Question

*** Please solve in R ***

You’re a newly hired analyst in Patriot Insurance Co. Your boss just asked to assess the relationship (if there is any) of the number of claims filed and length of time a policyholder hold his or her policy. You’ve randomly selected data on 20 policy holders: number of claims and length of time (in years) the policy holders held their policies. See “Prob 11-1 claims data.csv” for the data.

You decide that a Poisson distribution is a reasonable first choice as a data generating process that could have produced the counts of claims. Perform a Poisson regression to analyze the data; report your findings.

Claims	Years
1	1.024733
2	2.696719
3	3.626263
2	4.948643
4	6.024718
8	6.254113
1	6.526594
4	7.108674
5	9.215436
7	9.683853
3	9.85961
6	10.90103
9	11.35428
4	13.24533
6	13.38556
7	14.65962
13	15.61379
14	16.1935
10	18.62843
16	18.71313

Answer 1

Poisson Regression Analysis: Years versus Claims

Method

Link function	Natural log
Rows used	20

Deviance Table

Source	DF	Adj Dev	Adj Mean	Chi-Square	P-Value
Regression	1	31.61	31.607	31.61	0.000
Claims	1	31.61	31.607	31.61	0.000
Error	18	25.59	1.422
Total	19	57.20

Model Summary

Deviance R-Sq	Deviance R-Sq(adj)	AIC
55.26%	53.51%	109.13

Coefficients

Term	Coef	SE Coef	VIF
Constant	1.684	0.139
Claims	0.0869	0.0150	1.00

Regression Equation

Years

=

exp(Y')

Y'

=

1.684 + 0.0869 Claims

Goodness-of-Fit Tests

Test	DF	Estimate	Mean	Chi-Square	P-Value
Deviance	18	25.59001	1.42167	25.59	0.110
Pearson	18	23.87999	1.32667	23.88	0.159

The hypothesis being tested is:

H0: The data is of a Poisson distribution

Ha: The data is not of a Poisson distribution

The p-value is 0.159.

Since the p-value (0.159) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we can conclude that the data is of a Poisson distribution.