Continuing Exercise 26:
(a) Carry out the GOF test described on page 296 using the Post Fit GOF Test() function from Post Fit GOF Test. R on the book's website. Use the default number of groups, which is n/5 when n ≤ 100. State the hypotheses, test statistic and p-value, and interpret the results. Plot the Pearson residuals for the groups ($pear.res) and against the interval centers ($centers). Use this plot and the residual plots from Exercise 26 to explain the results.
(b) Perform a Monte Carlo simulation to investigate whether the GOF test holds the correct size for this model and data. Specifically, assuming that the model is contained in a glm object called mod.fit, use set.seed(1348765911), and repeat the following steps 1000 times:
i. Simulate Poisson response data from the estimated model. For example, use predict() to save the estimated means to an object called mean and simulate new responses using
y
ii. Estimate the Poisson regression model for the simulated data and obtain the predicted values from the model for each observation.
iii. Apply Post Fit LOF Test() to the simulated data and predicted values; obtain the p-value (pval component of the returned object)
The fraction of p-values that lie below 0.05 is the estimated size of the test when a type I error level of this same value is used. Is this fraction close to 5% as one would hope for if the test holds the correct size? Use a confidence interval to account for simulation variability in answering this question.
Exercise 26
Refer to Exercise 18 from Chapter 4 on the salamander count data.
(a) Compute deviance/df. What does this value suggest about the model?
(b) Examine the standardized residuals and perform an influence analysis. Interpret the results.
Exercise 18
The data in Table 4.8 are from a survey of salamander counts taken on plots of fixed size sampled from prairie regions in Manitoba, Canada.17 The prairies are subject to periodic controlled burning to reduce the chance of uncontrolled wildfires. The research question being investigated is how salamander populations react to these controlled burns. Use a Poisson regression model to address this question. Also comment on the apparent fit of the model.
