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.

 A: Agresti takes this data from the General Social Survey (GSS), and perhaps he treats it as a simple random sample(?) which is fine if he's simply using it as an illustration. However, the GSS is not a simple random sample; it's a multiple-stage cluster sample. You can find details here.
It is likely that respondents in the same cluster are not independent. If one respondent knows someone who was the victim of a homicide, it's likely that another respondent from the same cluster also knows someone who was the victim of a homicide, possibly the same victim.
I'm not sure that I follow your concern about repeated observations, but there is a problem here with using the chi-squared test as the standard test is not appropriate for data generated under a multi-stage cluster sample. P-values are likely to be too small. Several strategies exist for adjusting the chi-squared test. An early approach utilizes the Wald test. Rao and Scott introduce two additional approaches which are generally preferable. See here and here. All of these approaches are included in the survey package in R. See here.
A: 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.
A: Repetition would be when you measure the response within a single same unit while varying other variables.
For instance, if you have a cyclist and you let them cycle 6 times some distance while giving them either 0, 1, 2, 3, 4, 5, or 6 candies. Then you have six different measurements but all of them with the same cyclist rather than with six different cyclists. For each cyclist you repeated the measurement, under different conditions (this contrasts to having a different cyclist for each different condition).
The difference that repetition makes is that you can model the differences between the six measurements because they were done by the same cyclist. Or with a different type of analysis (in a mixed/random-effects model), you could separate the noise into two parts, one due to the cyclist and one due to the measurement.

From your description of the homicide victims dataset it does not seem like you have a repeated measurement. You have values of predictor variables that are repeated but that is everytime with a different person out of the 1308 persons. There is no change of a predictor variable within the same person.
Copy of the table from your reference
                 Race
           Black     White
Response
0            119      1070
1             16        60 
2             12        14
3              7         4
4              3         0
5              2         0
6              0         1

However, in some sense, you do have some sort of repeated measures here. The observations of the number of responses 0, 1, 2, 3, 4, 5, 6 are done within the same race category populations.
It would, for instance, be a difference when you where performing the survey in different cities and in city A you observe the number of 0 response among white respondents, in city B you observe the number of 1 response among white respondents, in city C you observe the number of 2 response among white respondents, etc.
