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Reading a paper, the authors use a "Conditional fixed effects Poisson with population exposure." I wasn't sure what "population exposure" meant, so I looked it up. It seems like the term comes from epidemiology (not my field). Could I check to see if I've understood it correctly?

Poisson models try to estimate a rate of event occurrence within a population. The relevant population is subject to interpretation. In epidemiology, the population might be called the "exposure" group -- e.g. those exposed to the disease risk. So, by saying "population exposure" a Poisson model with population exposure calculates the event rate as:

$\ \ \ \ \ \log(\frac{\text{event_ count}}{\text{region_population}}) = X\beta$

$\Rightarrow \log(\text{event_ count}) - \log(\text{region_population}) = X\beta$

$\Rightarrow \log(\text{event_ count}) = X\beta + \log(\text{region_population})$

I appreciate any help you can give toward understanding this.

gung - Reinstate Monica
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Dr. Beeblebrox
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1 Answers1

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The Poisson model actually estimates counts of events ($y$), like purchases at a grocery store or visits to the doctor. Let's use the latter as an example. You might reasonably assume that visits to the doctor for an area are proportional to population of that area. The more people live somewhere, the higher the expected number of visits. That's the exposure aspect.

The model says that \begin{equation}E[y \vert x]=\exp\{x'\beta+\ln pop\}=\exp\{x'\beta\} \cdot pop\end{equation} Let's take two areas with the same characteristics, except that the population of the second county is twice as large. The latter one has twice the expected number of doctor visits.

That is mathematically equivalent to \begin{equation}\frac{E[y \vert x]}{pop}=\exp\{x'\beta\},\end{equation} which is the rate of visits per person. Once you take the natural log of both sides, you get your expression above.

dimitriy
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