Overdispersion in a Motorcycle Frequency Filing

Poisson as fitted. z = −0.18/0.060 = −3.0, p ≈ 0.003. On its face the discount is highly significant - that is what the product manager is looking at.

But the Poisson variance assumption fails. Residual deviance/df = 1,842/412 = 4.47. For a correctly specified Poisson model this ratio is approximately 1, so φ̂ ≈ 4.47: the observed variability is about four and a half times Poisson. The reported standard errors are therefore too small by roughly √φ̂ = 2.11.

Quasi-Poisson correction. SE = 0.060 × 2.11 = 0.127, so z = −0.18/0.127 = −1.42, p ≈ 0.16. The coefficient estimate is unchanged (quasi-Poisson only rescales the variance), but the evidence is not there.

Negative binomial. z = −0.11/0.115 = −0.96, p ≈ 0.34. The point estimate itself has moved 40% toward zero, which is the more troubling result: the Poisson fit is not merely over-confident, it is biased on this covariate.

Conclusion. The 16% discount is not supported under either properly specified model. A 16% discount corresponds to a coefficient of −0.18, which is at the edge of (or outside) any reasonable interval once overdispersion is acknowledged. The defensible filing is no discount, or a substantially smaller one with an explicit caveat. Note also that a p-value near 0.16 is not evidence of no effect - the honest statement is “insufficient evidence at this credibility,” which argues for tracking the segment rather than pricing it.

What they do. Those eight cells are not 412 independent observations in the usual sense. Each is driven by one fleet, so its Pearson residual reflects that fleet's idiosyncratic risk plus whatever Poisson noise exists - not Poisson noise alone. A handful of such cells can be enough to move φ̂ from ~1 to ~4.5, because each contributes a squared residual on a small expected count and there is no averaging to damp it. The consequence is that a defect located in eight cells flattens the apparent significance of every covariate in the model, including ones that are genuinely predictive.

Quasi-Poisson vs negative binomial. They differ in what they assume and what they deliver:

• Quasi-Poisson keeps the Poisson mean structure and inflates the variance by a constant φ. It corrects standard errors and nothing else; there is no likelihood, so no AIC, no prediction intervals, no simulated counts.
• NB2 assumes Var = μ + αμ². It is a genuine likelihood, so it supports AIC/BIC, out-of-sample scoring and simulation, and its weighting of observations differs from quasi-Poisson because the variance is mean-dependent.

Here both point the same way (no significant discount), so the choice matters less for this decision than for the model's downstream use.

What I would do. Diagnose before choosing: plot Pearson residuals against fitted values and against exposure; check whether overdispersion persists after excluding the eight single-risk cells (as a diagnostic only, not as the filed model); consider a robust/sandwich variance estimator; and, better, refit at policy or fleet level with a fleet random effect so that between-fleet heterogeneity is modelled explicitly rather than dumped into φ. If the dispersion collapses once single-risk cells are down-weighted, the “overdispersion” is really heterogeneity of risk quality, and a fleet-level random effect is the right fix.

What changes technically. In a log-link Poisson GLM, an offset of log(exposure) constrains its exponent to exactly 1. Entering log(exposure) as a covariate estimates that exponent freely. So the proposal is a test of proportionality, not a modelling improvement per se.

If the estimated exponent is near 1, the offset was correct and the covariate version simply costs a degree of freedom. That is the expected outcome for claim counts.

If the estimated exponent is far from 1, the offset was not the problem - the model is. Plausible causes: exposure is correlated with the other covariates (territory, usage), so a free exponent absorbs their effect and then reports a meaningless “large fleets have fewer claims per unit exposure” coefficient; or a few high-exposure cells dominate the estimate, which is exactly the pathology already identified in this dataset. An exponent below 1 recovered from these data would more likely be a symptom of overdispersion and single-fleet leverage than a real economy of scale.

Recommendation. Keep the offset in the filed model. Run the free-exponent version as a diagnostic and report the estimate with its confidence interval; if it excludes 1 materially, investigate the high-exposure cells rather than changing the structure. Do not present a model without the offset to a regulator - it invites the question of why exposure proportionality is not enforced, and it makes the coefficients harder to interpret as frequency relativities.

A related and more useful diagnostic is to refit at policy level (so exposure is a per-policy scalar) and compare, which separates the exposure-scaling question from the cell-aggregation problem.