Canonical Links Versus Useful Links for Family Choice

Canonical Does Not Mean Automatically Suitable

A generalized linear model pairs a response distribution with a link relating its conditional mean to a linear predictor. Some links are called canonical because they make that predictor equal to the distribution’s natural parameter. This connection often simplifies likelihood calculations. It does not, by itself, make the resulting mean model scientifically sensible or the response distribution appropriate.

For positive outcomes, a log link is often useful because every real-valued linear predictor maps to a positive mean and coefficients have a multiplicative interpretation. A canonical link may instead restrict the range of the linear predictor or imply an awkward mean change for a unit change in a predictor. The choice is therefore not a contest in which “canonical” always wins: distributional assumptions and the scientific meaning of the mean relationship remain central.

Definition and the Canonical-Link Formula

For an exponential-dispersion model, write the density or mass function as

Here \(\theta\) is the natural parameter, \(\phi\) is a dispersion parameter, and \(b\) is the cumulant function. In the usual regular case, \(E[Y]=\mu=b'(\theta)\) and \(\operatorname{Var}(Y)=\phi b''(\theta)=\phi V(\mu)\). A GLM specifies a linear predictor \(\eta=x^\mathsf{T}\beta\) and a link \(g\) such that \(g(\mu)=\eta\).

The variance function gives a convenient way to identify the canonical link. Differentiating \(\mu=b'(\theta)\) gives \(d\mu/d\theta=b''(\theta)=V(\mu)\), so \(d\theta/d\mu=1/V(\mu)\) wherever \(V(\mu)>0\). Consequently, up to an additive constant,

This determines a link from the family’s variance structure; it does not determine whether that structure fits the response data or whether the induced predictor-to-mean relationship is useful.

Canonical Links for the Main Continuous Families

For a Gaussian model with constant variance, \(V(\mu)=1\), so the canonical link is the identity, \(g(\mu)=\mu\). In a gamma model, \(V(\mu)=\mu^2\), giving \(g(\mu)=-1/\mu\), up to an additive constant. The inverse-Gaussian variance function is \(V(\mu)=\mu^3\), giving \(g(\mu)=-1/(2\mu^2)\), again up to an additive constant.

For a Tweedie model with power variance function \(V(\mu)=\mu^p\), integration gives the canonical link below. In the semicontinuous setting, typically \(1<p<2\); the \(p=2\) gamma and \(p=3\) inverse-Gaussian cases are useful reference points in the broader power-variance family.

For \(1<p<2\), this canonical link takes only negative values for positive \(\mu\). The Gaussian, gamma, inverse-Gaussian, and Tweedie links are summarized here using natural-parameter conventions; common affine-equivalent forms are noted.

For example, the gamma canonical link is also often written \(1/\mu\), and the inverse-Gaussian canonical link as \(1/\mu^2\). Each differs from the natural-parameter version by a nonzero affine transformation, so the convention can be absorbed into the coefficients when the model is parameterized accordingly.

Worked Comparisons

Why the Canonical Link Simplifies the Score

The canonical link has a real computational advantage: it gives a particularly simple likelihood score for the regression coefficients. This is one reason it is useful, but the simplification concerns estimation algebra rather than scientific validity.

For the canonical link, \(d\eta_i/d\mu_i=1/V(\mu_i)\), so \(d\mu_i/d\eta_i=V(\mu_i)\), and the score reduces to

To derive the general expression, the log-likelihood contribution depending on \(\mu_i\) is \(\{Y_i\theta(\mu_i)-b(\theta(\mu_i))\}/\phi\). By the chain rule and \(b'(\theta)=\mu\), its derivative with respect to \(\mu_i\) is

Multiplying by \(d\mu_i/d\eta_i\) and \(d\eta_i/d\beta=x_i\), then summing independent contributions, yields the stated score. Under the canonical link, the variance factors cancel. This derivation assumes the regular family representation and differentiability stated above; it does not extend automatically to a parameter boundary where the derivatives or likelihood are not regular. \(\square\)

What the Link Does—and Does Not—Choose

The link specifies how covariates relate to the conditional mean: \(g(\mu_i)=x_i^\mathsf{T}\beta\). That relationship affects fitted means and therefore, through the family’s variance function, the implied conditional variances. But the link does not replace the family’s assumptions about response support or its conditional distributional shape. For example, choosing a log link does not turn a Gaussian response into a positive-only distribution; a Gaussian variable still has support on the real line.

Family and link choices should therefore answer distinct questions. The family concerns the response distribution and how its variability changes with the mean. The link concerns the mean scale on which predictors act. A link can be computationally convenient and still impose a mean relationship that is scientifically implausible. Conversely, a noncanonical link can be useful if it provides an appropriate mean relationship and yields valid fitted means.

Common Pitfall

Incorrect reasoning: “The gamma canonical link is mathematically natural, so it must be the best link for positive skewed data.” This confuses a property of the exponential-family parameterization with evidence about the mean relationship and the response distribution.

Correction: Canonical status provides a natural-parameter interpretation and can simplify the score equations. It does not establish that the gamma family fits the outcome, that the implied mean pattern is scientifically appropriate, or that all fitted means are valid. A log link may be preferable when a multiplicative mean effect and an unrestricted real-valued predictor are useful. The family and link should be evaluated for their separate roles.