Telematics Variable in a Personal Auto GLM Classification Plan

The AIC improvement of 41 on 7 degrees of freedom is statistically meaningful - roughly a chi-square statistic of 41, far beyond chance. But AIC is an in-sample fit criterion; it rewards adding parameters that improve likelihood, even when those parameters are estimated from very thin data. The −0.42 coefficient on level 8 rests on 1,150 car-years and 21 claims, three of which exceed $250k. A single large claim can move a Poisson coefficient substantially, because the GLM is fitting claim counts, not dollars; but the large claims also signal that the level may be heterogeneous. Before filing, refit with large losses capped or with a Tweedie distribution, and run a holdout or k-fold cross-validation. If the coefficient is stable across folds and after capping, the variable has signal. Then collapse the eight levels into no more than five groups so that every filed level meets the 2,500 car-year threshold, and apply credibility to the relativities. Do not file a level with 1,150 car-years as a standalone class; the regulator will reject it and the estimate is not reliable. The AIC gain does not, by itself, justify the eight-level structure; it justifies investigating telematics, not filing the raw output.

Use Bühlmann-Straub credibility on the indicated relativity for each level, with exposure as the volume measure. The complement is the overall average (1.00) or the average of adjacent levels if there is a monotonic pattern. Estimate K from the between-level and within-level variance of loss ratios; a common practical value in auto classification is K ≈ 1,500–3,000 car-years. For level 8, E = 1,150. If K = 2,000, Z = 1,150/(1,150+2,000) = 0.365. The indicated relativity is exp(−0.42) = 0.657. The credibility-weighted relativity is 0.365 × 0.657 + 0.635 × 1.00 = 0.875. The filed relativity should therefore be about 0.88, not 0.66. This protects against over-fitting and keeps the level within a defensible range. If level 8 is collapsed with level 7, the combined exposure may exceed 2,500 and Z rises; then the blended relativity can move closer to the indicated value. Document the K estimate and show sensitivity to K = 1,500 and 3,000. Also check monotonicity: if the telematics levels are ordered, use isotonic regression or a continuous score so the relativities do not zigzag.

Three options. First, replace the categorical enrollment variable with a continuous telematics score (for example, miles driven or hard-braking events per 1,000 miles) and fit a linear or spline term. This uses all 212,000 car-years and avoids thin cells; the trade-off is that the score must be available and auditable, and the relationship may not be linear. Second, pool the company's data with another state or affiliate that uses the same telematics definition, adding a state or company indicator and interacting it with the score. This raises exposures per level, but pooling requires that the underlying loss environment and program design be similar; otherwise the pooled relativity is biased. Third, interact telematics with a strong existing variable, such as annual mileage or vehicle symbol, if the telematics effect is concentrated in one segment. Interactions multiply the number of cells and can create new thin cells, so do this only where the data support it. A fourth option is a random-effects (GLMM) model that shrinks each level toward the mean automatically; this is statistically equivalent to credibility but easier to implement with many levels. None of these eliminates the need to validate out-of-sample.