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Derivation of likelihood-ratio tests for Guttman quasi-simplex covariance structures

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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.

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Authors and Affiliations

  1. Nagpur University, Nagpur, India

    Bishwa Nath Mukherjee

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  1. Bishwa Nath Mukherjee
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Additional information

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.

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Mukherjee, B.N. Derivation of likelihood-ratio tests for Guttman quasi-simplex covariance structures. Psychometrika 31, 97–123 (1966). https://doi.org/10.1007/BF02289462

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