Worked Severity Fit With Grouped Claim Counts

Why Grouped Severity Fits Require Care

Severity data are sometimes supplied as a count of claims and an average claim amount for each rating cell, rather than as one record per claim. The cell average is useful, but it does not represent the same amount of information as one individual claim. A cell based on 500 claims should contribute more to a fit than a cell based on 5 claims. In the Gamma log-link model, the claim count supplies that frequency weight.

The key result is specific: for coefficient estimation in a Gamma GLM with a log link, fitting the cell means with claim-count frequency weights gives the same coefficient score equations as fitting all individual claims, provided all claims in a cell have the same predictors and the cell mean is calculated over those claims. This does not mean that every aspect of the full claim-level data can be recovered from cell counts and means.

Definition: A grouped severity cell $j$ contains $n_j$ positive claim severities $Y_{j1},\ldots,Y_{jn_j}$ with the same model predictors. Its reported mean is $\bar{Y}_j=\sum_{k=1}^{n_j}Y_{jk}/n_j$. When fitting the Gamma log-link mean model to these cell means, $n_j$ is a frequency weight: it represents $n_j$ claims, not an exposure offset or an arbitrary measure of precision.

Model and Assumptions

Assume each individual claim severity $Y_i$ is positive and conditionally Gamma distributed given its predictors. Use the shape-scale parameterization $Y_i\mid x_i\sim\operatorname{Gamma}(1/\phi,\phi\mu_i)$, where $\mu_i=E[Y_i\mid x_i]$, $\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i^2$, and the common dispersion parameter $\phi>0$. Specify the mean with a log link:

\log(\mu_i)=x_i^\mathsf{T}\beta, \qquad \mu_i=\exp(x_i^\mathsf{T}\beta).

Here $x_i$ is the predictor vector, including an intercept where appropriate, and $\beta$ is the coefficient vector. This specification makes fitted means positive. A coefficient for a binary rating characteristic has a multiplicative interpretation: holding other predictors fixed, changing the characteristic from 0 to 1 multiplies the fitted mean by $\exp(\beta_r)$.

For grouped fitting, assume the claims within a cell share the same $x_j$, each cell mean is calculated from the $n_j$ claims represented, and the claims are conditionally independent under the model. The same Gamma mean-variance relationship and log link are used for every claim. Zero severities cannot be modeled directly by this positive-support Gamma response without changing the response definition or model.

Why Frequency-Weighted Cell Means Give the Same Coefficient Fit

The equivalence follows from the coefficient score, not from the claim that the cell mean contains all the original data. For one individual observation under the stated Gamma model, the part of the log likelihood that depends on $\mu_i$ is $-\log(\mu_i)/\phi-Y_i/(\phi\mu_i)$. Since $\log(\mu_i)=x_i^\mathsf{T}\beta$, differentiating with respect to $\beta$ gives the individual score.

\frac{\partial \ell_i}{\partial\beta} = \frac{Y_i-\mu_i}{\phi\mu_i}x_i.

For cell $j$, all claims have the same model mean $\mu_j$ and predictor vector $x_j$. Summing their scores and using $\sum_kY_{jk}=n_j\bar{Y}_j$ gives:

\sum_{k=1}^{n_j}\frac{Y_{jk}-\mu_j}{\phi\mu_j}x_j = \frac{n_j(\bar{Y}_j-\mu_j)}{\phi\mu_j}x_j.

This is exactly the contribution to the coefficient score from a grouped cell mean with frequency weight $n_j$. Summing over cells yields the same coefficient score equations as the claim-level fit. The equivalence requires the Gamma variance function and log link specified above, common predictors within each cell, and correctly calculated counts and means. It should not be assumed automatically for another response distribution or link.

The claim-level data contain within-cell variation that the means alone omit. In particular, cell means and counts are not generally sufficient to estimate Gamma dispersion or to assess within-cell residual behavior. A routine grouped-mean analysis may therefore produce different dispersion estimates, deviance summaries, or diagnostics from an analysis retaining all individual claims. Do not treat equality of the coefficient estimates as equality of all fitted-model information.

Worked Example: Fit Separate Territory Means

A hypothetical insurer groups uncapped claim severities by territory. The rural cell has 2 claims with a mean severity of $1,000 and 3 claims with a mean severity of $1,400. The urban cell has 4 claims with a mean severity of $1,800 and 1 claim with a mean severity of $2,200. Fit a Gamma log-link model with an intercept and a territory indicator, using rural as the reference. Calculate the urban severity relativity.

Method. With an intercept and a single binary territory indicator, this is a saturated two-group mean model. The Gamma log-link coefficient score for each group sets its fitted mean equal to that group’s claim-count-weighted arithmetic mean.

Rural fitted mean. The rural total represented by the grouped data is $2(1{,}000)+3(1{,}400)=2{,}000+4{,}200=6{,}200$ dollars, across $2+3=5$ claims. Thus $\hat{\mu}_{R}=6{,}200/5=\$1{,}240$.

Urban fitted mean. The urban total is $4(1{,}800)+1(2{,}200)=7{,}200+2{,}200=9{,}400$ dollars, across $4+1=5$ claims. Thus $\hat{\mu}_{U}=9{,}400/5=\$1{,}880$.

With rural as the reference, $\hat{\beta}_0=\log(1{,}240)$, and the urban coefficient is $\hat{\beta}_U=\log(1{,}880/1{,}240)$. The urban relativity is therefore $\exp(\hat{\beta}_U)=1{,}880/1{,}240=1.5161$, to four decimal places. The fitted urban severity is about 1.516 times the rural fitted severity, or about 51.61% higher.

Check. The grouped data represent $6{,}200+9{,}400=\$15{,}600$ over 10 claims, giving an overall mean of $1,560. The fitted group totals are $5(1{,}240)=\$6{,}200$ and $5(1{,}880)=\$9{,}400$, which sum to the same $15,600. A common mistake would be to average the two rural cell means equally, $(1{,}000+1{,}400)/2=\$1{,}200$; that ignores their unequal claim counts and is not the rural fitted mean.

Grouped and Claim-Level Representations

A frequency weight is not the same as repeating a cell mean as if it were a new, independently observed claim with no weighting. The weighted grouped fit represents the count of claims summarized by that mean. The next calculation checks the coefficient score directly against individual claims.

Worked Example: Verify a Grouped Fit Against Individual Claims

A hypothetical equipment insurer records three severities in class A: $80, $120, and $160. It records two severities in class B: $200 and $280. The corresponding grouped summaries are class A: $n_A=3,\bar{Y}_A=\$120$; class B: $n_B=2,\bar{Y}_B=\$240$. Fit an intercept-only Gamma log-link model and then an intercept-plus-class model. Show how the grouped and claim-level representations agree.

Intercept-only fit. The total severity is $80+120+160+200+280=\$840$, and the total count is $3+2=5$. The intercept-only fitted mean is $\hat{\mu}=\$840/5=\$168$. From grouped summaries, the same calculation is $[3(120)+2(240)]/5=(360+480)/5=\$168$.

The grouped coefficient score, apart from the common intercept predictor, is $\{3(120-168)+2(240-168)\}/(168\phi)$. Its numerator is $3(-48)+2(72)=-144+144=0$, so the intercept score is zero. From the five individual severities, the numerator is $(80-168)+(120-168)+(160-168)+(200-168)+(280-168)$ $=-88-48-8+32+112=0$. The claim-level score is also zero.

Intercept-plus-class fit. With separate fitted means for A and B, the grouped estimates are $\hat{\mu}_A=\$120$ and $\hat{\mu}_B=\$240$. The class B relativity against class A is $240/120=2.000$. The grouped score numerator for A is $3(120-120)=0$; for B it is $2(240-240)=0$. The claim-level within-class deviations also sum to zero: class A has $(80-120)+(120-120)+(160-120)=-40+0+40=0$, while class B has $(200-240)+(280-240)=-40+40=0$.

Check. The intercept-only estimate is $168 in both representations, and the two-class estimates are $120 and $240 in both. A frequent mistake is to average the two cell means for the intercept-only model, obtaining $(120+240)/2=\$180$. The correct claim-level mean is $168 because the cell means represent 3 and 2 claims, not one claim each.

Relativities in a Multiplicative Model

A fitted Gamma model may use several rating characteristics. Under a log link with main effects and no interactions, each coefficient defines a multiplicative change in the fitted conditional mean. The model’s no-interaction assumption means that the multiplier for one characteristic does not vary with the level of another. Grouped counts affect the fit through their frequency weights; they are not themselves severity relativities.

Worked Example: Interpret Two Severity Factors

A hypothetical portfolio is summarized by vehicle class (Standard or Premium) and territory (Rural or Urban). The four cells have counts and means as follows: Standard-Rural has 2 claims with mean $1,000; Standard-Urban has 3 claims with mean $1,500; Premium-Rural has 4 claims with mean $1,200; Premium-Urban has 1 claim with mean $1,800. Fit the main-effects log-mean model with Standard-Rural as the reference, and check the no-interaction structure.

Model. Let $P=1$ for Premium and $U=1$ for Urban. Specify $\log(\mu)=\beta_0+\beta_P P+\beta_U U$. The cell means are exactly multiplicative: relative to $1,000, Premium-Rural is $1.2$, Standard-Urban is $1.5$, and Premium-Urban is $1.8=1.2(1.5)$.

Therefore the fitted coefficients are $\hat{\beta}_0=\log(1{,}000)$, $\hat{\beta}_P=\log(1.2)$, and $\hat{\beta}_U=\log(1.5)$. The vehicle-class severity relativity is $\exp(\hat{\beta}_P)=1.200$; the territory relativity is $\exp(\hat{\beta}_U)=1.500$. The predicted Premium-Urban mean is $\$1{,}000(1.2)(1.5)=\$1{,}800$, matching its cell mean.

Check. The grouped total is $2(1{,}000)+3(1{,}500)+4(1{,}200)+1(1{,}800)$ $=2{,}000+4{,}500+4{,}800+1{,}800=\$13{,}100$ across $2+3+4+1=10$ claims. The observed overall mean is therefore $1,310. The model reproduces each of the four cell means, so its weighted total is also $13,100. A common error is to add the relativities, $1.2+1.5$, rather than multiply them when applying both factors.

Practical Checks and Common Errors

Before interpreting a grouped Gamma fit, confirm what each row represents. A row should identify a cell of claims sharing the modeled predictors; its count should be the number of claims whose severity average is reported. If rows instead summarize policies, sampled claims, or claims selected by coverage conditions, the target and appropriate weighting may differ. The interpretation here is for grouped claim severities under the stated assumptions.

Using equal weights by default. Equal weights give each cell equal influence, even if the cells represent very different claim counts. Use $n_j$ when the grouped mean summarizes $n_j$ claims under the assumptions stated here.
Confusing frequency weights with offsets. A count weight represents replicated severity observations. It is not a logarithmic exposure offset and should not be entered as one.
Assuming all data summaries are equivalent. Counts and means preserve the coefficient score for this Gamma log-link fit, but they do not preserve the individual residual pattern or generally determine dispersion diagnostics.
Interpreting a coefficient as a dollar change. In a log-link model, $\exp(\beta_r)$ is a multiplicative severity relativity. It is not a fixed dollar increase.
Ignoring model structure. A main-effects model imposes a multiplicative pattern across factor combinations. If observed cell means do not follow that pattern, the fitted values need not equal every cell mean; the model estimates coefficients using the weighted score equations.

These calculations are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus: identify the appropriate grouped-data weights, fit the specified severity mean model, and explain what the resulting relativities mean. State clearly whether a reported result is a coefficient-fit equivalence or a claim that the grouped data retain all claim-level information; only the first follows from the derivation above.

Check Your Understanding

For each problem, use claim counts as frequency weights where the grouped observations represent claims with common predictors.

A one-factor model has two severity cells: class A has 2 claims with mean $900, and class B has 3 claims with mean $1,100. Under an intercept-only Gamma log-link model, calculate the fitted mean. (Answer: $1,020)
A grouped portfolio has 4 claims with mean severity $150 and 6 claims with mean severity $210. Calculate the intercept-only fitted mean, then state why the unweighted average of the two cell means is not the appropriate estimate. (Answer: $186; the counts are unequal.)
A saturated two-class model has fitted mean $700 for the reference class and $840 for the other class. Calculate the second class’s severity relativity and percentage difference from the reference. (Answer: Relativity 1.200; 20% higher.)
A log-link main-effects model predicts $500 for the reference risk. A binary rating characteristic has coefficient $\log(1.3)$. Calculate the fitted mean when the characteristic is present, holding other predictors fixed. (Answer: $650)
A proposed two-factor main-effects fit has cell means $100 for the reference-reference cell, $180 for reference-second-factor, $130 for second-factor-reference, and $234 for second-factor-second-factor. Determine whether these means are exactly consistent with a no-interaction multiplicative model. (Answer: Yes; the two relativities are 1.8 and 1.3, and $100(1.8)(1.3)=234$.)

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