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I have a question regarding fitting a poisson model to non-independent observations when assessing the incidence of an outcome.

I am interested in the incidence of some outcome, O, over time. I have a cohort of N patients all of which are followed up for various lengths of time over the course of 20 years. i.e. some patients enter in year 0, some at year 5, 10, etc.

To assess the secular trend, I calculate the total time at risk in each year, and the number of events in each year (Individuals are no longer followed up after an event). For each year, I divide the number of events in the year by the total time at risk. I find there to be a secular trend in incidence over time. Alternatively, I could fit a Poisson model where the outcome = # events, offset = time at risk, predictor variable = calendar year.

I then would like to adjust for a potential confounding factor. That is, I hypothesize that changes in some covariate, C, may be responsible for the secular trend (my data is observational and I am aware of causality vs correlation, but it is still of interest).

To do this, I must break up my data into the individual level, as opposed to summarizing on the year level. The data would look like:

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| Patid | Followup | Event | Year   | C  |
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| 001     | 365      | 0     | 1    | 1  |
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| 001     | 100      | 1     | 2    | 1.5|
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| 002     | 150      | 0     | 5    | 1.3|
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| 002     | 365      | 0     | 6    | 1.3|
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|         |          |       |      |    |

Here the first individual is followed up for 365 days in year 1, doesn't not have the event of interest, and has a value of 1 for the confounding covariate ( measured on say the first day of the year). In year 2, the individual has an event after 100 days and then is no longer followed up, and the confounder when measured was 1.5.

I would then fit a Poisson model, outcome = # events, offset = time at risk, predictor variables = calendar year, and C.

My initial reaction was that this is invalid given the observations are not independent, rule 101 for most statistical procedures. However I have been advised that this is OK to do by someone I trust, but they weren't able to find a source for this. This person says it is something to do with the fact that once an individual has an event they are removed from the cohort, and the likelihood working out correctly, but could not give a concrete answer. I am planning to run ahead with the model, but would really appreciate any references on this subject as to why it is OK which I can use, and also for my own interest!

Many thanks.

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With non-independent observations, mixed-effect models are the way to go. Somebody asked a similar question very recently, the answers to which will provide you with some good reference books to look into. Cheers

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  • $\begingroup$ Thank you for your reply. They key difference between these examples is that I am looking at incdience (first event during follow up, then an individual is no longer followed up), whereas the linked example is looking at a rate (can have multiple events). I agree in the linked example that a multilevel model is applicable, however I do not think it is here. $\endgroup$ – AP30 Jul 27 '18 at 15:12
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Found this source, see paragraph 1 on page 4. Although the subject is about pooled logistic regression, as opposed to a poisson model, I believe the argument is the same.

"While there may be concern with the PLR regarding the dependence of multiple records within an individual contributing to several intervals, Allison (2010) has noted that in working with a dataset with multiple records for intervals within each individual there is no inflation of test statistics resulting from a lack of independence. This property is due to the fact that the likelihood factors into a distinct term for each interval. Allison also cautioned that this conclusion may not apply when the dataset includes multiple events for each individual."

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