(Solved): 2. (a) A logistic regression model is constructed to model the probability of being involved in a ...

2. (a) A logistic regression model is constructed to model the probability of being involved in a car accident in the next year, denoted $\mathrm{q}$. The model is: $\log \left[q_j/\left(1?q_j\right)\right]=?x+{?}_j$ where $x$ represents the driver's age and $?$ represents the driver's license status with learner $(L)$, newly passed $(N)$ or fully licensed $(F)$ being the 3 possible statuses. $?$ is estimated as 0.05 . ${?}_L,{?}_N$, and ${?}_F$ are estimated as $?6,?5.5$ and -4.5 respectively. (i) Calculate the probability of having an accident in the next year for a 25-year-old fully licensed driver. [2] (ii) Calculate the odds ratio of having an accident in the next year for a newly passed versus a learner driver. [4] (iii) Illustrate the nature of the fitted GLM graphically using a rough sketch and explain how the geometric features of the graph relate to the parameters of the fitted model. [4] (iv) An actuary is considering extending the original model to incorporate a quadratic effect of age. The original model has a deviance value of 22 with 20 degrees of freedom. The extended model has a deviance value of 18 . Perform a deviance test to investigate whether the quadratic term should be included in the model. [5]