The intelligibility of r or r2 as an effect size statistic: dichotomous variables

There have been differences in the use of the correlation coefficient (r) or the coefficient of determination (r2) for indexing the effect size (see Rosenthal and DiMatteo, 2001; Borenstein, 2009; Elis, 2010, for reviews). I intend to investigate this issue by considering it from the point of view of matching the findings with the implied prediction. In essence my argument follows from a simplification of the correlation coefficient to the case where both variables are dichotomous and where there are equal frequencies of each possible response on both variables. Based on this simplified case, the question is whether the correlation coefficient or the coefficient of determination most closely resembles the actual proportion of agreements (successes) between the two variables after controlling for chance. To flesh out the idea, suppose that there are two variables and each of these is dichotomous and scored 0 or 1. From the point of view of a researcher who believes that the relation between the two variables is important, each case of matching scores (0 on both variables or 1 on both variables) is a success whereas each case of mismatching (0 on one variable and 1 on the other, or the reverse) constitutes a failure. The straightforward way to index the ability of the two variables to produce successes (agreements with respect to zeroes and ones) would be to use the proportion of obtained successes. However, because a 50% success rate would be expected due to chance, this proportion likely would be misleading. I suggest controlling for chance by computing an adjusted proportion of successes or adjusted success rate (SA) using Equation (1) below, where s refers to the proportion of successes and C refers to the proportion of successes that would be expected based on chance alone.