# interpreting coefficient of poisson regression

I ran a Poisson regression in SPSS with hypothesis: increase in level of democracy leads to reduction in infant mortality (IMR). So democracy is the independent variable and IMR the dependent.

I got the coefficient of democracy -0.186.

Does this mean, one unit increase in democracy level will lead to a $exp(-0.186)=0.83$ case decrease in IMR?

Not exactly. When using Poisson regression with the default log link function, the prediction is

$$E(Y | X) = e^{\beta_0 + \beta_1 X_1 + \dots + \beta_k X_k} = e^{\beta_0} e^{\beta_1 X_1} \dots e^{\beta_k X_k}$$

so if, for example, $\beta_1 = -0.186$, then its effect on $X_1$ is $e^{\beta_1 X_1}$. It is not the same as $e^{\beta_1} X_1$, since given the properties of exponents this would lead to $e^{\beta_1 + \log X_1}$.

• Tim, thanks for your answer. what about natural log? is it the same as log? – hans Apr 25 '17 at 12:23
• @hans yes, in the above answer I used $\log$ for $\log_e$ – Tim Apr 25 '17 at 12:46