*Using R for Nonparametric Data Analysis: The Kruskal-Wallis Test*

A tutorial by Douglas M. Wiig

Analysis of variance(ANOVA) is a commonly used technique for examining the effect of an independent variable on three or more dependent variables. There are several types of ANOVA ranging from simple one-way ANOVA to the more complex multiple analysis of variance, MANOVA. ANOVA makes several assumptions about the sample data being used such as the assumption of normal distribution of the variables in the parent population, underlying continuous distribution of the variables, and interval or ratio level measurement of all variables. If any of these assumptions cannot be met a researcher can turn to a nonparametric counterpart to ANOVA for the analysis. This tutorial will discuss the use of the Kruskal-Wallis test, the nonparametric counterpart to analysis of variance.

In this tutorial I will explore a simple example and discuss entering the sample data into a data file using the R data editor. I will then discuss setting up the data for analysis and using the Kruskal-Wallis test.

I am going to assume that the reader has a working knowledge of ANOVA with parametric data. Since ANOVA uses sample means and variances as the basis of the statistical test interval or ratio level measurement is necessary to insure valid results in addition to the assumptions indicated above. With the nonparametric Kruskal-Wallis test the only assumptions to be met are ordinal or better measurement and the assumption of an underlying continuous measurement. The example to be used here is taken from a book on nonparametric statistics by Sidney Seigel.(Sidney Seigel, *Nonparametric Statistics for the Behavioral Sciences*, New York: McGraw-Hill, 1956, pp-184-196).

A researcher wishes to test the hypothesis that school administrators are typically more authoritarian than classroom teachers. He also believes that many classroom teachers are adminstration-oriented in their professional aspirations which may, in turn, have an effect on their authoritarianism. 14 subjects are selected and divided into three groups: teaching-oriented teachers (classroom teachers who wish to remain in a teaching position), administration-oriented teachers (classroom teachers who aspire to become administrators), and practicing administrators.(Seigel, p. 186). The level of authoritarianism of each subject is measured through a survey that assigns an authoritarianism score that is considered to be at least ordinal in nature. Higher scores indicate higher levels of authoritarianism. (Siegel, p. 186). The null hypothesis is that there is no difference in mean authoritarianism scores among the three groups. The alternative hypothesis is that the mean authoritarianism scores among the three groups are different. The alpha level for rejecting the null hypothesis is p = .05. (Seigel, p. 186).

Since we make no assumption about a normal distribution of scores, have a small sample size of n = 14, and ordinal measure we will use the nonparametric test which is based on median scores and ranks rather than means and variances as used in parametric ANOVA. The mathematical details of how this is done is beyond the scope of this tutorial. See Seigel, p. 187-189 for details. The authoritarian scores for the three groups are shown below:

Authoritarianism Scores of Three Groups of Educators

Teacher-Oriented Admin-oriented Administrators

teachers n=5 teachers n=5 n=4

—————————————————————————————-

96 82 115

128 124 149

83 132 166

61 123 147

101 109

—————————————————————————————-

(Seigel, p. 187)

The first task is to create an R data frame with the scores from the table. We will enter the scores using the R data editor. We will name the data frame ‘kruskal.’ Invoke the editor using the following commands:

** > kruskal <- data.frame()**

** > kruskal <- edit(kruskal)**

You should see the data entry editor open in a separate window. In order to process the data properly it needs to be entered into two columns. The first column will be the factors (which group the scores belong to), and the second column will contain the actual scores. Label column 1 ‘Group’ and column 2 ‘authscore.’ When the data are entered your editor should look like this:

———————-

Group authscore

1 1 96

2 1 128

3 1 83

4 1 61

5 1 101

6 2 82

7 2 121

8 2 132

9 2 135

10 2 109

11 3 115

12 3 149

13 3 166

14 3 147

———————-

Make sure that each column of numbers is of the data type “Real.” l Close the data editor by clicking ‘Quit’ and the data will be saved in the working directory for access. To see what has been entered in the data editor use the command:

**> kruskal**

Group authscore

1 1 96

2 1 128

3 1 83

4 1 61

5 1 101

6 2 82

7 2 121

8 2 132

9 2 135

10 2 109

11 3 115

12 3 149

13 3 166

>

You should see the output as above. If you need to make changes simple invoke the editor with:

**> kruskal <-edit(kruskal)**

The editor will open and you can make any changes you need to. Be sure to click on ‘Quit’ to save the changes to the working directory.

**Part Two will continue the analysis**