In this paper the author explains why using two standard deviation (sd) is better than using one sd in scaling the regression inputs. I have a couple of question regarding that.
1- What does it mean that the coefficient of a binary variable is directly interpretable as the comparison of the 0's to the 1's?
2- Here the author argues that because a binary variable with equal probability has mean 0.5 and sd 0.5, hence the range of the variable [0-1] is 2sd. So to keep the binary variable as it is we also need to scale the other variables by 2sd. In case the binary variable doesn't have an equal probability say, 0.9 for 0's and 0.1 for 1's we would still keep the 2sd rescaling. If I understood it correctly the reason is that in this case our samples of the binary predictor has more zeros and the coefficient of the regression represents (based on the definition) the mean change in the response given a two sd deviation change in the predictor. And since the probability of the 0's is higher than 1's, changing two sd would be still close to zero. Now if we want to have the full comparison from 0 to 1 then we need to divide the binary predictor by a larger sd, say 5sd. In the paper says that this could overstate the importance of the predictor in the regression, which I am guessing it means that the absolute value of its coefficient would be higher. But I don't understand how he gets to this point. Is it because now all those 0's in our sample will be 1 and that might affect the coefficient?
3- Also can someone please explain me mathematically why the coefficient of the regression represents the mean change in the response given a two sd deviation change in the predictor?