Fitting a Poisson GLM to Territory / Vehicle-Age Claim Frequency

Background

Meridian Mutual's pricing team is refreshing personal-auto claim-frequency relativities for two rating variables: Territory (Urban vs. Rural) and Vehicle Age (New vs. Old, where "Old" means model year 6+). A pricing analyst pulled one accident year of earned exposure and reported claim counts for the four cells formed by crossing these two variables:

Portfolio-wide frequency is $\bar f = 364/3600 = 0.0872$ claims per exposure. The raw cell frequencies suggest Rural business runs lighter than Urban, and Old vehicles run heavier than New, but the analyst wants a multiplicative structure that can be extended to many more variables without a combinatorial explosion of cells — the textbook motivation for a generalized linear model.

Model specification

The analyst fits a Poisson GLM with a log link and an exposure offset, so that expected claim counts are

$E[N_i] = e_i \cdot \exp(\beta_0 + \beta_1 \cdot \mathbb 1[\text{Rural}]_i + \beta_2 \cdot \mathbb 1[\text{Old}]_i),$

where $e_i$ is the exposure in cell $i$ and Urban/New is the baseline level absorbed into $\beta_0$. This is equivalent to $\log(E[N_i]/e_i) = \beta_0 + \beta_1\mathbb 1[\text{Rural}]_i + \beta_2\mathbb 1[\text{Old}]_i$ — the offset $\log e_i$ is added to the linear predictor with a coefficient fixed at 1, which is the standard GLM device for modeling a rate while keeping the response a Poisson count.

Fitting this saturated model (4 cells, 3 free parameters plus the fixed offset — one fewer than the number of cells, since a full interaction term would make it exactly saturated) by iteratively reweighted least squares gives:

The fitted expected counts are:

The fitted relativities say a Rural risk has $0.5462\times$ the frequency of an otherwise-identical Urban risk (a 46% discount), and an Old vehicle has $1.4318\times$ the frequency of an otherwise-identical New vehicle (a 43% surcharge), holding the other factor fixed — this is exactly the "holding all else equal" interpretation that a GLM provides and a one-way (univariate) analysis cannot, because a univariate Rural/Urban split would be confounded by the fact that Rural business happens to skew towards newer vehicles in this book.

Deviance and goodness of fit

The model's residual deviance is $D = 0.0154$ on $4 - 3 = 1$ degree of freedom (this model has one fewer parameter than cells, so it is nearly saturated and fits almost perfectly by construction). The null deviance (intercept-only model, i.e. every cell predicted at the overall frequency) is $D_0 = 34.08$ on $3$ degrees of freedom. The drop in deviance from adding Territory and Rural,

$\Delta D = D_0 - D = 34.08 - 0.0154 = 34.07 \text{ on } 3-1=2 \text{ d.f.},$

is large relative to a $\chi^2_2$ reference distribution (the 99th percentile of $\chi^2_2$ is about $9.21$), so both rating variables are statistically significant contributors to explaining claim frequency in this data — consistent with, but not proof of, causal risk differentiation (omitted-variable confounding with e.g. annual mileage or driver age is always possible with only two rating variables).

Your task

You are reviewing this GLM output for the rate filing. Treat the fitted coefficients, relativities and deviance figures above as given (do not refit). Answer the questions below using GLM theory and the specific numbers in this case.