Why Compare Predictor Effects Across Segments?
A rating effect that appears reasonable for the portfolio as a whole may not behave consistently across meaningful segments. For example, a coverage feature might be associated with a lower expected cost in one territory but a higher expected cost in another. Such variation may reflect a real interaction, sparse data, differences in portfolio mix, or a model specification that does not represent the intended rating structure. The task is to diagnose the pattern, quantify its uncertainty, and avoid treating every visible difference as established.
For a log-link generalized linear model, fitted rates are exponentials of linear predictors. Coefficients themselves are on the log scale, while their exponentials—or exponentials of coefficient contrasts—are relativities. Comparisons should therefore be made between the appropriate fitted rates or contrasts, not by comparing isolated coefficient values without checking their coding and reference levels. This tutorial develops that diagnostic approach, aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Set Up the Comparison Before Interpreting It
For a Poisson frequency model with a log link and exposure offset, one standard form is
where \(N_j\) is the claim count for observation \(j\), \(E_j\) is its exposure, \(\lambda_j\) is its expected frequency per exposure unit, and \(\eta_j\) is the linear predictor. The offset \(\log(E_j)\) accounts for exposure; it is not a coefficient to be interpreted as a rating factor. Segment comparisons can also be made for other response types and links, but the rate-ratio calculations below specifically use independent Poisson counts with fixed exposures.
For a binary predictor \(x\) and a binary segment indicator \(z\), a log-linear model can be written
With \(z=0\), changing \(x\) from 0 to 1 changes the log rate by \(\beta_x\), so the fitted rate ratio is \(\exp(\beta_x)\). With \(z=1\), the log-rate change is \(\beta_x+\beta_{xz}\), giving a rate ratio of \(\exp(\beta_x+\beta_{xz})\). The interaction contrast is \(\beta_{xz}\), and the ratio of the two segment-specific relativities is \(\exp(\beta_{xz})\).
Worked Example: Comparing Empirical Relativities Across Territories
Worked Example: Comparing Empirical Relativities Across Territories
A hypothetical insurer compares a selected vehicle class with a reference class in two territories. Each class has 1,000 policy-years of exposure in each territory. Territory A has 150 claims in the selected class and 120 in the reference class; Territory B has 60 and 40 claims, respectively. Calculate the empirical relativities, approximate 95% intervals, and the direct interaction comparison. Assume independent Poisson counts and fixed exposures.
Step 1: Calculate the rates and relativities. In Territory A, the rates are \(150/1000=0.150\) and \(120/1000=0.120\) claims per policy-year. The selected-to-reference relativity is \(0.150/0.120=1.25\). In Territory B, the rates are \(60/1000=0.060\) and \(40/1000=0.040\), so the relativity is \(0.060/0.040=1.50\).
Step 2: Calculate log-scale standard errors. For a Poisson rate ratio with fixed exposures, the approximate standard error of its logarithm is \(\sqrt{1/n_1+1/n_0}\), where \(n_1\) and \(n_0\) are the two counts. Therefore, for Territory A,
For Territory B,
Step 3: Form approximate 95% intervals. On the log scale, use \(\log(\widehat{R})\mathbin{\pm}1.96\operatorname{SE}\). Using unrounded intermediate values, Territory A has \(\log(1.25)=0.223144\) and margin \(1.96(0.122474)=0.240050\), giving the interval \([-0.0169,0.4632]\). Exponentiating the endpoints gives a relativity interval of approximately \([0.983,1.589]\).
For Territory B, \(\log(1.50)=0.405465\) and the margin is \(1.96(0.204124)=0.400083\). The log-scale interval is \([0.005382,0.805548]\). Exponentiating gives an interval of approximately \([1.005,2.238]\), to three decimals.
Step 4: Compare the effects directly. The ratio of the two empirical relativities is \(1.50/1.25=1.20\); the log interaction contrast is \(\log(1.20)=0.182322\). Its standard error, using independent counts in all four cells, is
The approximate 95% log-scale interval is \(0.182322\mathbin{\pm}1.96(0.238048)=[-0.2843,0.6489]\). Exponentiating gives an interval for the ratio of relativities of approximately \([0.753,1.914]\), to three decimals. Since it includes 1, these data do not establish a difference between the territory effects at this approximate 95% level. The point estimates differ, but the uncertainty is substantial, particularly for Territory B.
Method and mistake avoided. This is a direct interaction comparison under the independent-Poisson approximation. A common mistake is to infer a significant difference because one segment’s estimate appears larger, or to decide solely from whether the separate intervals overlap. The direct contrast targets the difference of interest; it does not turn a point-estimate difference into proof of a real segment effect.
Why the Log-Rate-Ratio Standard Error Has This Form
The approximation used above follows from the Poisson variance and a first-order Taylor approximation. Suppose two independent counts satisfy \(N_1\sim\operatorname{Poisson}(E_1\lambda_1)\) and \(N_0\sim\operatorname{Poisson}(E_0\lambda_0)\), with positive observed counts. The estimated rate ratio is \(\widehat{R}=(N_1/E_1)/(N_0/E_0)\). Its logarithm is
The exposures are fixed, so they contribute no sampling variance. For a Poisson count with mean \(\mu\), \(\operatorname{Var}(N)=\mu\). Applying the delta method to \(\log(N)\), whose derivative is \(1/N\), gives approximate variance \(\mu/\mu^2=1/\mu\), estimated by \(1/N\). Independence makes the count variances add, so
For an interaction contrast comparing two independent segment-specific rate ratios, the four count contributions add. This approximation is most useful when counts are not too small. A zero count makes the logarithm undefined, and very small counts can make the normal approximation unreliable; do not apply these interval formulas mechanically in those cases. Model-based methods, suitable likelihood intervals, or a defensible pooling strategy may be needed.
Worked Example: Translating Coefficients Into Segment Effects
Worked Example: Translating Coefficients Into Segment Effects
A hypothetical insurer fits a log-link model for expected loss cost per policy-year. The reference risk is an agent-distributed policy in East territory, with fitted loss cost of $480. Let \(x=1\) denote digital distribution and \(z=1\) denote West territory. The fitted coefficients are \(\beta_x=-0.10\), \(\beta_z=0.20\), and \(\beta_{xz}=0.15\). Interpret the distribution effect in each territory, calculate the four fitted costs, and assess the uncertainty of the West distribution effect. For this illustration, the estimated variances of \(\beta_x\) and \(\beta_{xz}\) are 0.0016 and 0.0064, and their estimated covariance is zero.
Step 1: Calculate the segment-specific relativities. In East, the digital-to-agent relativity is \(\exp(-0.10)=0.9048\). In West, it is \(\exp(-0.10+0.15)=\exp(0.05)=1.0513\). Thus the fitted distribution effect is a reduction of about 9.52% in East and an increase of about 5.13% in West.
Step 2: Calculate the fitted costs. East agent is the reference: $480.00. East digital is \(480\exp(-0.10)=\$434.32\). West agent is \(480\exp(0.20)=\$586.27\). West digital is \(480\exp(-0.10+0.20+0.15)=480\exp(0.25)=\$616.33\). These values are dollars per policy-year, rounded to the cent.
The ratio of West digital to West agent fitted costs is \(616.33/586.27\), approximately \(1.0513\), as obtained directly from the coefficients. The ratio of the West and East distribution relativities is \(\exp(0.15)=1.1618\). The interaction says the digital relativity is about 16.18% higher in West than in East on a multiplicative scale; it does not mean that the West digital cost itself is 16.18% higher than the East digital cost.
Step 3: Quantify uncertainty in the West distribution effect. Its log relativity is the contrast \(\beta_x+\beta_{xz}=0.05\). With the supplied covariance,
The approximate 95% interval on the log scale is \(0.05\mathbin{\pm}1.96(0.08944)=[-0.1253,0.2253]\). Exponentiating gives a rate-ratio interval of approximately \([0.882,1.253]\), to three decimals. It includes 1, so the estimated West distribution effect is not clearly distinguished from no effect by this interval.
Method and mistake avoided. The correct uncertainty calculation uses the variance of the full linear contrast, including its covariance term. A common mistake is to interpret \(\beta_x\) as the distribution effect in both territories. It is the effect only in the reference segment \(z=0\); the West effect is \(\beta_x+\beta_{xz}\).
Worked Example: Recognizing Sparse-Cell Instability
Worked Example: Recognizing Sparse-Cell Instability
A hypothetical portfolio compares a specialty vehicle class with a reference class. In a small segment, the specialty class has 8 claims in 400 policy-years and the reference class has 40 claims in 2,000 policy-years. A larger segment has 80 claims in 4,000 policy-years and 400 claims in 20,000 policy-years, respectively. Under independent Poisson counts and fixed exposures, compare the estimated relativities and their approximate 95% intervals.
Step 1: Calculate both point estimates. In the small segment, both rates are \(8/400=0.020\) and \(40/2000=0.020\), so the relativity is 1. In the larger segment, both rates are \(80/4000=0.020\) and \(400/20000=0.020\), so its relativity is also 1.
Step 2: Calculate the small-segment interval. Its log-rate-ratio standard error is
The log estimate is \(\log(1)=0\), so the approximate interval is \(0\mathbin{\pm}1.96(0.387298)=[-0.7595,0.7595]\). Exponentiation gives a relativity interval of approximately \([0.468,2.137]\), to three decimals.
Step 3: Calculate the larger-segment interval. Its standard error is
The log interval is \([-0.2401,0.2401]\), and the relativity interval is approximately \([0.786,1.271]\). The point estimates match, but the small segment’s interval is much wider because its counts are much lower. This is evidence of less precision, not evidence that the small segment necessarily has a different underlying effect.
Method and mistake avoided. This comparison uses the same rate-ratio approximation in each segment. Avoid treating a dramatic-looking coefficient or a wide interval as proof of a genuine pricing distinction. Sparse estimates may move substantially as individual claims enter or leave the experience. Any pooling or credibility adjustment should be supported by the modeling objectives and constraints, not chosen merely to make an inconvenient result disappear.
Practical Diagnostic Sequence
A useful segment review is systematic. It should separate the comparison being asked from the model’s parameterization, then examine the amount of information supporting each estimate.
Name the predictor levels, segment definitions, reference levels, and fitted quantity. For example, compare the selected-class-to-reference-class frequency ratio in each territory.
Use fitted rates or the relevant coefficient contrast. In a log-link model, exponentiate the contrast to return to the multiplicative rate scale.
Review exposure and claim counts in the contributing cells, along with a suitable standard error or interval. Small exposures or counts often imply unstable segment estimates.
Ask whether the sign, ordering, and magnitude of fitted effects make sense for the proposed rating structure. Investigate unexpected patterns rather than assuming they are either correct or erroneous.
Possible responses include retaining the effect, reconsidering an interaction, combining levels, or obtaining more experience. Each choice changes the model or its application and needs actuarial justification.
Common Mistakes and Interpretation Limits
Segment comparisons are diagnostic evidence, not automatic instructions to add or remove a rating factor. The following errors can lead to unjustified conclusions:
- Subtracting relativities instead of comparing them multiplicatively. If one relativity is 1.50 and another is 1.25, their ratio is \(1.50/1.25=1.20\), not a 0.25 interaction relativity. The corresponding log contrast is \(\log(1.20)=0.1823\). A subtraction may describe a difference on the ratio scale, but it is not the log-link interaction contrast.
- Comparing coefficients with different reference levels. A coefficient’s meaning depends on the coding and the other predictors in the model. Construct the fitted contrast for the same two risks in each segment instead.
- Ignoring exposure and portfolio composition. Raw claim counts are not rates. The offset or exposure denominator must be handled consistently. A segment with twice the exposure can have twice the expected count without having a higher frequency.
- Over-reading noisy estimates. Sparse cells can produce wide intervals and large changes in estimated effects. A point estimate alone does not show how stable it is.
- Treating separate intervals as a definitive interaction test. The direct contrast and its uncertainty address whether effects differ. Visual overlap of separate intervals is not a substitute for calculating the uncertainty of that contrast.
- Confusing statistical evidence with rating judgment. A statistically detectable difference may be too small or impractical to use, while a potentially important difference may be poorly estimated. Credibility, operational, and regulatory considerations also matter.
The Poisson calculations in the examples assume independent counts and fixed exposures. Dependence, overdispersion, clustering, changing exposure definitions, or a different response distribution can invalidate their standard errors. In a fitted GLM, the relevant contrast variance should be obtained from the model’s coefficient covariance matrix; for a contrast \(c^\mathsf{T}\hat{\beta}\), its estimated variance is \(c^\mathsf{T}\widehat{\operatorname{Cov}}(\hat{\beta})c\). Omitting covariance terms can misstate uncertainty.
Check Your Understanding
Work each problem from the stated comparison and assumptions. These questions use hypothetical rating data.
- A log-link model has a digital-channel coefficient of \(-0.08\) and a digital-by-West interaction coefficient of \(0.12\). Find the digital-to-agent relativity in the reference segment and in West, and the ratio of those two relativities. (Answer: \(0.9231\), \(1.0408\), and \(1.1275\), respectively.)
- With equal exposures within each segment, Segment U has 30 claims for the selected class and 20 for the reference class. Segment V has 18 and 15 claims, respectively. Find each empirical relativity and the ratio of the V relativity to the U relativity. (Answer: \(1.50\), \(1.20\), and \(0.80\), respectively.)
- With equal fixed exposures, a segment has 5 claims in one class and 20 in its reference class. Using the independent-Poisson approximation, calculate the estimated relativity and its approximate 95% interval. Give endpoints to three decimals. (Answer: relativity \(0.250\); interval approximately \([0.094,0.666]\).)
- A fitted segment contrast has estimate \(0.18\) on the log scale and standard error \(0.10\). Calculate its approximate 95% interval on the relativity scale and state whether it excludes 1. Give endpoints to three decimals. (Answer: \([\exp(-0.016),\exp(0.376)]\approx[0.984,1.456]\); it does not exclude 1.)