Noncanonical Links And Modeling Flexibility

Why Use a Noncanonical Link?

A generalized linear model connects a response mean to a linear predictor through a link. For a binary response, or a binomial count, the mean probability must lie between zero and one. The logit is the canonical link for the binomial family, but it is not the only link that can map this interval to the real line. Probit and complementary log-log links are two widely used alternatives.

Using a noncanonical link does not change the response distribution: a binary observation can still be modeled as Bernoulli, and a count of successes in a fixed number of trials can still be modeled as binomial. The choice changes the rule connecting the linear predictor to the mean. That flexibility can be useful when a particular link gives a more suitable shape or has a scientific interpretation for the application.

A link is suitable for this purpose when it gives a one-to-one mapping between probabilities strictly inside the unit interval and real-valued predictors. The three links considered here have that property. They differ in how probability values are spaced along the predictor scale.

Three Links for a Bounded Mean

Let \(\Phi\) denote the cumulative distribution function of a standard normal random variable. The following definitions use the same probability \(p\), but produce different predictor values.

The logit transforms the odds \(p/(1-p)\) by taking their logarithm. The probit uses the standard normal quantile. The complementary log-log link first transforms the probability of failure, \(1-p\), and then takes two logarithms, with the inner negative sign ensuring a positive quantity inside the outer logarithm.

Proof. For the logit inverse, \(e^\eta>0\), so \(e^\eta/(1+e^\eta)\) lies strictly between zero and one. As \(\eta\) increases, \(e^\eta\) strictly increases, and the function \(u/(1+u)\) strictly increases for \(u>0\). Its limits as \(u\) approaches zero and infinity are zero and one, respectively. Thus the logit inverse is strictly increasing with range \((0,1)\).

For the probit inverse, the standard normal cumulative distribution function \(\Phi\) is continuous and strictly increasing, with limits zero and one at negative and positive infinity. Its values at finite arguments are strictly between zero and one. Therefore \(\Phi\) maps the real line bijectively onto \((0,1)\).

For the complementary log-log inverse, \(\exp(\eta)>0\), so \(\exp[-\exp(\eta)]\) lies strictly between zero and one. Hence \(1-\exp[-\exp(\eta)]\) also lies strictly between zero and one. If \(\eta\) increases, \(\exp(\eta)\) increases, \(\exp[-\exp(\eta)]\) decreases, and the inverse-link value increases. Its limits are zero as \(\eta\) tends to negative infinity and one as \(\eta\) tends to positive infinity. This proves the claimed range and strict increase for all three inverse links. \(\square\)

Why the Logit Is Canonical for the Binomial

Canonicality is a property of the response family’s exponential-family representation, not a requirement that every GLM must satisfy. For a binomial count \(Y\) with \(m\) trials and success probability \(p\), the probability mass function can be written as follows.

For \(0<p<1\), set \(\theta=\log[p/(1-p)]\). Since \(p=e^\theta/(1+e^\theta)\) and \(1-p=1/(1+e^\theta)\), the same mass function becomes

Derivation. The exponential-family form has the term \(y\theta\), so comparing it with the binomial mass function identifies the coefficient of \(y\) as \(\theta=\log[p/(1-p)]\). The relationship between this natural parameter and the mean probability \(p\) is thus \(\theta=g(p)\) with \(g(p)=\log[p/(1-p)]\), exactly the logit link. The probit and complementary log-log links use different functions of \(p\), so they are noncanonical for the binomial family. This distinction does not make them invalid: it identifies which link matches the family’s natural parameter, not which links are permitted in a mean model. \(\square\)

Flexibility and Asymmetry

The links differ not only in the predictor value assigned to a particular probability, but also in their symmetry. The logit and probit assign opposite predictor values to probabilities \(p\) and \(1-p\). The complementary log-log link does not: it is asymmetric around probability one-half. This distinction can matter when the scientific setting suggests a transition that is not balanced in that way.

For example, the complementary log-log link is often considered for event probabilities that arise from an underlying process accumulating over time or exposure. The link choice can therefore encode a useful pattern in how probability changes across the predictor scale. That is a modeling rationale, not a guarantee that the link is appropriate for every such application; the suitability of a model still depends on the data and scientific context.

Using Alternatives Without Changing the Response Family

For independent binomial observations \(Y_i\) with trial counts \(m_i\), one can specify \(Y_i\sim\operatorname{Bin}(m_i,p_i)\) and choose any one of the three links for \(g(p_i)=x_i^\mathsf{T}\beta\). The inverse link ensures that each finite predictor gives \(0<p_i<1\), so the binomial probability model remains valid. What changes is the relationship imposed between the design row and the mean probability.

A common mistake is to assume that a GLM must use the canonical link, or that using probit or complementary log-log changes a binomial response into a different distribution. Neither is correct. Canonicality identifies a convenient match to the exponential-family natural parameter. A noncanonical link is still a legitimate choice when its inverse yields allowable means. Conversely, simply choosing a link because it is valid does not establish that it is scientifically appropriate or that it will fit well.