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Box Cox model R2 adjusted for original response

konstantin

Hello!
I built a multiple regression model that uses Box Cox response transformation and jmp reported R2_adjusted for my model, but now I need to find R2_adjusted for original not transformed response. Can I use standard R2_adjusted formula? I need somehow include Box Cox Lambda parameter into model DF. Should I just increase number of my model parameter by 1?
Thanks!

Accepted answers

Mark_Bailey

I have to think about this case. 'All bets are off' usually when you change the response between models and use most criteria such as AICc. They are only valid when you change the model.

But adjusted R square is a ratio of mean squares (variance) and I still think that the DF for the mean square does not include transformation parameter. You could also think about it this way: the original data also uses a lambda parameter for the Box-Cox transformation. Lambda is 1.

All comments

Mark_Bailey

Just use the same linear predictor with the original response.

konstantin

But my Box Cox Lambda came as optimization parameter from the previous model (function of jmp fit platform), how it is not a parameter?

Thanks for you response!

Mark_Bailey

It is a 'hyper-parameter' used in the optimization of the transformation. You fix the model terms and determine lambda. Now you fix lambda to determine the model parameters. This situation is common with predictive models that involve additional parameters of the fitting method, such as learning rate, et cetera.

Imagine that, independent of the model selection, it is determined that the measured response 'diameter' would not be linearly related to the predictors. The measured response is transformed as Pi() * (diameter/2)^2. The transformation and any 'parameters' it might use are not part of the model selection.

R squared is an estimate of model accuracy. Adjusted R squared accounts for the parameters in the linear predictor. That is all.

konstantin

Mark, this sounds correct when jmp calculates R2_adjusted for BoxCox Model. but I need new R2_adjusted when I do back transformation to get the original response, it is like Pi is not a constant but variable that came from jmp, should I still discard it from DF calculation?

The reason I need R2_adjusted is to compare with other model I did without Box Cox transformation.