Why is fitlm affected by variable scale?

Dear all,

My statistics is pretty solid and my understanding is that if you fit a linear regression the scale of the X and Y variables should not affect the resulting p-values. I am running fitlm on some data (see demo and data attached) and changing the scale of the variables by transfiorming them to z-scores has a profound effect on the resulting p values. In the attached (Demo.m) code I fit two models with the same model design on the same data (in the attached 'Data.mat' file). The only difference is that for model 1 the X and Y variables are normalised to z scores and in model 2 they are not. I then scatter the p-values. You can see in the upper left corner that two p values that were not significant for model 1 become signfiocant for model 2.

Sorry I cannot get the demo code embedded in this question, so I have attached it. If anyone has any insights into this that would be great :)

[Devendra]

Thank you very much for detailed explanation. I am getting wierd results of fitlm function used in my matlab code. I am attaching the code and input data file and request you to kindly have a look on code and suggest me how to get the correct results.

I would appreciate your kind cooperation.

Deva

Answer 1

[Ive J]

Well, the real question would be why not?

You have introduced interaction terms to the model. Two models test different hypotheses (except for the interaction terms). You can find a good explanation here. Clearly, when you remove the interaction terms, all t-stats would be the same for both models.

[Devendra]

thanks for valuable information.

Answer 2

[Jeff Miller]

Your understanding is correct for linear regression but your model is nonlinear because of the interaction terms. Consider:

zX = zscore(X);
corr(X(:,1),zX(:,1))
ans =
       1
corr(X(:,1).*X(:,2),zX(:,1).*zX(:,2))
ans =
       0.2421