I have learned - and taught - that to build a regression model for a binary outcome one should use a logistic regression, for a outcome that has discrete counts one should use the Poisson regression, and for count data with zero inflation or under-/overdispersed data one might choose negative binomial regression instead.

Now, I have come across this thread: Poisson regression to estimate relative risk for binary outcomes, and I'm devastated. Apparently, there are quite some people who do not adhere to the rules that I have learned, and actually apply Poisson regression to binary outcomes. My world seems to become more and more complex.

As I understand, when we apply Poisson regression, we model the parameter $\lambda$ of the Poisson distribution via a log-linear model, i.e. $\ln(\lambda)=\beta X$. This makes sense if we assume that our outcome $Y$ follows a Poisson distribution: $Y \sim Pois(\lambda)$, therefore $\lambda = E(Y|X) = e^{\beta X}$. The Poisson distribution does not have an upper bound, therefore its pmf $Pr(Y=k)=\frac{\lambda^k e^{-\lambda}}{k!}$ is supported $k \in \{0,1,...,\infty\}$. Now, when maximising the likelihood, we cannot restrict possible values for the outcome to $k \in \{0,1\}$, because $\sum_{k=0}^1 Pr(Y=k) \neq 1$, so it would not be a valid pmf anymore.

My question is, what assumption does one make, when one (correctly) applies Poisson regression to model a binary outcome?


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