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 :)