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 answer by @JonnyLomond answers your question as posed (+1), but also raises some questions about how you will do the modeling. You say:

For each interval of time, and for each set of similar age and other covariates, I compute the total exposition, and the number of deaths,

which suggests that you are binning continuous predictors like age, which is not usually a good idea. You generally are better off modeling individuals directly with their actual covariate values, then displaying results as desired for particular combinations of covariate values.
Also, it sounds like you might not be using time-interval-specific piecewise-constant hazards as in those notes, but rather a single constant baseline hazard for all your data (related to the intercept in the Poisson modeling). As the course notes you cite say (Section 7.2.2):

treating the deaths as Poisson conditional on exposure time leads to exactly the same estimates (and standard errors) as treating the exposure times as censored observations from an exponential distribution.

If you have a single intercept shared by all time intervals, you are thus assuming an exponential survival model, which doesn't always work well. Unless there's a compelling reason to force an exponential model onto the data, you might do better with a Cox model, which allows the data to tell you the form of the baseline survival function as well as the hazards associated with the covariates.
Your data seem already to be arranged in a way close to the counting-process format used to fit Cox models with time-varying covariates. From your description, each individual is evidently represented by multiple rows covering non-overlapping time intervals. During each time interval, the individual's covariate values, included in that row, are constant. Each row contains an indication of an event time. The only necessary change would seem to be to set the end time for the row with the event to the event time itself, with an indicator variable for the event.
The independence argument for the Poisson model does require that there be at most 1 event possible per individual* and that covariate values only change at time-interval boundaries. Under those assumptions, the argument holds more generally for proportional-hazards models. Under proportional hazards, an individual contributes to the (partial) likelihood at all event times in the data set while at risk. At each event time, however, only the instantaneous covariate values are used to estimate the relative hazards; covariate values at prior times are irrelevant, unlike for other approaches like accelerated failure time models. As the R time-dependent survival vignette explains (page 4):

One common question with this data setup is whether we need to worry about correlated data, since a given subject has multiple observations. The answer is no, we do not. The reason is that this representation is simply a programming trick. The likelihood equations at any time point use only one copy of any subject, the program picks out the correct row of data at each time.

If you need a fully parametric model for some reason, you can use tools provided by the R flexsurv or eha packages with counting-process data. The latter can take individual IDs into account in model types that require it.

*If an individual can experience more than 1 event, then you do have to take intra-individual correlations into account with methods like cluster variances, frailty terms, or a random-effects model.
