Negative estimated means in Poisson regression output Can anybody help me understand what the negative estimated means in a Poisson regression mean?
I am trying to model the count of damages during transit for 4 different logistics providers. To make it clearer, my dependent variable is the number of times Transit damages have been reported over the past 16 months for different Trucks in service. I have also included an offset variable which is the Logarithm of the number of times the Truck has been in service for the past 16 months(not all trucks have transported goods in all months). My predictor variable is just the category of Trucks, that is to which logistics company the trucks belong. There are 4 companies, which I have coded as 1=AAC,2=DSA,3=OMP and 4=TCI.My reference category is 1=AAC. 
I am including my detailed SPSS output as follows:

Estimated means are as follows:
 
Have I done the right thing after all? Also Omnibus and other preliminary tests have been done. Data is a bit under-dispersed with Deviance 0.767 and Pearson Chi Square 0.983. 
 A: If results reported by SPSS are on the scale of the linear predictor, then when you exponentiate the Mean values provided you would get expected values of your count outcome variable. 
For example, if your count outcome variable Y is number of headaches in the past month and your only predictor variable is Gender (with levels Male and Female, say), your SPSS output should provide enough information for you to retrieve the expected number of headaches in the past month for males and the expected number of headaches in the past month for females. 
If the model were to include Age (in years) in addition to Gender, then your SPSS output should provide enough information for you to retrieve the expected number of headaches in the past month for males with a "typical" age and the expected number of headaches in the past month for females for females with a "typical" age, where the "typical" age could be the average age of the subjects in your study (regardless of their gender). 
I don't know what SPSS considers to be "typical" for this type of calculations.
Someone who knows SPSS can provide further help, but you need to specify in your question further details about your Poisson regression model, including 1. what predictors it includes and how they are coded and 2. what outcome variable it includes. 
A: Do you have multiple trucks for each provider such that each truck may have experienced damage multiple times over the study period and as a consequence may have been in service multiple times?  (I'm guessing that, if the extent of damage is small, a truck does not need to be serviced.)  
If you do have multiple trucks per provider, then the data coming from these multiple trucks will likely be correlated, so you would need to account for that somehow when you compute your standard errors for the model estimates.  
I think what you need to do is perform post-hoc multiple comparisons of the expected number of times trucks have experienced damage during the study period per number of times they have been in service during that same period using something like Tukey's multiple comparisons. This will allow you to compare all possible pairs of providers with respect to this expected quantity and get associated p-values.  The comparison will likely be done on the log scale, though the scale may depend on the software.    
