Worked Diagnostics for a Severity Model

Why Severity Diagnostics Need Their Own Scale

A claim-severity model predicts a positive payment amount, usually conditional on a claim having occurred. Its residuals therefore need to be interpreted relative to the model’s variance, not just in dollars. A difference of $1,000 may be substantial for a model predicting $2,000 and unremarkable for one predicting $100,000. This scale issue is especially important for Gamma and inverse Gaussian (IG) GLMs, whose variances increase with the predicted mean.

The diagnostics in this tutorial are for individual, positive severity observations. Suppose $Y_i$ is the observed severity for claim $i$, $\mu_i$ is its fitted conditional mean, and $\phi$ is a common dispersion parameter. The usual assumptions are conditional independence given the rating variables, a correctly specified conditional mean and variance function, and observations on the scale modeled by the GLM. A claim payment of zero, censoring, or truncation may require a different likelihood or data treatment; the formulas below should not be applied mechanically to such observations.

Definition: For a Gamma GLM, $E[Y_i\mid x_i]=\mu_i$ and $\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i^2$. Equivalently, $Y_i$ has a Gamma distribution with shape $1/\phi$ and scale $\phi\mu_i$. For an inverse Gaussian GLM, $E[Y_i\mid x_i]=\mu_i$ and $\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i^3$; equivalently, its inverse Gaussian shape parameter is $\lambda=1/\phi$. Here $x_i$ denotes the claim’s rating variables.

Both models may use a log link, in which $\log(\mu_i)=x_i^\mathsf{T}\beta$, so fitted means remain positive. The link determines how rating variables affect the mean; the variance function determines how residual differences should be scaled. A severity GLM’s dispersion is not the same thing as a policy exposure weight: claim-level observations and any weighting scheme must be represented consistently with the model being fitted.

Pearson and Deviance Residuals

The unscaled Pearson residual divides the observed-minus-fitted difference by the square root of the variance function. A scaled Pearson residual also divides by $\sqrt{\phi}$. The unit deviance measures the likelihood-based discrepancy between an observation and its fitted mean; the signed deviance residual is its square root, with the sign of the ordinary residual. The scaled deviance residual divides by $\sqrt{\phi}$.

r_{P,i}=\frac{Y_i-\mu_i}{\sqrt{V(\mu_i)}}, \qquad r_{P,i}^{\,*}=\frac{Y_i-\mu_i}{\sqrt{\phi V(\mu_i)}}, \qquad r_{D,i}^{\,*}=\operatorname{sign}(Y_i-\mu_i)\sqrt{\frac{d_i}{\phi}}.

For Gamma, $V(\mu)=\mu^2$. For inverse Gaussian, $V(\mu)=\mu^3$. The unit deviance $d_i$ is likewise different for the two distributions:

d_{i,\mathrm{Gamma}} =2\left[\frac{Y_i-\mu_i}{\mu_i} -\log\left(\frac{Y_i}{\mu_i}\right)\right], \qquad d_{i,\mathrm{IG}} =\frac{(Y_i-\mu_i)^2}{Y_i\mu_i^2}.

If software reports an unscaled deviance residual, it uses $\operatorname{sign}(Y_i-\mu_i)\sqrt{d_i}$, without dividing by $\sqrt{\phi}$. Always check the convention before comparing reported residual magnitudes. The following examples use the scaled definition $r_D^*$, and show the scaled Pearson residual too.

Why These Deviance Formulas Have This Form

For a GLM with variance function $V(\mu)$, the unit deviance between an observed positive value $y$ and fitted mean $\mu$ can be written as $2\int_{\mu}^{y}(y-t)/V(t)\,dt$. This measures the model’s likelihood discrepancy from the saturated fit, which fits that observation exactly. Substituting each severity model’s variance function gives its corresponding deviance.

For Gamma, $V(t)=t^2$, so integration gives $2[y(1/\mu-1/y)-\log(y/\mu)]$, which simplifies to $2[(y-\mu)/\mu-\log(y/\mu)]$. For inverse Gaussian, $V(t)=t^3$, and integration gives $1/y+y/\mu^2-2/\mu$. Combining terms yields $(y-\mu)^2/(y\mu^2)$. In either case, the deviance is nonnegative and equals zero when $y=\mu$. The factor $1/\phi$ in a scaled deviance residual accounts for the dispersion in the observation variance.

Worked Example: Comparing Gamma Severity Residuals

A hypothetical insurer fits a claim-level Gamma severity GLM with a log link. Conditional on the fitted means, claims are independent and have Gamma distributions with shape $4$ and scale $\mu_i/4$, so $\phi=1/4=0.25$. For three claims, observed severities and fitted means, in dollars, are $(200,100)$, $(60,120)$, and $(100,100)$. Calculate each claim’s scaled Pearson and deviance residuals.

Method. For Gamma, $V(\mu)=\mu^2$, so $r_P^*=(y-\mu)/(\sqrt{0.25}\mu)$. Calculate $d=2[(y-\mu)/\mu-\log(y/\mu)]$, then give it the sign of $y-\mu$ and take $\sqrt{d/0.25}$.

For claim 1, $y/\mu=200/100=2$. Its scaled Pearson residual is $(200-100)/(\sqrt{0.25}\times100)=100/50=2.0000$. Its unit deviance is $2[1-\log(2)]=2[1-0.693147]=0.613706$. Thus its scaled deviance residual is $+\sqrt{0.613706/0.25}=+\sqrt{2.454824}=1.5668$.

For claim 2, $y/\mu=60/120=0.5$. Its scaled Pearson residual is $(60-120)/(\sqrt{0.25}\times120)=-60/60=-1.0000$. Its unit deviance is $2[-0.5-\log(0.5)]=2[-0.5+0.693147]=0.386294$. Its scaled deviance residual is $-\sqrt{0.386294/0.25}=-\sqrt{1.545176}=-1.2431$.

For claim 3, $y=\mu=100$. Both residuals are zero, because the observed and fitted severities agree.

Check. Claims 1 and 2 have positive and negative residuals, respectively, matching whether their observed severities are above or below their fitted means. The scaled Pearson residuals are $2.0000$, $-1.0000$, and $0.0000$; the scaled deviance residuals are $1.5668$, $-1.2431$, and $0.0000$. The method avoids comparing claim differences in dollars without accounting for the mean-dependent Gamma variance.

Interpreting Residuals Under Each Variance Function

For Gamma severity, the unscaled Pearson residual is simply $(y-\mu)/\mu$, the relative difference from the fitted mean. Its scaled version divides that relative difference by $\sqrt{\phi}$. This makes a relative discrepancy more informative than a dollar difference when the model assumes variance proportional to $\mu^2$. Under an inverse Gaussian model, the unscaled Pearson residual is $(y-\mu)/\mu^{3/2}$, and the scaled residual divides again by $\sqrt{\phi}$. The extra power of the mean matters: using the Gamma scaling for an IG model gives the wrong residual.

Worked Example: Reading an Inverse Gaussian Residual Pair

A hypothetical commercial-property portfolio uses an inverse Gaussian severity GLM. Each observation has an inverse Gaussian distribution with mean $\mu_i$ and shape $\lambda=25$, so $\phi=1/25=0.04$. Consider claims with observed and fitted severities $(200,100)$, $(50,100)$, and $(100,100)$, in dollars. Calculate the scaled residuals and compare what they indicate.

Method. For IG, use $V(\mu)=\mu^3$ and $d=(y-\mu)^2/(y\mu^2)$. The scaled Pearson denominator is $\sqrt{0.04\mu^3}$, and the scaled deviance residual is $\operatorname{sign}(y-\mu)\sqrt{d/0.04}$.

For $y=200,\mu=100$, the Pearson denominator is $\sqrt{0.04(100)^3}=\sqrt{40{,}000}=200$. Therefore $r_P^*=(200-100)/200=0.5000$. The unit deviance is $(200-100)^2/[200(100)^2]=10{,}000/2{,}000{,}000=0.005$. The scaled deviance residual is $+\sqrt{0.005/0.04}=+\sqrt{0.125}=0.3536$.

For $y=50,\mu=100$, the Pearson denominator is again $200$, so $r_P^*=(50-100)/200=-0.2500$. The unit deviance is $(50-100)^2/[50(100)^2]=2{,}500/500{,}000=0.005$. The scaled deviance residual is $-\sqrt{0.005/0.04}=-0.3536$. For $y=\mu=100$, both residuals are zero.

Check. The two nonzero deviance residuals have equal magnitude because these two choices of $y$ happen to give the same IG unit deviance at $\mu=100$; their Pearson residuals differ. The observed deviations are not being assessed against a constant dollar-scale variance. The method avoids applying a Gamma variance or deviance formula to an inverse Gaussian model.

Extreme Claims and Variance Patterns

A large claim can have a large residual, but “large in dollars” and “poorly fitted relative to the model” are different statements. Under Gamma, residual size depends in part on the claim’s relative deviation from its mean. Under IG, it depends on both the relative deviation and the cubic variance scaling. A claim that looks extreme on a raw-dollar plot may therefore be less unusual on a model-scaled diagnostic. Conversely, a claim can have an important residual even if its dollar amount is not among the largest in the portfolio.

Inspect residuals against fitted mean and relevant rating variables, and look for systematic patterns rather than relying on a single point. A spread that grows with fitted mean may be consistent with the model’s variance function; a persistent pattern in scaled residuals can signal variance misspecification, a missing rating effect, or another data or model issue. The earlier material on residuals against rating variables and response distributions gives the broader diagnostic context. Residual plots are not proof of a particular cause, and the distributional shape of severity residuals need not be normal in a small sample.

Worked Example: Separating an Extreme Claim From Influence

A hypothetical insurer fits an intercept-only Gamma severity model with a log link to four independent positive claim severities: $90, $100, $110, and $500. Treat the dispersion as known and fixed at $\phi=0.25$, so each observation follows a Gamma distribution with shape $4$ and scale $\mu/4$. Calculate the full-data fitted mean, the scaled residuals for the $500 claim, and the change in fitted mean when that claim is omitted.

Method. For an intercept-only Gamma model with fixed dispersion, the maximum-likelihood estimate of the common mean is the arithmetic average. First calculate that average; then calculate the claim’s residual using the Gamma formulas. Finally, refit after omission to assess how much the fitted prediction changes. This deletion comparison measures influence on the fitted mean in this simple model.

The total severity is $90+100+110+500=800$ dollars, so the full-data fitted mean is $800/4=$200$. For the $500 claim, the Pearson denominator is $\sqrt{0.25(200)^2}=\sqrt{10{,}000}=100$, giving $r_P^*=(500-200)/100=3.0000$. Its unit deviance is $2[300/200-\log(500/200)]=2[1.5-\log(2.5)]$ $=2[1.5-0.916291]=1.167418$. Thus its scaled deviance residual is $+\sqrt{1.167418/0.25}=+\sqrt{4.669672}=2.1610$.

After omitting the $500 claim, the remaining total is $90+100+110=300$, and the refitted mean is $300/3=$100$. The fitted mean therefore falls by $$200-$100)/$200=50\%$. On the log scale, the intercept changes from $\log(200)$ to $\log(100)$, a change of $\log(100/200)=\log(0.5)=-0.693147$.

Check. The $500 claim has both a large positive residual and a substantial effect on this fitted mean. That makes it influential for this particular fit, not automatically erroneous. Verify the record and the claim’s coverage and development treatment before considering any modeling response. The method avoids equating an extreme value with bad data or deleting it solely to reduce a residual.

Residual Size Is Not the Same as Influence

A residual measures how far an observation is from its fitted value on a model-specific scale. Influence concerns how much the fitted model changes when that observation is perturbed or removed. A large residual need not be influential if the observation has little leverage on the fitted parameters. An observation with unusual rating-variable values may have greater leverage, so even a moderate residual can have a material effect. In GLMs, leverage and influence measures depend on the fitted design and weights; residual size alone cannot settle the question.

For an initial screening, compare fitted predictions or coefficients with and without a flagged claim, and examine the relevant leverage or influence diagnostics. In a log-link severity model, coefficient changes are multiplicative effects on fitted means: a change of $\Delta$ in a linear predictor multiplies the corresponding mean by $e^\Delta$. The deletion calculation in the preceding example is deliberately simple; with several rating variables, deletion refitting is more informative than interpreting a single residual in isolation. Earlier material on leverage and Cook’s distance covers those general influence diagnostics.

Common exam trap: Using $(y-\mu)/\mu$ as the Pearson residual for both severity distributions is incorrect. That is the unscaled Gamma Pearson residual. For an IG model it is $(y-\mu)/\mu^{3/2}$, before dispersion scaling. Also distinguish scaled from unscaled residuals: if a definition includes $\phi$, a calculation omitting it changes the reported magnitude.

A Practical Diagnostic Sequence

Confirm the modeled response.
Check that each value is a positive severity on the intended payment basis, and that zero, censored, or truncated observations have not been treated as ordinary positive claims.

Identify the distribution and dispersion.
Use $V(\mu)=\mu^2$ for Gamma or $V(\mu)=\mu^3$ for IG, and verify whether residuals are scaled by $\phi$.

Calculate and examine residuals.
Use both Pearson and deviance residuals where useful. Compare signs and magnitudes against fitted means and rating variables, seeking patterns rather than relying on a fixed dollar cutoff.

Investigate flagged claims and influence.
Check data and claim handling, then assess leverage or refit without the observation. Document whether conclusions or predictions materially change; do not delete a valid large claim simply because it is extreme.

These diagnostics matter because severity predictions affect indicated rates and other ratemaking analyses. A variance function that does not describe the observed pattern can distort uncertainty assessments and make residual comparisons misleading. A single unusual claim can also influence a fitted mean or rating factor. The appropriate response is disciplined investigation: verify the observation, assess model fit on its stated scale, and evaluate the effect on fitted predictions.

Check Your Understanding

For each problem, state the relevant model and use the dispersion scaling specified.

A Gamma severity model has $\phi=0.25$, fitted mean $100, and observed severity $150. Calculate the scaled Pearson and signed scaled deviance residuals. (Answer: $r_P^*=1.0000$; $r_D^*=+0.8696$.)
A Gamma model has $\phi=0.25$, fitted mean $100, and observed severity $80. Calculate both scaled residuals. (Answer: $r_P^*=-0.4000$; $r_D^*=-0.4303$.)
An inverse Gaussian model has $\phi=0.01$, fitted mean $100, and observed severity $150. Calculate the scaled Pearson and signed scaled deviance residuals. (Answer: $r_P^*=0.5000$; $r_D^*=+0.4082$.)
An inverse Gaussian model has $\phi=0.04$, fitted mean $100, and observed severity $50. Calculate both scaled residuals, using the inverse Gaussian variance function. (Answer: $r_P^*=-0.2500$; $r_D^*=-0.3536$.)
A hypothetical intercept-only Gamma model with fixed $\phi=0.25$ is fit to severities $80, $100, $120, and $300. Find the full-data fitted mean and the mean after omitting the $300 claim. By what percentage does the fitted mean fall? (Answer: $150 initially and $100 after omission; a $33.33\%$ decrease.)

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