I have measurements $y_1$,...,$y_i$,...,$y_n$ taken from a set of replicates in a factorial designed experiment.

In order to use a linear regression I define my response $z_i = log(y_i + 1)$. The log is used in order to make the normality assumption hold and the model fit, and the + 1 is used since some $y_i$'s are 0.

If the model is ${z = a + \beta X}$, then the interpretation of $\beta$ is ${\beta = {\sum_{i=1}^nz_i}/n = {\sum_{i=1}^nlog(y_i + 1)}/n}$ so ${{\log(\pi_{i=1}^n(y_i + 1))} = \beta n}$, which gives that the geometric mean $\sqrt[n]{\pi_{i=1}^n(y_i + 1)} = e ^\beta$.

My question is whether there is some back transformation or some other way to get an interpretation of the geometric mean of $y_i$'s rather than of $(y_i + 1)$'s as a function of $\beta$. The reason I'm asking is that for cases where $y_i$'s are small (close to 0) the value of $\beta$ underestimates the magnitude of the effect.


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