Link-Scale And Mean-Scale Comparisons

From Linear-Predictor Changes to Expected Losses

In a generalized linear model, the linear predictor is transformed into a conditional mean by the inverse link. A change in the predictor therefore has a different meaning depending on the link: under a log link it changes the mean proportionally, while under an identity link it changes the mean additively. This distinction matters when comparing rating changes in dollars, percentages, or relativities.

Let $\eta$ denote the linear predictor and $\mu=E[Y\mid x]$ the conditional mean of the response. To make a comparison, hold all other predictor terms—including any fixed offset—constant, and let the predictor change from $\eta_0$ to $\eta_1=\eta_0+\Delta$. The starting mean is $\mu_0$, and the resulting mean is $\mu_1$.

Definition: A link-scale change is the difference $\Delta=\eta_1-\eta_0$ between two linear predictors. A mean-scale change is $\mu_1-\mu_0$. The ratio $\mu_1/\mu_0$ and the percentage change $(\mu_1-\mu_0)/\mu_0$ are additional ways to describe the change in the mean.

The Same Predictor Change Has Different Mean Effects

For a log link, $g(\mu)=\log(\mu)=\eta$, so the inverse link is $\mu=e^\eta$. If the predictor increases by $\Delta$, then the new mean is the old mean multiplied by $e^\Delta$. Thus the link-scale change determines a constant mean ratio, regardless of the starting mean.

\mu_1=e^{\eta_0+\Delta} =\mu_0 e^\Delta,\qquad \frac{\mu_1}{\mu_0}=e^\Delta,\qquad \mu_1-\mu_0=\mu_0(e^\Delta-1).

For an identity link, $g(\mu)=\mu=\eta$. The predictor and mean are on the same scale, so the predictor change is exactly the mean difference. The relative change, however, depends on the starting mean.

\mu_1=\eta_0+\Delta=\mu_0+\Delta,\qquad \mu_1-\mu_0=\Delta,\qquad \frac{\mu_1-\mu_0}{\mu_0}=\frac{\Delta}{\mu_0}.

These equations compare the inverse-link transformations; they do not require a particular response distribution. In the examples, the response is hypothetical positive loss cost $Y$, measured in thousands of dollars per exposure. For explicit distributional assumptions, take $Y\mid x$ to have a Gamma distribution with shape $k=20$ and scale $\mu/20$. Under this shape-scale parameterization, $E[Y\mid x]=\mu$ and $\operatorname{Var}(Y\mid x)=\mu^2/20$. The comparisons use the conditional means, not the variance. There is no interest or mortality assumption: neither discounting nor survival is involved.

Why the Log-Link Change Is Multiplicative

The result follows directly from the inverse link. Starting with $\log(\mu_0)=\eta_0$, add $\Delta$ to both sides of the predictor comparison and apply the exponential function:

Write the new predictor.
$\eta_1=\eta_0+\Delta$.

Apply the inverse link.
$\mu_1=e^{\eta_1}=e^{\eta_0+\Delta}$.

Separate the factors.
Using $e^{a+b}=e^ae^b$, $\mu_1=e^{\eta_0}e^\Delta=\mu_0e^\Delta$.

Read the two scales.
The ratio is $e^\Delta$, while the dollar or loss-cost difference is $\mu_0(e^\Delta-1)$.

For example, a log-link predictor change of $\Delta=\log(1.10)$ implies a 10% increase in the mean because $e^\Delta=1.10$. A change of $\Delta=0.10$ does not imply a 10% increase: its exact percentage effect is $e^{0.10}-1$, approximately 10.52%.

Worked Comparisons

Worked Example: Comparing Links at One Baseline

A hypothetical insurer models positive expected claim cost per exposure in thousands of dollars. For a risk profile, the baseline mean is $\mu_0=1.000$, or $1,000 per exposure. Compare an increase of $\Delta=0.200$ under a log link and an identity link, holding all other terms fixed. Assume $Y\mid x$ is Gamma with shape $20$ and scale $\mu/20$, so its mean is $\mu$ and variance is $\mu^2/20$. No interest or mortality is involved.

Log-link method. The mean ratio is $e^{0.200}=1.2214028$. Therefore,

\mu_1=1.000(1.2214028)=1.2214028

The mean-scale increase is $1.2214028-1.000=0.2214028$ thousand dollars per exposure. Converting to dollars gives $0.2214028\times \$1{,}000=\$221.4028$, or $221.40 to the nearest cent. The percentage increase is $100(1.2214028-1)=22.14028\%$, or 22.14%.

Identity-link method. The mean increases by $\Delta=0.200$ thousand dollars, so $\mu_1=1.000+0.200=1.200$. The mean-scale increase is $200.00, and the percentage increase is $0.200/1.000=20\%$.

Check and trap avoided. Both predictions are positive, as required for this Gamma mean model. Treating $0.200$ as a 20% log-link increase would give the wrong result: the correct log-link percentage increase is 22.14%. The identity-link increase is exactly $200.00.

Worked Example: The Same Change at Two Baseline Means

Two hypothetical risk profiles have baseline expected costs of $0.800$ and $1.600$ thousand dollars per exposure. Compare a predictor increase of $\Delta=0.100$ for each profile under both links. Hold all other model terms fixed. For each profile and link, assume a Gamma response with shape $20$ and scale $\mu/20$, hence mean $\mu$ and variance $\mu^2/20$. No interest or mortality assumption is needed.

Log link, lower baseline. The ratio is $e^{0.100}=1.1051702$. The new mean is $0.800(1.1051702)=0.8841361$ thousand dollars. The increase is $0.0841361$ thousand, or $84.14. Its percentage increase is 10.51702%, or 10.52%.

Log link, higher baseline. The same ratio applies. The new mean is $1.600(1.1051702)=1.7682723$ thousand dollars. The increase is $0.1682723$ thousand, or $168.27. The percentage increase is again 10.52%.

Identity link, lower baseline. The new mean is $0.800+0.100=0.900$ thousand dollars, an increase of $100.00. Relative to $0.800$, this is $0.100/0.800=12.5\%$.

Identity link, higher baseline. The new mean is $1.600+0.100=1.700$ thousand dollars, also an increase of $100.00. Relative to $1.600$, this is $0.100/1.600=6.25\%$.

Check and trap avoided. All four new means are positive. Under the log link, the percentage effect is the same but the dollar effect is larger for the higher-baseline profile. Under the identity link, the dollar effect is the same but the percentage effect is smaller for the higher-baseline profile. Do not assume that a fixed predictor shift has the same dollar or percentage interpretation under both links.

Worked Example: Finding the Predictor Change for a Target Increase

A hypothetical risk profile has expected loss cost $\mu_0=1.200$ thousand dollars per exposure. Find the predictor change needed to increase the mean by $0.300$ thousand dollars under an identity link and under a log link. Assume all other terms remain fixed and $Y\mid x$ is Gamma with shape $20$, scale $\mu/20$, mean $\mu$, and variance $\mu^2/20$. No interest or mortality is involved.

Identity link. Since $\mu_1-\mu_0=\Delta$, the required change is $\Delta=0.300$. The new mean is $1.200+0.300=1.500$ thousand dollars, a $300.00 increase.

Log link. The target mean is also $1.500$. From $\mu_1/\mu_0=e^\Delta$, solve for the predictor change:

\Delta=\log\left(\frac{\mu_1}{\mu_0}\right) =\log\left(\frac{1.500}{1.200}\right) =\log(1.25)=0.2231436.

Checking, $1.200e^{0.2231436}=1.200(1.25)=1.500$ thousand dollars. The increase is $300.00, or 25% of the baseline. Both predictions are positive. The Gamma distribution has the stated shape and scale for each mean; its variance does not change the required predictor calculation.

Check and trap avoided. Under the log link, the required predictor change is not the dollar increase $0.300$; it is the logarithm of the target-to-baseline mean ratio. Under the identity link, the predictor change is exactly $0.300$ in the chosen thousand-dollar units.

Units, Interpretation, and Common Traps

A numerical comparison of $\Delta$ across links needs care. With a log link, the predictor is the logarithm of a mean, so a predictor change is dimensionless. With an identity link, the predictor has the same units as the mean. If the mean is recorded in thousands of dollars rather than dollars, an identity-link predictor change of $0.200$ means $200, not 20 cents. Thus “the same numerical change” is a useful comparison of inverse-link behavior, not a claim that the two predictor changes have identical physical units or meaning.

For an identity-link model of positive loss cost, the fitted mean must remain positive over the cases being used. A predictor change that makes $\mu_0+\Delta\leq 0$ is not a valid positive Gamma mean. By contrast, the log inverse link produces a positive mean for every finite predictor. This does not by itself establish that either link is appropriate for a particular ratemaking target; the comparisons here describe how each specified link translates changes.

A frequent exam error is to interpret $\Delta$ as a percentage under every link. The incorrect calculation for a log link is “$\Delta=0.10$, therefore the mean rises 10%.” The correction is to calculate $e^{0.10}=1.1051702$, so the mean rises about 10.52%. For an identity link, in contrast, $\Delta=0.10$ is an absolute mean change in the units of the response. If the starting mean is $0.800$ thousand dollars, that is a 12.5% change; if it is $1.600$, it is a 6.25% change.

Key takeaway: With a log link, $\Delta$ determines the mean ratio $e^\Delta$, and the absolute change depends on the baseline mean. With an identity link, $\Delta$ determines the absolute mean change, and the percentage change depends on the baseline mean. Always identify the link, the response units, and the starting mean before interpreting a predictor change.

Check Your Understanding

For each problem, hold all unspecified predictor terms fixed. Loss costs are in thousands of dollars per exposure, and no interest or mortality is involved.

A hypothetical log-link model has baseline mean $2.000$ and predictor increase $\Delta=0.150$. Find the new mean and absolute increase in dollars. (Answer: New mean $2.3237$ thousand dollars; increase $323.67.)
A hypothetical identity-link model has baseline mean $1.400$ and predictor change $\Delta=-0.080$. Find the new mean and percentage change. (Answer: New mean $1.320$ thousand dollars; decrease 5.71%.)
Under a log link, a mean rises from $0.900$ to $0.990$. Find the predictor change and mean percentage increase. (Answer: $\Delta=\log(1.10)=0.09531$; increase 10.00%.)
Starting from $\mu_0=0.750$, compare the new means under an identity link and a log link when each has numerical predictor change $\Delta=0.120$. Give both mean changes in dollars. (Answer: Identity mean $0.8700$ thousand, increase $120.00; log mean $0.8456$ thousand, increase $95.62.)
A log-link model starts at $1.100$ thousand dollars and has predictor change $\Delta=-0.200$. Find the new mean and the percentage decrease. (Answer: New mean $0.9006$ thousand dollars; decrease 18.13%.)

What You'll Learn