Residuals on the GLM Response Scale

Why Response-Scale Residuals Matter

A GLM produces fitted values that can be compared with the observations used to fit or assess the model. The simplest such comparison is on the response scale: subtract the fitted outcome from the observed outcome. For ratemaking, the response might be a claim count, a claim severity, or a pure premium. The arithmetic is straightforward, but a useful interpretation depends on matching the units and level of aggregation.

For a policy-level count model, an observation may be the number of claims during a period, while the fitted value is an expected count for that same period. A severity model may instead use one record per claim, or a policy-level average conditional on claims. A pure-premium response is a loss amount per unit of exposure. These outcomes are not interchangeable: a fitted annual frequency is not a fitted count for a half-year policy, and a severity is not an expected loss for a policy.

Definition: For observation $i$, let $y_i$ be the observed response and let $\hat{\mu}_i$ be the fitted conditional mean on the response scale. The response residual is $$ r_i^{\mathrm{resp}}=y_i-\hat{\mu}_i. $$ A positive residual means the observed response exceeds its fitted mean; a negative residual means it is below its fitted mean. Its units are the same as those of $y_i$.

In a GLM, a fitted mean is obtained by applying the inverse link to the fitted linear predictor. Thus, with linear predictor $\hat{\eta}_i$ and link $g$, $\hat{\mu}_i=g^{-1}(\hat{\eta}_i)$. Response residuals compare $y_i$ with $\hat{\mu}_i$, not with $\hat{\eta}_i$. In a log-link model, for example, the predictor is on a logarithmic scale while the fitted mean and response residual are on the original outcome scale.

Counts: Fit the Same Period That Was Observed

For a count model with exposure $e_i$, a common specification uses a log link and an exposure offset. If the model estimates a claim frequency per exposure unit, then $\log(\mu_i)=\log(e_i)+\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}$, where $\mu_i$ is the fitted expected count for observation $i$. The offset fixes the coefficient of $\log(e_i)$ at one. The fitted frequency per exposure unit is $\hat{\lambda}_i=\exp(\mathbf{x}_i^\mathsf{T}\hat{\boldsymbol{\beta}})$, so the fitted count is $\hat{\mu}_i=e_i\hat{\lambda}_i$.

This formulation is appropriate when the expected count is proportional to exposure, conditional on the included predictors. A Poisson model is a standard count model; if it is used, the conditional count distribution is assumed to be Poisson with mean $\mu_i$. Regardless of whether Poisson is appropriate, the response residual for a count observation is the observed count minus the fitted expected count. It can be fractional because an expectation need not be an integer.

Worked Example: Policy Counts With Unequal Exposure

A hypothetical insurer fits a Poisson claim-count GLM with a log link and an exposure offset. Conditional on rating variables, $N_i\sim\operatorname{Poisson}(\mu_i)$, and the fitted annual claim frequencies are 0.20 for both policies. Policy A has 0.5 exposure-years and one claim; policy B has 1.25 exposure-years and no claims. Calculate each fitted count and response residual.

Method and assumptions. The model assumes counts are Poisson conditional on the rating variables, with expected count equal to exposure times fitted annual frequency. There is no interest or mortality assumption in this claim-count calculation.

For policy A, the fitted expected count is $\hat{\mu}_A=0.5(0.20)=0.10$ claim. The observed count is $y_A=1$, so $r_A^{\mathrm{resp}}=1-0.10=0.90$ claim.

For policy B, the fitted expected count is $\hat{\mu}_B=1.25(0.20)=0.25$ claim. The observed count is $y_B=0$, so $r_B^{\mathrm{resp}}=0-0.25=-0.25$ claim.

The signs are sensible: A had more claims than its fitted expectation, while B had fewer. As a check, fitted count equals fitted annual frequency multiplied by exposure in each case. Common mistake avoided: subtracting 0.20 directly from either observed count would compare a count with an annual rate and would ignore the policies' different exposures.

Severity: Keep the Claim or Policy Unit Clear

A severity model usually describes the amount paid per claim, conditional on a claim being present. If the data have one record per claim, $y_i$ may be an individual claim amount and $\hat{\mu}_i$ the fitted mean severity for that claim's rating characteristics. The residual is in dollars per claim. If the response is a policy-level average severity, the fitted value and observed response must both refer to that same policy-level average.

A claim-free policy does not have a zero claim severity: no claim exists whose severity is zero. Therefore, a zero-claim policy is ordinarily not assigned a zero response in a severity model. It contributes to frequency or pure-premium experience, while severity is modeled for claims or for an explicitly defined positive-claim aggregate. A severity residual also does not by itself measure the policy's total loss difference; the number of claims matters for that.

Worked Example: Policy-Level Average Claim Severity

A hypothetical commercial policy has three claims of $4,000, $6,000, and $8,000 during the experience period. A severity GLM fitted to claim records gives a conditional mean of $5,500 for each of these claims. Assume claim severities, conditional on the rating variables, follow an exponential distribution with mean $5,500 (equivalently, a Gamma distribution with shape 1 and scale $5,500). Find the policy's observed average severity and the residual comparing that average with the fitted mean.

Method and assumptions. Treat the three claims as the severity observations and compare their policy-level average with the common fitted conditional mean. The distributional assumption describes the hypothetical severity model; it does not change the response-residual subtraction. No interest or mortality assumption applies.

The observed total severity is $4{,}000+6{,}000+8{,}000=\$18{,}000$. There are three claims, so the observed average severity is $\$18{,}000/3=\$6{,}000$ per claim.

The fitted average is $5,500 per claim. Therefore, the response residual on the average-severity scale is $\$6{,}000-\$5{,}500=+\$500$ per claim. It indicates that the observed average exceeded the fitted mean by $500.

As a check, the observed average multiplied by three claims is $18,000, as given. The model's fitted aggregate severity across these three claim records is $3(\$5{,}500)=\$16{,}500$, a difference of $1,500 in aggregate, equal to three times the average-severity residual. Common mistake avoided: subtracting the fitted mean from the $18,000 total would mix a policy total with a per-claim fitted value. No rounding is needed.

Pure Premium: A Rate Difference and Its Dollar Equivalent

Pure premium is expected loss per unit of exposure. For policy $i$, let $L_i$ be the observed aggregate loss and $e_i$ its exposure. The observed pure premium is $P_i=L_i/e_i$, provided $e_i>0$. If the fitted pure premium is $\hat{P}_i$, the response residual on the pure-premium scale is $P_i-\hat{P}_i$, in dollars per exposure unit.

A frequency-severity model gives one route to the fitted pure premium. Under the usual decomposition, if the expected claim count per exposure unit is $\lambda_i$ and the expected severity per claim is $s_i$, then the expected loss per exposure unit is $\hat{P}_i=\hat{\lambda}_i\hat{s}_i$. This product follows because the expected aggregate loss is the expected number of claims times the expected amount per claim, when claim severity has the stated conditional mean and the count-severity structure is appropriate.

\hat{E}[L_i\mid \mathbf{x}_i]=e_i\hat{\lambda}_i\hat{s}_i, \qquad P_i-\hat{P}_i =\frac{L_i}{e_i}-\hat{\lambda}_i\hat{s}_i

Multiplying the pure-premium residual by exposure converts it to a difference in aggregate dollars: $e_i(P_i-\hat{P}_i)=L_i-e_i\hat{P}_i$. This identity is useful for interpretation, but the two residuals have different units. One is dollars per exposure unit; the other is dollars for the policy's actual exposure.

Worked Example: Pure-Premium Residual From Frequency and Severity

A hypothetical policy has 0.8 exposure-years and $4,800 of observed losses. A frequency model gives a fitted rate of 0.25 claims per year, and a severity model gives fitted mean severity of $20,000. For this illustration, assume claim count is Poisson with mean equal to exposure times fitted frequency, and claim amounts are independent Gamma variables with shape 2 and scale $10,000, giving mean severity $20,000. Calculate the observed and fitted pure premiums, the response residual, and its aggregate-dollar equivalent.

Method and assumptions. The frequency-severity model estimates expected loss per exposure-year as fitted frequency times fitted mean severity. The Poisson and Gamma assumptions specify the hypothetical component distributions; the fitted means are the values needed for this calculation. No interest or mortality assumption applies.

Observed pure premium is $P=\$4{,}800/0.8=\$6{,}000$ per exposure-year.

Fitted pure premium is $\hat{P}=0.25(\$20{,}000)=\$5{,}000$ per exposure-year. Hence the pure-premium response residual is $\$6{,}000-\$5{,}000=+\$1{,}000$ per exposure-year.

The fitted aggregate loss for 0.8 exposure-years is $0.8(\$5{,}000)=\$4{,}000$. The aggregate-dollar difference is $\$4{,}800-\$4{,}000=+\$800$. Checking the conversion gives $0.8(\$1{,}000)=\$800$, consistent with the aggregate calculation.

The positive residual says that the observed loss cost was above the fitted expectation for this policy. Common mistake avoided: comparing $4,800 of total loss directly with $5,000 per exposure-year would mismatch both the unit and the exposure period.

Interpreting and Comparing Response Residuals

Response residuals preserve the natural units of the response, which makes an individual comparison easy to explain: a count residual is in claims, a severity residual is in dollars per claim, and a pure-premium residual is in dollars per exposure unit. They also retain the original direction of the discrepancy. A positive value is above the fitted mean, and a negative value is below it.

Raw magnitude should not be treated as a universal measure of how unusual an observation is. A residual of 2 claims and a residual of $2,000 per claim are on different scales and have different levels of variability. Even within one response type, variability can differ by fitted mean, exposure, or other model conditions. Response residuals alone do not adjust for those differences or account for how much influence an observation has on the fitted model.

A response residual is also not an error known to be permanent. It is the observed outcome minus the model's fitted conditional mean, and an individual random outcome can lie above or below that mean even when the model is appropriate. For diagnosis, examine residuals in context and keep the response definition, exposure basis, and aggregation level visible. More specialized residuals address variance scaling and likelihood contributions.

A Reliable Calculation Sequence

Before interpreting a residual, identify the exact response represented by one observation. Then determine whether the fitted quantity is a count, a rate, an average, or an aggregate. The following sequence prevents the most common unit errors.

Name the observation and response.
State whether one record is a policy, policy period, claim, or policy-level average, and identify the units of its observed response.

Put the fitted value on the response scale.
Use the inverse link for the fitted mean. If the model uses an exposure offset, include exposure when converting a fitted rate into an expected count or total.

Subtract in matching units.
Calculate observed response minus fitted response. Do not subtract a frequency from a count, or a per-claim severity from a policy aggregate.

Interpret sign and scale.
Report whether the observation is above or below the fitted mean and state the residual's units. Convert to aggregate dollars only when the exposure multiplier is appropriate.

Key takeaway: A response residual is observed minus fitted on the original outcome scale. Its interpretation is only meaningful when the observed response and fitted mean describe the same unit, exposure basis, and level of aggregation.

Check Your Understanding

For each problem, calculate the requested response residual and keep its units explicit.

A hypothetical Poisson count model has a fitted annual frequency of 0.30 claims per exposure-year. A policy has 0.4 exposure-years and one observed claim. What is its fitted expected count and count response residual? (Answer: fitted count $=0.12$; residual $=+0.88$ claims.)
A hypothetical Poisson model has fitted annual frequency 0.08. A policy with 1.5 exposure-years has zero observed claims. Calculate its fitted count and response residual. (Answer: fitted count $=0.12$; residual $=-0.12$ claims.)
Two claims on a hypothetical policy are $3,200 and $7,100. The severity model's fitted mean is $4,800 per claim for both. Find the observed average severity and its response residual. (Answer: observed average $=\$5{,}150$; residual $=+\$350$ per claim.)
A policy has 0.6 exposure-years and $2,700 in observed losses. Its fitted frequency is 0.20 claims per year and its fitted mean severity is $18,000. Find the pure-premium residual and the aggregate-dollar difference. (Answer: observed pure premium $=\$4{,}500$, fitted pure premium $=\$3{,}600$, residual $=+\$900$ per exposure-year, aggregate difference $=+\$540$.)
Two hypothetical policies have exposures of 0.5 and 1.5 years, observed losses of $1,000 and $9,000, and fitted pure premiums of $4,000 and $5,000 per exposure-year, respectively. Calculate the combined observed and fitted pure premiums and the combined response residual. (Answer: combined observed pure premium $=\$5{,}000$, combined fitted pure premium $=\$4{,}750$, combined residual $=+\$250$ per exposure-year.)

What You'll Learn