Van Der Waerden Test for Comparing Group Locations Using Normal Scores

A nonparametric k-sample test based on normal scores (van der Waerden scores), serving as an alternative to the one-way ANOVA. It is more efficient than the Kruskal-Wallis test when the underlying data are approximately normally distributed.

Usage

vanWaerdenTest(x, ...)

# S3 method for class 'formula'
vanWaerdenTest(formula, data, subset, na.action, ...)

# Default S3 method
vanWaerdenTest(x, g, ...)

Arguments

x

a numeric vector of data values, or a list of numeric data vectors. Non-numeric elements of a list will be coerced, with a warning.

...

further arguments to be passed to or from methods.

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").

g

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

Value

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

statistic

the van der Waerden statistic.

parameter

the degrees of freedom of the approximate chi-squared distribution of the test statistic.

p.value

the p-value of the test.

method

the character string "Van-der-Waerden normal scores test".

data.name

a character string giving the names of the data.

Details

Performs a van der Waerden normal scores test.

vanWaerdenTest performs a van der Waerden test of the null that the location parameters of the distribution of x are the same in each group (sample). The alternative is that they differ in at least one.

The van der Waerden rank scores are defined as the ranks of data, i.e., \(R[i], i = 1, 2, ..., n\), divided by \(1 + n\) transformed to a normal score by applying the inverse of the normal distribution function, i.e., \(\Phi^(-1)(R[i]/(1 + n))\). The ranks of data are obtained by ordering the observations from all groups (the same way as kruskal.test() does it).

If x is a list, its elements are taken as the samples to be compared, and hence have to be numeric data vectors. In this case, g is ignored, and one can simply use vanWaerdenTest(x) to perform the test. If the samples are not yet contained in a list, use vanWaerdenTest(list(x, ...)).

Otherwise, x must be a numeric data vector, and g must be a vector or factor object of the same length as x giving the group for the corresponding elements of x.

References

Conover, W. J., Iman, R. L. (1979). On multiple-comparisons procedures, Tech. Rep. LA-7677-MS, Los Alamos Scientific Laboratory.

Conover, W. J. (1999). Practical Nonparametric Statistics (Third Edition ed.). Wiley. pp. 396–406.

See also

coin::normal_test() in package coin, where the test is implemented in a more general context.

Other test.location: brunnerMunzelTest(), hotellingsT2Test(), moodMedianTest(), signTest(), tTestA(), yuenTTest(), zTest()

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

vanWaerdenTest(list(x, y, z))
#> 
#> 	Van-der-Waerden normal scores test
#> 
#> data:  x
#> Van-der-Waerden chi-squared = 1.2014, df = 2, p-value = 0.5484
#> 

## Equivalently,
x <- c(x, y, z)
g <- factor(rep(1:3, c(5, 4, 5)),
            labels = c("Normal subjects",
                       "Subjects with obstructive airway disease",
                       "Subjects with asbestosis"))
vanWaerdenTest(x, g)
#> 
#> 	Van-der-Waerden normal scores test
#> 
#> data:  x and g
#> Van-der-Waerden chi-squared = 1.2014, df = 2, p-value = 0.5484
#> 

## Formula interface.
require(graphics)
boxplot(Ozone ~ factor(Month), data = airquality)

vanWaerdenTest(Ozone ~ factor(Month), data = airquality)
#> 
#> 	Van-der-Waerden normal scores test
#> 
#> data:  Ozone ~ factor(Month)
#> Van-der-Waerden chi-squared = 27.632, df = 4, p-value = 1.481e-05
#>