A Maximum Likelihood Method for Fitting the Wandering Vector Model | Psychometrika | Cambridge Core (original) (raw)

Abstract

After introducing some extensions of a recently proposed probabilistic vector model for representing paired comparisons choice data, an iterative procedure for obtaining maximum likelihood estimates of the model parameters is developed. The possibility of testing various hypotheses by means of likelihood ratio tests is discussed. Finally, the algorithm is applied to some existing data sets for illustrative purposes.

References

Akaike, H. On entropy maximization principle. In Krishnaiah, P. R. (Eds.), Applications of statistics . Amsterdam: North-Holland. 1977, 27–41.Google Scholar

Amemiya, T. The maximum likelihood, the minimum chi-square and the nonlinear weighted least-squares method in the general qualitative response model. Journal of the American Statistical Association, 1976, 71, 247–251.CrossRefGoogle Scholar

Bock, R. D., & Jones, L. V. The measurement of prediction of judgment and choice, San Francisco: Holden-Day, 1968.Google Scholar

Carroll, J. D. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., & Nerlove, S. B. (Eds.), Multidimensional scaling: Theory and Applications in the behavioral sciences (Vol. 1) . New York: Seminar Press. 1972, 105–155.Google Scholar

Carroll, J. D. Models and methods for multidimensional analysis of preferential choice (or other dominance) data. In Lantermann, E. D. & Feger, H. (Eds.), Similarity and choice . Bern: Huber. 1980, 234–289.Google Scholar

De Soete, G. On the relation between two generalized cases of Thurstone's Law of Comparative Judgment. Mathématiques et Sciences humaines, 1983, 21, 47–57.Google Scholar

Halff, H. M. Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 1976, 14, 244–246.CrossRefGoogle Scholar

Heiser, W., & De Leeuw, J. Multidimensional mapping of preference data. Mathématiques et Sciences humaines, 1981, 19, 39–96.Google Scholar

Jennrich, R. I., & Moore, R. H. Maximum likelihood estimation by means of nonlinear least squares. Proceedings of the American Statistical Association—Statistical Computing Section, 1975, 57-65.Google Scholar

Jones, L. V., & Jeffrey, T. E. A quantitative analysis of expressed preferences for compensation plans. Journal of Applied Psychology, 1964, 48, 201–210.CrossRefGoogle Scholar

Kruskal, J. B. Factor analysis and principal components. I. Bilinear models. In Kruskal, W. H. & Tanur, J. M. (Eds.), International encyclopedia of statistics. (Vol. 1) . New York: The Free Press. 1978, 307–330.Google Scholar

McFadden, D. Quantal choice analysis: a survey. Annals of Economic and Social Measurement, 1976, 5, 363–390.Google Scholar

Ramsay, J. O. Maximum likelihood estimation in multidimensional scaling. Psychometrika, 1977, 42, 241–266.CrossRefGoogle Scholar

Ramsay, J. O. The joint analysis of direct ratings, pairwise preferences, and dissimilarities. Psychometrika, 1980, 45, 149–165.CrossRefGoogle Scholar

Roskam, E. E., & Lingoes, J. C. MINISSA-I: Fortran IV(G) program for the smallest space analysis of square symmetric matrices. Behavioral Science, 1970, 15, 204–205.Google Scholar

Scheffé, H. An analysis of variance for paired comparisons. Journal of the American Statistical Association, 1952, 47, 381–400.Google Scholar

Schönemann, P. H., & Tucker, L. R. A maximum likelihood solution for the method of successive intervals allowing for unequal stimulus dispersions. Psychometrika, 1967, 32, 403–417.CrossRefGoogle Scholar

Sjöberg, L. Choice frequency and similarity. Scandinavian Journal of Psychology, 1977, 18, 103–115.CrossRefGoogle Scholar

Sjöberg, L., & Capozza, D. Preference and cognitive structure of Italian political parties. Italian Journal of Psychology, 1975, 2, 391–402.Google Scholar

Slater, P. The analysis of personal preferences. British Journal of Statistical Psychology, 1960, 13, 119–135.CrossRefGoogle Scholar

Takane, Y. A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical orderings are independent—Theory. Japanese Psychological Research, 1978, 20, 7–17.CrossRefGoogle Scholar

Takane, Y. Maximum likelihood estimation in the generalized case of Thurstone's model of comparative judgement. Japanese Psychological Research, 1980, 22, 188–196.CrossRefGoogle Scholar

Takane, Y. Multidimensional successive categories scaling: a maximum likelihood method. Psychometrika, 1981, 46, 9–18.CrossRefGoogle Scholar

Takane, Y., & Carroll, J. D. Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities. Psychometrika, 1981, 46, 389–405.CrossRefGoogle Scholar

Thurstone, L. L. A law of comparative judgment. Psychological Review, 1927, 34, 273–286.CrossRefGoogle Scholar

Tucker, L. R. Intra-individual and inter-individual multidimensionality. In Gulliksen, H. & Messick, S. (Eds.), Psychological scaling: theory and applications . New York: Wiley. 1960, 155–167.Google Scholar

Wilks, S. S. Mathematical Statistics, New York: Wiley, 1962.Google Scholar