Need an offset in GLMM even when it is almost invariable and an offset-free model yields a better fit?

Question

I am analyzing frequency data (response variable = no. of seeds produced per plant) by fitting GLMM with a logarithmic link function and a Poisson error distribution. I also have data on no. of ovules produced per plant, so technically I can calculate the proportion of seed set as (no. of seeds produced per plant) / (no. of ovules produced per plant).

In such situations, people usually include no. of ovules produced per plant as an offset term, while using the no. of seeds per plant as the response variable. In my case, however, I found that variation in no. of ovules produced hardly varied among plants. Moreover, an inclusion of the offset term into GLMM model deteriorate the model. That is, the offset-free model yields marginal R-square value (i.e., the variance explained by the fixed factor; see Nakagawa & Schielzeth 2013, MEE 4:133-142) that were 2-5 times greater than that of the offset model. I therefore prefer to omit the offset term and just include no. of seeds produced per plant as the response variable.

I present this result in a paper manuscript. Then someone commented that I should include the offset term into the model. I understand the logic, but I am not so happy about including any term into the model when it just decreases the model's explanatory power. People usually accept omitting independent variables when they are of no use in explain the variation in response variable. But how about the offset term? An inclusion of offset is obligation even when it is invariable and its inclusion just deteriorates the model? And is it incorrect to interpret results of offset-free models as almost the same with those about the proportion of seed set when the omitted offset little varied among individuals?

I would greatly appreciate any input. It would be especially helpful if you could guide me to any reference elaborating on this issue.

Answer 1

If your offset variable is almost constant then its role in the model will be determined by the values which are not constant which may be what is causing the problem. It may also be that if you were to estimate the coefficient you would get a different value from unity which might be problematic.

I would suggest exploring the data-set in more depth. Fit with your proposed offset variable (well, actually its log) as a covariate and explore the fit with respect to that variable bearing in mind it might be very affected by outliers or be non-linear. If you still cannot untangle things then explain it is almost invariant and do not include it.