## Fox Module 22: Generalized linear models, discrete and continuous data...

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NEAS
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Fox Module 22: Generalized linear models, discrete and continuous data

(The attached PDF file has better formatting.)

Homework assignment: Education and Auto Accidents

The homework assignment follows the discussion forum reading for this module.

We fit a linear model to three groups of drivers:

 Exposures Years of Schooling Auto Accidents per 100 Drivers 1,000 8 15 1,000 12 8 1,000 16 3

The X value is the years of schooling.

The Y value is the number of auto accidents per 100 drivers.

The table shows that drivers with

8 years of schooling (elementary school) have claim frequencies of 15%.

12 years of schooling (high school) have claim frequencies of 8%.

16 years of schooling (college) have claim frequencies of 3%.

We compare GLMs with different distributions of the error term.

Normal distribution with a constant variance.

Poisson distribution.

Assume each year of schooling has the same linear effect on claim frequency.

We fit a straight line to the three points.

The variance of the error term depends on the GLM.

A.     Which model gives the higher claim frequency for drivers with eight years of schooling?

B.     Which model gives the higher claim frequency for college educated drivers?

C.    Why might a linear model not be proper for these data? How does decreasing marginal utility affects the slopes? If a driver with 9 years of schooling has an expected claim frequency 1 percentage point less than a driver with 8 years of schooling, should the difference from 12 to 13 years of schooling be more or less than 1 percentage point?

D.    How do actuaries treat class dimensions like years of schooling? Do actuaries treat this as a quantitative or qualitative class dimension?

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joop
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i'm guessing part D should be qualitative? would appreciate some intuition on this

[NEAS: Yes; explain why. High school diploma is 8 years of school; college degree is 12 years; a PhD, law degree, or medical degree are about 15 to 18 years; but is the number of years a good estimator of a response variable?

cjlid
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We're comparing fitted GLM's with

(1) the normal distribution with a constant variance (which is the same as using Excel Regression like we have been in previous homework), and

(2) the Poisson distribution.

If you look in the Module 1 section, someone posted a combined PDF with all the module homework, practice problems, and readings.  Exercise 1.4: Line of Best fit, Part A, is similar to this homework.  I took the Poisson loglikelihood = Y Ln(u) - u - ln(y!) where y is the observation and u is the fitted mean and compared it with the classic Excel regression output to answer parts A and B of the homework.

Good luck...

CJLID

Kates
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I understand the idea behind using YLN(u) -u- ln(Y!), but for some reason I'm confused on how to actually use it.  When you say the observation y and fitted mean u, which values are you referring to in this example?

[NEAS: Y is the observation: 2, 3, or 10. ì is the fitted value at that point: 1.668, 5.000, or 8.333.]

Mbactuary
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I answered a and b by noting that the slopes between 8-12 and between 12-16 will be weighted differently by the poisson and normal distribution. The poisson weights the less steep slopes greater so it should result in a trend line that is a ccw rotation from the standard regression line. This logic says a) normal and b) poisson.

Is this good enough or do I need to use the poisson log likelihood method?

[NEAS: That is fine; the relation of the variance to the mean gives the solution.]

scomurphy
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Looking at the practice problems, excercise 1.5, for the exponential distribution, is f(y) = uexp(-uy) so the log likelikhood would then equal log(u) - uy, not -y/u - log(u), is that wrong?
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