Converting a circular outcome variable to a linear one I have a problem where I want to measure deviations from zero degrees. This outcome variable is a circular measure, since a deviation of -180 degrees is equivalent to 180 degrees.
However, I don't want to complicate my model (using a linear mixed effects model) by using circular statistics, so I was wondering if I can use the absolute deviation expressed as a percentage of 180?
So for instance, -180 and 180 degrees would both give 100% deviation, while 90 and -90 would both give 50% deviation. Is this a legitimate fix? Namely, would their be any caveats I need to be aware of by 'linear-izing' my circular outcome measure?
**Edit: To give more context into my problem. I am looking to predict peaks in a real-time signal. I make a prediction and see how far off I was to the closest peak (0 degrees corresponds to peaks, while 180 degrees corresponds to antipeaks). I'm interested in computing how 'accurate' my different prediction strategies are, so I was considering just looking at deviations from zero degrees. I'm not sure whether this is a completely circular problem in the first place, since an outcome measure of is bounded between 180 and -180 degrees.

You cannot validly linearize a circular measure which spans 360°, assuming the circularity of that measure is valid.
Any transformation which "linearizes" a circular measure must necessarily privilege some value as being maximally linearly distant from some other value by virtue of lying on the other side of whatever point the transformation uses as its point of either "unwinding" or "flattening" the circle. This maximal linear distance will be a fiction created entirely as an artifact of the transformation, and will not exist in the original circular measure. The same holds true in a continuous circular measure of degrees, radians, etc. In fact, by carefully choosing the privileged point in your transformation function, you could probably fabricate any relationship you wanted between outcome and predictor, by ensuring that certain linearized values become either the largest, smallest, or middlemost.
Modular measurements—whether discrete or continuous—have important characteristics which have no representation in linear forms. This is why treating the modular numbers adorning the face of a clock make no sense as truly (linear) natural numbers. For a simple example, as we actually read the 12 hour clock, $1 - 12 = 1$, $3 - 9 = 9 - 3 = 6$, etc. But linearizing the clock's hours into integers would mean that $1 - 12 = -11$, and that $3 - 9 \ne 9 - 3$.
