# question of a Poisson random effect model

Assume a Poisson random effect model (with a random intercept) $$\log E(Y_{it}|b_i,x_i)=\beta_0+\beta_1x_i+b_i$$ $$b_i \sim N(0,\tau^2)$$

How to derive the population-averaged effect of one unit change in $$x_i$$?

$$\log E(Y_{i}|b_i,x_i)=\beta_0+\beta_1x_i+b_i$$
The effect of a one-unit change in $$x_i$$ is easy to write: $$\log E(Y|x_i+1,b_i) - \log E(Y|x_i,b_i) = \beta_0+\beta_1(x_i+1) + b_i - (\beta_0+\beta_1x_i+b_i) = \beta_1$$
$${E(Y|x_i+1,b_i) \over E(Y|x_i,b_i)} = e^{\beta_1}$$
So a one-unit increase in $$x_i$$ causes an $$e^{\beta_1}-1$$ percent change regardless of $$x_i$$ and $$b_i$$, implying that the population-averaged effect of a one-unit increase in $$x_i$$ also causes an $$e^{\beta_1}-1$$ percent change in $$E(Y)$$.