Worked example

Poisson Regression Model Selection Worked Example

This example walks through a small Poisson-regression comparison the way an actuarial candidate should: identify the model choice, interpret the coefficients, and explain why one specification is more defensible than the other.

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Role: Worked Example
Level: Core
Time: Reference
Freshness: Stable

Search Intent

Poisson regression worked example

Problem

An insurer models claim counts using a Poisson GLM with a log link and an exposure offset. Model A includes only an urban indicator. Model B includes the same urban indicator plus prior-claim count. The output below is available from software.

Model A: beta-urban = 0.35, AIC = 412, residual deviance = 128 on 110 degrees of freedom. Model B: beta-urban = 0.22, beta-prior-claims = 0.28, AIC = 396, residual deviance = 109 on 109 degrees of freedom. Which model is preferred, and how should the prior-claims coefficient be interpreted?

Poisson regression form

\log(\mu_i)=\beta_0+\beta_1\,\text{urban}_i+\beta_2\,\text{priorClaims}_i+\log(\text{exposure}_i)

What This Example Is Testing

This is testing three habits at once: reading GLM output, using model-comparison statistics without overcomplicating them, and translating coefficients into actuarial rate language.

Step-By-Step Solution

Start with model comparison. Model B has the lower AIC, 396 instead of 412, and the lower residual deviance, 109 instead of 128, while using only one extra predictor. On the information given, Model B is the better-supported specification.

Now interpret beta-prior-claims = 0.28. In a log-link Poisson model, a one-unit increase in the predictor multiplies the expected claim count by exp(beta). Here exp(0.28) is about 1.32. That means one additional prior claim is associated with about a 32% increase in the expected claim rate, holding the other model inputs fixed.

Notice that the urban coefficient drops from 0.35 to 0.22 when prior claims are added. That suggests some of what looked like an urban effect in Model A may have been partly explained by prior-claim experience once the fuller model was used.

Rate ratio interpretation

e^{0.28}\approx 1.32

Final Answer

Model B is preferred on the provided evidence because it improves AIC and residual deviance. The prior-claims coefficient implies roughly a 32% multiplicative increase in expected claim count for each additional prior claim, holding the other covariates fixed.

Common Wrong Answer

A common mistake is to interpret the 0.28 coefficient as an additive increase of 0.28 claims. In a log-link Poisson model, the coefficient works multiplicatively on the expected rate, not additively on the raw count scale.