Is it possible to use the twin ACE model with paired variables (only one value for each twin pair)?
In a word, no. Such variables are inherently 100% shared environment (C) because regardless whether a twin pair is MZ or DZ, they still correlate perfectly for that variable.
Of course, it is possible that the variable in question has been poorly measured and is not, in fact, perfectly correlated between twins. Without the systematic error that gives both twins the same score, the variable in question might follow any pattern of ACE proportions.
Thank you for your response. Just to clarify: would it make a difference if the value was a measure of similarity between participants, and if we had that measure between every participant in the study?
Thank you again!
Yes, we could do something with a measure of twins' within pair similarity. This would not be as informative (i.e., have as much statistical power) as having separate measures on the two twins. By analogy, it is like the linkage analysis via regression of regressing intra-pair differences in phenotype on identity-by-descent sharing at a locus. A negative regression coefficient (high sharing of genetic identity by descent predicts low intra-pair phenotypic difference) is interpreted as evidence of linkage, but this approach is less powerful than regression of both pair sums and pair differences. The reason for this is that intra-pair differences (or similarities) differ in their informativeness according to where on the distribution they are found. Thinking of the elliptical scatter plot of pairs' scores, a large intra-pair difference is less likely to be observed for pairs at the tips of the ellipse than it would in the center (near the mean) of the distribution.
The following is conjecture. Your best approach is probably what Mike just said.
Similarity scores immediately lead me to think of multidimensional scaling (MDS). You could interpret similarity variables as distances. This is obviously more useful if you have multiple variables or constructs to analyze, each measured once per pair. This is not terribly useful for the OP's situation.
More usefully, one could do an ACE model with a single univariate measure per twin pair if and only if that similarity/disparity measure is transformable to the cross-product of the twin's individual responses. If your variable x can be considered equal to y*z, where y and z are the centered scores for twins 1 and 2, then dividing sum(x|mz)/n_mz is proportional to the mz_correlation, while sum(x|dz)/n_dz is proportional to the dz correlation. If you figured out possible scaling differences (i.e, what would r=1 and r=0 be in this metric), you could define the univariate ACE model, where a=2*(r_mz-r+dz), c=r_mz-a and e=1-r_mz. There are a lot of big ifs, though.