# Should I use the angle or the slope in my analysis?

In an aquatic vegetation study, one of the covariates of interest is the slope of the bank. This slope can be expressed as a ratio (1:1 or 1:5 for example) or as an angle (45 or 11 degrees same example). Which one should I use in a regression analysis, or does that depend on the type of regression?

Because there is NO linear relationship between both, and because angle caps at 90 degrees I would say it does matter, and am inclined to use the ratio. Am I right?

• Angles don't necessarily stop at 90 degrees, but the ratio 1:0 is undefined. I think asking which will be more interpretable is more important here. – gung - Reinstate Monica Nov 16 '15 at 22:01
• @gung posted the most important details here while I was writing my longer answer. – Nick Cox Nov 16 '15 at 22:15

Let's think about it analytically. Your slope is evidently from examples the tangent (rise/run). That term is better to avoid ambiguity, as slope is sometimes expressed as sine (rise/hypotenuse). We should think about both.

I mix here statistical considerations and some comments from experience with slope angles (e.g. Cox 1990).

Essentially tangent is a bad idea because it explodes as angles approach vertical and indeed is indeterminate for vertical angles. I can't see your banks, but I have no problem imagining banks that might be vertical or nearly so. Even if really steep banks do not occur in your dataset, it seems a bad idea to use a measure that could behave so badly.

Sine has a small virtue insofar as the downslope component of gravity is proportional to the sine of the angle, a fact directly relevant to whether particles will roll or slide off the bank and indirectly relevant to other kinds of slope failure. The mechanical interpretation here doesn't really trump a rather more mathematical (and statistical) fact. For quite a wide range of angles, the curvature of the sine function does not really bite and the approximation is good that sines are proportional to angles, which certainly implies a linear relationship. Traditionally in mathematics this is explained as an approximation that sines equal small angles (measured in radians), but an approximation of proportionality works well over a wide range. For example, if angles are uniformly distributed between $0$ and $45^\circ$ the correlation between angle and sine is about $0.999$. The correlation could be even higher if most of the angles were small. It could equally be lower if angles extended over a wider range.

For your data, plot sines versus angles and calculate the correlation to check on how well the linearity assumption works. It's likely that the approximation is so good that it really doesn't matter whether you use angle or sine. Coefficients in regression models, etc., will differ but not explanatory power. Consistency with previous literature in your field may be decisive here, especially if you have no interest in recapitulating this little analysis. Finally, if sines are unfamiliar, then it usually shouldn't matter if you use angles instead.

You don't mention overhangs, so it is convenient to follow suit.

Cox, N.J. 1990. Hillslope profiles. In A.S. Goudie (ed.) Geomorphological techniques (2nd edition). London: Unwin Hyman, 92-96. Translated as 1998. Hangprofile. In A.S. Goudie (ed.) Geomorphologie: Ein Methodenhandbuch für Studium und Praxis. Berlin: Springer, Berlin, 100-106.

• (+1) Many river banks not only approach vertical, they actually exceed verticality: this is the phenomenon of undercutting. It occurs both in young, small, mountain rivers as well as large, mature, oxbowed rivers of the alluvial plains. – whuber Nov 16 '15 at 23:07
• @whuber. Thanks and agreed; that's what I was referring to under the heading of overhangs. (Irrelevant to the question, but my geomorphologist persona is obliged to comment that the terminology of young and mature for rivers is some decades out-of-date, as it rests on a dubious analogy between the development of an organism and spatial changes downstream..) – Nick Cox Nov 17 '15 at 0:18
• Thanks for your excellent answer. Although the banks I was referring to aren't river banks but banks of lakes/ditches in peatlands (Is there a English word for the Dutch 'legakker'?), your answer applies to these too. – RHA Nov 17 '15 at 12:48
• You are naturally correct: you didn't say rivers, and I neglected the limnological interpretation, despite a brother who works on lakes. – Nick Cox Nov 17 '15 at 14:49