Gamma Canonical And Log-Link Predictions

Why the Link and Its Domain Matter

A severity model predicts a conditional mean loss, which must be positive. A generalized linear model (GLM) connects that mean to predictors through a link function. The chosen link determines how a linear predictor becomes a mean—and whether every possible predictor produces an admissible mean. For a Gamma response, the signed canonical link and the log link both produce positive predictions on their valid domains, but those domains are not the same.

This tutorial uses $Y$ for a positive individual loss and $\mu=E[Y\mid \mathbf{x}]$ for its conditional mean given predictors $\mathbf{x}$. The focus is on transforming the predictor and checking its validity, not on estimating coefficients from data. As in the earlier material on link functions, a link specifies $g(\mu)=\eta$, where $\eta$ is the linear predictor.

Definition: A Gamma GLM models a positive response $Y$ with conditional mean $\mu>0$ and variance $\operatorname{Var}(Y\mid\mathbf{x})=\phi\mu^2$, where the dispersion parameter $\phi>0$. Under the signed natural-parameter convention used here, the Gamma canonical link is $g(\mu)=-1/\mu$, so its linear predictor must satisfy $\eta<0$. The Gamma log link is $g(\mu)=\log\mu$, so its predictor may be any real number.

Deriving the Signed Gamma Canonical Link

The sign in the canonical link follows from the exponential-family representation. For a Gamma response with mean $\mu$ and dispersion $\phi$, the density can be written using the shape $1/\phi$ and scale $\phi\mu$. Its terms involving $y$ and $\mu$ are proportional to $-y/(\phi\mu)-\log(\mu)/\phi$. In the standard form $\exp\{[y\theta-b(\theta)]/\phi+c(y,\phi)\}$, matching the term multiplying $y$ gives $\theta=-1/\mu$.

\theta=-\frac{1}{\mu},\qquad b(\theta)=-\log(-\theta),\qquad b'(\theta)=-\frac{1}{\theta}=\mu.

The canonical link equates the linear predictor to the natural parameter: $\eta=\theta$. Therefore $\eta=-1/\mu$, and solving for the mean gives the inverse-link transformation below. This also explains the domain: a positive mean makes $-1/\mu$ strictly negative. A zero or positive canonical predictor cannot correspond to a finite positive Gamma mean.

\eta=-\frac{1}{\mu} \quad\Longleftrightarrow\quad \mu=-\frac{1}{\eta},\qquad \eta<0.

Some software or texts use a reciprocal link $1/\mu$ and call it an inverse link. That convention reverses the sign of the predictor relative to the natural parameter used in this derivation. A prediction is only meaningful when the specified predictor and inverse-link transformation use the same convention. Here, “signed canonical” means $\eta=-1/\mu$.

Gamma Log-Link Predictions

For a log link, $g(\mu)=\log\mu$. The inverse transformation is $\mu=\exp(\eta)$, which is positive for every real $\eta$. Thus, unlike the signed canonical specification, a log-link linear predictor does not need a sign restriction to yield a positive mean.

\eta=\log\mu \quad\Longleftrightarrow\quad \mu=\exp(\eta),\qquad -\infty<\eta<\infty.

If the predictor has the form $\eta=\beta_0+\beta_1x_1+\cdots+\beta_px_p$, calculate that entire expression first, then apply the inverse link. With a log link, a coefficient changes the mean multiplicatively: increasing $x_j$ by one unit multiplies the mean by $\exp(\beta_j)$, holding other predictors fixed. With a canonical link, the same coefficient changes the reciprocal-scale predictor; it is not a direct dollar change in the mean.

Worked Example: Signed Canonical Predictions for Two Classes

A hypothetical insurer models positive repair severity with the signed Gamma canonical predictor $\eta(z)=-1/400+(1/2000)z$, where $z=0$ denotes one vehicle class and $z=1$ another. Find each predicted mean and determine whether both predictors are valid. State the method and the common mistake to avoid.

Method. Evaluate the predictor for each class, check that it is strictly negative, and then use $\mu=-1/\eta$. The coefficients and predictor are on a per-loss dollar scale; no exposure offset is specified.

For $z=0$, $\eta(0)=-1/400=-5/2000=-0.0025$. This is negative, so the prediction is valid. The mean is $\mu(0)=-1/(-1/400)=\$400.00$.

For $z=1$, $\eta(1)=-1/400+1/2000=-5/2000+1/2000=-4/2000=-1/500=-0.002$. This is also negative, so the prediction is valid. The mean is $\mu(1)=-1/(-1/500)=\$500.00$.

Check. Substituting the predictions back into the link gives $-1/400=-0.0025$ and $-1/500=-0.002$, respectively, matching the two calculated predictors. Both means are positive. The common mistake is to lose the sign or miscombine the fractions for $z=1$; in particular, its predictor is $-1/500$, not $-1/1000$, so its mean is $\$500.00$, not $\$1{,}000.00$.

Worked Example: A Gamma Log-Link Severity Prediction

A hypothetical equipment insurer uses a Gamma log-link model with predictor $\eta(x)=\log(600)+x\log(1.5)$, where $x$ is a nonnegative integer indicator count. Find the predicted mean severity at $x=1$ and $x=2$, and check the predictions. The example assumes a Gamma response with positive losses and $\phi>0$; no exposure offset is included.

Method. Substitute each $x$ into the predictor and exponentiate. Since the log-link inverse is the exponential function, both predictions are positive without an additional sign-domain test.

At $x=1$, $\eta(1)=\log(600)+\log(1.5)=\log(900)$, so $\mu(1)=\exp(\log 900)=\$900.00$.

At $x=2$, $\eta(2)=\log(600)+2\log(1.5)=\log(600)+\log(1.5^2)=\log(600\times2.25)=\log(1350)$, so $\mu(2)=\exp(\log 1350)=\$1{,}350.00$.

Check. The mean ratio is $1350/900=1.5$, matching the multiplier $\exp(\log(1.5))=1.5$ for one additional indicator unit. The common mistake is to add the coefficient $\log(1.5)$ to the dollar mean; it must be added on the predictor scale, and the resulting predictor must then be exponentiated.

Checking the Canonical Domain Across Predictor Values

For a canonical model, checking one prediction is not enough if the model will be used over a range of covariates. Every predictor value in the intended rating territory must produce $\eta<0$. If the predictor is linear in a continuous covariate, its extrema over a closed interval occur at an endpoint, so checking both endpoints is sufficient to establish the sign throughout that interval. For categorical predictors, evaluate every modeled category.

Worked Example: Finding the Valid Range of a Canonical Predictor

A hypothetical commercial-property severity model uses the signed canonical predictor $\eta(x)=-1/1000+x/5000$, where $x$ is a building-age score. Determine the validity and predicted mean at $x=0$, $x=4$, $x=5$, and $x=6$. Identify the allowable range of $x$ and the mistake this check prevents.

Method. Put the terms over a common denominator to identify the sign of $\eta(x)$. Apply $\mu=-1/\eta$ only when the result is strictly negative.

At $x=0$, $\eta(0)=-1/1000$, so it is valid and $\mu(0)=\$1{,}000.00$.

At $x=4$, $\eta(4)=-1/1000+4/5000=-5/5000+4/5000=-1/5000$. It is valid, and $\mu(4)=-1/(-1/5000)=\$5{,}000.00$.

At $x=5$, $\eta(5)=-5/5000+5/5000=0$. The inverse $-1/\eta$ is undefined, so there is no finite predicted mean.

At $x=6$, $\eta(6)=-5/5000+6/5000=1/5000>0$. It is outside the canonical domain and would give $-\$5{,}000$ if one mechanically applied the inverse formula. That negative value is not a valid Gamma mean.

Check. The condition is $-1/1000+x/5000<0$, equivalent to $x<5$. Thus $x=0$ and $x=4$ are valid, while $x=5$ and $x=6$ are not. The common mistake is to apply the reciprocal formula without first checking its domain, thereby treating an undefined or negative result as a valid severity prediction.

Prediction Procedure and Practical Interpretation

The distinction between the two specifications can be summarized by their inverse links and domains. The Gamma assumption itself requires a positive conditional mean and, for the standard variance specification, a positive dispersion parameter. The link determines the additional restrictions on the linear predictor.

Specification	Predictor definition	Mean prediction	Valid predictor domain
Signed Gamma canonical	$\eta=-1/\mu$	$\mu=-1/\eta$	$\eta<0$
Gamma log link	$\eta=\log\mu$	$\mu=\exp(\eta)$	All real $\eta$

A reliable calculation follows the same order each time. If a known offset is part of the specified model, include it in the linear predictor before applying the inverse link. Do not confuse an offset with a coefficient estimated from the data; the earlier material on offsets distinguishes those roles. Then evaluate the predictor, check the link's domain, transform to the mean, and verify that the prediction satisfies the response model's requirements.

Write the full predictor.
Include the intercept, covariate terms, and any specified offset, using the model's stated sign convention.

Check the link-specific domain.
For the signed canonical Gamma link require $\eta<0$; for the log link any finite real $\eta$ is admissible.

Apply the matching inverse link.
Use $-1/\eta$ for the signed canonical specification or $\exp(\eta)$ for the log specification.

Check the resulting mean.
A valid Gamma conditional mean must be finite and strictly positive. If the canonical predictor is zero or positive, do not report a severity prediction from that specification.

The domain check is especially important when a canonical predictor is evaluated at new covariate combinations or near the edge of the observed rating territory. A linear predictor may cross zero even when all fitted observations had negative predictors. The log link avoids that particular problem because exponentiation always gives a positive mean, although the prediction can still be numerically extreme for a very large or very small predictor.

A central exam trap is to treat “inverse link” as a generic instruction to take a reciprocal. The signed canonical link requires a negative reciprocal, and its predictor must be negative. Conversely, a log-link prediction requires exponentiation, not a reciprocal. Always identify both the link and its predictor convention before calculating.

Key takeaway: For a Gamma model, positive means do not by themselves guarantee a valid canonical prediction. Under the signed natural-parameter convention, use $\mu=-1/\eta$ only when $\eta<0$. Under a log link, use $\mu=\exp(\eta)$, which is positive for every real predictor.

Check Your Understanding

For each problem, identify the link, apply its inverse transformation, and check any required domain before giving the prediction.

A signed canonical Gamma model has $\eta=-1/2500+z/10000$. For $z=3$, find the mean and state whether the prediction is valid. (Answer: $\eta=-1/10000<0$, so $\mu=\$10{,}000.00$.)
A Gamma log-link model has $\eta=\log(400)+2\log(1.2)$. Find its predicted mean. (Answer: $\mu=400(1.2)^2=\$576.00$.)
A signed canonical predictor is $\eta(x)=-0.004+0.0005x$, for $0\leq x\leq6$. Are all predictions valid, and what is the mean at $x=6$? (Answer: All are valid because the largest predictor is $-0.001<0$; at $x=6$, $\mu=\$1{,}000.00$.)
A Gamma log-link model has $\eta(x)=\log(300)+x\log(4/3)$. Find the mean at $x=2$, rounded to the nearest cent. (Answer: $\mu=300(4/3)^2=\$533.33$.)
A signed canonical predictor is $\eta(x)=-1/800+x/4000$, and $x$ is restricted to integer values from 0 through 5. Which values of $x$ give valid predictions, and what is the mean at the largest valid $x$? (Answer: $x=0,1,2,3,4$ are valid; $x=5$ gives $\eta=0$. At $x=4$, $\eta=-1/4000$, so $\mu=\$4{,}000.00$.)

Specification	Predictor definition	Mean prediction	Valid predictor domain
Signed Gamma canonical	\(\eta=-1/\mu\)	\(\mu=-1/\eta\)	\(\eta<0\)
Gamma log link	\(\eta=\log\mu\)	\(\mu=\exp(\eta)\)	All real \(\eta\)

What You'll Learn