You need a few angle measurements at each site. Then, to obtain the mean angle for each site, you calculate the arc tan (tan-1) of the ratio of the mean of the sine (S) of all angles to the mean of the cosine (C) of all angles i.e. the arc tan of the relative vector length, or mean vector direction. You can then compare resultant vector STRENGTH using (√ [S2+ C2])/n, where θ = angle and n = no. of angles. The '2's in the equation mean 'squared' btw. I used these equations in my PhD for seedling directions :)

Alternatively, you can break your directions into a north-south and an east-west aspect component.You require assignation of angles or compass directions in degrees. Depending on your study system, one of these slope aspect components might be of greatest interest (for example if working at a temperate latitude, most likely you would expect the degree of N-S orientation to matter more biologically, due to the difference in solar incidences). If you take the cos (angle) this will give you the N-S component as a numeric form ranging from 1 to -1, with 1 being N (0 or 360 degrees) and -1 being S (180), zero indicates a compete east or west exposition. The Sin(angle) is the E-W component, again ranging from 1 (East) to -1 (West). Then you can run correlations and linear regressions with your data.

I agree that compass direction would be a great way to go. However, like percentages, the distribution will lack tails. AN arcsine transformation would deal with that except that there's a bigger problem: the "ends" of the distribution are not true ends because 360 wraps back around to zero. A solution may be to use a non-parametric approach. We used such an approach to look at direction traveled by zebra mussels relative to a light source (or a strobe light). We compared observed direction to the null distribution of equal travel in all cardinal directions. This eliminates the issue of the wrap around: Article The Effects of Strobe Lights on Zebra Mussel Settlement and ...

Furthermore, because a chi square or similar test treats each direction as a category, there is no assumption of an order from 0 to 360 degrees....and therefore no problem with 0 falling next to 360. You could ask if each state is evenly distributed across all cardinal directions. Your expected numerical observations would be based on the number of sites in each direction. For simplicity, imagine you have 50 north-facing sites, 25 south-facing; 12 east; 13 west. And you have 200 observations of morphological state X. Your expected values would be 100 North; 50 South; 24 East; 25 West. Your observed would be your raw data; simple chi square test from there.

Having said all of that, there is a whole field devoted to "circular statistics" or directional statistics. We went with simplicity in our paper rather than delving into all of that.

Best of luck with your study.

Dear colleagues, many thanks to all of you. Now I have a clear picture on this matter.

Hi, look at

@book{pewsey2013circular, title={Circular statistics in R}, author={Pewsey, Arthur and Neuh{\"a}user, Markus and Ruxton, Graeme D}, year={2013}, publisher={Oxford University Press} }

It's a good reference an manual.

Enjoy :-)

Zar (Biostatistics) has a chapter on Circular Statistics. It likely includes some of the transformations high-lighted above.

Because 0 wraps back to 360, direction is problematic, even with appropriate transformations, as Declan McCabe said above. However, ranks are still a possibility. In our part of the world (northern hemisphere, semi-arid environment) it seems reasonable to rank directions like this: north-facing (NE through NW) - least likely to be water-stressed (for plants); followed by east-facing (NE through SE), then west-facing (NW through SW), and finally south-facing (SW through SE). West-facing is more likely to be water-stressed than east-facing, because it gets sun in the afternoon sun when air temperatures are higher. Obviously the southern hemisphere would be different.

You need Fisher (1995) "Statistical Analysis of Circular Data" Cambridge University Press (appropriate routines are implemented in R). The data distributions on circles and spheres are different from those on lines or flat surfaces so different tests are required. Transformations do not help you - they just transform one flat surface distribution to another flat surface distribution.

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