Poisson regression with time dependent covariates

Question

I am an epidemiologist and recently our team receive a data from nursing homes in our city. We have weekly data for each elderly person for about 3 years during their stay in nursing homes. The response variable of interest is flu infection (binary). There are many different covariates (some time-constant such as gender and chronic condition; some time-dependent such as age, vaccination, etc.) We also have the weekly nursing home staff vaccination proportion for each nursing home.

The research question is how the proportion of nursing home staff vaccination affect the risk of flu for the elderly in this city's nursing homes.

The data for one person looks like:

id week flu age gender nursing_home_id staff_vac_proportion
1 1 0 77 M 251 0.78
1 2 1 77 M 251 0.74
1 3 0 77 M 251 0.81
...
1 52 0 78 M 301 0.80
1 53 1 78 M 301 0.82
1 54 0 78 M 301 0.77
1 55 0 78 M 301 0.75
...

There are about 150 weeks for each person. Our team has been doing some brainstorming how to appropriately model this data. As the sample data above shows, a person can have re-infection of flu throughout the three years.

A teammate suggests using Cox model with time-varying covariates. But it doesn't seems appropriate since it is modeling time-to-event data, and the reinfection is not considered?

Another teammate suggests using Poisson regression. We agree that it is okay for the preliminary results such as the incidence rate of flu. But can this be extended to answer the research question? For example, Poisson regression with time dependent covariates? Like, from the model result we can have something like...for each unit increase in staff vaccination proportion, the rate ratio of flu for people in elderly nursing homes decreases by some amount?

Someone also suggests a very sophisticated multi-level mixed model. I personally am quite confused at this point, might someone be willing to share some of your insight?

Answer 1

As Maximilian Aigner suggested in a comment, this might fit quite well into a Cox framework. You might not even need a multi-state model, as this seems to fit the simple repeating events scenario described in section 3.2 of the main R survival vignette.

The example there involves multiple infections in children with chronic granulotomous disease, at multiple institutions. You have multiple flu infections in elderly residents of multiple institutions. There may be some subtle differences based on details of your situation, but from what you describe these seem to be identical scenarios.

If you use the Cox PH model you will have to adjust your data format a bit to include the start time and end time of each time period. That (startTime, stopTime, status) counting-process data format nicely incorporates time-varying covariate values like staff vaccination proportions and changes of nursing homes. Each row then holds the covariate values in place for the individual during that time period.

The regular weekly reporting of data might allow you to use a discrete-time survival model with the data format you currently have. That would be a binomial model of flu/no-flu that includes week as a predictor in some way (e.g., modeled with a regression spline). A binomial regression with this type of data and a complementary log-log link, instead of the more typical logistic link, is sometimes called a "grouped proportional hazards model." There's no restriction on the number of events an individual can experience. The critical assumption is that the status at the end of each time period is a function of the covariate values in place during that time period as recorded in that data row.