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I have pairs of measurements and need advice to select a measure of association between the measurements. The special aspect that has me confused is the symmetry: there is no reason to allocate a particular measurement to variable x rather than variable y.

I am working on hermaphrodite slugs, which swap spermatophores reciprocally when they mate. I have the sperm counts from both spermatophores from 25 couples. If I plot the sperm count of one arbitrarily-chosen partner against that of its partner, the 25 points show an obvious correlation, which of course I could measure using a standard Pearson correlation coefficient r. But if I randomly choose again which partner to plot on the x and y axes, I will get a different scatter of points, and a slightly different correlation coefficient. This does not seem totally satisfactory. Graphically a fair solution is to plot each individual on both axes, so that I end up with a scatterplot of 50 points, each pair represented by 2 points symmetrically placed either side of the x=y line. I could calculate a correlation coefficient based on these 50 points; but there is an issue with calculating the statistical significance, since a sample size of 50 is inappropriate when there are really only 25 independent data points.

I could calculate the significance of an r based on 50 non-independent points using simulation, generating a distribution for this r under the null hypothesis of no association by randomly reallocating partners between pairs. Or I could randomly allocate which partner of each pair I allocate to variable x, calculate an r based on the 25 independent points, repeat this 100 times, take the mean value of r, and then use standard tables of statistical significance. But is there a neater off-the-shelf solution?

I might want to go further to test whether factoring out the effect of other variables would explain away some of the correlation (e.g. late in the year, both partners might allocate more resources to sperm production). So it would be ideal to have a method that could extend to that sort of analysis.

Thank your for your advice.

      John Hutchinson
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I think I have found my own answer: the intra-class correlation coefficient based on a one-way analysis of variance. Helpful introductions are:

Müller, R. and Büttner, P. (1994). A critical discussion of intraclass correlation coefficients. Statist. Med., 13: 2465-2476. doi:10.1002/sim.4780132310

and

McGraw, K. O., & Wong, S. P. (1996). Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1(1), 30–46. https://doi.org/10.1037/1082-989X.1.1.30

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