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

Question

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?

Answer 1

Neither ANOVA nor MANOVA would work here because of the fact that your variables are dependent. Though I would say that the way you set up your variables seems a bit unusual for trying to solve this problem. My recommendation would be to, instead of summing up the number of times each participant selected the each type of face in general, count the number of times each participant chose the "more attractive option" in each decision and then use a chi-square test to compare the counts in the ug vs. neut decisions to the counts in the neut vs. att decisions, conditioned on the count of the ug vs. att decision (somewhat similar to a Cochran–Mantel–Haenszel test with a series of tables of ug vs. neut and neut vs. att counts indexed by the ug vs. att count).

If you need clarification, I'll be willing to work some more on this problem.