## On-line statistics

Multiple comparison tests

Data types that can be analysed with Multiple Comparison tests (MCT's)

the data points must be independent from each other

the distributions must be normal for the parametric MCT's (K-S or Shapiro-Wilke)

the variances of the samples must not be different for the parametric MCT's (F-test or homogeneity of variance test)

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

Multiple comparison tests can be used on their own, but should be used in conjunction with an ANOVA. The latter is a more robust test and will occasionally find differences where the multiple comparison tests will not.

Introduction to Multiple Comparison tests

The multiple comparison tests are a group of tests that follow on from one or two-factor ANOVA or the Kruskal-Wallis test, but only if significant differences have been found. It would appear that they could be used on their own but because they are not as powerful as ANOVA or Kruskal-Wallis, they can occasionally fail to find differences when the former succeed. They are used for exactly the same reasons that ANOVA and Kruskal-Wallis are used, but provide more information. ANOVA and Kruskal-Wallis can only tell you whether there is a difference between two or more of your groups and not which ones. The multiple comparison tests will tell you which groups are different. For example, you may compare the trunk diameters of oaks in four isolated woods with the following means:

Wood A: 400 mm

Wood B: 480 mm

Wood C: 320 mm

Wood D: 455 mm

You apply ANOVA and find that there is a significant difference between two or more of your means. The only conclusion that you can make from this is that woods B and C are different because they have the highest and lowest means. What you do not know is whether Wood D or A are different from C which would appear possible. Enter the multiple comparison tests. One of these tests will then make individual comparisons between all groups (tedium personified if carried out by hand) and tell you then that:

B and D are not significantly different from each other

B and D are significantly different to A and C.

A and C are significantly different from each other

It will then suggest that there are three sub-groups B and D, A, and C. Your conclusions would be that A comes from a different population to all the rest, so does C, but B and D come from the same population. This last point is a bit misleading because the only thing you can really say is that there is not enough evidence to indicate that they are different.

There are several parametric "multiple comparison tests", all of which carry out the same function. The most common two in biological studies are the Tukey test and the Student-Newman-Keuls test. The Tukey test is reputed to be the most powerful of the two (Zar, 1984) and would be the safer to use.

Where you can show that the requirements of ANOVA are not met and the need for a non-parametric multiple comparison test is needed the Tukey test with ranked sums can be used. This test is not available through all statistical programs and may need to be calculated by hand. You will need to consult Zar or Sokal and Rohlf for this.

Hypotheses

Data arrangement

Data arrangement for the multiple comparison tests is the same as for ANOVA and Homogeneity of Variance tests.

Results and interpretation

Degrees of freedom

between groups: No of samples/treatments - 1

within groups: n - No of samples/treatments,

e.g. Where there are four treatments with 17 data points per treatment (total n = 56), between group DF = 4-1=3, and within group DF = 56-4=52.

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