# Variables which sum to a constant - ANOVA, MANOVA or none of these?

I asked a group of subjects to make a series of 12 binary choices regarding preferences.

Let's say for arguments sake, these were between ugly (ug), attractive (att), and neutral (neut) faces. Hence, we have 4 ug vs att, 4 ug vs neut and 4 att vs neut choices. For each subject I summed the number of times each face was chosen. Hence, I have a 3 column table comprising a score (max 8) for Att, Ug and Neut for each subject. Each row sums to 12 hence the variables are negatively correlated.

My questions:

• Are attractive faces preferred to ugly and if so:
• Is this driven by an attraction to att or an aversion to ug or both? - this is why we have choices with the neutral faces.

I originally thought to do a repeated measures ANOVA followed by post hoc tests to look for differences in ratings but i'm wondering if the fact that the DVs all sum to a constant is problematic because in essence the third variable - say $neut = 12-(ug+att)$. If so, is MANOVA the way to go, or how about chi-square?

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