When is Linear Model not Good despite r^2 close to 1?

[Bacle]

Hi, All:
I was reading of cases in which linear models in least-squares regression were found to be
innefective, despite values of r, r^2 being close to 1 (obviously, both go together ).
I think the issue has to see with the distribution of the residuals being distinctively non-linear (and, definitely, not being normal), e.g., having a histogram that looks like a parabola, or a cubic, etc.
Just curious to see if someone knows of some examples and/or results in this respect, and of what other checks can be made to see if a linear model makes sense for a data set. Checks I know of are Lack-of-fit Sum of Squares F-test and inference for regression (with Ho:= Slope is zero.)

Thanks.

 

[mXSCNT]

Another way - suppose there is overfitting, or not enough data points for the number of dimensions. If you have 100 data points but are using a model with 100 different dimensions it doesn't matter how good your correlation is.

 

[Pyrrhus]

A high $R^{2}$ is not the only important statistic to check. I prefer adjusted $R^{2}$, because the more parameters you add to the former it'll tend to inflate it.

 

[Bacle]

Thanks, Pyrrhus:

What do I then do if the adjusted R^2 is low ? Do I start considering linear models on two-or-more variables, or do I consider quadratic, cubic, etc. models?

 

[Pyrrhus]

You could try adding square terms, and interaction terms, but if the r-squared is still low it might just be that the regressors don't do a good job to explain the dependent variable.

 

[Mapes]

Try this incredible free http://creativemachines.cornell.edu/eureqa" developed at Cornell. I've used it in my own research, rating fits by adjusted r² and Akaike Information Criterion values.

 

[Bacle]

Excellent, Thanks!.