Is there a test statistic for poisson regression? I compared the number of pregnant women receiving specific health service indicators (deliveries, antenatal attendances and malaria prophylaxis uptake) at 14 facilities during three specific time periods; pre, during and after a project intervention; using the pre-intervention data set as the reference in a Poisson regression model, I calculated the Incident rate ratios and their confidence intervals and p-values. I have now been asked to run a test statistic to show whether the data sets pre, during and after the intervention are different or not. Is showing the IRR not enough and is there another test statistic beyond what I have just done?
here is an example of the STATA read out that I obtained on running an analysis.
My challenge is in interpreting the print out especially the LR chi2(2) and prob > chi2 results. what do they mean?

 A: GLMs, including Poisson regression, have (asymptotic) tests that are akin to the corresponding familiar tests in regression.

*

*Tests of individual coefficients: there's an asymptotic z-test (some packages refer to the statistic as a t-statistic, but in my opinion there's not really a good justification for calling it that; we're not estimating a dispersion parameter). Here's an example of such statistics (which is obtained by running the first example in the help on glm in R - which fortunately doesn't call it a "t" statistic):
 Coefficients:  
               Estimate Std. Error z value Pr(>|z|)    
 (Intercept)  3.045e+00  1.709e-01  17.815   <2e-16 ***  
 outcome2    -4.543e-01  2.022e-01  -2.247   0.0246 *    
 outcome3    -2.930e-01  1.927e-01  -1.520   0.1285      
 treatment2   1.338e-15  2.000e-01   0.000   1.0000      
 treatment3   1.421e-15  2.000e-01   0.000   1.0000    

In this particular case, where each of the terms are levels of some factor, you wouldn't normally test them individually, but at least this gives an idea of where you look for the information.


*Overall tests of the model
There's an overall asymptotic chi-square test of the model, based on the deviance, which is akin to the normal-theory F-test.
For an example, if we take another part of the summary output from the same model from the glm help in R as above:
    (Dispersion parameter for poisson family taken to be 1)  
   
        Null deviance: 10.5814  on 8  degrees of freedom  
    Residual deviance:  5.1291  on 4  degrees of freedom  

then we obtain a chi-squared test for the model based on the difference, (10.5814-5.1291=5.4523) on (8-4=4) df


*Tests of subsets of variables. Again, this is asymptotically chi-square.
In the case of factors, of course, you want to test a whole factor at once:
           Df Deviance Resid. Df Resid. Dev   
 NULL                          8    10.5814   
 outcome    2   5.4523         6     5.1291  
 treatment  2   0.0000         4     5.1291  

So, for example, to test the 'treatment' variable, you'd test it with a chi-square of 0.0 on 2 df (which of course is not significant; it contributes essentially nothing).
Any difference of two nested models could be compared similarly.
