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Suppose we have $E (\log (Y)) = a+bx $ vs $\log (E (Y)) = a+bx $. Can $\exp (b) $ in both cases be interpreted as a geometric mean?

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I'm not sure this answers your question directly, but in general, the coefficients between the two approaches will have different interpretations. In one case (your first listed), it seems you'd like to transform your original data and run a least squares (linear) regression. In the other case (the second one listed), you'd be taking a generalized linear model approach, using log as your link function, which is called Poisson regression. The key difference is in the interpretation of the error, or variance, of the response data (Y), as a function of the independent data (x).

If, after transforming your original response data, your errors are close to normally distributed, then perhaps the former approach will work just fine. On the other hand, if your data is truly (or nearly) Poisson distributed (conditioned on each possible value of x), then Poisson regression will allow you to estimate variance in the domain of the original independent/predictor data.

There's already a discussion here touching on some of the related issues. And for a more general discussion about the differences between pre-transformation and using generalized linear models, this set of slides might be helpful (in the context of performing regression analyses on data that don't have nice, normally-distributed variances).

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  • $\begingroup$ (+1) Welcome to our site, and thank you for your contribution! $\endgroup$
    – whuber
    May 2 '14 at 18:47
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    $\begingroup$ @whuber, thanks! I've been a lurker for a while, but a sudden increase in free time means I can finally (try to) give back a bit! $\endgroup$
    – mmuurr
    May 2 '14 at 19:50

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