I have a Poisson model with two categorical variables. I am wondering if the model can only include the interaction term without the main effect terms? And how can I interpret it? Thanks!
Mathematically this should not be a problem, but as you mention it is hard to interpret. Say you have two variables A and B with both two levels ($a_1$ and $a_2$, and $b_1$ and $b_2$). Define $X_a = I(A=a_1)$ and $X_b = I(B=b_1)$. Then your mean model would be:
$$\log(E(Y_i)) = \beta_0 + \beta_1X_aX_b$$
The coefficient $\beta_1$ would then represent the difference in the logged expectation of Y between an individual with $A=a_1$ and $B=b_1$ and the other individuals (all other terms kept equal, if any).
If you think this is a useful model you can go on with it, but note that you cannot disentangle the effects of variables A an B with this model.