Kruskal-Wallis rank sum test can be performed to test the H$_0: F_1(x) = F_2(x) = \ldots = F_k(x)$ against the H$_\mathrm{A}: F_i (x) \ne F_j(x)~ (i \ne j)$ with at least one strict inequality.

Let $R_{ij}$ be the joint rank of $X_{ij}$, with $R_{(1)(1)} = 1, \ldots, R_{(n)(n)} = N, ~~ N = \sum_{i=1}^k n_i$, The test statistic is calculated as

$$H = \sum_{i=1}^k n_i \left(\bar{R}_i - \bar{R}\right) / \sigma_R,$$

with the mean rank of the $i$-th group

$$\bar{R}_i = \sum_{j = 1}^{n_{i}} R_{ij} / n_i,$$

the expected value

$$\bar{R} = \left(N +1\right) / 2$$

and the expected variance as

$$\sigma_R^2 = N \left(N + 1\right) / 12.$$

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

According to Conover and Imam (1981), the statistic $H$ is related to the $F$-quantile as

$$F = \frac{H / \left(k - 1\right)}{\left(N - 1 - H\right) / \left(N - k\right)}$$

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.

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)

## Median Test asymptotic
medianTest(x ~ g, dat)

## Median Test with simulated p-values
set.seed(112)
medianTest(x ~ g, dat, simulate.p.value = TRUE)

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.

See Also

kruskal.test, pKruskalWallis, Chisquare, FDist