I am trying to compare the results of three studies. All use linear regression with continuous predictors/independent variables and outcomes/dependent variables. The predictors (level of a chemical in urine) are log transformed in all the studies, but two use natural log and one uses a base 10. I would like to be able to directly compare the effect sizes (and ideally display them graphically). Do you have any suggestions for how I could do this?

A statistician I work with said that if the log transformation was on the outcome, I could just exponentiate them, but he did not believe that was appropriate for the predictor.

Thanks so much!


1 Answer 1


Log-transformation changes the interpretation on the coefficients. In a model $$y=\beta_0+\beta_xx+\beta_z\ln z+\beta_s\log_{10}s +error$$ You $\beta_0$ is per unit change of the level of $x$, while $\beta_z$ is per 100 percentage points change of $z$. So, if $z$ changes by 1% then the impact on $y$ is $0.01\beta_z$.

The base of a log matters only for the scale of the coefficient. So, one percent change in $s$ would lead to $\frac{\beta_s}{100\ln{10}} $ change in $y$.

  • $\begingroup$ Thanks for responding. My question really is about the scale of the coefficient, though. I can't compare the β from one study to the β from another study because of the different bases. If I compare them directly, one will look larger than the other just because of the base used. $\endgroup$ Mar 30, 2016 at 17:20
  • $\begingroup$ @OrangeTrout in my example you can't compare $\beta_x,\beta_z$ - one's for absolute and the other for relative change in independent variable. You can easily compare $\beta_z,\beta_s$ - the scaling factor is $\ln 10$. Of course, you have to make sure that the units of measure are the same $\endgroup$
    – Aksakal
    Mar 30, 2016 at 17:32
  • $\begingroup$ Indeed, but the problem yields to simple arithmetic. To get the coefficient for a log 10 transformed predictor from that for a ln transformed predictor, just multiply by ln 10. To go the other way, divide by that constant. This follows from the basic properties of logarithms. Note that the t statistic for each predictor is the same regardless of base. To see this, just use any data sensible to hand, and compare the regressions for some y and log 10 of some x and ln of some x. (Exponentiation is indeed irrelevant here.) $\endgroup$
    – Nick Cox
    Mar 30, 2016 at 17:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.