  This paper provides a theoretical and computational justification of the long
held claim that of the similarity of the probit and logit link functions often
used in binary classification. Despite this widespread recognition of the
strong similarities between these two link functions, very few (if any)
researchers have dedicated time to carry out a formal study aimed at
establishing and characterizing firmly all the aspects of the similarities and
differences. This paper proposes a definition of both structural and predictive
equivalence of link functions-based binary regression models, and explores the
various ways in which they are either similar or dissimilar. From a predictive
analytics perspective, it turns out that not only are probit and logit
perfectly predictively concordant, but the other link functions like cauchit
and complementary log log enjoy very high percentage of predictive equivalence.
Throughout this paper, simulated and real life examples demonstrate all the
equivalence results that we prove theoretically.
