Why doesn't Prism compute R² as part of Deming regression?

 Prism offers Deming linear regression, which fits a straight line when X, as well as Y, includes experimental error. In contrast, standard linear, and nonlinear, regression assumes that X is known precisely and all uncertainty (or scatter or variability) is in the  Y variable.

In Prism's Deming dialog,  you specify whether X and Y are in the same units with equal uncertainties (variation). If you choose this option, Deming regression minimizes the sum of the squars of the perpendicular distances of the points from the line. This is also called orthogonal linear regression. 

When Prism performs Deming regression, it reports the slope and intercepts with confidence intervals, and reports a P value testing the null hypothesis that the slope is really zero.

Prism does not report any measure of goodness-of-fit with Deming regression, and so does not report R² value. The reason is that we have been unable to find any paper or text that would explain how to compute or interpret such a value. In ordinary linear or nonlinear regression, R² is the fraction of the variation that is accounted for by the model. But with Deming regression, this definition doesn't really make sense, and it isn't obvious to us how to extend it. 

Please write to us if you know how to do this.