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GLM I: Introduction to Generalized Linear Models

GLM I: Introduction to Generalized Linear Models

By Curtis Gary DeanVice President and Actuary,

By Curtis Gary DeanVice President and Actuary,

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Classical Multiple Linear RegressionClassical Multiple Linear Regression

Yi = a0 + a1Xi1 + a2Xi2 …+ amXim + ei

Yi are the response variables and Xij are predictors

ei is random error term: Normally distributed

E[ei] = 0 and Var(ei) = σ2 is constant

Yi = a0 + a1Xi1 + a2Xi2 …+ amXim + ei

Yi are the response variables and Xij are predictors

ei is random error term: Normally distributed

E[ei] = 0 and Var(ei) = σ2 is constant

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Classical Multiple Linear RegressionClassical Multiple Linear Regression

μi = E[Yi] = a0 + a1Xi1 + …+ amXim

Yi is Normally distributed random variable with variance σ2

Want to estimate μi = E[Yi] for each i

μi = E[Yi] = a0 + a1Xi1 + …+ amXim

Yi is Normally distributed random variable with variance σ2

Want to estimate μi = E[Yi] for each i

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Response Yi has Normal Distribution

-12 -7 -2 3 8 μ i

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Problems with Traditional ModelProblems with Traditional Model

Number of claims is discrete

Claim sizes are skewed to the right

Probability of an event is in [0,1]

Variance is not constant across data points i

Nonlinear relationship between X’s and Y’s

Number of claims is discrete

Claim sizes are skewed to the right

Probability of an event is in [0,1]

Variance is not constant across data points i

Nonlinear relationship between X’s and Y’s

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Generalized Linear Models - GLMsGeneralized Linear Models - GLMs

Fewer restrictions

Y can model number of claims, probability of renewing, loss severity, loss ratio, etc.

Large and small policies can be put into one model

Y can be nonlinear function of X’s

Classical linear regression model is a special case

Fewer restrictions

Y can model number of claims, probability of renewing, loss severity, loss ratio, etc.

Large and small policies can be put into one model

Y can be nonlinear function of X’s

Classical linear regression model is a special case

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Generalized Linear Models - GLMsGeneralized Linear Models - GLMs

g(μi )= a0 + a1Xi1 + …+ amXim

g( ) is the link function

E[Yi] = μi = g-1(a0 + a1Xi1 + …+ amXim)

Yi can be Normal, Poisson, Gamma, Binomial, Compound Poisson, …

Variance can be modeled

g(μi )= a0 + a1Xi1 + …+ amXim

g( ) is the link function

E[Yi] = μi = g-1(a0 + a1Xi1 + …+ amXim)

Yi can be Normal, Poisson, Gamma, Binomial, Compound Poisson, …

Variance can be modeled

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Exponential Family of DistributionsExponential Family of Distributions

assumption commona is/)(

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Normal Distribution in Exponential FamilyNormal Distribution in Exponential Family

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Some Math Rules: RefresherSome Math Rules: Refresher

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Poisson Distribution in Exponential FamilyPoisson Distribution in Exponential Family

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Poisson Distribution in Exponential FamilyPoisson Distribution in Exponential Family

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Compound Poisson DistributionCompound Poisson Distribution

Y = C1 + C2 + . . . + CN

N is Poisson random variable Ci are i.i.d. with Gamma distribution

This is an example of a Tweedie distribution

Y is member of Exponential Family

Y = C1 + C2 + . . . + CN

N is Poisson random variable Ci are i.i.d. with Gamma distribution

This is an example of a Tweedie distribution

Y is member of Exponential Family

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Members of the Exponential FamilyMembers of the Exponential Family

• Normal• Poisson• Binomial• Gamma• Inverse Gaussian• Compound Poisson

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Variance StructureVariance Structure

E[Yi] = μi = b' (θi) → θi = b' (-1)(μi )

Var[Yi] = a(Φi) b' ' (θi) = a(Φi) V(μi)

Common form: Var[Yi] = Φ V(μi)/wi

Φ is constant across data but weights applied to data points

E[Yi] = μi = b' (θi) → θi = b' (-1)(μi )

Var[Yi] = a(Φi) b' ' (θi) = a(Φi) V(μi)

Common form: Var[Yi] = Φ V(μi)/wi

Φ is constant across data but weights applied to data points

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V(μ) Normal 0

Poisson Binomial (1-) Tweedie p, 1<p<2 Gamma 2

Inverse Gaussian 3

Recall: Var[Yi] = Φ V(μi)/wi

Variance Functions V(μ)

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Variance at Point and Fit Variance at Point and Fit

A Practioner’s Guide to Generalized Linear Models: A CAS Study Note

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Why Exponential Family?Why Exponential Family?

Distributions in Exponential Family can model a variety of problems

Standard algorithm for finding coefficients a0, a1, …, am

Distributions in Exponential Family can model a variety of problems

Standard algorithm for finding coefficients a0, a1, …, am

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Modeling Number of ClaimsModeling Number of Claims

Number ofPolicy Sex Territory Claims in 5 Years

1 M 02 02 F 01 03 F 01 04 F 02 15 F 01 06 F 02 17 M 02 28 M 02 29 M 02 1

10 F 01 1: : : :

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Assume a Multiplicative ModelAssume a Multiplicative Model

μi = expected number of claims in five years

μi = BF,01 x CSex(i) x CTerr(i)

If i is Female and Terr 01

→ μi = BF,01 x 1.00 x 1.00

μi = expected number of claims in five years

μi = BF,01 x CSex(i) x CTerr(i)

If i is Female and Terr 01

→ μi = BF,01 x 1.00 x 1.00

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Multiplicative ModelMultiplicative Model

μi = exp(a0 + aSXS(i) + aTXT(i))

μi = exp(a0) x exp(aSXS(i)) x exp( aTXT(i))

i is Female → XS(i) = 0; Male → XS(i) = 1

i is Terr 01 → XT(i) = 0; Terr 02 → XT(i) = 1

μi = exp(a0 + aSXS(i) + aTXT(i))

μi = exp(a0) x exp(aSXS(i)) x exp( aTXT(i))

i is Female → XS(i) = 0; Male → XS(i) = 1

i is Terr 01 → XT(i) = 0; Terr 02 → XT(i) = 1

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Natural Log Link FunctionNatural Log Link Function

ln(μi) = a0 + aSXS(i) + aTXT(i)

μi is in (0 , ∞ )

ln(μi) is in ( - ∞ , ∞ )

ln(μi) = a0 + aSXS(i) + aTXT(i)

μi is in (0 , ∞ )

ln(μi) is in ( - ∞ , ∞ )

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Poisson Distribution in Exponential FamilyPoisson Distribution in Exponential Family

eb

yy

yY

)(

ln

)!ln(1

lnexp]Pr[

eb

yy

yY

)(

ln

)!ln(1

lnexp]Pr[

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Natural Log is Canonical Link for PoissonNatural Log is Canonical Link for Poisson

θi = ln(μi)

θi = a0 + aSXS(i) + aTXT(i)

θi = ln(μi)

θi = a0 + aSXS(i) + aTXT(i)

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Estimating Coefficients a1, a2, .., amEstimating Coefficients a1, a2, .., am

Classical linear regression uses least squares

GLMs use Maximum Likelihood Method

Classical linear regression uses least squares

GLMs use Maximum Likelihood Method

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Likelihood and Log LikelihoodLikelihood and Log Likelihood

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Find a0, aS, and aT for PoissonFind a0, aS, and aT for Poisson

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Iterative Numerical Procedure to Find ai’sIterative Numerical Procedure to Find ai’s

Use statistical package or actuarial software

Specify link function and distribution type

Use statistical package or actuarial software

Specify link function and distribution type

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Solution to Our ExampleSolution to Our Example

a0 = -.288 → exp(-.288) = .75 aS = .262 → exp(.262) = 1.3 aT = .095 → exp(.095) = 1.1

μi = exp(a0) x exp(aSXS(i)) x exp( aTXT(i))

μi = .75 x 1.3XS(i) x 1.1XT(i)

i is Male,Terr 01 → μi = .75 x 1.31 x 1.10

a0 = -.288 → exp(-.288) = .75 aS = .262 → exp(.262) = 1.3 aT = .095 → exp(.095) = 1.1

μi = exp(a0) x exp(aSXS(i)) x exp( aTXT(i))

μi = .75 x 1.3XS(i) x 1.1XT(i)

i is Male,Terr 01 → μi = .75 x 1.31 x 1.10

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Other Common Link FunctionsOther Common Link Functions

Identity link: canonical link for Normal

Logit link ln[ μ / (1- μ) ]: canonical link for Binomial maps (0,1) onto ( -∞, ∞)

Reciprocal 1/μ: canonical link for Gamma

But, you do not have to match link and distribution. Choose link and distribution that make sense!

Identity link: canonical link for Normal

Logit link ln[ μ / (1- μ) ]: canonical link for Binomial maps (0,1) onto ( -∞, ∞)

Reciprocal 1/μ: canonical link for Gamma

But, you do not have to match link and distribution. Choose link and distribution that make sense!

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Measuring Goodness of FitMeasuring Goodness of Fit

Deviance is measure of goodness of fitCompare different modelsCompare effect of dropping/adding Xij’s

Also called log likelihood ratio statistic

Deviance is measure of goodness of fitCompare different modelsCompare effect of dropping/adding Xij’s

Also called log likelihood ratio statistic

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More on DevianceMore on Deviance

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Generalized Linear Models - GLMsGeneralized Linear Models - GLMs

Response Y can be discrete or continuous

Y can be number of claims, probability of renewing, loss severity, loss ratio, etc.

Large and small policies can be put into one model.

Y can be nonlinear function of X’s

Measure goodness of fit using Deviance

Response Y can be discrete or continuous

Y can be number of claims, probability of renewing, loss severity, loss ratio, etc.

Large and small policies can be put into one model.

Y can be nonlinear function of X’s

Measure goodness of fit using Deviance

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