Admissibility of Identity and Inverse Links

Why Link Admissibility Matters

For a positive-response model, the conditional mean must itself be positive. A Gamma or inverse-Gaussian distribution, for example, cannot have a nonpositive mean parameter. The link function connects that mean to the linear predictor, so the link and the range of the predictor determine whether the model can produce allowable fitted means.

This is especially important for the identity and reciprocal links. Neither link gives positive means for every possible real-valued linear predictor. A fit can yield plausible estimates at the observed covariate values yet produce an invalid prediction elsewhere in the range where the model is intended to apply. The right check is therefore not merely whether the fitted values look reasonable in the data table, but whether the mean remains admissible over the relevant covariate range.

The family and the link impose different requirements. The family determines which distributions and parameter values are available for the response; the link determines how the mean is related to covariates. A valid positive mean does not establish that the chosen family describes the data well. Conversely, a sensible family cannot make a negative fitted mean allowable.

Identity Link: Check the Predictor Itself

With the identity link, \(g(\mu)=\mu\), so the inverse link is also the identity function and \(\mu_i=\eta_i=x_i^\mathsf{T}\beta\). For a positive-response family, every relevant value of this linear predictor must be strictly greater than zero. A zero prediction is not enough: the mean parameter must be positive, not merely nonnegative.

For one covariate \(x\) in an interval \([a,b]\), an identity-link predictor has the form \(\eta(x)=\beta_0+\beta_1x\). Its minimum occurs at an endpoint. Thus, checking both endpoints suffices to establish positivity throughout the interval. With multiple predictors, checking only the observed rows may fail to check the full range or region of intended use.

Proof. If \(\eta(x)>0\) for every \(x\in P\), then it is positive at each vertex because each vertex belongs to \(P\). For the converse, take any \(x\in P\). By the definition of a convex hull, there are weights \(\lambda_1,\ldots,\lambda_m\) with \(\lambda_j\geq 0\), \(\sum_{j=1}^m\lambda_j=1\), and \(x=\sum_{j=1}^m\lambda_jv_j\). Affineness gives \(\eta(x)=\sum_{j=1}^m\lambda_j\eta(v_j)\). If each vertex value is strictly positive, this weighted average is strictly positive. Therefore \(\eta(x)>0\) throughout \(P\). \(\square\)

Reciprocal Link: Keep the Predictor Away from Zero

Here, “inverse link” means the reciprocal link, \(g(\mu)=1/\mu\), not simply the inverse function \(g^{-1}\) of an arbitrary link. Under this convention, the fitted mean is \(\mu=1/\eta\). For a positive mean, the predictor must be strictly positive. If \(\eta=0\), the expression is undefined; if \(\eta<0\), the resulting mean is negative.

Some texts or software use a signed reciprocal convention, such as \(g(\mu)=-1/\mu\). Under that convention, the predictor must instead be strictly negative to yield a positive mean. The sign condition must be matched to the stated link; the word “inverse” alone is not precise enough to identify it.

Proof. If \(\eta(x)>0\), then its reciprocal exists, is finite, and is positive, so \(\mu(x)=1/\eta(x)>0\). If \(\eta(x)=0\), the reciprocal is undefined. If \(\eta(x)<0\), its reciprocal is negative. These cases prove the first claim. Applying the pointwise equivalence at every \(x\in P\) proves the second. In addition, if \(\eta\) is continuous and strictly positive on compact \(P\), it attains a minimum \(\delta\) there. Since it is positive everywhere, \(\delta>0\), and hence \(0<\mu(x)\leq1/\delta\) throughout \(P\). \(\square\)

A reciprocal predictor can fail even when its values appear moderate at some observed covariate values. For example, if \(\eta(x)=0.6-0.1x\) on \(0\leq x\leq8\), then \(\eta(0)=0.6\), but it reaches zero at \(x=6\) and is negative for \(x>6\). The mean is \(1/0.6=5/3\) at \(x=0\), undefined at \(x=6\), and negative at \(x=8\), where it equals \(-5\). The zero is a singularity, not a very large but valid prediction.

What an Invalid Prediction Does—and Does Not—Tell You

A nonpositive mean under either link indicates that the proposed mean structure cannot be used as specified across that covariate range for a positive-response family. The problem may be addressed by changing the link, restricting the prediction domain, or changing the mean structure, but the appropriate choice depends on the scientific purpose and data. A log link is one common way to enforce positivity because its inverse, \(\mu=\exp(\eta)\), is positive for every finite real \(\eta\); positivity alone, however, does not make a model otherwise appropriate.

It is important not to confuse an inadmissible prediction with a family mismatch. For example, negative means from an identity-link Gamma model do not show that Gamma variability or positive skewness is unsuitable. They show that the fitted predictor conflicts with the family’s positive mean domain. The reverse distinction matters too: if all means are positive, that does not establish that the family captures the response distribution or its variance pattern.

Common Mistakes

• Checking only fitted values at observed rows. A model intended for a wider covariate range may predict zero or negative means between or beyond those rows. Define the range of use and check it explicitly.
• Accepting a zero mean for a positive family. The condition is strict: \(\mu>0\). A zero identity-link prediction is not admissible for the positive mean domain considered here.
• Assuming a reciprocal link is positive for any predictor. With \(g(\mu)=1/\mu\), \(\eta=0\) is undefined and \(\eta<0\) gives a negative mean.
• Calling an inadmissible link prediction a family failure. It is first a conflict between the link-generated mean and the family’s parameter domain. A separate assessment is needed to judge the family’s fit.