Re: r-Squared Question

Reef Fish wrote: 

Jerry Dallal wrote:

Reef Fish wrote:

Jerry Dallal wrote: 

Rather it is usually defined
as 1-ResSS/TSS (or RegSS/TSS),



No.  But it's equivalent to the usual RegSS/TotSS because
RegSS + SSE (your ResSS) = TotSS.


Isn't that what "or" means, as in "3/6 or 1/2"?


If one uses the formal definition of R^2
to calculate it for this example, R^2 turns out to be -0.03, which says
the problem is with the model, not R^2.



This is your ERROR, Jerry.

The definition of Multiple R^2 CANNOT lead to a negative value!


I'm not sure what the issue is here. R^2 cannot lead to a negative value in the land of sanity and least squares.

The poster was getting an R^2 of 1 for his ill-fitting model, not obtained by any least squares procedure, by calculating it as the square of the correlation between observed and predicted and thought it showed a weakness in R^2 as a summary measure.

The problem was not with R^2, but with the poster's definition of it. One can calculate RegSS, ResSS, and TotalSS. The poster's model was worse (in terms of least squared errors) than no model at all, that is, ResSS was greater than TotalSS. If one blindly plugs these numbers into a formula for R^2 one gets -0.03. The point is that R^2 is not, in fact, deficient for suggesting the model is perfect. Rather, it is saying that something is very wrong with the model because it gives a negative value where such a thing should be impossible. One would hope that a measure of goodness-of-fit would go off the scale when assessing a model that(a) was derived under methods different from those the measure was designed to assess and (b) is worse than no model at all.
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