Kruskal-Wallis Test

Menu location: Analysis_Analysis of Variance_Kruskal-Wallis.

This is a method for comparing several independent random samples and can be used as a nonparametric alternative to the one way ANOVA.

The Kruskal-Wallis test statistic for k samples, each of size n_i is:

- where N is the total number (all n_i) and R_i is the sum of the ranks (from all samples pooled) for the ith sample and:

The null hypothesis of the test is that all k distribution functions are equal. The alternative hypothesis is that at least one of the populations tends to yield larger values than at least one of the other populations.

Assumptions:

random samples from populations
independence within each sample
mutual independence among samples
measurement scale is at least ordinal
either k population distribution functions are identical, or else some of the populations tend to yield larger values than other populations

If the test is significant, you can make multiple comparisons between the samples. You may choose the level of significance for these comparisons (default is a = 0.05). All pairwise comparisons are made and the probability of each presumed "non-difference" is indicated (Conover, 1999; Critchlow and Fligner, 1991; Hollander and Wolfe, 1999). Two alternative methods are used to make all possible pairwise comparisons between groups; these are Dwass-Steel-Critchlow-Fligner and Conover-Inman. In most situations, you should use the Dwass-Steel-Critchlow-Fligner result.

By the Dwass-Steel-Critchlow-Fligner procedure, a contrast is considered significant if the following inequality is satisfied:

- where q is a quantile from the normal range distribution for k groups, n_i is size of the ith group, n_j is the size of the jth group, t_b is the number of ties at rank b and W_ij is the sum of the ranks for the ith group where observations for both groups have been ranked together. The values either side of the greater than sign are displayed in parentheses in StatsDirect results.

The Conover-Inman procedure is simply Fisher's least significant difference method performed on ranks. A contrast is considered significant if the following inequality is satisfied:

- where t is a quantile from the Student t distribution on N-k degrees of freedom. The values either side of the greater than sign are displayed in parentheses in StatsDirect results.

An alternative to Kruskal-Wallis is to perform a one way ANOVA on the ranks of the observations.

StatsDirect also gives you an homogeneity of variance test option with Kruskal-Wallis; this is marked as "Equality of variance (squared ranks)". Please refer to homogeneity of variance for more details.

Technical Validation

The test statistic is an extension of the Mann-Whitney test and is calculated as above. In the presence of tied ranks the test statistic is given in adjusted and unadjusted forms, (opinion varies concerning the handling of ties). The test statistic follows approximately a chi-square distribution with k-1 degrees of freedom; P values are derived from this. For small samples you may wish to refer to tables of the Kruskal-Wallis test statistic but the chi-square approximation is highly satisfactory in most cases (Conover, 1999).

Example

From Conover (1999, p. 291).

Test workbook (ANOVA worksheet: Method 1, Method 2, Method 3, Method 4).

The following data represent corn yields per acre from four different fields where different farming methods were used.

Method 1	Method 2	Method 3	Method 4
83	91	101	78
91	90	100	82
94	81	91	81
89	83	93	77
89	84	96	79
96	83	95	81
91	88	94	80
92	91		81
90	89
	84

To analyse these data in StatsDirect you must first prepare them in four workbook columns appropriately labelled. Alternatively, open the test workbook using the file open function of the file menu. Then select Kruskal-Wallis from the Nonparametric section of the analysis menu. Then select the columns marked "Method 1", "Method 2", "Method 3" and "Method 4" in one selection action.

For this example:

Adjusted for ties: T = 25.62883 P < 0.0001

All pairwise comparisons (Dwass-Steel-Chritchlow-Fligner)

Method 1 and Method 2 , P = 0.1529

Method 1 and Method 3 , P = 0.0782

Method 1 and Method 4 , P = 0.0029

Method 2 and Method 3 , P = 0.0048

Method 2 and Method 4 , P = 0.0044

Method 3 and Method 4 , P = 0.0063

All pairwise comparisons (Conover-Inman)

Method 1 and Method 2, P = 0.0078

Method 1 and Method 3, P = 0.0044

Method 1 and Method 4, P < 0.0001

Method 2 and Method 3, P < 0.0001

Method 2 and Method 4, P = 0.0001

Method 3 and Method 4, P < 0.0001

From the overall T we see a statistically highly significant tendency for at least one group to give higher values than at least one of the others. Subsequent contrasts show a significant separation of all groups with the Conover-Inman method and all but method 1 vs. methods 2 and 3 with the Dwass-Steel-Chritchlow-Fligner method. In most situations, it is best to use only the Dwass-Steel-Chritchlow-Fligner result.

P values

analysis of variance