R² and nonlinear regression

Key points about R²

• The value R² quantifies goodness of fit.
• It is a fraction between 0.0 and 1.0, and has no units. Higher values indicate that the model fits the data better. 
• When R² equals 0.0, the best-fit curve fits the data no better than a horizontal line going through the mean of all Y values. In this case, knowing X does not help you predict Y. 
• When R²=1.0, all points lie exactly on the curve with no scatter. If you know X you can calculate Y exactly. 
• You can think of R² as the fraction of the total variance of Y that is explained by the model (equation). With experimental data (and a sensible model) you will always obtain results between 0.0 and 1.0. 
• There is really no general rule of thumb about what values of R² are high, adequate or low. If you repeat an experiment many times, you will know what values of R² to expect, and can investigate further when R² is much lower (or higher) than the expected value.
• By tradition, statisticians use uppercase (R²) for the results of nonlinear and multiple regression and lowercase (r²) for the results of linear regression, but this is a distinction without a difference.

Problems with R² and nonlinear regression

• Use of R² in nonlinear regression is not standard. In linear regression, the R² compares the fits of the best fit regression line with a horizontal line (forcing the slope to be 0.0). The horizontal line is the simplest case of a regression line, so this makes sense. With most models used in nonlinear regression, the horizontal line is not a simple case and can't be generated at all from the model with any set of parameters. So comparing the fits of the chosen model with the fit of a horizontal line doesn't quite make sense. For this reason, Minitab does not  report R² with nonlinear regression and SAS labels the value "Pseudo R²". 
• The R² value can be very high, yet the fit can be essentially useless if all the parameters have very wide confidence intervals. Read why. 
• With nonlinear regression, R² can be negative. Really!
• Computing R² when the points are unequally weighted is tricky, and there doesn't appear to be a standard method. Prism 6 uses a different (better) method than prior versions. 
• R² (and even the adjusted R²) should not be used to compare the fits of alternative models. Why?  Because models that fit very differently as assessed by AICc may have R² values that differ only in the third to fifth digit after the decimal (1). 

Is R² from nonlinear regression useful at all?

The people at Minitab think that R² is not a useful value so don't report it with results of nonlinear regression. 

GraphPad Prism does report the R² (and adjusted R² if you check an option in the Diagnostics tab). Why?  In my opinion, R² is really is useful in only one way, as a reality check for evaluating repeated experiments. Say you repeat an experiment many times with some variations of course) so come to learn that R² values alre always between 0.6 and 0.8. If one experiment gives instead R² of 0.2, you should be suspicious and look carefully to see if something went wrong with the methods or reagents used in that particular experiment. And if a new employee brings you results showing R² of 0.95 using that same system, you should look carefully at how many"outliers" were removed, and whether some data was made up. 

1.    Spiess, A.-N. & Neumeyer, N. An evaluation of R2 as an inadequate measure for nonlinear models in pharmacological and biochemical research: a Monte Carlo approach. BMC Pharmacol 10, 6–6 (2010).