Variables A and B are shown to have a weak to modest correlation (r = .31, r2 = .096). Suppose I have good reason to believe that the correlation is negatively affected by heteroscedasticity, knowing from previous research that those with high levels of A tend to have high levels of B, but (described here in a visual sense) the rest of the distribution opens up like a funnel as one goes down the A and B axes, though those with low levels of A also tend to have low levels of B, but the effect is not as apparent. As a result, I do a t-test to compare the highest quartile of A with the lowest quartile of A with B as my IV. The effect size correlation (biseriel r = .69, r2 = .476). Knowing that they are two different correlation coefficients, Could I compare these two coefficients of determination, or is that statistical apples to oranges?
Correlation coefficient (r) is the measure of linear relation between two variables (A and B). Low value of r only means that the relation can not be explained completely by a linear measure. Funneling out of data indicates that one of the variables is not Gaussian (normal). Most probably, it has come from either binomial or multi-nomial or log-normal distribution. You should test the data to check which distribution it has come from. If it is binomial, you can use logistic regression. If it is multinomial, you can use multinomial logistic regression. If it is log-normal, then take log-transform of the variable and then estimate correlation coefficient (r).
Based on your descriptions, the relationship between A and B is nonlinear and therefore you can not use Pearson correlation for comparing degree of dependencies between them. You have two choices:
1- As Somenath suggested, try to find an operator (i.e log) to transfer one of two variable to make their relationships linear (close to linear)
2- Using nonlinear criteria such as Ranking correlations, or joint probabilities to investigate the real inter-dependencies.
"As a result, I do a t-test to compare the highest quartile of A with the lowest quartile of A with B as my IV. The effect size correlation (biseriel r = .69, r2 = .476)"
I don't this is a sensible procedure - the correlation is artificially inflated by selecting the extremes of the distribution. Transformations or a more robust measure are more sensible approaches.
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