Kruskal Wallis procedure demonstrated with an example

  1. The Wilcoxon signed-rank test
  2. Example: analyze the data with Wilcoxon signed-rank test
  3. Output and interpretation

The Kruskal-Wallis test

Kruskal-Wallis test can be considered as a backup method for ANOVA where the independent variable is categorical but the dependent variable are not normally distributed. On the other hand, Kruskal-Wallis test can also be considered an alternative method for Mann-Whitney test where it is a nonparametric test but the independent variable could have more than two categories. In summary, Kruskal-Wallis test is preferred when:

  1. The independent variable has three or more categories and randomly selected;
  2. For each sample (that determined by each categories of the independent variable), there is a least 5 observations. If the sample size is fewer than 5, refere to special tables of credival values, such as CRC standard Probability and Statistics Tables and Formulae, published by CRC press;
  3. There is no requirement that the populations have a normal distribution or any other particular distribution.

Analyzing the data with Kruskal-Wallis test

For example, suppose weights of poplar trees are different based on treatments (none treatment, fertilizer, irrigation, or fertilizer and irrigation). Each weight samples that determined by the treatments is independent and random, and each sample size is 5. But the weight samples are not normally distributed. The research question is to test whether the poplar tree weights are different under the four treatments.

The data is "poplar.csv".

A Kruskal-Wallis test is performed as following.

Open the data set from SAS. Or import with the following command.

 
   data poplar;
	infile "H:\sas\data\poplar.csv" dlm=',' firstobs=2;
	input weight treat $;
    run;

 

    proc npar1way data=poplar;
	 class treat;
	  var weight;
	run;

The key output and intepretation

Under the Wilcoxon Analysis output, the test results can be found and shown here. 

        			The NPAR1WAY Procedure

                         Wilcoxon Scores (Rank Sums) for Variable weight
                                  Classified by Variable treat

                                   Sum of      Expected       Std Dev          Mean
              treat       N        Scores      Under H0      Under H0         Score
              _____________________________________________________________________
              no          5         45.00         52.50     11.443511          9.00
              fert        5         37.50         52.50     11.443511          7.50
              irrig       5         42.50         52.50     11.443511          8.50
              f_i         5         85.00         52.50     11.443511         17.00

                               Average scores were used for ties.


                                       Kruskal-Wallis Test

                                    Chi-Square         8.2329
                                    DF                      3
                                    Pr > Chi-Square    0.0414

The p-value is 0.0414 (<0.05), so we reject the null hypothesis, and there is sufficient evidence to reject the claim that the populations of poplar tree weights from the four treatments have equal medians. At least one of the medians appears to be different from the others.

Comparing with F test, Kruskal-Wallis test is easier without checking assumptions of Normal distribution. However, the Kruskal-Wallis test is not as efficient as the F test, so it might require more dramatic differences for the null hypothesis to be rejected.