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I am trying to regress the death rate for a particular cause of death for 25-30 year olds. My dependent variable of interest is the crude rate, i.e., y = number of deaths of 25-30 year olds/number of people 25-30 year old in a county.

If I were to run this as a Poisson regression, would I run this on the count of deaths of 25-30 year olds (the numerator) on my regressors, and include the log of the denominator (number of people 25-30) as a regressor as an offset?

Is the idea thinking of the link function, let deaths = number of 25-30 year olds dying from the cause, and n = the number of 25-30 year olds

\begin{align} \newcommand{\deaths}{{\rm deaths}} \frac{\deaths}{n} &= e^{x'\beta} \\[5pt] \deaths &= ne^{x'\beta} \\[5pt] \deaths &= e^{x'\beta + \ln(n)} \end{align}

and this last equation is Poisson link function?

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    $\begingroup$ I can't quite parse a couple of your sentences. I'm not sure if you're asking if you should use ln(n) as a regressor or an offset. I'm also not sure if you're asking if the derivation is right or what, exactly. Either way, I think you'll find the info you need in the linked thread. $\endgroup$ – gung - Reinstate Monica Nov 19 '20 at 19:50