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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!

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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.

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  • $\begingroup$ Thanks for your reply! Just wondering if you know how to code this in R? $\endgroup$ – J. Zhu Oct 26 '17 at 13:41
  • $\begingroup$ See ?formula, something like glm(y~a:b, family="poisson") $\endgroup$ – Knarpie Oct 26 '17 at 13:55

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