A Framework for Quantifying Qualitative Responses in Pairwise Experiments

  • A. H. Al-Ibrahim


Suppose an experiment is conducted on pairs of objects with outcome response a continuous variable measuring the interactions among the pairs. Furthermore, assume the response variable is hard to measure numerically but we may code its values into low and high levels of interaction (and possibly a third category in between if neither label applies). In this paper, we estimate the interaction values from the information contained in the coded data and the design structure of the experiment. A novel estimation method is introduced and shown to enjoy several optimal properties including maximum explained variance in the responses with minimum number of parameters and for any probability distribution underlying the responses. Furthermore, the interactions have the simple interpretation of correlation (in absolute value), size of error is estimable from the experiment, and only a single run of each pair is needed for the experiment. We also explore possible applications of the technique. Three applications are presented, one on protein interaction, a second on drug combination, and the third on machine learning. The first two applications are illustrated using real life data while for the third application, the data are generated via binary coding of an image.


Quantification Positive semidefinite programming Networks Protein interaction Drug synergy Machine learning 



  1. Al-Ibrahim, A. H. (2015). The analysis of multivariate data using semi-definite programming. Journal of Classification, 32, 3. Scholar
  2. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. UK: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  3. Burt, C. (1950). The factorial analysis of qualitative data. British Journal of Psychology, 3, 166–185.Google Scholar
  4. Chou, T. C., & Talalay, P. (1984). Quantitative analysis of dose-effect relationships: the combined effects of multiple drugs or enzyme inhibitors. Advances in Enzyme Regulation, 22, 27–55.CrossRefGoogle Scholar
  5. Cokol, M., Chua, H., Tasan, M., Mutlu, B., Weinstein, Z. B., Yo, S., Nergiz, M., Costanzo, M., Baryshnikova, A., Giaever, G., Nislow, C., Myers, C. L., Andrews, B., Boone, C., & Roth, F. (2011). Systematic exploration of synergistic drug pairs. Molecular Systems Biology, 7, 544 online Pub.CrossRefGoogle Scholar
  6. Cox, T., & Cox, M. (2001). Multidimensional Scaling (2nd ed.). Boca Raton: Chapman Hall.zbMATHGoogle Scholar
  7. Fisher, R. A. (1940). The precision of discriminant functions. Annals of Eugenics, 10, 422–429.MathSciNetCrossRefGoogle Scholar
  8. Globerson A., Roweis S. (2007). Visualizing pairwise similarity via semidefinite programming. Pro. of the 11 th Inter. Conf. on Artif, Intelligence and Stat., pp. 139–146.Google Scholar
  9. Guttman, L. (1941). The quantification of a class of attributes: a theory and method of scale construction. In Horst et al. (Ed.), The prediction of personal adjustment (pp. 319–348). New York: Social Science Research Council.Google Scholar
  10. Horst, P. (1935). Measuring complex attitudes. Journal of Social Psychology, 6, 369–374.CrossRefGoogle Scholar
  11. Jackups, R., & Liang, J. (2005). Interstrand pairing patterns in β-barrel membrane proteins: the positive-outside rule, aromatic rescue, and strand registration prediction. Journal of Molecular Biology, 354, 979–993.CrossRefGoogle Scholar
  12. Joreskog, K. G., & Moustaki, I. (2001). 2007 factor analysis of ordinal variables: a comparison of three approaches. Multivariate Behavioral Research, 36(3), 347–387.CrossRefGoogle Scholar
  13. Li Z., Liu J. and Tang X. (2008). Pairwise constraint propagation by semidefinite programming for semi-supervised classification. Pro. of the 25th Conf. on Mach. Learn. pp. 576583.Google Scholar
  14. Mandel, J. (1969). The partitioning of interaction in analysis of variance. Journal of Research National Bureau Standards - B, V, 738(4).Google Scholar
  15. Mandel, J. (1970). A new analysis of variance model for non-additive data. Technometrics, 13(1), 1–18.Google Scholar
  16. McKeon, J. J. (1966). Canonical analysis: Some relations between canonical correlation, factor analysis, discriminant function analysis and scaling theory. [Monograph No. 13]. Psychometrika.Google Scholar
  17. Mignon, A., & Jurie, F. P. C. C. A. (2012). A new approach for distance learning from sparse pairwise constraints. Computer Vision and Pattern Recognition. IEEE Conference on Biometrics Compendium.Google Scholar
  18. Nishisato, S. (1980). Analysis of categorical data: dual scaling and its applications. Toronto: University of Toronto Press.zbMATHGoogle Scholar
  19. Shepard R.N. (1962). The analysis of proximities: multidimensional scaling with an unknown distance functions I, Psychometrika, V 27, no 2, 125–140.Google Scholar
  20. Tenenhaus, M., & Young, F. W. (1985). An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis, and other methods for quantifying categorical multivariate data. Psychometrika, 50(1), 91–119.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.Google Scholar

Copyright information

© The Classification Society 2019
corrected publication 2019

Authors and Affiliations

  1. 1.SafatKuwait

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