# Very different scale parameter estimates in Poisson regression

The background: I'm analysing survival data using a Poisson model. I've splitted the data on 2 time-scales (attained age and calendar year). Attained age is modelled using flexible parametric functions, calendar year is a categorical variable and I've also included the interaction terms between attained age and calendar year.

The problem: looking at the goodness-of-fit criteria, I see that the deviance divided by the degrees of freedom (df) is 0.03, while the Pearson's $\chi^2$ divided by the df is 5.84. The fact that the former is <1 and the latter is >1 implies that the standard errors associated to my parameter estimates are deflated, if I estimate the scale parameter using Pearson's $\chi^2$ / df, or inflated, if I estimate the scale parameter using Deviance / df. This obviously leads to completely different results.

What could be wrong here?

Edit (04 Jul 2012): I think this question can be reformulated also as: with clustered data (i.e. dependent data within each cluster), are the goodness-of-fit criteria for overdispersion (namely Pearson's $\chi^2$ divided by the residual degrees of freedom and Deviance divided by the residual degrees of freedom) still interpretable? (With iid data, >1 indicates overdispersion, while <1 indicates underdispersion).

Edit (19 Aug 2013): After delving into this topic a little bit more, I no longer think that in this particular scenario the use of those goodness-of-fit criteria makes sense. In fact, we're only exploiting the GLM framework to fit a survival model characterized by a non-constant baseline incidence rate (see also: piecewise exponential model). We're not assuming that the outcome has a Poisson distribution. For this reason, the question as reformulated in my last edit (04 Jul 2012) is also wrong, as we don't have clustered data here. See this answer for more information.

• A couple questions, not directly related to your question, but which may be helpful: I am not sure what you mean by "splitted the data". Why is calendar year categorical? Feb 22, 2012 at 17:11
• This is what I mean by "splitting the data" as explained by R help (survSplit function of the survival pkg): "Given a survival data set and a set of specified cut times, split each record into multiple subrecords at each cut time. The new data set will be in ‘counting process’ format, with a start time, stop time, and event status for each record." Calendar year is categorical because I want to plot a graph of the incidence rate for every calendar year (1998...2009). Something like the panel B of this picture. Feb 22, 2012 at 20:37
• It might be useful to move your later edit to an answer and accept it as it essentially answers the question. Jul 20, 2018 at 9:30