Why Predictor Constraints Matter
An inverse-Gaussian response has a positive mean, so a regression model must produce a fitted mean \(\mu_i>0\) wherever it is used. A link function maps that mean to a linear predictor, \(\eta_i=\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}\). The link therefore determines which values of \(\eta_i\) can correspond to valid means. If a fitted predictor falls outside that range, the implied mean is not a positive real number.
The restriction applies to the predictor at each covariate vector of interest. It does not generally require every coefficient to have a particular sign. The three links considered here are the canonical link, the identity link, and the inverse link \(g(\mu)=1/\mu\).
From Positive Means to Predictor Ranges
For the inverse-Gaussian family, the variance function is \(V(\mu)=\mu^3\), and the canonical link is \(g(\mu)=-1/(2\mu^2)\). The identity and inverse links are \(g(\mu)=\mu\) and \(g(\mu)=1/\mu\), respectively. In each case, the admissible predictor range follows by evaluating the link for every \(\mu>0\).
The ranges and inverse-link formulas are:
| Link | Link formula | Admissible predictor | Mean from the predictor |
|---|---|---|---|
| Canonical | \(\eta=-1/(2\mu^2)\) | \(\eta<0\) | \(\mu=\sqrt{-1/(2\eta)}\) |
| Identity | \(\eta=\mu\) | \(\eta>0\) | \(\mu=\eta\) |
| Inverse | \(\eta=1/\mu\) | \(\eta>0\) | \(\mu=1/\eta\) |
For the canonical link, \(\mu^2>0\), so \(-1/(2\mu^2)<0\); conversely, if \(\eta<0\), then \(-1/(2\eta)>0\), and its positive square root is a valid mean. Under the identity link, the predictor equals the mean. Under the inverse link, the reciprocal of a positive mean is positive, and a positive predictor has a positive reciprocal. In all three cases the boundary \(\eta=0\) is excluded: it does not correspond to a finite, positive mean under that link.
Canonical link: \(\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}<0\).
Identity link: \(\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}>0\).
Inverse link: \(\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}>0\).
These are strict conditions on each fitted predictor, not separate sign conditions on the components of \(\boldsymbol{\beta}\).
To prove the theorem, write \(\eta_i=\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}\). For the canonical link, the inverse formula is real and positive exactly when \(-1/(2\eta_i)>0\), which holds exactly when \(\eta_i<0\). For the identity link, \(\mu_i=\eta_i\), so \(\mu_i>0\) exactly when \(\eta_i>0\). For the inverse link, \(\mu_i=1/\eta_i\), which is finite and positive exactly when \(\eta_i>0\). Thus the stated conditions are both necessary and sufficient. \(\square\)
Working with Coefficients and Design Points
At a specified design point, the condition is a linear inequality in the regression coefficients. For example, with an intercept and one covariate \(x_i\), the predictor is \(\eta_i=\beta_0+\beta_1x_i\). The identity and inverse links require \(\beta_0+\beta_1x_i>0\); the canonical link requires \(\beta_0+\beta_1x_i<0\). A coefficient may be positive or negative in either case. What matters is the sum at the covariate value being considered.
For a finite collection of design points, all fitted means are valid precisely when all the corresponding strict inequalities hold. A useful special case is a model with an intercept when the slopes are held fixed.
For the identity and inverse links, requiring \(\beta_0+s_i>0\) for every \(i\) is equivalent to requiring \(\beta_0>-s_i\) for every \(i\). The strongest of those lower bounds is \(-\min_i s_i\). For the canonical link, the requirements are \(\beta_0<-s_i\) for every \(i\), whose strongest upper bound is \(-\max_i s_i\). Because the set of design points is finite, these bounds are finite whenever the \(s_i\) are finite. Thus, with a free intercept, one can choose an intercept that satisfies the requirements at a finite set of design points. This does not guarantee validity at every possible covariate value, especially outside the observed design range. \(\square\)
Worked Example: Canonical Link and an Invalid Extrapolation
Consider \(\eta_i=\beta_0+\beta_1x_i\) with \(\beta_0=-0.5\), \(\beta_1=0.2\), and the canonical link. Calculate the predictors and means at \(x=0,1,2\), then check the extrapolation to \(x=3\).
At \(x=0\), \(\eta=-0.5+0.2(0)=-0.5\), so \(\mu=\sqrt{-1/(2(-0.5))}=\sqrt{1}=1\).
At \(x=1\), \(\eta=-0.5+0.2(1)=-0.3\), so \(\mu=\sqrt{-1/(2(-0.3))}=\sqrt{5/3}\approx1.291\).
At \(x=2\), \(\eta=-0.5+0.2(2)=-0.1\), so \(\mu=\sqrt{-1/(2(-0.1))}=\sqrt{5}\approx2.236\). All three predictors are negative, so these are valid positive means.
At \(x=3\), \(\eta=-0.5+0.2(3)=0.1\). The expression for the squared mean would be \(-1/(2(0.1))=-5\), which has no real square root. The fitted mean is invalid there. The positive slope is not itself the problem: validity depends on the complete predictor and the covariate range.
Worked Example: Identity Link and a Negative Fitted Mean
Let \(\eta_i=\beta_0+\beta_1x_i\), with \(\beta_0=1\), \(\beta_1=-0.4\), and the identity link. Evaluate the fitted means at \(x=0,1,2,3\).
Because the identity link gives \(\mu_i=\eta_i\), calculate each predictor directly:
At \(x=0\), \(\eta=1-0.4(0)=1\), so \(\mu=1>0\). At \(x=1\), \(\eta=1-0.4=0.6\), so \(\mu=0.6>0\). At \(x=2\), \(\eta=1-0.8=0.2\), so \(\mu=0.2>0\). These three fitted means are valid.
At \(x=3\), \(\eta=1-1.2=-0.2\), so \(\mu=-0.2\). This violates the positive-mean requirement. The predictor reaches zero at \(x=2.5\), where the mean is also zero and is therefore not admissible. The model is valid at the first three listed design points but not at the fourth.
Worked Example: Inverse Link and the Zero Boundary
Suppose \(\eta_i=\beta_0+\beta_1x_i\), with \(\beta_0=0.5\), \(\beta_1=0.3\), and the inverse link \(g(\mu)=1/\mu\). Evaluate the fitted means at \(x=0,1,-2\).
At \(x=0\), \(\eta=0.5+0.3(0)=0.5\), so \(\mu=1/0.5=2\). At \(x=1\), \(\eta=0.5+0.3=0.8\), so \(\mu=1/0.8=1.25\). Both predictors are positive and both means are valid.
At \(x=-2\), \(\eta=0.5+0.3(-2)=-0.1\), so the inverse formula gives \(\mu=1/(-0.1)=-10\). Although this reciprocal is finite, it is not a positive mean and is therefore invalid. At \(x=-5/3\), the predictor is exactly \(0.5+0.3(-5/3)=0\); the reciprocal is undefined there. Thus neither a zero nor a negative predictor is admissible for the inverse link.
Common Mistakes and Practical Checks
A common mistake is to reason that a positive coefficient guarantees a positive mean, or that a negative coefficient makes a model invalid. Neither conclusion follows from the coefficient sign alone. For instance, under the identity link, \(\beta_1=-0.4\) gave positive means at \(x=0,1,2\), even though it produced an invalid mean at \(x=3\). The correction is to calculate or bound \(\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}\) at every design point where the model will be used, then compare it with the link's admissible range.
This check is important both when fitting a model and when using it for prediction. Validity at the observed covariate values does not automatically extend to new values. For example, a linear predictor that is negative over the observed range can become positive when extrapolated, violating the canonical-link constraint. Similarly, an identity- or inverse-link predictor can cross zero. A numerical routine producing a value does not make an out-of-range predictor admissible.
Check Your Understanding
Use the link formulas and strict predictor conditions to answer the following.
- For the canonical link, what fitted mean corresponds to \(\eta=-0.25\), and why is \(\eta=0\) not allowed?
- With the identity link and \(\eta=0.7\), what is the fitted mean? What happens if \(\eta=-0.7\)?
- For the inverse link, determine the fitted mean at \(\eta=2\), and state whether \(\eta=-2\) is admissible.
- Explain why a positive slope coefficient does not by itself guarantee valid means under the canonical link.
- For \(\eta_i=\beta_0+s_i\) at finitely many design points, give the intercept condition that ensures positive means under the identity link.