Could you, please, help me with the following problem?

Suppose we have a one-way ANOVA with a single 2-level factor. The dependent variable is a logarithmized value: $y_i = log(Y_i)$.

$y_{iz} = \mu + a_i + \epsilon_{iz}$

$z$ is the number of observations in each $i$ group, different for different $i$.

We want to estimate the contrast:

$C = a_1 - a_2$

If we exponentiate this contrast we get a ratio of geometric means of the dependent variable in the two treatment groups on the original scale.

$C = \frac{\sum_{h=1}^{z} log(Y_{1h})}{z} - \frac{\sum_{h=1}^{z} log(Y_{2h})}{z}$

$C = log(\prod_{h=1}^{z} Y_{1h}^{\frac{1}{z}})- log(\prod_{h=1}^{z} Y_{2h}^{\frac{1}{z}})$

$C = log(\frac{(\prod_{h=1}^{z} Y_{1h})^{\frac{1}{z}}}{(\prod_{h=1}^{z} Y_{2h})^{\frac{1}{z}}})$

Now, suppose we add additional factors in the model. Importantly, we do not add interaction terms in the model.

What do we get if we exponentiate the same contrast from the model with additional variables? Do we still get an adjusted ratio of geometric means of some sort? Have you seen any literature on this?

I would appreciate any insights!