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Least-Squares Approximation of an Improper Correlation Matrix by a Proper One

Published online by Cambridge University Press:  01 January 2025

Dirk L. Knol and
Jos M. F. ten Berge
Dirk L. Knol*
Affiliation:
University of Twente
Jos M. F. ten Berge*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Dirk Knol, University of Twente, Department of Education, PO Box 217, 7500 AE Enschede, THE NETHERLANDS or to
Jos ten Berge, University of Groningen, Vakgroep Psychologic, Grote Markt 31-32, 9712 HV Groningen, THE NETHERLANDS.
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Abstract

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based upon a solution for Mosier's oblique Procrustes rotation problem offered by ten Berge and Nevels. A necessary and sufficient condition is given for a solution to yield the unique global minimum of the least-squares function. Empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely. A possible drawback of the optimal solution is that it is a singular matrix of necessity. In cases where singularity is undesirable, one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be δ, where δ is an arbitrary small positive constant. Finally, it may be desirable to weight the squared errors of estimation differentially. A generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting. The generalized solution can readily be obtained from the standard “unweighted singular” solution by transforming the observed improper correlation matrix in a suitable way.

Keywords

missing value correlationtetrachoric correlationindefinite correlation matrixconstrained least-squares approximation

Information

Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 1 , March 1989, pp. 53 - 61
Copyright
Copyright © 1989 The Psychometric Society

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References

Beale, E. M. L., Little, R. J. A. (1975). Missing values in multivariate analysis. Journal of the Royal Statistical Society, Series B, 37, 129–145.10.1111/j.2517-6161.1975.tb01037.xCrossRefGoogle Scholar
de Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 22, 113–137.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M., Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.10.1111/j.2517-6161.1977.tb01600.xCrossRefGoogle Scholar
Devlin, S. J., Gnanadesikan, R., Kettenring, J. R. (1975). Robust estimation and outlier detection with correlation coefficients. Biometrika, 62, 531–545.CrossRefGoogle Scholar
Devlin, S. J., Gnanadesikan, R., Kettenring, J. R. (1981). Robust estimation of dispersion matrices and principal components. Journal of the American Statistical Association, 76, 354–362.CrossRefGoogle Scholar
Dong, H. K. (1985). Non-Gramian and singular matrices in maximum likelihood factor analysis. Applied Psychological Measurement, 9, 363–366.10.1177/014662168500900404CrossRefGoogle Scholar
Frane, J. W. (1976). Some simple procedures for handling missing data in multivariate analysis. Psychometrika, 41, 409–415.CrossRefGoogle Scholar
Frane, J. W. (1978). Missing data and BMDP: Some pragmatic approaches. Proceedings of the Statistical Computing Section (pp. 27–33). Washington, DC: American Statistical Association.Google Scholar
Gleason, T. C., Staelin, R. (1975). A proposal for handling missing data. Psychometrika, 40, 229–252.CrossRefGoogle Scholar
Gnanadesikan, R., Kettenring, J. R. (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 28, 81–124.10.2307/2528963CrossRefGoogle Scholar
Knol, D. L., ten Berge, J. M. F. (1987). Least-squares approximation of an improper by a proper correlation matrix using a semi-infinite convex program, Enschede, The Netherlands: University of Twente, Department of Education.Google Scholar
Luenberger, D. G. (1984). Introduction to linear and nonlinear programming 2nd ed.,, Reading, MA: Addison-Wesley.Google Scholar
Mosier, C. I. (1939). Determining a simple structure when loadings for certain tests are known. Psychometrika, 4, 149–162.CrossRefGoogle Scholar
Mulaik, S. A. (1972). The foundations of factor analysis, New York: McGraw-Hill.Google Scholar
Orchard, T., Woodbury, M. A. (1972). A missing information principle: Theory and applications. Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, 6, 697–715.Google Scholar
Shapiro, A. (1985). Extremal problems on the set of nonnegative definite matrices. Linear Algebra and its Applications, 67, 7–18.CrossRefGoogle Scholar
ten Berge, J. M. F., Nevels, K. (1977). A general solution to Mosier's oblique Procrustes problem. Psychometrika, 42, 593–600.CrossRefGoogle Scholar
Timm, N. H. (1970). The estimation of variance-covariance and correlation matrices from incomplete data. Psychometrika, 35, 417–437.CrossRefGoogle Scholar