Independence in Poisson regression when used for rates estimation I'm interested in using Poisson regression for estimating mortality rates. I've observations for a rather long period, and have to face the fact that covariates often vary over time. This is especially the case when the age of the people comes into play and the observation period exceeds a few months.
In this case, here is my way to deal with this issue : the observation period is divided into intervals where the covariates are constant. Each observation then becomes a set of observations and the exposures are calculated for each interval. However, the log-likelihood in Poisson regression models assume independence between the observations.
But, as I have, say for a person named Jack, a period where he is forty and another one where he is forty one, I will have for Jack two observations in my model, and I can't really say that those two observations are independent.
Can I just assume that the death rates are independent anyway, or are there specific methods to deal with this issue in a better way?
EDIT
In this lesson, at the paragrah 7.4.3 The Equivalent Poisson Model there is something that could let think my approach is not invalid. Nevertheless, as far as I'am aggregating data in each bunch of pseudo-observation (and not as in the text , one for each combination of individual and interval) I wonder if it still stand.
EDIT
About grouping data :
I have a set of thousands of observations. Each one relate to an individual, and for each one I have the date of entry in the study, the date of exit, the date of death, an indicator for smoking or not, the birth date, the gender (and a few other one but that don't matter here).
I cut the whole interval of time covered form the first entry to the last at each time something happen (someone entre or leave the study, or a covariate changes).
For each interval of time, and for each set of similar age and other covaraiates, I compute the total exposition, and the number of deaths. And I this way I get "pseudo-observations" with a number of death, an exposition, and covariates.
Then I fit a Poisson Regression (generalised linear model, with expostion as offset).

The notes you posted explain why creating these pseudo-observations and fitting a Poisson log-linear regression (Poisson GLM with log-link) is equivalent to a piece-wise exponential survival model. They also note that it is valid to aggregate the pseudo-observations with equal covariates. This is mentioned at the very end of the "The Equivalent Poisson Model" section. So I will stick to the question about independence as it relates to these intervals.
Why doesn't creating many small interval observations out of one big interval observation in Poisson regression violate some independence assumption?
The key thing is that by assuming your response is Poisson distributed, you already assume something about the independence of these smaller intervals.
Suppose your data set were simpler, and all observations represented a single unit interval of time, say January - December 2020, and each row represented a different subject. And suppose your response variable were a discrete count of something for each observation within this interval of time. You decide to do some form of Poisson regression, having made the following assumptions about the data:

*

*Your response variable is a Poisson distributed random variable. That is, you are counting events which occur independently of one another at a fixed rate in the interval.

*Your observations are independent of one another. That is, you have pairs $(X_i, y_i)$, and each pair is independent.

You then decide to split Jack's row into two sub-intervals. You partition it into a January - June sub-interval, and a July - December sub-interval, and distribute the count of something between the sub-intervals based on whether each something happened in Jan-Jun, or Jul-Dec. Maybe you are doing this for fun, or maybe it is because there is some covariate you think is important which changes half-way through the year (which, note, implies that you think the rate changes midway through the original interval, and also that it wouldn't be possible to include this covariate in a regression using the original data).
You started with one observation $(X_j, y_j)$ for Jack, now you have two observations $(X_{j1}, y_{j1})$ and $(X_{j2}, y_{j2})$. It sure "feels like" these aren't independent. Indeed, $X_{j1}$ and $X_{j2}$ are definitely not independent, they will be necessarily identical on all the covariates which you could have included before splitting into sub-intervals. But crucially, $y_{j1}$ and $y_{j2}$ are independent, by assumption. It is a property of the Poisson distribution that the occurrences of something we are counting in the original interval are occurring independently from one another. So necessarily, the count of something for Jack in the Jan-Jun sub-interval is independent of the count of something for him in the Jul-Dec sub-interval. Again, you got this for free once you assumed the response was Poisson.
What about the fact that $X_{j1}$ and $X_{j2}$ are not independent? Aren't we pretending we have two observations, when really it is only one? Well, this is part of what the exposure offset is for. When we maximize likelihood in a Poisson regression without exposure offsets (constant duration intervals), the log-likelihood function we're maximizing looks like this:
$$\sum_i{y_i \log{\mu_i} - \mu_i}.$$
But when we have differing interval lengths and exposure offsets, it looks like this:
$$\sum_i{y_i \log{(\mu_ie_i)} - \mu_ie_i} = \sum_i{y_i\log{(\mu_i)} + y_i\log(e_i)} - \mu_ie_i.$$
Which rearranges as:
$$\sum_i{e_i\left(\frac{y_i}{e_i}\log{(\mu_i)} - \mu_i\right) + y_i\log(e_i)}$$
The term on the right doesn't depend on $\mu$ and so is irrelevant. So, our likelihood function is equivalent to the likelihood function of a Poisson regression with $y_i/e_i$ as the response, and $e_i$ as the weight. Conceptually we are standardizing the observed rate of occurrence so that all observations have the same measurement unit (observations per unit time), and then we are weighting each observation by the amount of time it represents.
All that is to say, if we partition Jack's interval into two equal length sub-intervals, then each interval is only being treated as half an observation.
