MLR Models: Log Transformation Interpretations and Collinearity

#1
Hi everyone,
According to this link,
https://data.library.virginia.edu/interpreting-log-transformations-in-a-linear-model/,
"Only the dependent/response variable is log-transformed. Exponentiate the coefficient, subtract one from this number, and multiply by 100. This gives the percent increase (or decrease) in the response for every one-unit increase in the independent variable. Example: the coefficient is 0.198. (exp(0.198) – 1) * 100 = 21.9. For every one-unit increase in the independent variable, our dependent variable increases by about 22%."
However, I'd like to know: What is the formula that gives the percent increase or decrease in the response for every one-unit decrease in the independent variable, when only the outcome variable is log-transformed in a multiple linear regression model? Also, if there are two similar temperature variables that are just at different time points in the same multiple linear regression model, could there be issues of collinearity? What about: if there are standardized height variable and standardized body mass index variable (which includes height in its calculation) in the same multiple linear regression model, could there be issues of collinearity?
Any and all help would be very much appreciated. Thank you.
 

hlsmith

Less is more. Stay pure. Stay poor.
#2
Welcome to the forum!

You can look at tolerance and VIF stats for collinearity, to see if it may be an issue. Collins a rich doesn't impact estimate, but impacts SEs.

I will note that formula is an approximation, there is a better formula. Will post tomorrow. If I followed, your example is the same thing but for a decrease, correct? Just put a negative sign in front of the estimate.
 

hlsmith

Less is more. Stay pure. Stay poor.
#4
I usually reference the following article for this:

Barrera-Gomez, Basagana. Models with transformed variables: interpretation and software. Epidemiology;26(2):e16-e17.