# Poisson regression: how do number of observations and offset affect variance of betas?

Background: In Poisson regression with an offset, like in this answer, @Hong Ooi writes

Your underlying random variable is still $Y$, but by dividing by $\varepsilon$ we've converted the LHS of the model equation to be a rate of events per unit exposure. But this division also alters the variance of the response, so we have to weight by $\varepsilon$ when fitting the model.

Question: Since the exposure $\varepsilon$ is accounted for when fitting the model, does that mean one can divide a (multiple) observation into multiple observation(s) with new exposure $\varepsilon_i$, where $\sum_i\varepsilon_i=\varepsilon$, without this affecting the variance of the parameter estimates $\beta$?

Attempt: Intuitively I would think that doing this would give me more "observations", which would decrease the variance of each $\beta$ estimate, even though we weigh by $\varepsilon$ when fitting the model.

By dividing a observation into multiple observation I mean the following:

UPDATE: A while back @Scortchi wrote this in Ten fold:

If you're using the full likelihood formulation (Poisson + offset) nothing will change - 5 counts in 10 hours is the same as 1 count in 6 hours & 4 counts in 4 hours. If you're using the quasi-Poisson formulation you'll get the same point estimates but the standard error will change when your estimate of the dispersion parameter changes.

• I don't understand what you are proposing with "divide each observation into multiple observation". Could you elaborate? – Memming Aug 22 '16 at 13:22
• @Memming: thank you for pointing that out, updated the Q. Note that I divided only one observation in my example into multiple observations. – Erosennin Aug 22 '16 at 13:34

$$\text{var}(\hat{\beta}) = \left( \mathbf{X}^T \text{diag}(\hat{y}) \mathbf{X} \right) ^T$$
Note there is no use of the offset here. The variance structure is given by the usual $\mathbf{A}$ matrix formulation of exact likelihood based inference, where the variance is the predicted model variance (the variance is the mean for Poisson models).