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I'm studying sensitivity analysis. I know that to use PCC and SRC, linearity and/or monotonic must be assumed.

So I'm trying to calculate R² for this, and my question for this is:

  1. Can I use GLM to calculate R²?
  2. Is there a lower bound on the R² value to assume that the model is linear or monotone?

thank you

Dave
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S. Jeon
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  • At https://stats.stackexchange.com/a/13317/919 I provide an example where $R^2$ can be made arbitrarily close to $1$ for a radically non-monotone relation (namely, one with infinitely many twists). Contemplating the graphs there might help you answer these and any related questions you might have. – whuber Jul 01 '22 at 13:46

1 Answers1

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  1. GLMs don’t usually care about square loss, so $R^2$ is of limited use. You’re always allowed to calculate a sum of squares, but it might not be meaningful.

UCLA has a nice page on $R^2$-style metrics for logistic regression. The McFadden variant is how I would think about it: as a ratio of loss functions (or likelihoods).

  1. $R^2$ can be very low without violating linearity, and $R^2$ can be very high without having linearity. For the former, imagine $\hat y_i =x_i+\epsilon_i$ with a very high variance for each $\epsilon_i$. For the latter, imagine slight curvature, such as a $y_i=x_i^3+\epsilon_i$ With a low variance for each $\epsilon_i$. Add in some lower-order polynomial terms to make that non-monotonic, if you want.
Dave
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  • Thank you for your comment Dave! Can I ask one more? Thus which parameter is best thing for determine whether linearity and monotonic or not? Can you recommend it? – S. Jeon Jul 04 '22 at 07:26
  • @S.Jeon I think that’s a good topic for a new question to post (e.g., “Sure, we can plot regression residuals, but how do we quantify the extent to which a relationship is linear or monotonic?”). It might be best to post two questions, even, one for linearity and another for monotonicity. – Dave Jul 04 '22 at 09:45