Overdispersion in Group Dental Claim Counts
A group dental carrier has 125 employer groups observed over four policy years, for 500 group-years of experience. The number of claims per group-year has sample mean 2.40 and sample variance 5.10. The pricing team fitted a Poisson frequency model, and on that basis intends to offer a 15% no-claim discount to any group-year with zero claims, arguing the discount is self-funding because it will apply to only about 9% of exposure. The data team notes that the 500 group-years are not 500 independent observations: groups renew and their claim patterns persist year to year. Group sizes vary by a factor of six.
1. Test the Poisson (equidispersion) assumption formally and state the practical conclusion.
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Dispersion test. Under equidispersion, the statistic
T = sum (x_i - xbar)^2 / xbar = (n - 1) s^2 / xbar
is approximately chi-square with n - 1 degrees of freedom.
T = 499 x 5.10 / 2.40 = 1060.4 on 499 df.
Mean 499, standard deviation sqrt(2 x 499) = 31.6, so z = (1060.4 - 499)/31.6 = 17.8, p far below any reasonable threshold.
Conclusion. Equidispersion is rejected decisively. The variance-to-mean ratio is 5.10/2.40 = 2.125, so the Poisson model understates the variance by more than a factor of two. Any premium, discount, or capital figure that depends on the Poisson shape - not just its mean - is exposed.
Caveat that must be stated before acting. The chi-square reference distribution assumes independent observations. With 125 groups tracked over four years the effective sample size is closer to 125 than 500, so the p-value is overstated; the direction of the conclusion survives, but the magnitude of the evidence does not.
2. Fit a negative binomial by moments and use it to evaluate the proposed 15% no-claim discount.
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Moment fit (NB2 form). Var = m + m^2/r with m = 2.40 and Var = 5.10:
5.10 = 2.40 + 5.76/r => 5.76/r = 2.70 => r = 2.133.
Zero-claim probability. P(0) = (r/(r+m))^r = (2.1333/4.5333)^2.1333 = 0.4706^2.1333. ln(0.4706) = -0.7538; x 2.1333 = -1.6081; exp = 0.200.
Poisson would have given exp(-2.4) = 0.091.
Consequence for the discount. The carrier plans to grant the 15% discount to 20% of exposure while funding it as if only 9% of exposure qualified. The funding shortfall is 15% x (0.200 - 0.091) = 1.6% of the base rate, which must be loaded back into every group's rate. The discount is not self-funding; it is a 1.6-point subsidy from claim-making groups to claim-free groups.
Second-order effect. The NB also raises the probability of high counts (the heavy tail), so the largest groups - the ones whose experience drives the book - are under-priced even after the load. Before filing, check that the fitted r is stable across group size bands.
3. The data team's warning about within-group correlation: how does it change the model you would fit, and what does that mean for the discount in practice?
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Two competing explanations for the same variance.
- Apparent contagion: each group-year really is overdispersed relative to Poisson (clustering of procedures, plan changes).
- Unobserved heterogeneity: each group has its own stable frequency level; pooling groups with different lambdas produces overdispersion even when each group is exactly Poisson.
Both give mean 2.40 and variance 5.10, so the moments cannot distinguish them. The observable difference is the within-group vs between-group decomposition: if year-to-year counts within a group correlate strongly and group means differ more than sampling error allows, heterogeneity dominates.
Modeling consequence. If heterogeneity dominates, a marginal NB misattributes the variance to every group: it inflates the zero probability and the tail for all groups uniformly. The better structure is a mixed model - a group-level random effect, or equivalently a credibility-weighted group relativity - so that large, well-credible groups keep their own experience and only small groups are pulled toward the book mean.
Pricing consequence. Under the mixed model the uniform 15% no-claim discount is indefensible twice over: it is under-funded on average, and it is wrong in the cross-section, since the groups most likely to be claim-free are the small ones whose zero is least informative. Replace it with a credibility-graded discount, or accept the flat discount and fund it explicitly.