# Interpretation of log transformed predictors in logistic regression

One of the predictors in my logistic model has been log transformed. How do you interpret the estimated coefficient of the log transformed predictor and how do you calculate the impact of that predictor on the odds ratio?

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Also perhaps of interest: Should quantitative predictors be transformed to be normally distributed? –  Macro Jul 18 '12 at 0:35
A very clear, comprehensive treatment of this question is jthetzel's answer at stats.stackexchange.com/questions/18480/… –  rolando2 Jul 18 '12 at 12:25
Thanks for all your help. A further clarification. Actually if I transform to log base 2- then as per the earlier answer it makes intuitive sense that a doubling in the predictor results in a x% change in the outcome. –  mp77 Jul 20 '12 at 12:50

If you exponentiate the estimated coefficient, you'll get an odds ratio associated with a $b$-fold increase in the predictor, where $b$ is the base of the logarithm you used when log-transforming the predictor.

I usually choose to take logarithms to base 2 in this situation, so I can interpet the exponentiated coefficient as an odds ratio associated with a doubling of the predictor.

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Interesting. I always use natural logs because many of the coefficients tend to be close to zero and then can be interpreted as proportional (relative) differences. That's not possible in any other base of logarithm. I see some merit in using other bases, but I think you need to clarify your response, because prima facie your interpretation doesn't use the value of the coefficient at all! –  whuber Mar 15 '11 at 15:12
@whuber sorry what does prima facie mean? First face?? –  onestop Mar 16 '11 at 7:56
–  whuber Mar 16 '11 at 14:26

@gung is completely correct, but, in case you do decide to keep it, you can interpret the coefficient has having an effect on each multiple of the IV, rather than each addition of the IV.

One IV that often should be transformed is income. If you included it untransformed, then each (say) \$1,000 increase in income would have an effect on the odds ratio as specified by the odds ratio. On the other hand, if you took log(10) of income, then each 10 fold increase in income would have the effect on the odds ratio specified in the odds ratio. It makes sense to do this for income because, in many ways, an increase of \$1,000 in income is much bigger for someone who makes \$10,000 per year than someone who makes \$100,000.

One final note - although logistic regression makes no normality assumptions, even OLS regression doesn't make assumptions about the variables, it makes assumptions about the error, as estimated by the residuals.

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+1, good points. I suppose I could have been more complete. In addition, I turned off the inadvertent mathjax by putting a backslash "\" immediately before the dollar signs. I hope you don't mind. –  gung Jul 18 '12 at 1:05
Thanks for fixing that! –  Peter Flom Jul 18 '12 at 10:30