Why is fitlm affected by variable scale?

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Harold Matthews
Harold Matthews on 1 Dec 2021
Answered: Jeff Miller on 1 Dec 2021
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 :)

Accepted Answer

Ive J
Ive J on 1 Dec 2021
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.

More Answers (1)

Jeff Miller
Jeff Miller on 1 Dec 2021
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

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