Worked Translation From GLM Output to Rating Factors

From Fitted Coefficients to a Rating Indication

A fitted GLM output table reports quantities on the model’s parameter scale, but a rate indication is usually needed on the original scale and for a particular risk profile. The translation requires more than exponentiating a coefficient: first identify the response and link, the reference levels, any offset, and the coefficients that apply to the profile. Then combine the applicable effects and, when requested, carry their uncertainty through the calculation.

This tutorial uses a log-link model. Assume the conditional mean of the response is positive and has the form \(\mu=\exp(\eta)\), where \(\eta\) is a linear predictor. A log link makes additive changes in \(\eta\) multiplicative changes in the modeled mean. The calculations below interpret fitted output; they do not establish that the model is appropriate for a particular portfolio.

For a categorical factor coded with a reference level, the reference level’s effect is included in the intercept rather than shown as a separately estimated coefficient. A nonreference coefficient is therefore a log-relative effect against that factor’s reference, holding other modeled terms fixed. If the model includes interactions, the relevant effects can depend on the other factor levels; the examples here explicitly use models without interactions.

For a pure-premium model with annual exposure \(e\), one common specification is \(\log E[X\mid \text{risk profile}]=\log(e)+\eta\). Here \(\log(e)\) is the offset, and \(\exp(\eta)\) is the expected amount per exposure unit. If the response is already a rate per exposure unit, the offset may not appear in the predictor. Always check whether the modeled quantity is a rate or an exposure-scaled total before producing an indication.

Why Exponentiation Produces Relativities

The central translation follows directly from the log link. Suppose a profile’s linear predictor is \(\eta\), and changing one rating characteristic adds coefficient \(\beta\), with all other terms unchanged. The original-scale means before and after the change are \(\exp(\eta)\) and \(\exp(\eta+\beta)\). Their ratio is

Thus the intercept and all unchanged factors cancel in the comparison. For a profile with several additive effects and no interactions, the effects sum on the log scale and multiply on the original scale:

This is why the right workflow is to identify the model terms that apply, add their fitted coefficients, and exponentiate the resulting linear predictor. Multiplying rounded relativities can produce a slightly different answer from exponentiating the unrounded coefficient sum, so retain full precision until the final step.

Offsets and the Quantity Being Indicated

An offset changes the linear predictor but is not an estimated rating effect. For an exposure-scaled total, an exposure of \(e\) contributes \(\log(e)\), so the fitted expected total is \(e\exp(\eta)\). The per-unit rate remains \(\exp(\eta)\). A policy with twice the exposure has twice the modeled expected total under this specification, but not twice the expected rate per exposure unit.

A practical translation should therefore distinguish three quantities: the base rate implied by the intercept, the multiplicative rating factors implied by the other terms, and the exposure amount used to scale a total. The relevant output table may also show a response-specific scale or other model information; do not mistake such a quantity for an additional rating coefficient.

Comparing Two Levels and Using Covariance

When comparing two nonreference levels of the same factor, subtract their fitted coefficients on the log scale. If the estimates are \(\hat\beta_A\) and \(\hat\beta_B\), the fitted relativity of level A to level B is \(\exp(\hat\beta_A-\hat\beta_B)\). The uncertainty calculation must use the variance of the difference, including covariance:

For a combined log-scale effect \(c^\mathsf{T}\hat\beta\), where \(c\) selects and signs the coefficients in the contrast, its estimated variance is \(c^\mathsf{T}\widehat{\operatorname{Var}}(\hat\beta)c\). The standard error is the square root of that variance. A Wald interval is formed on the log scale and then exponentiated to obtain an interval for the multiplicative comparison. This procedure requires the relevant covariance estimates; adding standard errors or ignoring covariance is generally incorrect.

A Reliable Translation Process

When given a compact GLM output table, use a fixed sequence rather than jumping directly to exponentiation. This keeps the model interpretation, the selected profile, and the uncertainty calculation aligned.

These translations matter because a rating indication is often presented as a base rate multiplied by a set of selected relativities, while the fitted model output is expressed in coefficient form. The conversion makes the implied comparison and the modeled unit visible. It also makes assumptions easier to audit: a missing reference level, an omitted offset, or an unsupported independence assumption can materially change the result.

For standard errors, the displayed coefficient standard error alone is enough for a single coefficient’s Wald interval, but not necessarily for a combined profile or level-to-level contrast. Those require the estimated covariance matrix, and a confidence interval for a fitted mean can require uncertainty in the intercept as well as in the factor coefficients. The uncertainty examples above are specifically for the stated coefficient or contrast, not a full allowance for model selection, process variation, or uncertainty in the model specification.