Admissible Links For Binary Means

Why Link Admissibility Matters

For a binary response \(Y\), the conditional mean is a probability: \(\mu=P(Y=1\mid x)\), so \(0\leq\mu\leq1\). In a generalized linear model, a link \(g\) relates this mean to a linear predictor \(\eta=x^\mathsf{T}\beta\) through \(g(\mu)=\eta\). The practical question is whether solving this equation gives a legitimate probability for every predictor the model permits.

Some familiar links transform the entire open probability interval \((0,1)\) onto the whole real line. Their inverse links therefore produce a probability strictly between zero and one for every finite \(\eta\). Other links have a smaller range: they can still be used, but only if the predictors are restricted to that range. Distinguishing these cases is important when specifying a model and checking its fitted values.

The word “globally” matters. A link may define a perfectly valid model on a restricted predictor space, even when its inverse does not give a probability for every real number. Also, the usual open mean space excludes the boundary probabilities 0 and 1. A model with a globally admissible link gives probabilities strictly inside those boundaries at every finite predictor; the limiting probabilities 0 and 1 can be approached as the predictor tends to infinity in magnitude.

A Criterion Using Range and Endpoint Limits

For the common increasing links, a convenient way to check global admissibility is to examine the link’s range or its endpoint limits. The next result makes this check precise.

Proof. First suppose the two endpoint limits hold. Let \(\eta\) be any real number. By the first limit, there is a \(\mu_0\in(0,1)\) sufficiently close to 0 such that \(g(\mu_0)<\eta\). By the second limit, there is a \(\mu_1\in(\mu_0,1)\) sufficiently close to 1 such that \(g(\mu_1)>\eta\). Continuity on \([\mu_0,\mu_1]\) and the intermediate value theorem give a \(\mu\in[\mu_0,\mu_1]\) with \(g(\mu)=\eta\). Strict increase ensures that no second value of \(\mu\) has the same image. Thus \(g\) is one-to-one and onto \(\mathbb{R}\).

Conversely, suppose \(g\) is continuous, strictly increasing, and onto \(\mathbb{R}\). For any real \(A\), surjectivity gives a \(\mu_A\in(0,1)\) with \(g(\mu_A)=A\). If \(0<\mu<\mu_A\), strict increase gives \(g(\mu)<A\). Since \(A\) can be arbitrarily negative, these inequalities imply that \(g(\mu)\to-\infty\) as \(\mu\to0^+\). Similarly, for any real \(B\), choose \(\mu_B\) with \(g(\mu_B)=B\). If \(\mu_B<\mu<1\), then \(g(\mu)>B\), and allowing \(B\) to be arbitrarily large shows that \(g(\mu)\to+\infty\) as \(\mu\to1^-\). This proves the criterion. \(\square\)

For a decreasing link, the corresponding endpoint limits are reversed: the limit at 0 must be \(+\infty\), and the limit at 1 must be \(-\infty\). More generally, global admissibility is a question about being one-to-one and onto, regardless of whether the link increases or decreases. The increasing convention is convenient for comparing the links considered here.

Proof. Let \(\eta_1<\eta_2\), and write \(\mu_1=g^{-1}(\eta_1)\) and \(\mu_2=g^{-1}(\eta_2)\). If \(\mu_1\geq\mu_2\), monotonicity of \(g\) would imply \(\eta_1=g(\mu_1)\geq g(\mu_2)=\eta_2\), a contradiction. Hence \(\mu_1<\mu_2\), so the inverse is strictly increasing. To prove the first limit, fix any \(\varepsilon\in(0,1)\). For \(\eta<g(\varepsilon)\), monotonicity implies \(g^{-1}(\eta)<\varepsilon\); also \(g^{-1}(\eta)>0\). This is the definition of convergence to 0 as \(\eta\to-\infty\). For the other limit, fix \(\varepsilon\in(0,1)\). For \(\eta>g(1-\varepsilon)\), we have \(g^{-1}(\eta)>1-\varepsilon\) and \(g^{-1}(\eta)<1\). Therefore \(g^{-1}(\eta)\to1\) as \(\eta\to+\infty\). \(\square\)

Five Links: Domains, Ranges, and Limits

The following table uses \((0,1)\) as the mean domain. A range of \(\mathbb{R}\) means the link can accommodate every real-valued linear predictor. A smaller range means the predictor must be restricted if the inverse is to stay in \((0,1)\).

For the probit link, \(\Phi\) denotes the standard normal cumulative distribution function. It is continuous and strictly increasing from 0 to 1, so its inverse on \((0,1)\) is strictly increasing and has limits \(-\infty\) and \(+\infty\) at the respective endpoints.

For the complementary log-log link, the overall composition is increasing even though some individual operations are decreasing. As \(\mu\) increases, \(1-\mu\) decreases, so \(-\log(1-\mu)\) increases; applying the increasing logarithm preserves that increase. As \(\mu\to0^+\), \(-\log(1-\mu)\to0^+\), and its logarithm tends to \(-\infty\). As \(\mu\to1^-\), \(-\log(1-\mu)\to+\infty\), and its logarithm tends to \(+\infty\).

For the log link, \(\log(\mu)\) increases from \(-\infty\) to 0 as \(\mu\) traverses \((0,1)\). Its inverse is \(\exp(\eta)\), which lies in \((0,1)\) only when \(\eta<0\). The identity link has inverse \(g^{-1}(\eta)=\eta\), so its predictor must satisfy \(0<\eta<1\).

For the cauchit link, the inverse is \(g^{-1}(\eta)=\tfrac12+\pi^{-1}\arctan(\eta)\). The arctangent increases from \(-\pi/2\) to \(\pi/2\), so this inverse increases from 0 to 1. Correspondingly, the link increases from \(-\infty\) to \(+\infty\) over the open mean interval.

Worked Checks

What the Classification Does—and Does Not—Say

Probit, complementary log-log, and cauchit links pass the global admissibility criterion: every finite real predictor maps to a mean in \((0,1)\). Log and identity links do not. This does not make the latter links unusable. It means that the model must ensure that its predictors stay in the relevant range, such as by restricting the parameter space or covariate profiles. Without such a restriction, some fitted means may fall outside the probability interval.

The distinction is particularly important when a linear predictor varies with covariates. Checking one fitted profile is not enough if other covariate values can produce a predictor outside the allowed range. A model specification, its parameter constraints, and the intended covariate domain should be considered together.

Common Mistake

• Wrong reasoning: “The log link is valid for binary data because \(\exp(\eta)\) is always positive.” Positivity alone does not make a number a probability; it must also be no greater than 1.
• Correction: Under the log link, \(\exp(\eta)<1\) exactly when \(\eta<0\). Thus the predictor must be restricted to negative values to obtain means in \((0,1)\). Likewise, the identity link requires \(0<\eta<1\), not merely a defined formula.