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I am trying to interpret a Poisson regression without being very interested in the mean. As a complication, I also have an exposure variable. Let $y_i$ be a count variable, and $p$ the offset, for example the population at risk and $x_i$ some treatment indicator taking value $0$ or $1$.

I estimate:

$$y_i=exp(log(p_i)+\beta x_i+\gamma z_i+u_i)$$

So:

$$E(y_i|x_i=1)=exp(log(p_i)+\beta+\gamma z_i)$$ $$E(y_i|x_i=0)=exp(log(p_i)+\gamma z_i)$$

and:

$$E(y_i|x_i=1)-E(y_i|x_i=0)=p_i*exp(\gamma z_i)*(exp(\beta)-1)$$

Now, $\beta$ can be interpreted as the difference of the logs of the conditional expectations as:

$$E(y_i|x_i)=exp(log(p_i)+\beta x_i+\gamma z_i)$$

and:

$$log(E(y_i|x_i))=log(p_i)+\beta x_i+\gamma z_i$$

so that:

$$log(E(y_i|x_i=1))-log(E(y_i|x_i=0))=\beta$$

Now assume that i am not interested in the effect on the expectation but only on the probability that $\frac{y_i}{p_i}>=1/1000$. For $y_i$ only, finding the probability $Pr(y_i>=1|p_i, x_i, z_i)$ is easy. Given $p_i, x_i, z_i$ we can find $E(y_i|p_i, x_i, z_i)$. Using the cdf of a Poisson distribution:

$$\begin{align*} Pr&(y_i>=1|p_i, x_i, z_i) \\ &=1-Pr(y_i=0|p_i, x_i, z_i)\\ &=1-exp(-exp(log(p_i)+\beta x_i+\gamma z_i)) \end{align*} $$

However, what I care about is:

$$Pr \left( \frac{y_i}{p_i}>=\frac{1}{1000} \mid p_i, x_i, z_i \right)$$

How can I apply the Poisson cdf in such a case?

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