Abstract
The numbers in each column of ann ×m matrix of multivariate data are interpreted as giving the measured values of alln of the objects studied on one ofm different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D.Statistical inference under order restrictions. New York: Wiley, 1972.
Bennett, J. F. Determination of the number of independent parameters of a score matrix from the examination of rank orders.Psychometrika, 1956,21, 383–393.
Birkhoff, G., and MacLane, S.A survey of modern algebra. New York: Macmillan, 1953, (revised edition).
Carroll, J. D. Polynomial factor analysis.Proceedings of the 77th Annual Convention of the American Psychological Association, 1969,4, 103–104. (Abstract)
Carroll, J. D., & Chang, J.-J. Non-parametric multidimensional analysis of paired-comparisons data.Paper presented at the joint meeting of the Psychometric and Psychonomic Societies, Niagara Falls, October 1964.
Coombs, C. H.A theory of data. New York: Wiley, 1964.
Coombs, C. H., and Kao, R. C. Nonmetric factor analysis.Engineering Research Bulletin No.38, University of Michigan Press, Ann Arbor, 1955.
Coombs, C. H., and Kao, R. C. On a connection between factor analysis and multidimensional unfolding.Psychometrika, 1960,25, 219–231.
Eckart, C., and Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1936,1, 211–218.
Harman, H. H.Modern factor analysis. Chicago: University of Chicago Press, 1967, (revised edition).
Kelley, H. J. Method of gradients. In G. Leitman (Ed.),Optimization techniques. New York: Academic Press, 1962.
Klemmer, E. T., and Shrimpton, N. Preference scaling via a modification of Shepard's proximity analysis method.Human Factors, 1963,5, 163–168.
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 1964,29, 1–27. (a).
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method.Psychometrika, 1964,29, 28–42. (b)
Kruskal, J. B. Analysis of factorial experiments by estimating monotone transformations of the data.Journal of the Royal Statistical Society, Series B, 1965,27, 251–263.
Kruskal, J. B. Monotone regression: Continuity and differentiability properties.Psychometrika, 1971,36, 57–62.
Kruskal, J. B., and Carroll, J. D. Geometric models and badness-of-fit functions. In P. R. Krishnaiah (Ed.),Multivariate analysis II. New York: Academic Press, 1969. Pp. 639–670.
Lazarsfeld, P. F. Latent structure analysis. In S. Koch (Ed.),Psychology: A study of a science, Vol. 3. New York: McGraw-Hill, 1959. Pp. 476–543.
Levine, M. V. Transformations that render curves parallel.Journal of Mathematical Psychology, 1970,7, 410–443.
Levine, M. V. Transforming curves into curves with the same shape.Journal of Mathematical Psychology, 1972,9, 1–16.
Lingoes, J. C. A general survey of the Guttman-Lingoes nonmetric program series. In R. N. Shepard, A. K. Romney, and S. Nerlove (Eds.),Multidimensional scaling: Theory and application in the behavioral sciences (Volume I, Theory). New York: Seminar Press, 1972. Pp. 49–68.
Lingoes, J. C., and Guttman, L. Nonmetric factor analysis: A rank reducing alternative to linear factor analysis.Multivariate Behavioral Research, 1967,2, 485–505.
McDonald, R. P. A general approach to nonlinear factor analysis.Psychometrika, 1962,27, 397–415.
McDonald, R. P. Nonlinear factor analysis.Psychometric Monograph No. 15, 1967.
Rosen, J. B. Gradient projection method for non-linear programing.Journal SIAM, 1960,8, 181–217.
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function (I & II).Psychometrika, 1962,27, 125–139, 219–246.
Shepard, R. N. Extracting latent structure from behavioral data. InProceedings of the 1964 symposium on digital computing. Bell Telephone Laboratories, May, 1964.
Shepard, R. N. Approximation to uniform gradients of generalization by monotone transformations of scale. In D. Mostofsky (Ed.),Stimulus generalization. Stanford, California: Stanford University Press, 1965. Pp. 94–110.
Shepard, R. N. Metric structures in ordinal data.Journal of Mathematical Psychology, 1966,3, 287–315.
Shepard, R. N. A taxonomy of some principal types of data and of multidimensional methods for their analysis. In R. N. Shepard A. K. Romney, and S. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences (Volume I, Theory). New York: Seminar Press, 1972. Pp. 21–47.
Shepard, R. N., and Carroll, J. D. Parametric representation of nonlinear data structures. In P. R. Krishnaiah (Ed.),Multivariate Analysis. New York: Academic Press, 1966. PP. 561–592.
Shepard, R. N., and Kruskal, J. B. Nonmetric methods for scaling and for factor analysis,American Psychologist, 1964,19, 557–558. (abstract)
Thurstone, L. L.Multiple factor analysis. Chicago: University of Chicago Press, 1947.
Torgerson, W. S.Theory and methods of scaling. New York: Wiley, 1958.
Author information
Authors and Affiliations
Additional information
Although the method described here existed as an operational computer program toward the end of 1962 and although the tests reported here were completed by 1966, this is the first full description of the method and report of the results of the test application. The preparation of this paper was in part supported by NSF grants GS-1302 and GS-2283 to the second author and was completed during the second author's tenure as a John Simon Guggenheim Fellow at the Center for Advanced Study in the Behavioral Sciences, Stanford. The authors are indebted to J. D. Carroll, Mrs. J.-J. Chang, Mrs, C. Brown, and the former Miss M. M. Sheenan, all of the Bell Telephone Laboratories, for their assistance in connection with the test application.
Rights and permissions
About this article
Cite this article
Kruskal, J.B., Shepard, R.N. A nonmetric variety of linear factor analysis. Psychometrika 39, 123–157 (1974). https://doi.org/10.1007/BF02291465
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02291465