Hi all,

A week ago, I asked a question regarding the possibility of using Bayesian stats to test for equivalence of circular data.  Thank you all for your responses (copied in below) -- there was a wide variety of suggestions, and I've alot to chew on now!  Unfortunately, there was some disagreement about whether a useful Bayesian test can be applied here.  I think that I should try rephrasing the question.

The data are circular: angles around a circle, from zero to 2*pi.  Example: a sample of 40 current flow compass headings.  Generally, I have 2 samples of headings which I want to compare and test whether the mean headings for their underlying populations are the same (a test of equivalence, with H0: means are different; Ha: means are the same).
Example 1: paired samples of 40 current headings, one measured by a hideously laborious manual method versus the other measured by an easy-breezy automatic method.  I'm testing for equivalence of the 2 methods.
Example 2: a paired sample of 40 current headings, one measured at one place underwater versus another measured about 3 meters away. Test: are the current headings statistically the same at the 2 locations?  Again, the goal is to test for statistical equivalence, NOT test for statistical difference, because I want to be able to use the conclusion that the current flow is the _same_ across 3 meters of space in my subsequent work.

Some responders suggested that Bayesian stats would work here, because they compute both the probabilities of equivalence & difference.  Others said the opposite, or that Bayesian stats wouldn't necessarily work so easily, and that alternative methods might be more appropriate.
Anyone feel like wading in?  My personal impression is that 1) Bayesian stats ought to work here, 2) no one's done this before [dammit] 3)I need to read up on how to use Bayesian stats!

I've summarized the references I received below -- if anyone has anything to add, please let me know, and the original responses are also at the bottom of the msg.

Thanks for all the input,
Russell

________________________________________________________________________________________________
Suggested References:
For Circular Data Analysis
[I'm not familiar with Fisher, but the other 3 do not do equivalence tests]
N. I. Fisher, Ed., Statistical analysis of circular data, Cambridge University Press, 1993
KV Mardia and PE Jupp Directional Statistics (now in 2nd edition)
Zar 1998, Biostatistical Analysis.  [suggested by me]
Oriana Software (metasearch pulled a couple websites selling it)

For Alternative Equivalence Testing Possibilities
Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet 1986;I:307-310.
Kelsall, J and Diggle P. Statistics in Medicine,1995 14:2335-2342

________________________________________________________________________________________________
ORIGINAL RESPONSES:
But seriously, this is Bayesian statistics.  Make your own test.
Perhaps the simplest place to start is to reformulate the PINES example to fit your problem.  Let me know if you run into problems.
Finn Krogstad

I'm no expert, but I have done some reading about circular distributions.  I think I understand what was meant by the suggestion.  In Fisherian analysis, you just assume the null hypothesis for convenience, when you really want to reject it in favor the alternative.  But, in Bayesian analysis, you compute the probability of both the null hypothesis and the alternative.  Therefore, the Bayesian approach would be similar to what you have already been reading about.
Except that you can reject the alternative in favor of the null, which you couldn't do before.
Rodney Sparapani

I am a relatively new BUGS user so cannot help with implementation but wonder if you have looked at the Directional Statistics book by KV Mardia and PE Jupp.  It has two sample test for mean and concentration parameter.
Jack
John H Schuenemeyer

This is not a Bayesian solution but might give you something to try.
Kelsall, J and Diggle P published two papers in 1995 (one in Statistics in Medicine the other in Bernoulli) that compared two density functions.  You might be able to do this on a circle rather than a line.  We used the approach to compare densities of sea turtle nesting sites http://www.pnl.gov/statenvi/ssenews/ENVR_5_1.pdf and you could use the same "random labeling" Monte Carlo tests to find which directions differ between data sets.  Again (as you said), this is a test of difference but the Kelsall and Diggle approach gives pointwise confidence bands, so if the ratio of directional densities is always within the random labeling bands, you conclude no significant difference between the two distributions.
Lance Waller

Not sure what you mean by "circular data". And I'm just beginning BUGS. But I won't let this stop me answering your question on sea slugs! It strikes me that an equivalence test is entirely straightforward from the Bayesian route (and surprisingly torturous otherwise).
You have a bunch of parameters which includes d = the difference (ratio, whatever) between one method (expensive measurement) and the other (cheap measurement). You want to test whether d is "about zero". Define what this means (i.e. define your equivalence interval, [a,b] say. Now calculate the posterior for d, and see what proportion of the area is over H0: d in [a,b] versus H1: d not in [a,b].
Patrick Johnston.

I was just thinking about it (sea slugs...even cooler than turtles?).
Equivalence testing is a tough problem that I'm not sure becomes easier if you are a Bayesian.  I would suggest a variant of Kelsall and Diggle around a circle rather than along a line.
[Statistics in Medicine,1995 14:2335-2342.]
Lance Waller (via Elizabeth Hill)

Russell, your problem may be analogous to one we run into in doing psychological studies.  There are times when we want to test whether  two groups are equivalent (perhaps before beginning some intervention), which is different than the traditional test of differences between groups.  Someone published a procedure for doing this in Psychological Bulletin, probably ten or more years ago.  I don't remember more and only know about it because I was the associate editor handling the manuscript.  I'm not sure whether this will be helpful or not.  I've had some neuroscience friends who studied slugs, and always thought it would be fun to show up at the lab in the morning by shouting "you slugs!"  Good luck on your project.--Rick Wagner

Russell,
Equivalence testing is hard, which is why you don't find much useful literature.  Here is I method that I discovered (as far as I know, anyway).  It works for any estimate x whose sampling standard deviation s can be obtained (just read it off the computer output, mostly).  Assuming that 0 is an appropriate null hypothesis value (as in comparing two means), set up the hypotheses:
H-  the parameter is < -s
H0  the parameter is between -s and +s
H+  the parameter is > +s
Choose H- if x<-s, H+ if x>+s, H0 otherwise.  This has three handy features.  (1) the 0.05 level test of H- vs. H+ has power 0.95, (2) the strength of the assertion being made is estimated by the data (essentially it is 2s, the width of the H0 interval), (3) if your data don't have much to say, you end up saying nothing (H0).  I have a paper under review that explains this more generally.
Mikel Aickin, Ph.D.

Hi Russel:
I am not sure if I understand your question. You mention that you want to test for equivalence - does that mean that you want to treat equivalence as the alternative hypothesis.
If that is the case, it is not a problem in the Bayesian setting as many Bayesian hypothesis  testing procedures treat the null and the alternative hypotheses interchangeably.
I did some work on Bayesian analysis of circular data and have interest in this area.
Sanjib Basu

Dear Russell,
I'm not really an expert in this field, but there are books by Mardia and by Fisher on circular statistics.
Many years ago I worked with wind direction statistics and as far as I remember you can do very nice things with the von Mises distribution: c * exp(-a * cos(x - mu)), where mu is the mean direction, a an equivalent for the inverse of the variance and c a normalizing constant.
See N. I. Fisher, Ed., Statistical analysis of circular data, Cambridge University Press, 1993. and references there.
There exists a package (non-free) called Oriana.
Also see: http://geography.lancs.ac.uk/cemp/resources/software.htm for applications in a different field.
I hope this will help.
Paul Eilers

Hi Russell,
This is not a direct answer to your question about Bayesian methods, and you may already be aware of what I'm suggesting, but there has been some work on measuring agreement between different methods of clinical measurement in the medical field - probably in others too. I have no idea whether it might be adaptable to your data.
The source paper is Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet 1986;I:307-310. I've used the method once in a published study (Desai SP, Sivakumar S, Fryers PT. Evaluation of a disposable prism for applanation tonometry. Eye 2001;15:279-282.)
It's not so clear-cut as traditional hypothesis tests to identify a difference, but is fairly intuitive.
Paul Fryers


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Russell Wyeth
Dept of Zoology
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University of Washington
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