Gaussian log-response model ($\log Y=\eta +\epsilon$)

For a positive response, the model $\log Y=\eta +\epsilon$ with $\epsilon \sim N(0,{\sigma }^2)$ gives conditional median $e^{\eta }$ and conditional mean $e^{\eta +{\sigma }^2/2}$. The latter follows from the moment-generating function of a normal distribution. Additive factor effects in $\eta$ become multiplicative effects on these response summaries. Its constant log-scale variance corresponds to original-scale variance proportional to the square of the mean; direct exponentiation of fitted log means incurs retransformation bias.