I have found many useful posts about standardized independent variables and centered independent variables on stats.stackexchange.com, but I am still a bit confused. I am asking you an evaluation of what I have understood. Also, if what follows is not correct, could you please correct me?
- How to standardize. Standardized variables are obtained by subtracting the mean of the variable and by dividing by the standard deviation of that same variable.
- How to center. Centered independent variables are obtained just by subtracting the mean of the variable.
- The reason for standardizing. You standardize variables to facilitate the interpretation of the estimated coefficients when the variables in your regression have different units of measurement. When you want to standardize, you have to standardize all the variables in the regression--which implies you won't get an estimate of the constant (i.e., the B0 or intercept).
- The reason for centering. You center variables if you want to gain a meaningful interpretation of the estimated constant. In this case, you can center the amount of variables you want to; you do not need to center all the independent variables in the model.
- The dependent variable, Y. (plain question) Do you ever center or standardize the Y?
- Natural logarithm utilization. If one or more of your variables are not normally distributed, you can transform them using the natural logarithm. Only AFTER this transformation you may either standardize all the variables or center those that you need to center. In general, whatever transformation of a variable has to happen before standardizing or centering (here I speak about natural logarithm, but you could square a variable or divide a variable by another one, e.g., population/km2)
- Interpretation coefficients standardized variables. "An increase by 1 standard deviation in X1 will increase (or decrease) Y by -number-."
- Interpretation coefficients centered variables. Coefficients of random variables: "An increase in X1 by -number- from its mean will increase (or decrease) Y by -number-." Constant: "It represents the expected value of Y when the non-centered variables are zero and when the centered variables are at their mean."
- Interaction terms. The interpretation of the coefficient of an interaction term should not be problematic, whether you have standardized your variables, or centered them (either only one variable of the interaction or both of them). Basically, the interpretation is that that you normally give to an interaction term (e.g., you are interested in the effect of X1 on Y and X1 is interacted with X2, the total effect of X1 is given by its coefficient + coeff. of the interaction term when X2 is fixed), just remember to contextualize the interpretation following either point 7 or 8, depending on the type of transformation you did.