When to not use R squared I recently graduated graduate school and am looking for a proof on R squared.  Specifically when to not use it.  I really remember a professor impressing upon me multiple times not to report R squared due to the fact it is unreliable.  I did a proof on the matter, I just do not remember it.  I've looked all over the internet, and can not seem to remember the solution.
Of course do not use R square when the data is non-linear.  And do not add variables in order to increase R square.
Here are some links to back me up:
http://jackman.stanford.edu/classes/ssmart/2011/rsquared.pdf
Specifically section 3.1 #6
http://blog.minitab.com/blog/adventures-in-statistics/why-is-there-no-r-squared-for-nonlinear-regression
Do not use R sq for non-linear regression
 A: I don't know if you should never use it, but like all metrics it simplifies things and thus has its limitations.
One caveat is that you should always use adjusted R-squared, not R-squared, as the latter will increase every time you add a new explanatory variable. This behavior may have been what you remember as a "proof" that you should never use it. Adjusted R-squared tries to compensate for this.
The courses that I have had teach that you should look at the adjusted R-squared, and then look at how the residuals are distributed with respect to the explanatory and explained variables, QQ-plots, etc. This will give you a sense not only for how well your model is doing, but if you are maybe missing some important explanatory variables, or if you your predictions are failing in particular areas of your parameter space. Essentially you are testing some hypothesis about your model, your data, and the residuals and there are different ways you can approach this, for example "eyeballing it" (like I describe above), or using sophisticated tests of normality.
Modeling is as much an art as a science, so any statement with "always" or "never" in it is bound to be wrong.
