A Theorem on the Rank of a Product of Matrices with Illustration of Its Use in Goodness of Fit Testing

• Published in: Psychometrika Vol. 80; no. 4; pp. 938 - 948
• Main Authors: Satorra, Albert, Neudecker, Heinz
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2015; Springer Nature B.V
• Subjects: Algebra; Algorithms; Assessment; Behavioral Science and Psychology; Chi-Square Distribution; Goodness of Fit; Humanities; Hypotheses; Law; Matrices; Psychology; Psychometrics; Psychometrics - statistics & numerical data; Statistical Theory and Methods; Statistics for Social Sciences; Structural equation modeling; Structural Equation Models; Testing and Evaluation; wald test; augmented moment structures; rank deficiency; structural equation modeling; chi-square goodness of fit test; matrix algebra
• ISSN: 0033-3123; 1860-0980
• DOI: 10.1007/s11336-014-9438-5

This paper develops a theorem that facilitates computing the degrees of freedom of Wald-type chi-square tests for moment restrictions when there is rank deficiency of key matrices involved in the definition of the test. An if and only if (iff) condition is developed for a simple rule of difference of ranks to be used when computing the desired degrees of freedom of the test. The theorem is developed exploiting basics tools of matrix algebra. The theorem is shown to play a key role in proving the asymptotic chi-squaredness of a goodness of fit test in moment structure analysis, and in finding the degrees of freedom of this chi-square statistic.