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.
- 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)
- 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)
- What are the odds of a male rating L2 or less with the sporty setup? (2 marks)
- 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)
- 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)
- Construct a Wald statistic to test the hypothesis that the car setup significantly affects the cumulative odds. What do you conclude? (2 marks)
- 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)