Choosing Among Inverse-Gaussian Links

Why Link Choice Matters

Inverse-Gaussian regression models a positive continuous response, but positivity of the response alone does not determine the link. The link decides how a linear predictor maps to a mean, what a coefficient means, and which predictor values produce valid fitted means. Those decisions matter both for interpretation and for whether predictions remain sensible across the covariate range of interest.

The inverse-Gaussian mean is \(\mu>0\), and its variance function is \(V(\mu)=\mu^3\), so \(\operatorname{Var}(Y)=\phi\mu^3\) for dispersion \(\phi>0\). For this family, the natural parameter is \(\theta=-1/(2\mu^2)\). The four links compared here are the canonical link, identity link, log link, and reciprocal link. Their central difference is the set of linear predictors they permit and the way those predictors move the fitted mean.

The Four Links and Their Domains

Definition: In an inverse-Gaussian GLM, let \(\eta=\mathbf{x}^{T}\boldsymbol{\beta}\) denote the linear predictor and \(\mu=E[Y\mid\mathbf{x}]>0\) the conditional mean. The canonical link is \(g_c(\mu)=\theta=-1/(2\mu^2)\). The identity, log, and reciprocal links are respectively \(g_I(\mu)=\mu\), \(g_L(\mu)=\log\mu\), and \(g_R(\mu)=1/\mu\). The fitted mean is obtained by solving \(g(\mu)=\eta\).

Inverting each link gives the fitted mean and the predictor domain needed for a positive finite mean:

Link	Inverse link \(\mu=g^{-1}(\eta)\)	Admissible \(\eta\)	Effect of increasing \(\eta\)
Canonical	\(\sqrt{-1/(2\eta)}\)	\(\eta<0\)	Increases \(\mu\)
Identity	\(\eta\)	\(\eta>0\)	Increases \(\mu\) one-for-one
Log	\(e^\eta\)	All real \(\eta\)	Increases \(\mu\) multiplicatively
Reciprocal	\(1/\eta\)	\(\eta>0\)	Decreases \(\mu\)

The canonical link takes its value in the negative half-line because the inverse-Gaussian natural parameter is negative. A commonly used alternative is \(1/\mu^2\). Since

\frac{1}{\mu^2}=-2\theta,

this alternative multiplies the natural parameter by \(-2\), a negative rescaling. If the canonical predictor is \(\mathbf{x}^{T}\boldsymbol{\beta}=\theta\), the alternative predictor is \(\mathbf{x}^{T}\boldsymbol{\gamma}=1/\mu^2\) with \(\boldsymbol{\gamma}=-2\boldsymbol{\beta}\). The sign and scale of the coefficients therefore change, while the represented fitted means can remain the same. This is not a positive rescaling of the natural parameter.

What the Coefficients Say

The link determines the scale on which covariates have additive effects. If \(x_j\) increases by \(\Delta\) while the other predictors stay fixed, the linear predictor changes by \(\beta_j\Delta\). The corresponding mean change depends on the inverse link.

Identity: \(\mu\) changes by exactly \(\beta_j\Delta\). This is a constant additive change in the response mean.
Log: the mean is multiplied by \(e^{\beta_j\Delta}\). Thus \(\beta_j\) describes a log mean ratio per unit increase.
Reciprocal: \(1/\mu\) changes additively by \(\beta_j\Delta\). The coefficient is an effect on the reciprocal mean, not a constant change in \(\mu\).
Canonical: \(-1/(2\mu^2)\) changes additively by \(\beta_j\Delta\). The effect on \(\mu\) depends on its current value.

These are link-scale interpretations. They do not imply that an observational association is causal, and they do not by themselves establish that a proposed predictor range is scientifically appropriate.

Worked Example: Comparing Coefficient Interpretations

Suppose \(x\) is a continuous predictor and consider a one-unit increase in \(x\). For each link, calculate the resulting mean at a specified starting value.

Log link. Let \(\eta=\log\mu=\log 2+(\log 1.5)x\). At \(x=0\), \(\mu=e^{\log 2}=2\). At \(x=1\),

\mu=e^{\log 2+\log 1.5}=2(1.5)=3.

The coefficient \(\log 1.5\) gives a 1.5-fold mean, or 50% increase, for this one-unit change.

Identity link. Let \(\eta=\mu=2+0.5x\). The means are \(2\) at \(x=0\) and \(2.5\) at \(x=1\), an additive increase of \(0.5\).

Reciprocal link. Let \(\eta=1/\mu=0.5+0.1x\). The mean at \(x=0\) is \(1/0.5=2\), while at \(x=1\) it is \(1/0.6=5/3\). The reciprocal mean increases by \(0.1\), but the mean itself decreases by \(2-5/3=1/3\).

Canonical link. Let \(\eta=-1/(2\mu^2)=-0.125+0.025x\). At \(x=0\), inversion gives \(\mu=\sqrt{-1/(2(-0.125))}=2\). At \(x=1\), \(\eta=-0.1\), so

\mu=\sqrt{\frac{-1}{2(-0.1)}}=\sqrt{5}\approx2.236.

The canonical coefficient changes the natural parameter by \(0.025\); it does not specify a constant additive or percentage change in the mean.

Predictor Constraints and Fitted-Mean Behavior

Theorem: For a finite linear predictor \(\eta\), the canonical inverse-Gaussian, identity, log, and reciprocal inverse links produce a positive finite mean exactly on the domains \(\eta<0\), \(\eta>0\), \(\eta\in\mathbb{R}\), and \(\eta>0\), respectively. On those domains, their derivatives with respect to \(\eta\) are \(\mu^3\), \(1\), \(\mu\), and \(-\mu^2\), respectively.

Proof. For the canonical link, \(\eta=-1/(2\mu^2)\) with \(\mu>0\), so necessarily \(\eta<0\). Conversely, if \(\eta<0\), then \(-1/(2\eta)>0\), and \(\mu=\sqrt{-1/(2\eta)}\) is positive and finite. Differentiating \(\eta=-1/(2\mu^2)\) with respect to \(\mu\) gives \(d\eta/d\mu=\mu^{-3}\), hence \(d\mu/d\eta=\mu^3\). For the identity link, \(\mu=\eta\), so positivity and finiteness require \(\eta>0\), and \(d\mu/d\eta=1\). For the log link, \(\mu=e^\eta\) is positive and finite for every finite real \(\eta\), with derivative \(e^\eta=\mu\). For the reciprocal link, \(\mu=1/\eta\) is positive and finite exactly when \(\eta>0\), and its derivative is \(-1/\eta^2=-\mu^2\). These calculations establish the domains and derivatives. \(\square\)

The derivatives clarify how fitted means respond locally. Under the canonical link, an increment \(d\eta\) produces approximately \(\mu^3d\eta\) in the mean; under the log link it produces approximately \(\mu d\eta\). The reciprocal link moves in the opposite direction: increasing its predictor decreases its fitted mean. Large changes need not be well described by these local derivatives, so exact inverse-link calculations are preferable when the change is substantial.

Worked Example: Checking Predictor Admissibility

Suppose the predictor is \(x\in[0,2]\), and a proposed identity-link model is \(\eta=0.4+0.3x\). The endpoint values are \(\eta(0)=0.4\) and \(\eta(2)=1.0\). Since this linear function increases across the interval, it remains positive throughout, and all fitted means are positive. In particular, \(\mu(0)=0.4\) and \(\mu(2)=1.0\).

Now change only the intercept to \(-0.2\). The predictor becomes \(\eta=-0.2+0.3x\), which is negative at \(x=0\). The identity-link fitted mean at that covariate value would be \(-0.2\), so the model does not define a valid positive mean over the stated range. The problem is not fixed by the fact that the response distribution has positive support: the fitted conditional mean itself must also be positive.

For comparison, a log-link model with \(\eta=-0.2+0.3x\) is admissible over the whole range. Its endpoint means are \(e^{-0.2}\approx0.819\) and \(e^{0.4}\approx1.492\), both positive. The canonical link also requires an explicit range check, but with the opposite sign: its predictor must stay strictly below zero.

A Practical Link-Selection Comparison

A useful choice begins with the scientific meaning of a predictor effect, then checks the predictor domain over all covariate values where fitted means will be used. No link is automatically best merely because its name is canonical, familiar, or easy to fit.

Specify the intended effect scale.
Use a log link when multiplicative mean changes are scientifically natural; use identity when additive mean changes are meaningful; use reciprocal when additive changes in inverse mean have a credible interpretation. The canonical link makes the natural parameter linear in the predictors.

Check the entire predictor range.
For identity and reciprocal links, verify \(\eta>0\); for the canonical link, verify \(\eta<0\). The log link accepts every finite real predictor and always gives a positive mean.

Inspect fitted means and predictions.
Evaluate inverse-link predictions at observed covariate combinations and at the planned prediction range. Check for implausible magnitudes, sharp changes, or invalid means rather than relying only on coefficients.

A domain check for a linear predictor is more than checking its values at the observed rows if the model will be used elsewhere. For example, a fitted identity model may give positive means at every training observation while becoming negative at an extrapolated covariate value. Likewise, a canonical model can approach \(\eta=0\) from below, where \(\sqrt{-1/(2\eta)}\) grows without bound. That behavior may be numerically and scientifically problematic even though each fitted mean remains finite when \(\eta<0\).

Worked Example: Choosing a Link for Positive Biomarker Levels

Suppose a biomarker concentration is positive, and investigators expect a treatment contrast to act approximately as a constant percentage change in the conditional mean. Consider two treatment groups with a log-link model \(\log\mu=\beta_0+\beta_1T\), where \(T=0\) for control and \(T=1\) for treatment. Let \(\beta_0=\log 4\) and \(\beta_1=\log(0.75)\).

For controls, \(\eta=\log4\), so \(\mu=e^{\log4}=4\). For treated subjects, \(\eta=\log4+\log(0.75)=\log3\), so \(\mu=3\). The treatment-to-control mean ratio is \(3/4=0.75\), exactly \(e^{\beta_1}\); the fitted treatment mean is 25% lower. Both means are positive without a separate constraint on \(T\)'s predictor range.

If the same scientific question is represented by a reciprocal-link model \(1/\mu=\gamma_0+\gamma_1T\), its treatment coefficient instead describes an additive change in reciprocal mean. To produce the same two group means, set \(\gamma_0=1/4\) and \(\gamma_1=1/3-1/4=1/12\). Then the control mean is \(1/(1/4)=4\), and the treatment mean is \(1/(1/4+1/12)=1/(1/3)=3\). This model matches these two fitted values, but its coefficient is not a 25% mean reduction parameter. The log link is the more direct interpretation for the stated multiplicative scientific expectation.

Common Mistakes

Assuming every link preserves positivity. The inverse-Gaussian response is positive, but identity predictions can be nonpositive, and canonical predictors must remain negative.
Calling \(1/\mu^2\) a positive rescaling of the natural parameter. Because \(1/\mu^2=-2\theta\), it is a negative rescaling; the corresponding regression coefficients change sign and scale.
Interpreting all coefficients as mean differences. Only the identity-link coefficient gives a constant additive mean change. Log-link coefficients give multiplicative changes, while reciprocal and canonical coefficients act on their respective transformed scales.
Checking validity only at observed data points. If predictions are required for a wider covariate range, the link-domain condition must hold there as well.

Key takeaway: Choose among inverse-Gaussian links by matching the scientific effect scale and verifying valid, plausible fitted means throughout the intended prediction domain. The canonical link has a special natural-parameter role, but it is not automatically the most interpretable or practical choice.

Check Your Understanding

For each question, distinguish the link-scale statement from the fitted-mean consequence.

What is the canonical inverse-Gaussian link, and why must its linear predictor be negative?
A reciprocal-link predictor increases from \(0.4\) to \(0.5\). Calculate both fitted means and describe the direction of change.
Under a log link, what mean ratio corresponds to a one-unit increase in a covariate with coefficient \(\beta_j\)?
Explain why a positive-support response distribution does not prevent negative fitted means under the identity link.
How does the coefficient vector change if the canonical predictor \(-1/(2\mu^2)\) is replaced by \(1/\mu^2\) while retaining the same fitted means?