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

1. Tests of individual coefficients: there's an asymptotic z-test (some packages refer to it as a t-statistic, but there's not really a good justification for calling it that). 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.

2. 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

3. 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.

• Here is the STATA read out for deliveries that I obtained. I should point out that statistics is not something I excel at and clear lay-man interpretation of this print out would help me understand the read-out better. – user41816 Mar 14 '14 at 8:40
• user41816: The LR chi-square and prob>chi-square would presumably be the chi-square (and corresponding p-value) for the test of the model I describe in 2. above. (You should check the documentation of your software to be absolutely certain.) – Glen_b -Reinstate Monica Mar 15 '14 at 1:24