## Poisson regression in R software

Questions
(1) Use Poisson regression to estimate the main effects of car, age, and dist (each treated as categorical and modelled using indicator variables) and interaction terms.
(2) It was determined by one study that all the interactions were unimportant and decided that age and car could be treated as though they were continuous variables. Fit a model incorporating these features and compare it with the best model obtained in (1). What conclusions do you reach?

n is the number of insurance policies
y is the number of claims
car is the car in an insurance category
age is the age of policy holder
dist is the district where the policy holder lived (1 for London and other major cities, and 0 otherwise)

Data:

car age dist y n
1 1 0 65 317
1 2 0 65 476
1 3 0 52 486
1 4 0 310 3259
2 1 0 98 486
2 2 0 159 1004
2 3 0 175 1355
2 4 0 877 7660
3 1 0 41 223
3 2 0 117 539
3 3 0 137 697
3 4 0 477 3442
4 1 0 11 40
4 2 0 35 148
4 3 0 39 214
4 4 0 167 1019
1 1 1 2 20
1 2 1 5 33
1 3 1 4 40
1 4 1 36 316
2 1 1 7 31
2 2 1 10 81
2 3 1 22 122
2 4 1 102 724
3 1 1 5 18
3 2 1 7 39
3 3 1 16 68
3 4 1 63 344
4 1 1 0 3
4 2 1 6 16
4 3 1 8 25
4 4 1 33 114

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I need help finding the correct R code to construct models. According to the solution manual, the model in (2) "is simpler than (1), fits well (deviance = 53.11, d.f. = 60, p-value = 0.72) and gives coefficients (standard errors): AGE, – 0.177 (0.018); CAR, 0.198 (0.021); DIST, 0.210 (0.059)."

As of the first model, I think that I should use this code, but not sure:

> firstmodel<-glm(y~factor(age)*factor(car)*factor(dist),family= poisson)

As of the second model, I used this code, but it produces results that contradict what the solution manual says (and deleting intercept does not help):

> secondmodel<-glm(y~age+car+factor(dist),family=poisson)
> summary(secondmodel)

Call:
glm(formula = y ~ age + car + factor(dist), family = poisson)

Deviance Residuals:
Min 1Q Median 3Q Max
-14.0258 -3.3200 -0.6296 2.0575 18.1442

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.08222 0.08127 37.92 <2e-16 ***
age 0.83664 0.02067 40.48 <2e-16 ***
car -0.16723 0.01612 -10.37 <2e-16 ***
factor(dist)1 -2.15937 0.05849 -36.92 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 5660.6 on 31 degrees of freedom
Residual deviance: 1154.5 on 28 degrees of freedom
AIC: 1330.8

Number of Fisher Scoring iterations: 5