Estimates of intraclass correlations can be negative, yet they are a ratio of two variances -- the variance of the means of the classes to the variance of the entire set of values. What is a neat (and correct) way to explain this paradox?
p.s. Intraclass correlation refers to a number of quantities, but the question refers to the simplest form, namely, the usual linear (Pearson product-moment) correlation among a set of pairs of values when the order in each pair is arbitrary. That would be the case if we wanted to know the correlation of heights in same sex couples or the agreement of two independent ratings of an exam done by a set of students (where the two raters differed from one student to the next).
Interclass correlation is actually the ratio of a covariance and a variance, therefore it can take negative values.