Thanks Andrew, Your measure sd(y|x)/sd(y) (or 1 minus that) is what I was suggesting. The reason for my post is that I have an Excel package that I jointly wrote and I am planning to change the standard summary of model accuracy to (a) adjusted r, not squared (b) sd(y|x) and (c) 1-sd(y|x)/sd(y).

Another point of confusion for students is that for non-regression models the standard relation does NOT hold. Yet, standard pacakges report the r^2 statistic calculated as 1-MSE/VAR(Y). I find good students calcualte the correlation of fitted and observed and get a different answer. I then have to explain to them that the softeware calculates it using an alterantive formula that is known to be wrong for the non-regression model!

It makes our profession look stupid and quixotic. At the very least, we could label it variation explained rather than r^2.

Hi Chris,

I imagine that this application comes from the irrational passion for the additivity of the sums of squares that are associated with model components, and its triumphant but irrelevant invocation of Pythagoras.  

R^2 is a sad example of preferring the elegant answer to the wrong question over the messy answer to the right question.  I have found it more useful to report and compare, sd_y to sd_y|x.

In its sacrifice of interpretability in favour of geometry, R^2 is little more than Theory virtue signalling, not unalike the worship of the canonical link (or, indeed, the conjugate prior); canonised for algebraic elegance at the expense of statistical utility. 

Warm wishes,

Andrew