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Does the appliance of R-squared to non-linear models depends on how we calculate it? $R^2 = \frac{SS_{exp}}{SS_{tot}}$ is going to be an inadequate measure for non-linear models since an increase of $SS_{exp}$ doesn't necessarily mean that the variance is decreasing, but if we calculate it as $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$, then it's as much meaningful for non-linear models as it is for linear ones. I asked a similar question here where I showed that R-squared is no worse for non-linear models

So, what is the particular reason to say that in the case of non-linear models $\mathbf{R^2}$ loses its interpretation of proportion of variance explained? In both cases (I mean linear and non-linear models) we learn by how much your model's variance decreased with respect to its initial (total) variance. If $R^2 = 0.86 \text %$ then your model's variance decreased by $0.86 \text %$ (no matter whether it's linear or not).


EDIT:

  1. $SS_{tot} = \|y - \bar y\|^2$
  2. $SS_{exp} = \|\hat y - \bar y\|^2$
  3. $SS_{res} = \|y - \hat y\|^2$.

Where:

  1. $y$ is a vector of true answers;
  2. $\bar y$ is a vector whose elements are mean of $y$;
  3. $\hat y$ is a vector with our model's predictions.
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  • $\begingroup$ What is $SS_{exp}?$ $\endgroup$
    – Dave
    Commented Oct 11, 2021 at 23:48
  • $\begingroup$ @Dave, Hello! I've edited. $\endgroup$
    – mathgeek
    Commented Oct 12, 2021 at 0:33
  • $\begingroup$ In a linear model, is $SS_{tot}=SS_{res}+SS_{exp}?$ What about in a nonlinear model? stats.stackexchange.com/q/427390/247274 stats.stackexchange.com/questions/494274/… $\endgroup$
    – Dave
    Commented Oct 12, 2021 at 0:38
  • $\begingroup$ @Dave, Yes, I see that. But do you actually calculate $SS_{exp}$ when you use $R^2$? No. The formula you use is $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$ which doesn't care about $SS_{exp}$ at all. It just calculate the difference $SS_{tot} - SS_{res}$ in the nominator and after that divides it by $SS_{tot}$. Why is it worse for nonlinear models, then? We still learn by how much your model's variance decreased with respect to its initial (total) variance. $\endgroup$
    – mathgeek
    Commented Oct 12, 2021 at 0:47
  • $\begingroup$ Doesn’t the “exp” mean “explained”? $\endgroup$
    – Dave
    Commented Oct 12, 2021 at 0:48

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