Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2025-01-04T03:37:20.103Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivation of Likelihood-Ratio Tests for Guttman Quasi-Simplex Covariance Structures

Published online by Cambridge University Press:  01 January 2025

Bishwa Nath Mukherjee
Bishwa Nath Mukherjee*
Affiliation:
Nagpur University, Nagpur, India
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, maximum-likelihood estimates have been obtained for covariance matrices which have the Guttman quasi-simplex structure under each of the following null hypotheses: (a) The covariance matrix Σ, can be written asTΔT′ + Γ where Δ and Γ are both diagonal matrices with unknown elements andT is a known lower triangular matrix, and (b) the covariance matrix Σ*, is expressible asTΔ*T′ + γI where γ is an unknown scalar. The linear models from which these covariance structures arise are also stated along with the underlying assumptions. Two likelihood-ratio tests have been constructed, one each for the above null hypotheses, against the alternative hypothesis that the population covariance matrix is simply positive definite and has no particular pattern. A numerical example is provided to illustrate the test procedure. Possible applications of the proposed test are also suggested.

Type
Original Paper
Information
Psychometrika , Volume 31 , Issue 1 , March 1966 , pp. 97 - 123
Copyright
Copyright © 1966 Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Adapted from portions of the author's dissertation under the same title submitted to the Department of Psychology, University of North Carolina, in partial fulfillment of the requirements for the Ph.D. degree. The author wishes to express his gratitude to his thesis chairman Dr. R. Darrell Bock and to his committee members Professors Samarendra Nath Roy, Lyle V. Jones, Thelma G. Thurstone, and Dorothy Adkins. Indebtedness is also acknowledged to Dr. Somesh Das Gupta who was quite helpful during the initial stage of the study.

†

Formerly at the Department of Psychology, Indiana University. The author is grateful both to Indiana University and University of North Carolina for the support extended to him during his doctoral studies.

References

Anderson, T. W. Introduction to multivariate statistical analysis, New York: Wiley, 1958.Google Scholar
Anderson, T. W. Some stochastic process models for intelligence test scores. In Arrow, K. J., Karlin, S., Suppes, P. (Eds.), Mathematical methods in the social sciences, Stanford, Calif.: Stanford Univ. Press, 1960.Google Scholar
Bilodeau, E. A., Jones, M. B., andLevy, C. M. Long-term memory as a function of retention time and repeated recalling. J. exp. Psychol., 1964, 67, 303–309.CrossRefGoogle ScholarPubMed
Bock, R. D. Analysis of a class of covariance structures. Invited paper presented in honor of Harold Hotelling on his 65th birthday. American Statistical Association, Palo Alto, California, 1960.Google Scholar
Bock, R. D. Component of variance analysis as a structural and discriminal analysis of psychological tests. Brit. J. statist. Psychol., 1960, 13, 151–163.CrossRefGoogle Scholar
Bodewig, E. Matrix calculus (2nd rev. ed.), New York: Interscience, 1959.Google Scholar
DuBois, P. H. An analysis of Guttman's simplex. Psychometrika, 1960, 25, 173–182.CrossRefGoogle Scholar
Dwyer, P. W. Linear computations, New York: Wiley, 1951.Google Scholar
Gabriel, K. R. The simplex structure of progressive matrices test. Brit. J. statist. Psychol., 1954, 7, 9–14.CrossRefGoogle Scholar
Gabriel, K. R. Ante-dependence analysis of an ordered set of variables. Ann. math. Statist., 1962, 33, 201–222.CrossRefGoogle Scholar
Gantmacher, F. R. The theory of matrices (2 vols.), New York: Chelsea Publishing Co., 1959.Google Scholar
Gibson, W. A. The latent structure of the simplex. Amer. Psychologist, 1961, 16, 416–416.Google Scholar
Greenberg, B. G. and Sarhan, A. E. Matrix inversions, its interest and application in analysis of data. J. Amer. statist. Ass., 1959, 54, 755–766.CrossRefGoogle Scholar
Guttman, L.. A new approach to factor analysis: the radex. In Lazarsfeld, P. (Eds.), Mathematical thinking in the social sciences, Glenco, Ill.: Free Press, 1954.Google Scholar
Guttman, L. A generalized simplex for factor analysis. Psychometrika, 1955, 20, 173–191.CrossRefGoogle Scholar
Guttman, L. Empirical verification of the radex structure of mental abilities and personality traits. Educ. psychol. Measmt, 1957, 17, 391–407.CrossRefGoogle Scholar
Guttman, L. What lies ahead for factor analysis?. Educ. psychol. Measmt, 1958, 18, 497–515.CrossRefGoogle Scholar
Horst, P. Matrix factoring and test theory. In Frederiksen, N. and Gulliksen, H. (Eds.), Contributions to mathematical psychology, New York: Holt, Rinehart, and Winston, 1964.Google Scholar
Humphreys, L. G. Investigation of the simplex. Psychometrika, 1960, 25, 313–323.CrossRefGoogle Scholar
Jones, M. B. The use of simplex theory in training research. Amer. Psychologist, 1958, 13, 398–398.Google Scholar
Jones, M. B. Simplex theory, Pensacola, Fla.: U. S. Naval School of Aviation Medicine, 1959.Google Scholar
Jones, M. B. Molar correlational analysis, Pensacola, Fla.: U. S. Naval School of Aviation Medicine, 1960.Google Scholar
Jones, M. B. Practice as a process of simplification. Psychol. Rev., 1962, 69, 274–294.CrossRefGoogle ScholarPubMed
Kaiser, H. Scaling a simplex. Psychometrika, 1962, 27, 155–162.CrossRefGoogle Scholar
Kinzer, J. R. and Kinzer, L. G. Predicting grades in advanced college mathematics. J. appl. Psychol., 1953, 37, 182–184.CrossRefGoogle Scholar
Lask, E. A. A critique of Jones' simplical approach to decisions of task order. Unpublished Master's thesis submitted to Univ. Illinois, Department of Psychology, Urbana, Illinois, 1961.Google Scholar
Mukherjee, B. N. Derivation of likelihood ratio tests for Guttman quasi-simplex covariance structure. Unpublished Ph.D. dissertation, Univ. North Carolina, Chapel Hill, N. C., 1963.Google Scholar
Mukherjee, B. N. Invariance of the Guttman quasi-simplex linear model under selection. Submitted for publication inBrit. J. math. statist. Psychol..Google Scholar
Nishisato, S. and Mukherjee, B. N. Maximum likelihood estimates of a Jacobi matrix: A computer program for the UNIVAC 1105. Working paper, Psychometric Laboratory, Univ. North Carolina, Chapel Hill, N. C. 1965.Google Scholar
Rogers, C. R. and Dymond, R. F. Psychotherapy and personality changes, Chicago, Ill.: Univ. Chicago Press, 1954.Google Scholar
Scarborough, J. B.. Numerical mathematical analysis (2nd ed.), Baltimore, Md.: Johns Hopkins University Press, 1950.Google Scholar
Ukita, Y. Characterization of 2-type diagonal matrices with an application in order statistics. J. Hokkaido College of Arts and Literature, 1957, 6, 66–75.Google Scholar
Wald, A. Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Amer. math. Statist., 1943, 17, 257–281.Google Scholar
Wilks, S. S. Mathematical statistics, New York: Wiley, 1962.Google Scholar