For the horseshoe crab data from Exercise 23, there may be signs of over dispersion.

(a) Repeat the analysis from the example on page 316 to see whether a quasi-Poisson or a negative binomial is a better option. In particular, create a plot like Figure 5.10 and compute the quadratic regression as shown in the example. What do you conclude about the two model options?

(b) Fit the quasi-Poisson model to the data and test the model effects again. How do the results compare to those from

Exercise 23?

(c) Plot standardized residuals against the estimated means. Comment on the results.

Exercise 23

Agresti (2007) provides data on the social behavior of horseshoe crabs. These data are contained in the HorseshoeCrabs.csv file available on our website. Each observation corresponds to one female crab. The response variable is Sat, the number of "satellite" males in her vicinity. Physical measurements of the female-Color (4-level ordinal), Spine (3-level ordinal), Width (cm), and Weight (kg)-are explanatory variables.

(a) Fit a Poisson regression model with a log link using all four explanatory variables in a linear form. Test their significance and summarize results.

(b) Compute deviance/df and interpret its value.

(c) Examine residual diagnostics and identify any potential problems with the model.

(d) Carry out the GOF test described on page 296 using the PostFitGOFTest() function available in the Post Fit GOF Test. R program of the same name from our website. Use the default number of groups. (M/5 when M ≤ 100)

i. State the hypotheses, test statistic and p-value, and interpret the results.

ii. Plot the Pearson residuals for the groups against the interval centers (available in the pear.res and centers components, respectively, of the list returned by the function). Use this plot and the residual plots from part (c) to explain the results.