# How to interpret transformed variables in multiple regression?

I am a Danish medical doctor doing multiple regression on a clinical study, but I am running into some challenges as both independent and dependent variables have been log-transformed due to skewed distribution.

It is not too hard to find information online when it comes to interpreting my log-transformed results in a multiple regression when only the independent variables are transformed.

How do I interpret the results when both the dependent variable and the independent variables are log-transformed ?

What would be the easiest-to-interpret transformation (log, Log(1.1) etc) ?

My significant log-transformed (log(x)) parameter estimates are 0.2767198895 (95%CI 0.05855987 - 0.49487992) and 0.241749740 (95%CI 0.019216904 - 0.464282577)

I am using SAS.

• Possible duplicate of How do I interpret interaction effects in a log-log regression model? – Arne Jonas Warnke Dec 19 '16 at 9:18
• I know this is not what you asked but it is not necessary to worry about the distribution of the predictors, what matters is the distribution of the residuals from your model. In fact it is not necessary to worry about the distribution of your outcome either. Having said that if you do want to carry on with a log-log model for theoretical reasons the link posted by @ArneJonasWarnke may help. – mdewey Dec 19 '16 at 13:35
• Instead of transforming $Y$, you could look into some generalized linear model (GLM). If you want help with that, you need to give us more details. – kjetil b halvorsen Dec 19 '16 at 15:21

## 1 Answer

If the outcome is log(y) and the regressor is log(x), then the coefficient is an "elasticity". It measures the relative change in y for a given relative change in x. If the coefficient is 0.25, it means that when x (no log) goes up by 1%, then y (no log) will go up by 0.25%. We say that the elasticity is 25% in that case.

• +1. $y = ax^b$, implying $\log y = \log a + b \log x$, is often called a power function or power law. – Nick Cox Dec 19 '16 at 17:15