Re: r-Squared Question

Reef Fish wrote: 

Jerry Dallal wrote:

I wrote:


It depends how you defined R2.  If you define it as the square of the
correlation between observed and predicted, then it's a weakness.



What do you mean "a weakness"?

For OLS fitted regression, R^2 is ALWAYS the correlation between
the observed Y and the fitted Y.


You have to read the thread.

The values "Y-Yhat" below are my own calculation. The rest is from the person posing the question, to wit: The correlation between Y and Yhat is 1. If one defines R^2 for any model (not necessarily linear LS) as the square of the correlation between observed and predicted, then R^2 for this example is 1. Was that an indication of a weakness in R^2 as a summary measure?

My point was that while, as you say, "For OLS fitted regression, R^2 is ALWAYS the [square of the] correlation between the observed Y and the fitted Y.", R^2 is not defined that way. Rather it is usually defined as 1-ResSS/TSS (or RegSS/TSS), which, for OLS, *happens* to be the square of the correlation between Y and Yhat. [I realize there are *many* way to approach R^2.] If one uses the formal definition of R^2 to calculate it for this example, R^2 turns out to be -0.03, which says the problem is with the model, not R^2.




However if you define it as 1 - ResSS/TSS, then, for an arbitrary model
fitting procedure, R2 isn't even constrained to the interval [0,1],
since ResSS might exceed TSS.

Here
> X   Y     YHat Y-Yhat
> 1   101    97    4
> 2   102    99    3
> 3   103   101    2
> 4   104   103    1
> 5   105   105    0
> 6   106   107   -1
> 7   107   109   -2
> 8   108   111   -3
> 9   109   113   -4
> 10  110   115   -5

Here, TSS=82.5 and ResSS=85, so R^2 = 1-85/82.5 = -0.03, and the fitted
line predicts worse than always using the sample mean.


.

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