Josh Tabor wondered why the TI-83 reported r and r^2, or R^2, for certain non-linear regression models. Dick Scheaffer replied as follows:
> With regard to Josh Tabor's question, the TI treats only nonlinear models > that can be linearized by an appropriate transformation. The r and R^2 terms > are calculated (I think) from the linearized version of the data. When the > results are transformed back to the original scale, the r and R^2 are > inappropriate measures. In my opinion, they should not be given. > > It should be pointed out to students that fitting a least squares line > to the transformed data is not the same as fitting a nonlinear > model directly by least squares.
which, of course, is from an authority on statistics. Perhaps it could use a bit of elaboration, however.
First of all, the TI-83
(a) reports r and r^2 for LinReg, LnReg, ExpReg, and PwrReg
(b) reports R^2 for the three polynomial regression models QuadReg, CubicReg, and QuarticReg
(c) reports none of the above for Logistic regression and for SinReg
The regression models in (a) all can be linearized by an appropriate transformation. For example, the model for ExpReg is y = a*b^x. Take the log of both sides and you have ln y = ln a + x(ln b). This equation is linear in the variables ln y and x. Suppose you have x in L1 and y in L2. So put ln y in L3 and do a LINEAR regression on L1 and L3. You will find that the r and r^2 reported from that regression is precisely the r and r^2 reported by the TI-83 when you do ExpReg on the original variables (which is what Scheaffer said above--the "I think" part). I have verified this for all three non-linear models in (a).
As for (b), polynomial models are just special cases of multiple regression, where R^2 makes sense, because R^2 = SSR/SST is the proportion of the total variation SST in y that is "explained" by the variables X, X^2, X^3, etc. But sqrt(R^2) makes no sense here, which is what Josh indicated.
The confusing thing, perhaps, is that the TI-83 gives the r and r^2 for the LINEARIZED model in the window that reports the parameters of the NON-LINEAR model. The last sentence in Sheaffer's first paragraph suggests that this may not be wise.
As for (c), there are no linearizing transformations for these models (none that I know of, anyway), so no r, r^2 or R^2 are reported.
I think I have that right. If not, someone will come to the rescue...
============================================== Bruce King Department of Mathematics and Computer Science Western Connecticut State University 181 White Street Danbury, CT 06810 (email@example.com)