Motor claim frequency GLM with a sparse telematics level
A motor insurer fits a Poisson GLM to one year of claim counts for 18,000 comprehensive policies. The linear predictor uses log(policy-years) as an offset, plus vehicle group (A–D), driver age band (17–24, 25–34, 35–54, 55+) and a new telematics factor with six levels. The fitted model has residual deviance 1,840 on 1,600 degrees of freedom. Telematics level T6 contains only 35 policies and zero claims; its coefficient is reported as −12.4 with a standard error of 310. The pricing committee wants to use the model to produce relativities for all levels and to test whether telematics should be retained. The data include several policies with multiple claims, and the largest claim count is 9.
1. Diagnose the two main problems in the fitted model, explaining the statistical cause and the effect on the proposed use.
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First, residual deviance/df = 1840/1600 = 1.15. For a Poisson model the deviance should be roughly chi-square with 1,600 df; the standard deviation is about √(2×1600)=56.6, so 1,840 is about four standard deviations above the mean. There is significant overdispersion, possibly from unobserved heterogeneity, clustering of claims within policies, or a missing interaction. Poisson standard errors will be too small and AIC comparisons unreliable. Second, telematics level T6 has only 35 policies and zero claims. In a log-link Poisson GLM the MLE for a zero-claim level is driven towards minus infinity; the coefficient −12.4 with standard error 310 is a symptom of separation. The implied relativity exp(−12.4)≈4×10⁻⁶ is not credible and should not be used for pricing. The low exposure also means the level cannot be estimated reliably even with a different link.
2. Propose a revised modelling and validation strategy. Include how you would treat T6, how you would allow for overdispersion, and how you would decide whether telematics is retained.
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Treat T6 before interpreting telematics. Merge it with the closest comparable level or with an 'other' category, or impose a weakly informative prior/penalty, or use a random effect for telematics level. Do not report a free-standing coefficient from 35 policies and no claims. For overdispersion, fit a quasi-Poisson or negative-binomial model, or a Poisson GLMM with policy or employer random effects. Use dispersion-adjusted tests, such as quasi-likelihood F tests, rather than raw likelihood-ratio tests; if using AIC, compare only models fitted under the same distributional assumption and be cautious. Validate by holdout or cross-validation, check Pearson residuals and influence, and test whether telematics improves out-of-sample prediction and is stable across levels. Retain telematics only if the improvement is material, interpretable and robust to the treatment of T6.
3. Interpret the offset and explain what a coefficient of +0.25 for the 17–24 age band would mean. State the assumption required for that interpretation.
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The offset is log(policy-years) with a fixed coefficient of 1. With a log link, the model is therefore for the claim rate per policy-year: log(E[claims]) = log(exposure) + β′x. A coefficient of +0.25 for age band 17–24 means that, all else equal, the expected claim rate for that band is exp(0.25)=1.284, or about 28.4% higher than the baseline band. This is a multiplicative relativity, not an additive number of claims. The interpretation requires that the log link is appropriate, that exposure enters only through the offset with coefficient 1, that there are no interactions involving age band, and that the other covariates are held fixed. It does not require the Poisson variance assumption, but if overdispersion is present the standard error of +0.25 must be dispersion-adjusted.