Why Link Domains Matter
A Gamma response has positive support, so its mean must be strictly positive. In a generalized linear model, the mean is obtained by applying the inverse link to a linear predictor. Whether that produces an admissible mean depends on the link and on the predictor value. A formula for a fitted mean is not automatically valid merely because it can be evaluated algebraically.
This issue is especially visible with the identity and reciprocal links: both require restrictions on the linear predictor. The log link, by contrast, maps every finite predictor to a positive mean. These are mathematical domain conditions, not preferences about which link is easier to interpret or estimate.
For observation \(i\), write the linear predictor as \(\eta_i=\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}\), where \(\mathbf{x}_i\) is a finite real-valued covariate vector and \(\boldsymbol{\beta}\) is a finite coefficient vector. Thus \(\eta_i\) is real. The fact that a linear predictor is real does not guarantee that every link accepts it.
Domains of the Three Links
For clarity, “reciprocal link” here means the positive reciprocal link \(g(\mu)=1/\mu\). The mean domain for all three links is \((0,\infty)\), but their predictor domains differ.
| Link | Link function | Predictor domain | Inverse link and resulting mean |
|---|---|---|---|
| Log | \(g(\mu)=\log\mu\) | All real \(\eta\) | \(\mu=e^\eta\), always positive and finite for finite \(\eta\) |
| Identity | \(g(\mu)=\mu\) | \(\eta>0\) | \(\mu=\eta\) |
| Positive reciprocal | \(g(\mu)=1/\mu\) | \(\eta>0\) | \(\mu=1/\eta\) |
The identity and positive reciprocal links have the same predictor domain, but they give different means for the same positive predictor. For the identity link, the mean equals the predictor; for the reciprocal link, the mean is its reciprocal. At \(\eta=0\), the identity link would give a zero mean and the reciprocal inverse is undefined. For \(\eta<0\), both would give an inadmissible mean. The restrictions are therefore strict.
The log link has no such restriction on a finite real predictor: \(e^\eta>0\) for every real \(\eta\). It does not assign a finite mean to an infinite predictor, so the usual model assumes finite coefficients and finite covariates when making this claim.
Worked Example: Log-Link Predictions Across a Covariate Range
Suppose the fitted predictor is \(\eta=0.2+0.3x\). Calculate the means at \(x=0\), \(x=2\), and \(x=-5\), and check their admissibility under the log link.
At \(x=0\), \(\eta=0.2+0.3(0)=0.2\), so \(\mu=e^{0.2}\approx1.221\). At \(x=2\), \(\eta=0.2+0.3(2)=0.8\), so \(\mu=e^{0.8}\approx2.226\). At \(x=-5\), \(\eta=0.2+0.3(-5)=-1.3\), so \(\mu=e^{-1.3}\approx0.273\). Each predictor is finite and each mean is positive and finite, so all three predictions are mathematically admissible.
A negative predictor is not a problem for the log link. It gives a mean between zero and one, not a negative mean.
Worked Example: Identity-Link Validity on and Beyond the Observed Range
Suppose an identity-link model has \(\eta=1.5+0.4x\), and the observed covariate values range from \(0\) to \(5\). Check the fitted means at the endpoints, then assess a proposed prediction at \(x=-5\).
At the lower endpoint \(x=0\), \(\eta=1.5+0.4(0)=1.5\), giving \(\mu=1.5>0\). At the upper endpoint \(x=5\), \(\eta=1.5+0.4(5)=3.5\), giving \(\mu=3.5>0\). Because the predictor increases with \(x\), its minimum on the observed interval \([0,5]\) is \(1.5\). All fitted means on that interval are therefore positive.
At \(x=-5\), however, \(\eta=1.5+0.4(-5)=-0.5\), so the identity-link mean would be \(-0.5\), which is inadmissible for a Gamma response. Validity over the observed range does not guarantee validity for extrapolation.
Worked Example: Positive Reciprocal-Link Predictions
Let a positive reciprocal-link model have predictor \(\eta=0.5+0.1x\). Evaluate the fitted means at \(x=0\), \(x=4\), and \(x=-8\).
At \(x=0\), \(\eta=0.5\), so \(\mu=1/0.5=2\). At \(x=4\), \(\eta=0.5+0.1(4)=0.9\), so \(\mu=1/0.9=10/9\approx1.111\). Both predictors are positive, and both means are finite and positive.
At \(x=-8\), \(\eta=0.5+0.1(-8)=-0.3\), so the inverse-link formula gives \(\mu=1/(-0.3)=-10/3\), an inadmissible value. The reciprocal expression can be calculated for a negative predictor, but that calculation does not make the resulting mean a valid Gamma mean.
Validity as a Constraint on Coefficients
For a fixed data set with \(n\) observations, a link must produce an admissible mean at every design point. Under either the identity or the positive reciprocal link, this means \(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}>0\) for every \(i=1,\ldots,n\). These are strict linear inequalities in the coefficients. The inequalities depend on the design vectors; they do not imply that each coefficient must itself be positive.
- for the log link, no additional sign restriction on the finite predictors is required;
- for the identity link, \(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}>0\) for every \(i\);
- for the positive reciprocal link, \(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}>0\) for every \(i\).
Proof. For the log link, each finite predictor gives \(\mu_i=e^{\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}}\). The exponential of a finite real number is finite and strictly positive, so every fitted mean is admissible. For the identity link, \(\mu_i=\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}\), which is finite and admissible exactly when that predictor is strictly positive. For the positive reciprocal link, a predictor of zero does not have an inverse-link value, a negative predictor gives a negative mean, and a positive finite predictor gives the finite positive mean \(1/(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta})\). This proves each direction of the stated conditions. \(\square\)
The constraints can be checked at the design points where predictions are required. If the goal is validity over an entire covariate region rather than just the observed rows, the inequalities must hold throughout that region. For example, checking only the smallest and largest \(x\) suffices for a one-dimensional linear predictor on an interval, since a linear function attains its minimum at an endpoint. In higher dimensions or more complicated regions, the relevant minimum must be established for that region; checking the data rows alone does not establish it.
Worked Example: Feasible Coefficients for Two Links
Consider observations at \(x=-1\) and \(x=1\), with predictor \(\eta=\beta_0+\beta_1x\). Find the conditions on the coefficients for valid identity-link and positive reciprocal-link means at both observations.
At \(x=-1\), the predictor is \(\beta_0-\beta_1\); at \(x=1\), it is \(\beta_0+\beta_1\). Both links require \(\beta_0-\beta_1>0\) and \(\beta_0+\beta_1>0\). Together these inequalities are equivalent to \(\beta_0>|\beta_1|\): the first says \(\beta_0>\beta_1\), the second says \(\beta_0>-\beta_1\), so \(\beta_0\) must exceed the larger of \(\beta_1\) and \(-\beta_1\), which is \(|\beta_1|\).
For example, \((\beta_0,\beta_1)=(2,1)\) satisfies the condition. The predictors are \(1\) and \(3\), yielding identity means \(1,3\) and reciprocal means \(1,1/3\); all are positive. In contrast, \((\beta_0,\beta_1)=(0,1)\) gives predictors \(-1,1\), so it fails the requirement for both links. The coefficient \(\beta_1\) need not be positive in general: \((2,-1)\) also satisfies \(\beta_0>|\beta_1|\).
Feasible Regions and Prediction Scope
For a fixed finite design, the admissible coefficient set for the identity and positive reciprocal links is the intersection of the strict half-spaces \(\{\boldsymbol{\beta}:\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}>0\}\). This set may be empty, and when it is nonempty, it is convex.
To see convexity, take two admissible vectors \(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2\) and any \(t\) with \(0\leq t\leq1\). At design point \(i\), \[ \mathbf{x}_i^\mathsf{T}\{t\boldsymbol{\beta}_1+(1-t)\boldsymbol{\beta}_2\} =t\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}_1+(1-t)\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}_2>0. \] The last inequality holds because both endpoint predictors are strictly positive (and when \(t=0\) or \(t=1\), the predictor is one of those positive endpoints). Thus every point on the line segment remains admissible, proving the claim. \(\square\)
Nonemptiness should not be assumed. If the model has no intercept and the observed scalar covariates are \(x=-1\) and \(x=1\), the predictor is \(\eta=\beta_1x\). Validity would require \(-\beta_1>0\) and \(\beta_1>0\) simultaneously, which is impossible. No coefficient value can yield valid means at both design points under either of these two links.
For the log link, every finite coefficient vector is admissible at a finite design, because no predictor sign condition is needed. That fact is about the domain of the inverse link; it does not guarantee a good fit, reliable extrapolation, or numerical stability in software. Very large predictors can cause computational overflow even though their exponential is mathematically finite.
Common Mistake: Checking the Formula but Not the Mean
A common error is to conclude that an identity- or reciprocal-link prediction is valid because the formula can be evaluated. For example, computing \(1/(-0.3)\) is algebraically possible, but its value is negative and cannot be a Gamma mean. The correction is to check the predictor domain first, then verify that the inverse link yields a finite, strictly positive mean.
Another mistake is to demand positive coefficients instead of positive predictors. The required condition is \(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}>0\) at each prediction point. Coefficients combine with the design values, so individual coefficient signs alone do not determine whether the fitted means are valid.
Check Your Understanding
Use the link domains and strict positivity conditions to answer these questions.
- For a log-link model, is a finite predictor of \(-2\) admissible? What is the corresponding mean?
- An identity-link model has predictor \(\eta=3-0.5x\). For which values of \(x\) is its fitted mean admissible?
- For the positive reciprocal link, what goes wrong at \(\eta=0\), and what goes wrong at \(\eta=-2\)?
- A model is valid at every observed covariate value. Does that establish validity at a new value outside the observed range? Explain.
- For an identity-link model with predictor \(\eta=\beta_0+\beta_1x\) and observed values \(x=0,2\), state the inequalities required for both fitted means to be admissible.