Problem 1

The data set Claims (available on Blackboard) gives the number of policyholders (npolicies) of an insurance company that were "exposed to risk", and the number of car insurance claims (claims) made in the third quarter of a given year by these policyholders, tabulated by three factors:
- car the engine capacity of the car (1: < 1 liter, 2: 1-1.5 liters, 3: 1.5-2 liters. 4: > 2 liters)
- age the age of the policyholder (1: < 25, 2: 25-29, 3: 30-35, 4: > 35)
- dist the district in which the policyholder lived (1: London and other major cities, 0: otherwise) We are interested in modeling the number of claims using the other variables.

1. Identify the exposure input, i.e,, which variable in Claims gives the exposure? Explain your answer.

2. What is N, i.e., what is the total number of categories for counts? Explain your answer.

3. Calculate the rate of claims (i.e„ claims/npolicies) for each category and plot these rates by age and car. using different colors for dist in each plot. Also produce a plot of the claim rates against dist. Comment on what the plots say about the main effects of these factors (age, car, dist ). Do there appear to be any interaction effects?

4. Fit a Poisson regression model to estimate the main effects of the factors (each treated as cate-gorical variables). Do not include any interaction terms in this model, Interpret the results.

5. Is there evidence of overdispersion in the model fit in #4? Perform the appropriate test of signifi¬cance and provide visual evidence to support your answer.

6. Add pairwise interaction terms to the model fit in #4. Explain why none of these interactions are important.

7. Fit a Poisson regression model now treating age and car as numeric variables. How does the model compare to the model fit in #4?

8. What conclusions about the Claims data set can you draw based on the above results?

Problem 2:

The data set GractSchool contains information on a sample of 400 college juniors to deter¬mine what factors influence the decision of whether to apply to graduate :schoc.pl. The following variable,q are included!
- apply responses when asked if student is "unlikely", "somewhat likely'', or "very likely" to apply to graduate school.
- pared indicatur 'variable fur at Yeah VDU parent 1-um ving a graduate degree.
- pub 1 i c indicator variable for public college,
- qpa student's GPA on a 4.0 scale

(a) Fit an ordinal logistic model with response apply and the other variables predictors_ A bit of preprocessimg is necessary betore fitting, a model to the data set. Namely, the levels of apply need to be pi' perly ordered, which Gan be accomplished as fellows:
GradSchoolapply - factor{GradSclool$apply,
Levela-c("unlikely", "somowhat likely*,"very like]y"),
ordered-I)

(b) Remove any predictors that were not significant in the model Ili in (a) (explain how you know a predictor is not significant) and refit an ordinal logistic model, Row does this model compare to the model fit in (a)?

(c) Using the mode] lit in (b). estimate (and interpret) the following probabilities for a %hider!! with a GPA, at a public undergraduate institution, whose parents did not earn a graduate degree:

i. L the cumulative probability of -somewhat likely" applying

ii. IL the probability of "unlikely" applying

iii. i„ the probability of ".somewhat likely" applying

iv. the probability of "likely" applying

(d) Repeat (e), for a student with the same GPA, at a public institution, but at least one parent earning a graduate degree (i,e„ increase pa by 1 unit)? Given the results what can, you
say about the effect of pared on the likelihood of applying to graduate school'?

(e) Is the model tit in (II) good for classification? Answer this by estimating the test error,

(i) Using the same predictors in the model fit in (b), fit an ordinal pr it mode] and compare to the. ordinal logistic model. Specifically, compare the probability estimates found in (e) to estimate found with the fitted ordinal probit model.

Problem 3

Suppose Y₁, ......Y_N are independent random variables each with the normal distribution and

E[Y_i] = (β₀ + β₁x_i)²

Is this a generalized linear model. Give reasons for your answer. If it is a GLM, give the link function.

Attachment:- stats -data.rar