ON THE PREDICTIVE PROPERTIES OF BINARY LINK FUNCTIONS

Year 2017, Volume: 66 Issue: 1, 1 - 18, 01.02.2017

Necla Gündüz Ernest Fokoué

https://doi.org/10.1501/Commua1_0000000770

Cited By: 6

Abstract

This paper provides a theoretical and computational justification
of the long held claim 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 Örmly all the aspects of the similarities and diffierences.
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

References

• A. Armagan and R. Zaretzki. A note on mean-…eld variational approximations in bayesian probit models. Computational Statistics and Data Analysis, 55:641–643, S. Basu and S. Mukhopadhyay. Bayesian analysis of binary regression using symmetric and asymmetric links. Sankhya: The Indian Journal of Statistics, 62(3):372–387, 2000.
• S. Chakraborty. Bayesian binary kernel probit model for microarray based cancer clas- si…cation and gene selection. Computational Statistics and Data Analysis, 53:4198– , 2009.
• E.A. Chambers and D.R. Cox. Discrimination between alternative binary response models. Biometrika, 54(3/4):573–578, 1967.
• L. Csat´ o, E Fokou´ e, M Opper, B. Schottky, and O. Winther. E¢ cient approaches to gaussian process classi…cation. In S. A. Solla, T. K. Leen, and eds. K.-R. M¨ uller, editors, Advances in Neural Information Processing Systems, number 12. MIT Press, W. Feller. On the logistic law of growth and its empirical veri…cation in biology. Acta Biotheoretica, 5:51–66, 1940.
• W. Feller. An Introduction to Probability Theory and Its Applications, volume II. John Wiley and Sons, New York, second edition, 1971.
• G. D. Lin and C. Y. Hu. On characterizations of the logistic distribution. Journal of Statistical Planning and Inference, 138:1147–1156, 2008.
• S. Nadarajah. Information matrix for logistic distributions. Mathematical and Com- puter Modelling, 40:953–958, 2004.
• M. M. Nassar and A. Elmasry. A study of generalized logistic distributions. Journal of the Egyptian Mathematical Society, 20:126–133, 2012.
• M. Schumacher, R. Robner, and W. Vach. Neural networks and logistic regression: Part i. Computational Statistics and Data Analysis, 21:661–682, 1996.
• K. A. Tamura and V. Giampaoli. New prediction method for the mixed logistic model applied in a marketing problem. Computational Statistics and Data Analysis, 66:202– , 2013.
• A. van den Hout, P. van der Heijden, and R. Gilchrist. The Logistic Regression Model with Response Variables Subject to Randomized Response. Computational Statistics and Data Analysis, 51:6060–6069, 2007.
• D. Zelterman. Order statistics for the generalized logistic distribution. Computational Statistics and Data Analysis, 7:69–77, 1989.