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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!

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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 $y$.

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  • $\begingroup$ 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

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