Journal of Statistical Theory and Practice

, Volume 1, Issue 2, pp 265–278 | Cite as

Priority and Choice Probability Estimation by Ranking, Rating and Combined Data

  • Stan LipovetskyEmail author


Ranking data is commonly used in marketing and advertising research for priority estimation among the compared items by Thurstone scaling. Rating data is also often used in TURF, or total unduplicated reach and frequency analysis to find the best items. Both ranks and rates data sets can be elicited and utilized simultaneously to obtain a combined preference estimation. This work develops several techniques of priority evaluation. It considers maximum likelihood of the order statistics for the ranking data with the probit, logit, and multinomial links for the Thurstone scale. Non-linear optimization with the least squares or maximum likelihood objective is introduced for TURF modeling. Combined estimation by both rank and rate data is suggested in singular value decomposition and Geary-Khamis equation approaches. The proposed methods produce priorities among the compared items and probabilities of their choice.

AMS Subject Classification



ranking rating Thurstone scale TURF order statistic nonlinear optimization maximum likelihood choice probability SVD Geary-Khamis equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arminger G., Clogg C.C., Sobel M.E., eds. 1995. Handbook of Statistical Modeling for the Social and Behavioral Sciences, Plenum Press, New York, London.zbMATHGoogle Scholar
  2. Bartholomew D.J., Knott M., 1999. Latent Variable Models and Factor Analysis, Arnold, London.zbMATHGoogle Scholar
  3. Bock R.D., Jones L.V., 1968. The Measurement and Prediction of Judgment and Choice, Holden-Day, San Francisco, CA.Google Scholar
  4. Boschman M.C., 2001. DifScal: A tool for analyzing difference ratings on an ordinal category scale, Behavior Research Methods, Instruments, & Computers, 33, 10–20.CrossRefGoogle Scholar
  5. Bradley R.A., Terry M.E., 1952. Rank analysis of incomplete block designs: I. The method of paired comparisons, Biometrika, 39, 324–345.MathSciNetzbMATHGoogle Scholar
  6. Bradley R.A., 1954. The rank analysis of incomplete block designs: II. Additional tables for the method of paired comparisons, Biometrika, 41, 502–537.MathSciNetzbMATHGoogle Scholar
  7. Cohen E., 1993. TURF Analysis, Quirk’s Marketing Research Review, June/July, 10–13.Google Scholar
  8. Conklin M., Lipovetsky S., 2005. Marketing Decision Analysis by TURF and Shapley Value, Information Technology and Decision Making, 4, 5–19.CrossRefGoogle Scholar
  9. Daniels H.E., 1950. Rank correlation and population models, Journal of the Royal Statistical Society, ser. B, 12, 171–181.MathSciNetzbMATHGoogle Scholar
  10. David H.A., 1988. The Method of Paired Comparisons, 2nd ed., London, Griffin.zbMATHGoogle Scholar
  11. Ennis D.M., Johnson N.L., 1993. Thurstone-Shepard similarity models as special cases of moment generating functions, Mathematical Psychology, 37, 104–110.CrossRefGoogle Scholar
  12. Ferrando P.J., 2004. Person reliability in personality measurement: an item response theory analysis, Applied Psychological Measurement, 28, 126–140.MathSciNetCrossRefGoogle Scholar
  13. Geary R.C., 1958. A note on the comparison of exchange rates and purchasing power between countries, J. of Royal Statistical Society, ser. A, 121, 97–99.CrossRefGoogle Scholar
  14. Glenn W.A., David H.A., 1960. Ties in paired-comparisons experiments using a modified Thurstone-Mosteller models, Biometrics, 16, 86–109.CrossRefGoogle Scholar
  15. Green P.E., Tull D.S., 1978. Research for marketing decisions. Prentice-Hall, New Jersey.Google Scholar
  16. Hambleton R.K., Swaminathan H., Rogers H.J., 1991. Fundamentals of Item Response theory, Sage, Newbury Park, CA.Google Scholar
  17. Harville D.A., 2003. The selection of seeding of college basketball or football teams for postseason competition, Journal of the American Statistical Association, 98, 17–27.MathSciNetCrossRefGoogle Scholar
  18. Hogg R.V., Craig A.T., 1969. Introduction to Mathematical Statistics, Macmillan, New York.zbMATHGoogle Scholar
  19. Johnson V.E., Albert J.H., 1999. Ordinal Data Modeling, Springer, New York.zbMATHGoogle Scholar
  20. Khamis S.H., 1972. A new system of index numbers for national and international purposes, J. of Royal Statistical Society, ser. A, 135, 96–121.CrossRefGoogle Scholar
  21. Kravis I.B., Kenessey Z., Heston A.W., Summers, R., 1975. A System of International Comparisons of Gross Product and Purchasing Power, Baltimore, J. Hopkins University Press.Google Scholar
  22. Kreiger A., Green P., 2000. TURF revisited: enhancements to total unduplicated reach and frequency analysis, Marketing Research, 12, 4, 30–36.Google Scholar
  23. Lazarsfeld P.F., 1950. The logical and mathematical foundations of latent structure analysis (Ch. 10); Some Latent Structures (Ch. 11). In: Stouffer S.A. et al. (Eds.), Measurement and Prediction, vol. IV of The American Soldier: Studies in Social Psychology in World War II, Princeton University Press, NJ. Reprinted (1973) by P. Smith, Gloucester, MA.Google Scholar
  24. Lazarsfeld P.F., 1959. Latent structure analysis, in: Koch S. (ed.) Psychology: A Study of a Science, vol. 3, 476–535, McGraw-Hill, New York.Google Scholar
  25. Lazarsfeld P.F., 1966. Latent structure analysis and test theory. In: Lazarsfeld P.F. and Henry N.W. (eds) Readings in Mathematical Social Science, 78–88, Science Research Assoc, Chicago.Google Scholar
  26. Lazarsfeld P.F., Henry N.W., 1968. Latent Structure Analysis, Houghton Mifflin, Boston.zbMATHGoogle Scholar
  27. Lieberman M., 2005. Unearthing TURF, Quirk’s Marketing Research Review, April, 18–24.Google Scholar
  28. Lipovetsky S., 1992. Cost-benefit analysis in decision making, Proceedings of the 10th International Conference in Multiple Criteria Decision Making, 4, 441–448, Taipei, Taiwan.Google Scholar
  29. Lipovetsky S., Conklin M., 2004. Thurstone scaling via binary response regression, Statistical Methodology, 1, 93–104.MathSciNetCrossRefGoogle Scholar
  30. Lipovetsky S., Conklin M., O’Donnell K., 2005a. SURF — structural unduplicated reach and frequency, JSM’05, Proceedings of the Joint Statistical Meeting, 2442–2448, Minneapolis, MN.Google Scholar
  31. Lipovetsky S., Conklin M., 2005b. Singular value decomposition in additive, multiplicative, and logistic forms, Pattern Recognition, 38, 1099–1110.CrossRefGoogle Scholar
  32. Lipovetsky S., 2007. Thurstone scaling in order statistics, Mathematical and Computer Modelling, 2007, 45, 917–926.MathSciNetCrossRefGoogle Scholar
  33. Lord F.M., 1953. The relation of test score to the trait underlying the test, Educational and Psychological Measurement, 13, 517–549.CrossRefGoogle Scholar
  34. Lord F.M., 1980. Applications of Item Response Theory to Practical Testing Problems, Erlbaum, Hillsdale, NJ.Google Scholar
  35. Lord F.M., Novick M.R., 1968. Statistical Theories of Mental Test Scores, Addison-Wesley, Readings.zbMATHGoogle Scholar
  36. Luce R.D., 1959. Individual Choice Behavior: A Theoretical Analysis. Wiley, New York.zbMATHGoogle Scholar
  37. Luce R.D., Suppes P., 1965. Utility, preference and subjective probability. In: Luce R.D., Bush R.R. and E. Galanter (eds.) Handbook of Mathematical Psychology, 3, 249–410, Wiley, New York.Google Scholar
  38. Mease D., 2003. A penalized maximum likelihood approach for the ranking of college football teams independent of victory margins, The American Statistician, 57, 241–248.MathSciNetCrossRefGoogle Scholar
  39. Miaoulis G., Free V., Parsons H., 1990. TURF: A new planning approach for product line extensions, Marketing Research, March, 28–39.Google Scholar
  40. Mosteller F., 1951., Remarks on the method of paired comparisons, Psychometrika, 16, 3–9, 203–218.CrossRefGoogle Scholar
  41. Mullet G., 1997. TURP: total unduplicated reach and profit, Canadian Journal of Marketing Research, 16, 90–95.Google Scholar
  42. Myerson R.B., 1991. Game Theory: Analysis of Conflict. Harvard University Press, Cambridge, MA, London, England.zbMATHGoogle Scholar
  43. Owen G., 1982. Game Theory. Academic Press, New York.zbMATHGoogle Scholar
  44. Rao P., Selvanathan E.A., 1992. Computation of standard errors for Geary-Khamis parities and international prices: a stochastic approach, J. of Business & Economic Statistics, 10, 109–115.Google Scholar
  45. Roth A.E., (ed.) 1988. The Shapley Value — Essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, MA.zbMATHGoogle Scholar
  46. Roth A.E., 1997. The Shapley Value as a Von Neumann-Morganstern Utility, Econometrica, 45, 657–664.CrossRefGoogle Scholar
  47. Shapley L.S., 1953. A Value for n-Person Games. In: Kuhn H.W. and Tucker A.W. (Eds.), Contribution to the Theory of Games, II, 307–317, Princeton Univ. Press, Princeton, NJ.MathSciNetzbMATHGoogle Scholar
  48. Skrondal A., Rabe-Hesketh, S., 2004. Generalized Latent Variable Modeling, Chapman & Hall/CRC, London, New York.CrossRefGoogle Scholar
  49. Stern H., 1990. Models for distributions on permutations, Journal of the American Statistical Association, 85, 558–564.CrossRefGoogle Scholar
  50. Thompson M., 1975. On any given Sunday: fair competitor orderings with maximum likelihood methods, Journal of the American Statistical Association, 70, 536–541.Google Scholar
  51. Thurstone L.L., 1927. A law of comparative judgment, Psychological Reviews, 34, 273–286.CrossRefGoogle Scholar
  52. Thurstone L.L., 1959. The Measurement of Values. University of Chicago Press, Chicago.Google Scholar
  53. Thurstone L.L., Jones L.V., 1957. The rational origin for measuring subjective values, Journal of the American Statistical Association, 52, 458–471.CrossRefGoogle Scholar
  54. Williams V.S.L., Pommerich M., Thissen, D., 1998. A comparison of developmental scales based on Thurstone methods and item response theory, J. of Educational Measurement, 35, 93–107.CrossRefGoogle Scholar
  55. Wolfe E.W., Chiu C.W.T., 1999. Measuring change across multiple occasions using the Rasch rating scale model, Journal of Outcome Measurement, 3, 360–381.Google Scholar
  56. Wolfe E.W., Chiu C. W. T., Myford C. M., 2000. Detecting rater effects in simulated data with a multi-faceted Rasch rating scale model. In: Wilson M. and Engelhard G., Jr. (Eds.), Objective measurement: Theory into practice, 5, 147–164. Ablex Publishing, Stamford, CT.Google Scholar
  57. Wrenn B., 1997. The market orientation construct: measurement and scaling issues, J. of Marketing Theory & Practice, 5, 31–54.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  1. 1.GfK Custom Research North AmericaMinneapolisUSA

Personalised recommendations