Count models - do you need to 'impute' absent covariate patterns?

Question

I have individual level data (an observation is a person presenting to an ED with an injury). I am interested in looking at a time trend. There are several categorical covariates - gender, age, type of injury, etc.

My approach was to aggregate the data into distinct covariate patterns (i.e each unique combination of factor variables) and tally the count of injuries within that pattern. Then look at model options - Poisson, Neg Bin, etc.

But my question really relates to how to deal with the 'absent' covariate patterns - i.e. those people that simply didn't present with an injury with a particular combination of characteristics. I would assume (as I have done) that they need to be imputed with a count recorded as zero? Otherwise a bias will be introduced?

Would appreciate any tips.

Answer 1

NO, you do not need that. Say you have independent sampling, and a model function $f_\theta(y_i \mid x_i)$ with covariate pattern $x_i$, response variable $y_i$, and each pattern $i$ observed $w_i$ times (in practice, $w_i$ equals 0 or 1).

Then the likelihood can be written $$ L= \prod_i f_\theta(y_i \mid x_i)^{w_i}, $$ and for the unobserved patterns the contribution is $$ f_\theta(y_i \mid x_i)^0 = 1, $$ that is, the same as if that pattern was not included into the data.