# Analysing circular (angular) data - how to find significant differences?

#### DEW

##### New Member
Dear users,

this is a problem from my PhD thesis. I am grateful for suggestions.

I have measured angular data from several individuals, each specimen only once. This data is clustered in 5° from 1-180°. No angle above 180° are recorded. I have calculated the frequency of angles for each 5°-cluster, therefore my data looks like following:

Specimen1 1-5°: 2.34% 6-10°: 4.30% ...
Specimen2 1-5°: 3.77%, 6-10°: 2.12%...

The question of my research is, if these specimens differ in their distribution of angles from each other. Which test is appropriate to use? I have come across the Wilcoxon usigned rank test, but I am not convined that this is correct.

So far I have no access to literature about analysing circualar data (e.g. N.I. Fisher).
I came across the software Oriana, but have no experience how to use it.

Hoping for replies

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Not at all familar with circular data analyses, but can you consider it as ordinal data?

#### DEW

##### New Member
No, I do not think it is ordinal. The frequencies are continou´s and can have any value between 0-100% (in theory).

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Well can you use non-parametric as you previously referenced. I am not totally sure you rule out interval when you have a limited range that the values can take.

I would recommend going to the literature and seeing what procedures everyone else uses in these cases!

#### bugman

##### Super Moderator
This is a good starting point:

#### DEW

##### New Member
Thank you very much for the PDF. I will sort through it and post in this thread as soon as I have found a solution

#### GretaGarbo

##### Human
What is it that is pointing in a special direction? Is it birds or insects or what that is oriented in a specific direction? I am “totally confused” as they say! It is a pity that all data are so secret!

Just by googling on your title I find the wikipedia: http://en.wikipedia.org/wiki/Directional_statistics

and the R package “circular”: http://cran.r-project.org/web/packages/circular/circular.pdf

I have never heard of software oriana.

I don't think that the Mann-Whitney-Wilcoxon is appropriate here.

#### DEW

##### New Member
What is it that is pointing in a special direction? Is it birds or insects or what that is oriented in a specific direction? I am “totally confused” as they say! It is a pity that all data are so secret!
Indeed! It is a shame that I have to try to avoid stating what I am actually researching
I will try to see what R can do for me, thanks for the reference to the package!

#### TheEcologist

##### Global Moderator
Dear users,

this is a problem from my PhD thesis. I am grateful for suggestions.

I have measured angular data from several individuals, each specimen only once. This data is clustered in 5° from 1-180°. No angle above 180° are recorded. I have calculated the frequency of angles for each 5°-cluster, therefore my data looks like following:

Specimen1 1-5°: 2.34% 6-10°: 4.30% ...
Specimen2 1-5°: 3.77%, 6-10°: 2.12%...

The question of my research is, if these specimens differ in their distribution of angles from each other. Which test is appropriate to use? I have come across the Wilcoxon usigned rank test, but I am not convined that this is correct.

So far I have no access to literature about analysing circualar data (e.g. N.I. Fisher)

I came across the software Oriana, but have no experience how to use it.

Hoping for replies
Hi DEW,

Welcome to talkstats. What is it you are exactly trying to prove? I 'm referring to point 6 of this.

You are stating that you want to test whether these groups "differ in their distribution of angles from each other" but your suggestion of a Wilcoxon would lead me to think you want to know whether the central tenancy of both groups differ. Whether the means or medians differ is a different question from whether distributions differ. So what are you trying to learn about the data? Or better even what are you trying to understand about these bugs?

I can recommend; N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press. However there are also copies to be found of http://books.google.nl/books/about/...istics_Vol_5.html?id=sKqWMGqQXQkC&redir_esc=y on certain websites.

And this book is really fun: http://books.google.nl/books/about/Circular_Statistics_in_Biology.html?id=ip5kQgAACAAJ&redir_esc=y

But sadly I cant get it at my university (its fun because it got lots of biology examples). Maybe your does?

Good luck!

TE

#### GretaGarbo

##### Human
Indeed! It is a shame that I have to try to avoid stating what I am actually researching
It is tempting to say that since it it so secret what it is, I am not going to bring any suggestions for how it can be analysed.

Let's see now, I remembered that I had seen that in an other package how to analyse it and a great software to show the graphs for it. I remember that I thought that it was interesting but could not imagine what it could be used for in practice. It can not only be used for Buffons needle problem, can it?

It is the combination of applied examples and analysis schemes that is the strength of this site. But there is to much “dry swim” here!

#### bugman

##### Super Moderator
...and there is also "circstat" and "CircNNTSR"

If you don't and you should really learn how at least just to utilise the awesome functions avaiable to you in this package. Or just take a look at the package documetation, this lists a whole load of different analysis functions like : watsons test, anova circular etc... that might lead you in the right direction.

#### DEW

##### New Member
Hi DEW,

Welcome to talkstats. What is it you are exactly trying to prove? I 'm referring to point 6 of this.

TE
Hi TE,

thanks for your reply. Well, as I had never before thought of circular statistics, I was not even sure which statistical procedure I wanted to use. But I have made progress. I learned how to compute the mean angle(jipeee), which can now help me to compare means (jipee again!). The book of N.I. Fisher is gladly in our library, and I already started on it.

Now I wonder what other tests are possbile, and which I need for my questions. So before I post more, I have to go back to the data and ask myself what exactly I want to show. But for now...means are great!

#### DEW

##### New Member
It is tempting to say that since it it so secret what it is, I am not going to bring any suggestions for how it can be analysed.

Let's see now, I remembered that I had seen that in an other package how to analyse it and a great software to show the graphs for it. I remember that I thought that it was interesting but could not imagine what it could be used for in practice. It can not only be used for Buffons needle problem, can it?

It is the combination of applied examples and analysis schemes that is the strength of this site. But there is to much “dry swim” here!
Lol, but I am afraid my supervisor would kill me if I talked about my research before publishing. Though it is not really a topic of general interest...more a tiny niche...but anyway ;-)

#### DEW

##### New Member
Hello again,

with the help of the literature I got on and am now facing new questions:

My data is not always distributed unimodally, but in many cases multimodal. Therefore the most common tests (Raleigh etc) do not seem applicable. Has anyone suggestions what test is appropriate? I am afraid that a test for comparing means will not help, because the mean is not too helpful if there are, for example, two clusters of data facing in different directions.
Therefore I am now looking for a measure of dispersion and a way to compare how dispersed the data is, say, around a given angle.
I am grateful for ideas.

In a statistic script I found that there is an "angular dispersion" r, which ranges between 0 (uniform dispersion) and 1 (complete concentration). Has anyone a reference for me which gives reliable values for the values between 0 and 1, so that I can say "With a dispersion of r=0.7, the data is still unimodal...."? Or does this not work?

#### GretaGarbo

##### Human
I did not know when one could apply the theory of “angular data”. Not only for the angular orientation of a birds nest (I assume) round a tree.

But I realized that events that happen in regular cycles, like over the day-and-night cycle or the yearly cycle, could be interpreted like angular data. Just like a sine curve repeat it self again and again, the year dates can be scaled to 0 to 360 degrees.

Examples could be: “at what time did you wake up today” or at what time did the first raptor arrive in the spring? (I assume that at least eagles are migratory birds.)

Does anyone else have application examples?
- - - -

“Lol, but I am afraid my supervisor would kill me if I talked about my research before publishing. “
Dew does not want to cooperate with us. I can understand that his employer (the KGB?) thinks of it as very secret. But right now he is talking about his research before publishing.

For other data I have speculated about if a user could add a constant and maybe scale it by multiplying with a constant. (Such linear transformations does not generally affect the statistical properties.) Also a random sample of the data could be selected. For angular data maybe the values can be “shifted” like 20 degrees. Such real data would make all discussions more realistic and interesting. Now, most threads are not answered because they are not understood.

(Maybe Dew could estimate bi-modal distributions (maybe with the EM-algorithm). That would be a little bit difficult. But why not be an ice-breaker?)

#### ichbin

##### New Member
The specific test you want is called the Kuiper test. It is a variant of the Kolmogorov-Smirnov test suitable for circular data.

#### DEW

##### New Member
Thank you for your suggestion. Could you give me more info about the test, because I find the search on google does not enlighten me...
Is the Null-Hypothesis that the data is uniformly distributed? And is therefore the Null-Hypothesis accepted if the p-value is significant?
SOrry for these dumb questions, but I can't find any references to the Kuiper Test

#### ichbin

##### New Member
The null hypothesis is that the data is distributed according to whatever distribution you care to specify: a uniform distribution, a normal distribution or circular normal distribution, etc. You specify the CDF of the comparison distribution as an input to the test. There is also a two-sample variant that directly tests whether two samples have been drawn from the same distribution without you having to specify what that distribution is. The null hypothesis that the distributions match is rejected if the upper-tailed probability of the test statistic value is too large.

You can find more on the Kuiper test in Numerical Recipies (http://www.amazon.com/dp/0521880688/), a book which is less well-known among statisticians but which I would expect to be on every working scientist's shelf. There is a Wikipedia article on Kuiper's test (http://en.wikipedia.org/wiki/Kuiper's_test) which describes how to calculate the test statistic V. The circstats package for R implements the test (http://cran.r-project.org/web/packages/CircStats/index.html), as dpes the Meta.Numerics package for .NET (http://www.meta-numerics.net/), and there is code for it in the Numerical Reciepies book.

#### DEW

##### New Member
great. that helps a lot! I came across another test, the v-test, where I can test against a predicted mean.
now I think I can solve all issues.
thanks to everyone who replied to my thread!