Link Function

Definition

In a generalized linear model (GLM; Nelder-Wedderburn 1972) the link function $g$ connects the mean of the response to the linear predictor, $g(\mu_i)=\eta_i=x_i'\beta$, so that the systematic part of the model is linear on the transformed (link) scale while the response stays in its natural range. For binary data $\mu_i=\Pr(y_i=1)=\theta_i\in(0,1)$ and the link maps the probability onto the whole real line; the choice of $g$ determines the shape of the response curve and, unlike the linear-predictor coefficients, is itself a modeling assumption that can be tested.

Key Ideas

• Standard binary links. The logit $g(\theta)=\log\{\theta/(1-\theta)\}$ (canonical, symmetric), the probit $g(\theta)=\Phi^{-1}(\theta)$ (Gaussian-latent, symmetric), and the complementary log-log $g(\theta)=\log\{-\log(1-\theta)\}$ (asymmetric, extreme-value). Logit and probit are close over the central range but differ in the tails; cloglog is skewed.
• The link is an assumption, not a given. A misspecified link biases fitted probabilities and covariate effects; because it is tentative "as for all models," it warrants a check (Aranda-Ordaz 1981).
• Parametric / flexible link families. Embed the candidate links in a one-parameter family and estimate the extra parameter. The Aranda-Ordaz symmetric family nests logit ($\lambda\to0$) and a linear scale; its asymmetric family nests logit ($\lambda=1$) and complementary log-log ($\lambda=0$). Prentice (1976) gives a two-parameter generalization for dose-response curves.
• Heavy-tailed links. Robit regression (Liu 2006) uses a Student-$t$ CDF link: $\nu\to\infty$ recovers probit and $\nu\approx7$ approximates logit, while a small $\nu$ bounds the influence of outliers — a flexible-link idea aimed at robustness rather than skewness.
• Link selection and testing. Profile likelihood over the family's shape parameter yields an MLE and an acceptable-fit interval; score (Lagrange-multiplier) tests detect departures from a baseline link (e.g., from logit toward probit or cloglog) without refitting the full extended model.

How It Works

1. Choose a baseline link (often logit) and embed it in a parametric family $g_\lambda$ with $g_{\lambda_0}=$ baseline.
2. For a grid of $\lambda$, fit $\beta$ by maximum likelihood and record the maximized log-likelihood $\ell(\lambda)$.
3. Read off $\hat\lambda=\arg\max_\lambda \ell(\lambda)$ and an interval of $\lambda$ giving acceptable fit; a baseline value outside the interval is evidence against that link.
4. Alternatively, compute a score test at $\lambda_0$ to detect symmetric or asymmetric departures cheaply.
5. Report covariate effects on the selected scale, noting that their interpretation shifts with $\lambda$.

Why It Matters

• Model adequacy. The link is one of the few GLM choices with no default that is always right; flexible families turn "which link?" into an estimable, testable question rather than a convention.
• Discriminating logit vs. probit vs. cloglog. Embedding families can separate links that fit almost identically in the middle of the range but diverge in the tails — where extreme-probability predictions are most consequential.
• A unifying view. Robit, Aranda-Ordaz, and Prentice generalizations are the same instinct — a shape parameter that indexes a family of links — applied for robustness, symmetry, and dose-response flexibility respectively.

Open Questions

• Weak identification of the shape parameter. When responses are not near 0 or 1, the link parameter $\lambda$ (or robit $\nu$) is poorly identified and the profile likelihood is flat.
• Interpretability. Coefficients change meaning as the estimated scale changes, complicating communication of effect sizes.
• Bayesian link selection. Placing a prior on the link-family parameter and averaging over it is natural but sensitive to the prior, echoing the Bayesian model averaging caveats.

Related

• Bayesian Logistic Regression
• Binary Probit
• Robit Regression
• Ordinal Regression
• Generalized Linear Mixed Model
• Generalized Additive Model
• Discrete Choice
• Francisco J. Aranda-Ordaz
• Aranda-Ordaz (1981) On Two Families of Transformations to Additivity for Binary Response Data
• Liu (2006) Robit Regression