Aranda-Ordaz (1981) On Two Families of Transformations to Additivity for Binary Response Data

Summary

This paper introduces two one-parameter families of link transformations for binary-response probabilities, each embedding the logistic model as a special case, so that the adequacy of the logit link can be tested rather than assumed. A symmetric family (indexed by $\lambda$, built around the logit) captures departures that treat successes and failures interchangeably, while an asymmetric family (built around the complementary log-log) captures skewed departures relevant to extreme-value problems. The transformation parameter is estimated by maximum likelihood using profile-likelihood over $\lambda$ in GLIM, and score tests are derived to detect symmetric and asymmetric departures from the logit.

Key Claims

• Symmetric family. A family $T_\lambda(\theta)$ of transformations of the success probability $\theta$ that satisfies $T_\lambda(\theta)=-T_\lambda(1-\theta)$ (symmetric treatment of successes/failures). It reduces to the logit as $\lambda\to0$ and to a linear transformation at $\lambda=1$, giving a continuum of symmetric link scales bracketing the logistic.
• Asymmetric family. Based on $W(\theta)=\{(1-\theta)^{-\lambda}-1\}/\lambda$ with $\log W(\theta)=\eta$ linear in covariates. It reduces to the logistic at $\lambda=1$ and to the complementary log-log at $\lambda=0$, so those two standard links are compared through a single parameter and asymmetric departures are made testable.
• Estimation by profile ML. Fit the linear predictor by ML across a grid of $\lambda$; the maximized log-likelihood as a function of $\lambda$ yields $\hat\lambda$ and an acceptable-fit interval, while keeping the covariate coefficients interpretable on the chosen scale.
• Score tests for adequacy. Score (Lagrange-multiplier) tests detect symmetric and asymmetric departures from the logit without fully refitting the extended model, indicating not only inadequacy but the direction of a useful modification.
• Empirical discrimination of links. On the Milicer-Szczotka (1966) Warsaw-girls menarche data, the symmetric family gives $\hat\lambda=0.325$; the probit scale lies inside the confidence interval (acceptable) whereas the logit ($\lambda=0$) is excluded — so the flexible family discriminates logit from probit.

Concepts Introduced or Extended

• Link Function
• Bayesian Logistic Regression
• Binary Probit

Entities Mentioned

• Francisco J. Aranda-Ordaz

Quotes

"Two families of power transformations for probabilities are introduced to model symmetric and asymmetric departures from the logistic model."

"An appealing and informative way to achieve our objective is to construct extended models which include the logistic and the alternatives of interest as special cases."

My Take

The lasting contribution is the embedding strategy: rather than pick a link and hope, nest the candidate links in a one-parameter family and let the data locate $\lambda$, with a score test that flags departures cheaply. The symmetric family nests logit and linear; the asymmetric family nests logit and complementary log-log — together they turn "which link?" into an estimable, testable question. The wiki's later flexible-link ideas are the same instinct by other means: robit regression nests probit and (approximately) logit through the Student-$t$ degrees of freedom, and Prentice's (1976) generalization plays the analogous role for dose-response curves. The caveats are practical — $\lambda$ is weakly identified when the data are not extreme, and interpretation of coefficients shifts with the estimated scale, so the method is best used as a diagnostic for link adequacy rather than as a final model.