Think about if assumptions for Poisson regression is fulfilled. That is, persons should be independent from each other, and the rate should be piecewise constant. Further, for log-linear Poisson regression (the one with log as link-function), you also have to assume the multiplicative structure of the rate of events. But, you can not use deviance to justify Poisson regression as "Poisson regression" in this case is not used to fit Poisson distributed data, but only as a trick to maximize the right likelihood function.
... View more
