# Repeated measures ANOVA for circular / angular / directional data

I am looking for a test for circular data that is equivalent to linear repeated measures ANOVA (I have an experiment using human participants where the same sample of participants perform multiple experimental conditions and the dependent variable is an angle).

I have noticed 3 tests for performing ANOVA on circular data:

1. Watson & Williams (1956),
2. Harrison & Kanji (1986),
3. Anderson & Wu (1995) http://fmatoolbox.sourceforge.net/API/FMAToolbox/General/CircularANOVA.html

Do these tests require independent samples (as I suspect is the case for the Watson-Williams test) or can they be used on repeated measures data? If the former, is there another test that does what I desire?

• As repeated measures anova is simply a special case of a factorial design, it should be possible to implement. However, as far as I can tell, the coding in matlab linked to above only works for balanced cases of 2x2 - so it might need some jiggery pokery to expand it to a repeated measures design, which is essentially a 3-way ANOVA. I'm trying to do similar, using R. Its brain stretching! – user8512 Jan 13 '12 at 12:31

I don't know enough about the mentioned tests to tell you how they work in your case, but one option would be to do a permutation test:

1. Compute one of the above statistics for the original data
2. Randomly permute values within an individual but between the experimental condition groups
3. Compute the same statistic for the permuted data and store it
4. Repeate steps 2 and 3 a bunch of times (1999 or so)
5. Look at where the statistic on the original data falls in the distribution of the permuted statistics to do the test.

This tests the null that all the experimental condition groups are the same and the only differences seen are due to the sampling/randomization. By permuting within subject you are accounting for the lack of independence that the repeated measures ANOVA does.

• Greg, thanks for your answer. I don't really understand how permutting within subject accounts for lack of independence. Could you point me to a book or paper exaplaining why it is acceptable to use a test statistic designed for independent groups. I have tried to find some information on permutation tests for repeated measures data and in the example linked to below, the test statistic used seems to be the F value from a repeated measures ANOVA, not the F value from between-group ANOVA: uvm.edu/~dhowell/StatPages/Resampling/RandomRepeatMeas/… – omian Mar 20 '12 at 10:13
• The permutation test will work with any statistic, though some will work better than others. The correlation in your data means that the original distributions of those statistics for independent data do not hold for your case, but the permutation test does not use those distributions, rather computes its own based on exchangeability. Permuting within individual matches the null idea that subjects can be different from each other, but under the null the angles within subject are exchangeable. There are several references from the wikipedia page for "permutation tests" that may help. – Greg Snow Mar 20 '12 at 20:57
• I have now read a bit about permutation tests and understand the shuffling procedure required for repeated measures designs and why this is appropriate. But I remain stuck on the idea that it is OK to use F values designed for between-groups effects as the test-statistic within a repeated-measures permutation test. Therefore I ran a permutation test (1999 permutations) on simulated data (linear data, not angular data) containing 3 experimental conditions (columns) and 13 subjects (rows), permutting across columns (independently for each subject) but not across rows. – omian Mar 23 '12 at 18:44
• FOLLOWING ON FROM PREVIOUS COMMENT........I ended up with different p-values depending on whether I used standard between-group or repeated-measures ANOVA as the test statistic to reveal effects across columns (permutations used were the same for each test-statistic). Here is the simulated experimental data I used (spaces separate columns and colons separate rows): [4 3 6; 8 3 4; 10 4 3; 9 2 4; 6 3 4; 4 6 2; 5 4 5; 8 3 6;7 4 2;3 4 9;8 7 7;5 3 5;4 5 7] – omian Mar 23 '12 at 18:44