Hostname: page-component-5f745c7db-8qdnt Total loading time: 0 Render date: 2025-01-06T07:00:41.450Z Has data issue: true hasContentIssue false
  • English
  • Français

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Published online by Cambridge University Press:  01 January 2025

Peter M. Bentler  and
Ke-Hai Yuan
Peter M. Bentler*
Affiliation:
University of California, Los Angeles
Ke-Hai Yuan
Affiliation:
University of Notre Dame
*
Requests for reprints should be sent to Peter M. Bentler, University of California, Los Angeles, Los Angeles, USA. E-mail: bentler@ucla.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Indefinite symmetric matrices that are estimates of positive-definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based methods for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.

Keywords

indefinite matrixeigenvaluesminimum trace factor analysis
Type
Original Paper
Information
Psychometrika , Volume 76 , Issue 1 , January 2011 , pp. 119 - 123
Copyright
Copyright © 2010 The 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

This research was supported by grants DA00017 and DA01070 from the National Institute on Drug Abuse. The first author acknowledges a financial interest in EQS and its distributor, Multivariate Software.

References

Bentler, P.M. (1972). A lower bound method for the dimension-free measurement of internal consistency. Social Science Research, 1, 343–357.CrossRefGoogle Scholar
Bentler, P.M., Tanaka, J.S. (1983). Problems with EM algorithms for ML factor analysis. Psychometrika, 48, 247–251.CrossRefGoogle Scholar
Bonett, D.G., Price, R.M. (2005). Inferential methods for the tetrachoric correlation coefficient. Journal of Educational and Behavioral Statistics, 30, 213–225.CrossRefGoogle Scholar
Della Riccia, G., Shapiro, A. (1982). Minimum rank and minimum trace of covariance matrices. Psychometrika, 47, 443–448.CrossRefGoogle Scholar
Hoerl, A.E., Kennard, R.W. (1970). Ridge regression: biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.CrossRefGoogle Scholar
Jamshidian, M., Bentler, P.M. (1998). A quasi-Newton method for minimum trace factor analysis. Journal of Statistical Computation and Simulation, 62, 73–89.CrossRefGoogle Scholar
Knol, D.L., ten Berge, J.M.F. (1989). Least squares approximation of an improper correlation matrix by a proper one. Psychometrika, 54, 53–61.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y., Bentler, P.M. (1995). A two-stage estimation of structural equation models with continuous and polytomous variables. British Journal of Mathematical and Statistical Psychology, 48, 339–358.CrossRefGoogle ScholarPubMed
Wothke, W. (1993). Nonpositive definite matrices in structural modeling. In Bollen, K.A., Long, J.S. (Eds.), Testing structural equation models (pp. 256–293). Newbury Park: Sage.Google Scholar
Yuan, K.-H., Chan, W. (2008). Structural equation modeling with near singular covariance matrices. Computational Statistics & Data Analysis, 52, 4842–4858.CrossRefGoogle Scholar
Yuan, K.-H., Wu, R., & Bentler, P.M. (2010). Ridge structural equation modeling with correlation matrices for ordinal and continuous data. British Journal of Mathematical and Statistical Psychology.Google Scholar