## On-line statistics

Kruskal-Wallis non-parametric ANOVA

Data types that can be analysed with Kruskal-Wallis

the data points must be independent from each other

the distributions do not have to be normal and the variances do not have to be equal

you should ideally have more than five data points per sample

all individuals must be selected at random from the population

all individuals must have equal chance of being selected

sample sizes should be as equal as possible but some differences are allowed

Limitations of the test

if you do not find a significant difference in your data, you cannot say that the samples are the same

if significant differences are found when comparing more than two samples there are non-parametric multiple comparison tests available but they are only found in UNISTAT and otherwise have to be performed manually or calculated long-hand in Excel.

Introduction to Kruskal-Wallis

Kruskal-Wallis compares between the medians of two or more samples to determine if the samples have come from different populations. For instance it is a well known aspect of natural history that the littorinid species (snails) that are found on sheltered and exposed shores have different shell morphologies. This could be tested by measuring the shell thickness of each individual in samples taken from a sheltered, an exposed and an intermediate shore. If the distributions prove not to be normal and/or the variances are different then the Kruskal-Wallis should be used to compare the groups. If a significant difference is found then there is a difference between the highest and lowest median. A non-parametric multiple comparison test must then be used to ascertain whether the intermediate shore also is significantly different. These are found in UNISTAT but must be set up on a spreadsheet in Excel or done by hand from the examples given in Zar (1984).

In the above example only one factor is considered (level of shore exposure) and so is termed a one-way Kruskal-Wallis. There are examples in Zar (1984) of a two-way Kruskal-Wallis test but again must be set up in Excel or done by hand.

Hypotheses

Data arrangement

Once you have established that your data suits Kruskal-Wallis, your data must be arranged thus for use in one of the statistical packages (SPSS, UNISTAT):

Results and interpretation

(Degrees of Freedom = number of samples/treatments - 1)

On completion of the 1-way Kruskal-Wallis the results will look something like this:

Although it looks a bit daunting do not be worried. There is only one value that concerns the selection of one of the hypotheses. The Right-Tail Probability (0.0052) is the probability of the differences between the data sets occurring by chance. Since it is lower than 0.05 the HO must be rejected and the HA accepted.

Two-way Kruskal-Wallis results would appear in a similar format to the two-way ANOVA.

Graphical representation

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Ted Gaten  Department of Biology  gat@le.ac.uk
Entry approved by the Head of Department. Last Updated: May 2000