# Interpretation difference between log link and log transformation

I have a question about the interpretation difference between log link of GLM and log transformation of LM. I know that log transformation is for target variable but log link is for mean .But related with the interpretation of the model seems they are similar ? both need e^coefficent to measure the influence of the target variable? The question may be convert to if we want to get 1) estimation of Y or 2)the interpretation of coefficient, how these two affect the result ?

I have some clue that:

1. For the interpretation of coefficient : using mean to measure the impact ? so for the log of LM, nothing changing with normal distribution , so u=p0+p1x. For log link will be u=e^(po+p1x)
2. For the estimation of y, using signal function and estimator of p to get y.For log transform y=e^(po+p1x), for log link is only depend on the which exponential family choose.

But I am not very clear why using mean to moniter the impact of coefficient. I originally thought is same as the estimation of y. So I am confused there

• It is not clear what your question is. As I see it, the log transform results in the model: $\mathbb{E}[\log(y)] = Xb$ whereas in a GLM with log link we would have $\log(\mathbb{E}[y]) = Xb$ which are not the same - does that help ? May 23, 2021 at 18:17
• I wrote a blog post about a related topic you might find useful dpananos.github.io/posts/2020/10/blog-post-25 May 24, 2021 at 3:24
• thanks！！！ I see the difference May 24, 2021 at 19:55

Those models are similar, but the key different thing is we model $$\log{E(Y)}$$ for GLM and $$E(\log Y)$$ for LM. Thus, we can estimate $$Y$$ directly in GLM and $$\log Y$$ in LM.
GLM: We can directly say $$E(Y)=\exp(\beta_0+\beta_1X)$$. In this case, $$\beta_1$$ catches the effect of a unite change in $$X$$ on $$Y$$ (acutually, log of ratio of $$Y$$).
LM: we only say $$E(\log Y)=\beta_0+\beta_1X$$. Then, $$\log Y$$ increases by $$\beta_1$$ when $$X$$ increases by 1.
Critical note: $$E(\log Y)\neq \log E(Y)$$.