Is Homicide Victims dataset repeated? I am studying an example where a Poisson model is applied to the Homicide Victims dataset. The dataset contains survey results of 1308 people who where asked about the number of homicide victims they knew. The dataset includes the response of a person (the number of homicide victims he/she knows) and his/her race. The response has minimum and maximum values of 0 and 6. The goal is to model the response using a Poisson model and assess the goodness of fit of the model.
I have two questions.

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*Is the dataset repeated? I understand that a repeated measure is a study where for each predictor variable there are repeated number of observations. In this case, I can redesign the dataset as a seven by two table (response and race) where each cell contains the number of people who know a specific number of homicide victims and have a specific race. It is important to identify if the data is repeated or not to make sure that the results of the Deviance and Pearson's Chi-Square tests are valid.

*The results of the Deviance and Pearson Chi-Square tests are as follows:
Deviance Test: Deviance = 844.7073, df = 1306, p-value = 1 and
Pearson's Chi-Square Test: Chi-Square=2279.873, p-value = 0.
So, in this case I am not sure why the p-values are critically different.


I think the concern motivating this question is that the example shows up in the chapter on Other Mixture Models for Categorical Data.  That chapter follows a series of chapters on Models for Matched Pairs (10), Analyzing Repeated Categorical Response Data (11), and Random Effects... (12), which establishes a section dedicated to repeated measures data.  The dataset's existence within this section implies it should be that type somehow (prototypically, repeated measures or hierarchically nested data).  In contrast, however, the data do not appear to have that character.
I think what Agresti is doing is using these data to illustrate a different issue.  The simplest model for count data is a Poisson model.  There is a sense in which Poisson regression 'should' be the right model to use.  However, the Poisson distribution makes a very strong assumption that the conditional variance is always the same as the condition mean.  That assumption is typically violated in the real world.  If so, we can ask why and we can ask what we're supposed to do now.
One reason the variance can be 'too high' is that the data come from heterogeneous units (e.g., patients) with different attributes (e.g., age, sex, family history, etc.) that are relevant to the outcome but which are not included in the model.  These attributes could average out, in which case the mean can still be the same, but the data will still spread out further (i.e., the higher conditional variance).
There are various ways to handle such a situation.  One is to use a 'quasi-Poisson' likelihood with an ad-hoc adjustment.  Another is to use the negative binomial distribution instead of the Poisson, which allows the variance to be independent of the mean.  But another way is to have a latent normal distribution of units (i.e., random effects).  This can be done even though there is only one response value per unit.  Agresti uses this dataset to illustrate this approach and how it compares to the default Poisson and the negative binomial.  Thus, a GLiMM is appropriate (and these data make sense to show in this chapter) even though the data are neither repeated nor nested.
