Guidelines: Please remember to show your work and explain your answers as necessary. Answers that are not supported by reasoning will not receive full credit.
(5.1) For the horseshoe crabs data file, fit a model using weight and width as explanatory variables for the probability of a satellite.
Conduct a likelihood-ratio test of H_0: β_1=β_2=0. Interpret. Use R
Conduct separate likelihood-ratio tests for the partial effects of each variable. Why does neither test show evidence of an effect when the test in (a) shows very strong evidence?
(5.5) Exercise 4.12 introduced four scales of the Myers–Briggs personality test. Table 5.6 shows SAS output for fitting a model using the four scales as predictors of whether a subject drinks alcohol frequently.
Conduct a model goodness-of-fit test and interpret. If you were to simplify the model by removing a predictor, which would you remove? Why?
Software reports AIC values of 642.1 for the model with the four main effects and the six interaction terms, 637.5 for the model with only the four binary main effect terms, and 648.8 for the model with no predictors. According to this criterion, which model is preferred? Explain the rational for using AIC.
When six interaction terms are added, the deviance decreases to 3.74. Show how to test the hypothesis that none of the interaction terms are needed and interpret.
Using the MBTI data file at the website www.stat.ufl.edu/~aa/introcda/data, use model-building methods to select a model for this alcohol response variable. Use R
(5.9) The following table shows a 2×2×6 contingency table for y = whether admitted to graduate school at the University of California, Berkeley, for fall 1973, by gender of applicant for the six largest graduate departments.
Admitted, Male Admitted, Female
Department Yes No Yes No
1 512 313 89 19
2 353 207 17 8
3 120 205 202 391
4 138 279 131 244
5 53 138 94 299
6 22 351 24 317
Total 1198 1493 557 1278
Fit the logistic model that has department as the sole explanatory variable for y. Use the standardized residuals to describe the lack of fit. Use R.
When we add a gender effect, the estimated conditional odds ratio between admissions and gender (1 = male, 0 = female) is 0.90. The marginal table, collapsed over department, has odds ratio 1.84. Explain what causes these associations to differ so much.
Refer to Table 2.7 on mother’s drinking and malformations.
Fit the logistic regression model using scores {0, 0.5, 1.5, 4, 7} for alcohol consumption. Check goodness of fit. Use R.
Test independence using the likelihood-ratio test for the model in (a). (The trend test of Section 2.5.1 is the score test for this model.) Use R.
Fit the model and conduct the test of independence for all the data using scores {1, 2, 3, 4, 5}. Compare the results with (b). (Results for highly unbalanced data can be sensitive to the choice of scores.) Use R.
(5.12) Suppose y=0 at x=0,10,20 ,30 and y=1 at x=70,80,90,100. Use RExplain intuitively why β ̂=∞ for the model, logit(P(Y=1))=α+βx. Report β ̂ and its SE for the software you use.
Add two observations at x = 50, y = 1 for one and y = 0 for the other. Report β ̂ and its SE. Do you think these are correct? Why? What happens if you replace the two observations by y = 1 at x = 49.9 and y = 0 at x = 50.1?
(5.10) The Lungs data file at the text website summarizes eight studies in China about smoking and lung cancer. Analyze these data and prepare a short report that summarizes your analyses and interpretations using the two variables City and Smoking. Use R
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