Group income protection claim counts and overdispersion
A pricing team is reviewing group income protection experience for 40 small employers. For each employer you have the number of claims in a three-year period and the total policy-years of exposure. The portfolio totals are 312 claims and 1,240 policy-years. The mean number of claims per employer is 7.8 and the sample variance of the employer claim counts is 23.4. The team initially proposes a Poisson GLM with a log link and log(exposure) as offset, using employer size band as the only rating factor. A colleague notes that several employers with similar exposure have zero claims while two large employers have more than 30 claims each. The data are to be used to set premium rates and to estimate the standard error of the size-band relativities. The team asks whether the Poisson assumption is defensible and how the ratemaking should proceed.
1. Carry out a dispersion check for the Poisson assumption using the portfolio summaries. State the null and alternative hypotheses, compute a suitable statistic, and interpret it. What does the result imply for the proposed Poisson GLM?
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The Poisson model implies Var(Y_i)=E(Y_i). A quick moment check is the dispersion index s²/ȳ = 23.4/7.8 = 3.0. More formally, if the fitted mean per employer is near 7.8, the Pearson statistic is approximately Σ(y_i−7.8)²/7.8 = (n−1)s²/7.8 = 39×23.4/7.8 ≈ 117 on 39 df. The upper-tail probability is far below 0.001. The null is equidispersion; the alternative is overdispersion or a misspecified mean. Reject the Poisson variance assumption. The coefficient estimates may still be consistent if the mean structure is correct, but Poisson standard errors, confidence intervals and likelihood-ratio tests are too optimistic. The dispersion is about 3, so naive standard errors are too small by roughly √3≈1.7. That matters directly for the size-band relativities and for any premium or capital loading based on those standard errors.
2. Recommend how to model the data. Compare at least two approaches and explain the effect on premium relativities and their standard errors.
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Use a quasi-Poisson or negative-binomial model. Quasi-Poisson keeps the log link and offset, estimates the same mean structure, and multiplies the covariance matrix by the estimated dispersion φ≈3.0; it is simple but does not change fitted relativities. A negative-binomial model specifies Var(Y)=μ+μ²/k; it changes the weighting of large and small employers and is often more natural when claims cluster within employers. Either way, keep log(exposure) as an offset so the model is for claim rate per policy-year. For ratemaking, shrink sparse size-band relativities towards the portfolio mean using credibility or a random effect; do not quote a relativity from a band with very little exposure as if it were fully credible. Report dispersion-adjusted standard errors and show the sensitivity of the premium to the overdispersion treatment. Check the two very large employers: they may be genuine large risks, data errors, or a sign that employer-level random effects are needed.
3. Explain why the two employers with more than 30 claims each should not automatically be removed, and how you would assess their influence.
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Large counts are not necessarily outliers; in a skewed claims distribution they may be genuine. Removing them without a documented data reason biases the portfolio experience downwards and understates the risk of similar large employers. First verify the exposure and claim records. Then assess influence using Pearson or deviance residuals, Cook's distance, dfbeta for the size-band coefficients, and refit with and without those employers. If the coefficients or standard errors change materially, report both fits and consider a random effect or a robust/quasi-likelihood approach. Also consider whether the size bands are too coarse: two employers with similar total exposure can have very different numbers of lives or industry mix, which the current model ignores. The decision to retain or remove must be based on data quality and model stability, not simply on the size of the count.