# interpretation of coefficients from linear regressions with log dependent variable

I have a seemingly trivial yet troublesome question. Let's consider the following model:

$$\ln(y_i)=\alpha + \beta D_i + \epsilon_i$$

where $D_i$ is a binary variable that indicates whether treatment was assigned to patient $i$.

I understand that $\exp\left(\hat{\beta}\right)-1$ represents the percentage point variation in $y_i$ when treatment is assigned.

However, in a lot of papers I read (and these papers have been subject to peer-review and are very well published), when describing their results, authors write: "from our estimations, we can infer that the assignment of treatment leads to a $\hat{\beta}$ percent variation in $y_i$.

I understand that this is linear approximation but when coefficients are large this approximation becomes inaccurate. So why don't authors simply report the percentage variation from the exponentiated coefficient? Am I missing something here?

Any comment is welcome. Thanks a lot!

• You are right. In your example citation however, it is not the only problem: not every single patient benefits by $\hat \beta$. It is only a (geometric) mean benefit. Furthermore, the word "variation" is strange. – Michael M Jul 23 '15 at 19:35
• Thanks for the answer. When I write "variation" I just used a generic term for increase or decrease. The mystery is still why people do not bother to calculate this percentage point increase accurately. – user81018 Jul 23 '15 at 19:58

For example, in Cox survival analysis where there is a similar issue with regression coefficients and exponentiation, results are almost always presented in terms of hazard ratios ($\exp\left(\hat{\beta}\right)$, as you would prefer) rather than percentage-point change in hazard.