Why Link Ranges Must Fit the Response Family
A generalized linear model assigns a conditional mean through a linear predictor and an inverse link. The inverse link must do more than produce a number: that number must be a mean allowed by the response distribution. This is a compatibility check between the link and the response family, not a requirement that the mean itself be a possible observed response.
For a covariate value \(x\), write the linear predictor as \(\eta(x)\), and let \(h=g^{-1}\) denote the inverse link. The model specifies \(\mu(x)=E[X\mid x]=h(\eta(x))\). A finite linear predictor is a real number. Thus, when checking compatibility for every possible finite predictor, the relevant question is whether every value of \(h(\eta)\), for \(\eta\in\mathbb{R}\), is an admissible conditional mean for the chosen response family.
The family matters when defining \(\mathcal{M}\). For example, a Bernoulli response has mean \(p\), where \(0\leq p\leq1\). If the family is restricted to nondegenerate Bernoulli distributions, then its mean parameter space is instead \(0<p<1\). A Poisson distribution with rate zero is a degenerate distribution at zero, so the set of means for all Poisson distributions is \([0,\infty)\); the usual regular Poisson family has positive rates and hence mean space \((0,\infty)\). In either convention, a negative proposed mean is not admissible.
Means Are Not Observed Values
A response family also has a support: the values the response can take. Support and mean parameter space describe different things. A Bernoulli response can only be observed as 0 or 1, while its mean can be any value between 0 and 1. A Poisson response takes nonnegative integer values, while its mean can be any nonnegative real number. The mean need not be one of the values the response could actually take.
This distinction follows directly from expectation. For a Bernoulli random variable \(X\) with \(P(X=1)=p\), its mean is
For instance, a Bernoulli variable with mean \(0.7\) still produces only 0 or 1 on any single observation. The value 0.7 describes its expected value over repetitions, not a possible outcome. Conversely, a link that returns a number outside \([0,1]\) cannot specify the mean of any Bernoulli distribution.
Proof. Since \(a\leq X\leq b\) almost surely and \(X\) is integrable, monotonicity of expectation gives \(E[a]\leq E[X]\leq E[b]\). Because \(a\) and \(b\) are constants, this is \(a\leq E[X]\leq b\). For the strict claim, \(X-a>0\) almost surely and \(b-X>0\) almost surely. A nonnegative random variable that is positive almost surely has strictly positive expectation: if its expectation were zero, then it would equal zero almost surely. Therefore \(E[X-a]>0\) and \(E[b-X]>0\), which imply \(E[X]>a\) and \(E[X]<b\). \(\square\)
The proposition gives a necessary constraint on a response mean from the response’s bounded support. The response family can impose further constraints or define which boundary distributions are included, so checking the family’s mean parameter space remains essential.
A Range Check for Inverse Links
The basic compatibility test can be stated directly in terms of the range of the inverse link. It concerns the entire set of predictor values allowed by the question being asked, not just a few fitted values computed from one data set.
Proof. Suppose first that every \(\eta\in A\) produces an admissible mean. By definition, this means \(h(\eta)\in\mathcal{M}\) for each \(\eta\in A\). Every element of \(h(A)\) has the form \(h(\eta)\) for some \(\eta\in A\), so every element of \(h(A)\) lies in \(\mathcal{M}\). Hence \(h(A)\subseteq\mathcal{M}\). Conversely, if \(h(A)\subseteq\mathcal{M}\), then for each \(\eta\in A\), \(h(\eta)\in h(A)\) and therefore \(h(\eta)\in\mathcal{M}\). Thus every predictor in \(A\) produces an admissible mean. \(\square\)
For the check “every finite predictor,” take \(A=\mathbb{R}\). This theorem gives a necessary and sufficient condition for mean admissibility, but it does not say that the link can represent every admissible mean. A range contained in \(\mathcal{M}\) is valid; it may still be a strict subset of \(\mathcal{M}\).
Specify which conditional distributions are allowed, including any relevant boundary conventions.
Determine which finite conditional means are attainable in that family.
Consider all real predictor values if the requirement is validity for every finite predictor.
The link is admissible if its range is contained in the mean parameter space. If the containment is strict, the model is valid but restricts the means it can express.
Worked Link Checks
Worked Example: Bernoulli Response and Three Inverse Links
Suppose \(X\mid x\) is Bernoulli, so the conditional mean is a success probability \(p(x)\) in \([0,1]\). Consider identity, exponential, and logistic inverse links, each applied to a real predictor \(\eta\).
Identity inverse link. Here \(h(\eta)=\eta\), so \(h(\mathbb{R})=\mathbb{R}\). At the finite predictor \(\eta=-0.4\), the proposed mean is \(-0.4\), outside \([0,1]\). At \(\eta=1.2\), the proposed mean is \(1.2\), also outside \([0,1]\). Thus the identity inverse link is not admissible for every real predictor in a Bernoulli model.
Exponential inverse link. Here \(h(\eta)=e^\eta\), so its range is \((0,\infty)\). At \(\eta=1\), it gives \(e^1\approx2.718\), which is greater than 1 and cannot be a Bernoulli mean. The link fails the range check.
Logistic inverse link. Here \(h(\eta)=1/(1+e^{-\eta})\). For any finite \(\eta\), \(e^{-\eta}>0\), so \(1+e^{-\eta}>1\) and \(0<h(\eta)<1\). Its range is contained in the Bernoulli mean space, including when that space is restricted to nondegenerate distributions. For example, if \(\eta=\log(3)\), then \(e^{-\eta}=1/3\) and \(h(\eta)=1/(1+1/3)=3/4\), an admissible success probability.
In every case, a Bernoulli observation remains either 0 or 1. The link specifies the mean, not the set of individual outcomes.
Worked Example: Poisson Response and a Valid but Restrictive Link
Suppose \(X\mid x\) follows a Poisson distribution. Its support is \(\{0,1,2,\ldots\}\), and its mean parameter \(\mu(x)\) is a nonnegative real number (positive in the regular family). Compare the exponential and logistic inverse links.
For the exponential inverse link, \(h(\eta)=e^\eta\). Since \(e^\eta>0\) for every finite \(\eta\), all its predictions are valid positive Poisson means. At \(\eta=-1\), the prediction is \(e^{-1}\approx0.368\), which is not a possible observed count but is a valid Poisson mean.
For the logistic inverse link, \(h(\eta)=1/(1+e^{-\eta})\). Its range \((0,1)\) is also contained in the positive Poisson mean space, so it is mean-admissible. At \(\eta=2\), it gives \(1/(1+e^{-2})\approx0.881\), a valid mean. However, a Poisson mean of 2 is also admissible, and this inverse link cannot produce it for any real \(\eta\), because every output is below 1. Thus the logistic inverse link is valid but restricts the model to a strict subset of the possible Poisson means.
The noninteger value \(0.881\) does not mean that the response can be observed as \(0.881\). The Poisson response is integer-valued; its conditional expectation need not be.
Worked Example: Gaussian Response and the Identity Inverse Link
Suppose \(X\mid x\) is normal with an unrestricted real mean and a fixed positive variance. Its response support is the real line, and its mean parameter space is \(\mathbb{R}\). With the identity inverse link \(h(\eta)=\eta\), the range is \(\mathbb{R}\), exactly the mean parameter space. Every finite predictor therefore gives an admissible Gaussian mean.
For example, \(\eta=-1.2\) gives \(\mu=-1.2\). A distribution \(X\mid x\sim N(-1.2,4)\) is well defined; the second argument is the variance. Its response can take positive or negative real values, and its mean \(-1.2\) is allowed. This contrasts with the Bernoulli case, where a negative proposed mean would be invalid.
Using the exponential inverse link for this same Gaussian family would also produce valid means, because its positive outputs are real numbers. It would nevertheless exclude all negative Gaussian means. As in the Poisson example, admissibility does not imply that the link can express every mean in the family.
Common Pitfall: Comparing the Mean to the Support
A frequent error is to require the inverse link’s output to be one of the response’s possible observed values. That confuses the response support with the mean parameter space. A Poisson response cannot be observed as \(0.4\), but a Poisson distribution with mean \(0.4\) is valid. A Bernoulli response cannot be observed as \(0.7\), but a Bernoulli distribution with success probability \(0.7\) is valid.
The opposite error is to check only whether a prediction is a plausible response value and forget the mean constraint. A negative number is a possible Gaussian response and can be a Gaussian mean, but it cannot be a Bernoulli mean or a Poisson mean. Always check against the response family’s mean parameter space, not just its support.
A second related mistake is to treat “contained in the mean parameter space” as “covers the whole mean parameter space.” Containment ensures that each prediction is valid. Equality of the range and the mean parameter space would additionally ensure that every admissible mean can be represented by some predictor. A strict subset indicates a modeling restriction, not necessarily an invalid model.
Check Your Understanding
For each question, distinguish the response’s possible observed values from its admissible conditional means.
- A Bernoulli response has mean \(0.35\). Is \(0.35\) a possible observed value, and is it an admissible mean?
- For a Poisson response, explain why the identity inverse link is not admissible for every finite predictor.
- The logistic inverse link has range \((0,1)\). Is it admissible for a Poisson mean, and does it represent every positive Poisson mean?
- A Gaussian response has an unrestricted real mean. Is the exponential inverse link admissible for every finite predictor? Which Gaussian means does it omit?
- State the range-containment condition that checks a proposed inverse link against a response family.