A car manufacturer conducts a survey to see how appealing its new model is to its existing customer base. It selects current customers at random, and gives them a new vehicle with two different car setups, a conventional city setup (A) and a sporty setup

Question (16 marks)

A car manufacturer conducts a survey to see how appealing its new model is to its existing customer base. It selects current customers at random, and gives them a new vehicle with two different car setups, a conventional city setup (A) and a sporty setup (B) and, following a period of one month, asks them to rate the vehicle’s performance on a scale of L0 (not satisfied) to L3 (very satisfied). The data is provided in the file market.csv.

1. State why a proportional odds model is appropriate for these data. Fit a proportional odds model to the data, both with a logic link and a profit link and comment on the respective fits. Use the pole function in the MASS package for fitting the model. (2 marks)
2. From now on consider only the proportional odds model with the logistic link. Use the estimated ? parameters to fifind the probability of a male rating L2 or less with the sporty setup. Verify your results using the predict function. (Note: polr assumes a model of the form g(?? ij ) = ?0j ? x1i?1 ? x2i?2 ? . . .). (2 marks)
3. What are the odds of a male rating L2 or less with the sporty setup? (2 marks)
4. Using the estimated ? parameters, compute the odds of a female rating L2 or less with the city setup relative to those of a male rating L2 or less with a sporty setup. Verify your answer by computing the probabilities (and from those the odds) using the predict function. (2 marks)
5. By exponential confidence intervals on a linear combination of the ? parameters, provide 99% confidence intervals for the odds ratio in 1d) (Hint: Use the function vcov to get the covariance matrix of the ? parameters). Is the observed odds ratio significant? (4 marks)
6. Construct a Wald statistic to test the hypothesis that the car setup significantly affects the cumulative odds. What do you conclude? (2 marks)
7. Refit the data using a proportional odds model without the car setup as an explanatory variable. Using the difference in deviance, construct a likelihood-ratio test statistic for the null hypothesis that the effect of car setup is not significant. What do you conclude? (2 marks)