Graduate Interpreting Poisson Regression Estimates across groups

[FallenApple]

Say for example I want to see the rate of injury for firefighter vs police vs soldier.

$InjuryCount_{i}$ The number of injuries recorded for the ith person over time
$T_{i}$ Time the person was followed. Varies from person to person.
$I(f)_{i}$ indicator for ith person of being a firefighter or not, police is baseline
$I(s)_{i}$ indicator for the ith person of being a soldier or not, police is baseline

Then I would model $log(InjuryCount_{i}/T_{i})=\beta_{0} +\beta_{1}I(f)_{i}+\beta_{2}I(s)_{i}.$

Where the regression model is either a poisson, negative binomial, or quasi poisson.

Now how would I intepret the coefficients?

Is $exp(\beta{0} )$ the estimated rate of injury for the baseline group. Or is it the estimated mean rate for the baseline group. I'm not sure which.

If we look at individuals, then I can say that it is the estimated rate of injury for someone belonging in the baseline group.

But if I look at the group, I can say that it is the estimated mean rate for the baseline group as a whole.

Not sure which one is right.

 

[Dale]

indicator for ith person of being a firefighter or not, police is baseline...
indicator for the ith person of being a soldier or not, police is baseline

Do you have subjects where the same subject is both a firefighter and a soldier?

 

[FallenApple]

Do you have subjects where the same subject is both a firefighter and a soldier?

No, but what might happen if there is overlap?

The way I set up the regression equation would result in log(response)=B_0+0+0 for the police(baseline group) since I suppose that would have to be the result from the categories being mutually exclusive.