kruskalTest

Compute the Kruskal-Wallis test for equal medians.

The Kruskal-Wallis rank sum test evaluates for multiple samples the null hypothesis that their medians are identical. The Kruskal-Wallis test is a nonparametric test which does not require the data to be normally distributed.

To carry out the test, the rank sums S_h of the individual groups are calculated. The test statistic is then calculated as

upper H equals StartStartFraction StartFraction 12 Over upper N left-parenthesis upper N plus 1 right-parenthesis EndFraction sigma-summation Underscript h Endscripts StartFraction upper S Subscript h Superscript 2 Baseline Over n Subscript h Baseline EndFraction minus 3 left-parenthesis upper N plus 1 right-parenthesis OverOver 1 minus StartFraction 1 Over left-parenthesis upper N cubed minus upper N right-parenthesis EndFraction sigma-summation t Subscript r left-parenthesis i right-parenthesis Superscript 3 Baseline minus t Subscript r left-parenthesis i right-parenthesis Baseline EndEndFraction

where N denotes the total number of observations and t_{r(i)} are the number of tied observations with rank i.

Usage

var kruskalTest = require( '@stdlib/stats/kruskal-test' );

kruskalTest( a[,b,...,k][, opts] )

For input arrays a, b, ... holding numeric observations, this function calculates the Kruskal-Wallis rank sums test, which tests the null hypothesis that the medians in all k groups are the same.

// Data from Hollander & Wolfe (1973), p. 116:
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = kruskalTest( x, y, z );
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'df': 2,
        'pValue': ~0.68,
        'statistic': ~0.771,
        ...
    }
*/

The function accepts the following options:

  • alpha: number in the interval [0,1] giving the significance level of the hypothesis test. Default: 0.05.
  • groups: an array of group indicators. If set, the function assumes that only a single numeric array is provided holding all observations.

By default, the test is carried out at a significance level of 0.05. To choose a custom significance level, set the alpha option.

var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = kruskalTest( x, y, z, {
    'alpha': 0.01
});
/* returns
    {
        'rejected': false,
        'alpha': 0.01,
        'df': 2,
        'pValue': ~0.68,
        'statistic': ~0.771,
        ...
    }
*/

The function provides an alternate interface by supplying an array of group indicators to the groups option. In this case, it is assumed that only a single numeric array holding all observations is provided to the function.

var arr = [
    2.9, 3.0, 2.5, 2.6, 3.2,
    3.8, 2.7, 4.0, 2.4,
    2.8, 3.4, 3.7, 2.2, 2.0
];
var groups = [
    'a', 'a', 'a', 'a', 'a',
    'b', 'b', 'b', 'b',
    'c', 'c', 'c', 'c', 'c'
];
var out = kruskalTest( arr, {
    'groups': groups
});

The returned object comes with a .print() method which when invoked will print a formatted output of the results of the hypothesis test. print accepts a digits option that controls the number of decimal digits displayed for the outputs and a decision option, which when set to false will hide the test decision.

var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = kruskalTest( x, y, z );
console.log( out.print() );
/* =>
    Kruskal-Wallis Test

    Null hypothesis: the medians of all groups are the same

        pValue: 0.68
        statistic: 0.7714    df: 2

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

Examples

var kruskalTest = require( '@stdlib/stats/kruskal-test' );

// Data from Hollander & Wolfe (1973), p. 116:
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = kruskalTest( x, y, z );
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'df': 2,
        'pValue': ~0.68,
        'statistic': ~0.771,
        ...
    }
*/

var table = out.print();
/* returns
    Kruskal-Wallis Test

    Null hypothesis: the medians of all groups are the same

        pValue: 0.68
        statistic: 0.7714    df: 2

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/