Poisson regression on parts of a total count My dataset contains data like the total amount of deaths in different countries. Then I have as columns indicating smaller parts of the total amount specifying the cause of the death. I have another column with a continuous variable indicating the wealth of the nation. For example:




Nation
Total
By_car_incident
Suicide
Homicide
Nation_Wealth




France
50
20
10
20
30


Italy
70
35
20
15
27




Now, I am interested in the effect of the nation wealth on each cause of death. I was thinking to first make the dataset long and have one column with cause and number of deaths.




Nation
Cause
Number
Nation_Wealth




France
By_car_incident
20
30


France
Suicide
10
30


France
Homicide
20
30


Italy
By_car_incident
35
27


Italy
Suicide
20
27


Italy
Homicide
15
27




Hence, I was thinking of a Poisson regression to predict Number ~ Nation_Wealth:Cause using glm(),  but I get that every interaction is significant with extremely low sig. levels, there is a Nagelkerke R^2 = 100% and the assumption of over-dispersion is violated.  Do you see any potential problem on testing in my reasoning?  Am I reasoning wrongly? Can I find other modellization strategies to describe the effect on Nation_Wealth on each cause of death? I apologize in advance for the non-reproducibility of the dataset, but I just want to know if the test and modeling strategy is correct.

Some thoughts:

*

*You'd likely need an offset term in your model for population, as the more persons you have, the more deaths. In a standard glm in R the offset would be log(population).

*Poisson overdispersion can come from a variety of sources. Here I suspect it's partly due to missing an offset term. It can also come from missing a predictor, or simply that Poisson is not a good descriptor of this count data - you could try a negative binomial instead.

*I'm not convinced that using Cause of Death as a predictor is very useful, or meaningful. Perhaps three models, one for Homicide, one for Car etc.?

