How Overdispersion Changes Relativity Uncertainty

Why Overdispersion Changes Rating-Factor Uncertainty

A frequency model can describe the expected claim count well while the observed counts vary more than a Poisson model allows. This distinction matters: if the mean structure is retained but the variance is larger, a Poisson model-based standard error can make a rating factor look more precisely estimated than it is. This tutorial quantifies that effect for fitted coefficients and multiplicative relativities, aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

For example, with exposure \(e_j\) and a log link, the expected count may be written \(\mu_j=e_j\exp(\mathbf{x}_j^\mathsf{T}\boldsymbol{\beta})\). The model-based Poisson standard errors use the Poisson variance assumption. Under the overdispersed specification, the coefficient estimates can remain the same as the Poisson estimates under the usual quasi-Poisson estimating equations, but their estimated covariance is scaled by \(\phi\). Consequently, when this scalar adjustment applies, standard errors are scaled by \(\sqrt{\phi}\), not by \(\phi\).

From a Coefficient to a Relativity Interval

With a log link, a coefficient for a binary rating indicator represents a log relativity. If \(\beta_k\) is the coefficient for a class compared with its reference class, the fitted relativity is \(\widehat{R}_k=\exp(\widehat{\beta}_k)\). Uncertainty is first assessed on the coefficient scale, where an approximate normal interval is formed, and then transformed to the positive relativity scale.

For a 95% interval, \(z_{0.975}\) is approximately 1.96. Exponentiate both endpoints to obtain the interval for the relativity. Because the exponential function is increasing, the lower coefficient limit remains the lower relativity limit. The resulting interval is generally asymmetric around \(\widehat{R}_k\), even though the coefficient interval is symmetric around \(\widehat{\beta}_k\).

Why the Standard Error Is Multiplied by the Square Root

The scaling follows from the variance of the coefficient estimator. Let \(A\) denote the information-like matrix in the Poisson estimating equations. Under the Poisson variance assumption, the model-based covariance is approximately \(A^{-1}\). If the actual conditional variance is instead \(\phi\mu_j\), the variance of the estimating equations is multiplied by \(\phi\). The sandwich covariance then has the form below, when the same mean structure and scalar dispersion assumption apply.

Thus each coefficient variance is multiplied by \(\phi\), and each standard error, being the square root of a variance, is multiplied by \(\sqrt{\phi}\). This derivation also explains why multiplying a standard error directly by \(\phi\) is incorrect. In practice, \(\phi\) is usually estimated, for example from a Pearson statistic divided by residual degrees of freedom. The resulting interval is a plug-in interval: it uses the estimated dispersion but does not separately account for uncertainty in that estimate.

Intervals for Contrasts Between Rating Classes

A comparison of two non-reference classes is a contrast of coefficients. Its uncertainty depends on both coefficient variances and their covariance. If the Poisson model-based covariance matrix is available and the scalar dispersion assumption applies, multiply the variance of the contrast by \(\widehat{\phi}\), or equivalently multiply its standard error by \(\sqrt{\widehat{\phi}}\). Do not calculate the contrast uncertainty by adding standard errors.

The contrast \(\widehat{\beta}_b-\widehat{\beta}_a\) is the log of the factor for class \(b\) relative to class \(a\). Exponentiating its confidence limits gives an interval for that ratio of rating factors.

Estimating Dispersion and Knowing the Adjustment’s Limits

A common estimate of dispersion for count data is the Pearson chi-square statistic divided by its residual degrees of freedom. In a fitted model, Pearson residuals compare observed counts with fitted means while accounting for the Poisson variance. This estimate is a diagnostic and a scale estimate; it does not by itself identify why the data are overdispersed.

The scalar adjustment has a defined scope. It is appropriate when the mean model is retained and the variance is treated as a common multiple of the Poisson variance. It is not a universal replacement for refitting a different count distribution. For example, a negative binomial model specifies a particular mean-variance relationship and generally has its own fitted covariance matrix; one should use that model’s uncertainty estimates rather than automatically applying a Pearson-based multiplier to Poisson standard errors. Likewise, robust sandwich standard errors need not equal a model-based covariance multiplied by one scalar, especially when the variance inflation differs across observations or the mean model is misspecified.

Exam Traps and Practical Interpretation

• Scaling the wrong quantity. If \(\widehat{\phi}=2.25\), the standard error multiplier is 1.5, while the variance multiplier is 2.25.
• Leaving the interval on the wrong scale. A coefficient interval describes log relativities. Exponentiate both limits to report a multiplicative-factor interval.
• Dropping the covariance in a contrast. Variances of differences include \(-2\operatorname{Cov}\); separate standard errors cannot simply be added or subtracted.
• Assuming all overdispersion has the same structure. A scalar quasi-Poisson adjustment is not automatically the correct uncertainty calculation for every alternative count model or for every robust covariance estimate.
• Confusing uncertainty with the fitted indication. The adjustment changes standard errors and intervals, not the coefficient estimate or the fitted factor under the stated quasi-Poisson setup.