Abstract
In connection with a least-squares solution for fitting one matrix,A, to another,B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure,e, and a refined similarity measure,e s, which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix,B,e.g.,e(A,kB) ≠e(A,B) fork ≠ 1. Therefore, both measures are useless for answering questions of the type: “DoesA fitB better thanA fitsC?”. In this note two new measures of fit are suggested which do not depend upon the norms ofA andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.
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References
Lingoes, J. C.The Guttman-Lingoes Nonmetric Program Series. Ann Arbor, Michigan: Mathesis Press, 1973.
Schönemann, P. H. & Carroll, R. M. Fitting one matrix to another under choice of a central dilation and a rigid motion.Psychometrika, 1970,35, 245–255.
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This research in nonmetric techniques is supported in part by a grant from the National Science Foundation (GS-2850) to the University of Michigan.
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Lingoes, J.C., Schönemann, P.H. Alternative measures of fit for the Schönemann-carroll matrix fitting algorithm. Psychometrika 39, 423–427 (1974). https://doi.org/10.1007/BF02291666
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DOI: https://doi.org/10.1007/BF02291666