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I asked a question yesterday (Better function to fit log-like data?) and the accepted answer got me thinking.

For non-linear data, Is it better/more recommended to asses the goodness of fit on the original data or on the transformed linear data?

Here's an example of what I mean using the data from my previous question. The $R^2$ metric says that the logit fit is better on the transformed data (top row), but the MSE says that the arctan fit is better on the original data (bottom row).

enter image description here

Gabriel
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  • What does the $R^2$ say on the original data and the MSE say on the transformed data? – wzbillings Mar 27 '24 at 15:23
  • These are related but not identical fit statistics, your question seems to boil down to whether you want to use $R^2$ or $\text{MSE}$ to decide which is the best one. – PBulls Mar 27 '24 at 15:27
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    The MSEs measure entirely different things and therefore are not suitable for comparison. After all, if (say) the original variable is a length, then its arctangent is an angle, so you are comparing lengths to angles, which is nonsensical. The $R^2$ are irrelevant, too, because they measure a combination of goodness of fit and range of the data, while the arctangent likely changed the ranges substantially. Thus, you ought to be asking how to assess goodness of fit rather than asking which of two meaningless methods to choose from! – whuber Mar 27 '24 at 15:58

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