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Notes
Contributions from: Willem Heiser, Leiden University; Lawrence Hubert, University of Illinois; Henk Kiers, University of Groningen; Hans-Friedrich Köhn, University of Illinois; Charles Lewis, Fordham University; Jacqueline Meulman, Leiden University; John McArdle, University of Southern California; Klaas Sijtsma, Tilburg University; Yoshio Takane, University of Victoria.
Acknowledgements: We thank the National Institute on Aging (AG0713720 to the author) for support of this research and to Carol Prescott (University of Southern California) for helpful comments on the draft. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institute on Aging or the National Institutes of Health.
We actually used \(n_{m}^{\prime } = N - 1 -(n+5)/6 - 2m/3,\) where n is the number of variables, but all subsequent work uses \(N-1,\) so that is what I do here as well.
For more details, see the commentary on Kaiser (1958).
It must have been intended as a quick fix, because the rationales for KMO-MSA and IFS are quite different, and only the label “unacceptable” for a value of 0.50 or less was substantiated (albeit with a different argument). Further attempts to calibrate the KMO-MSA scale came later (see Meyer, Kaiser, Cerny, and Green, 1977, or Cerny and Kaiser, 1977), but too late to have much impact.
According to Google Scholar, Kaiser and Rice (1974) has 1578 citations as of 4/1/2016. This is a substantial number but nowhere near the 6043 citations for Kaiser (1974), almost all of which should be reallocated to Kaiser and Rice (1974).
This note was written while the author was Visiting Professor at the Department of Statistics, Stanford University, in June–July 2016.
This comment about being “unsolved” is somewhat of an understatement. The task of finding the “closest” ultrametric falls within the class of optimization problems now known as NP-hard, which includes all the old combinatorial chestnuts, such as the traveling salesman problem. Although Johnson was a computer scientist, he cannot be faulted for not proving or knowing this in 1967. The notion of a problem being NP-hard was not even introduced into the computer science literature until the early 1970s.
The Horan paper was submitted to Psychometrika in September 1964, but because the author was killed in a road accident before the completion of his PhD work, his supervisor John Ross took care of finalizing its publication.
I am indebted to Pieter Kroonenberg for pointing out this reference to me.
The author wishes to thank Roger Schmitz for providing the data in Table 2.
For another Psychometrika paper on approximate methods for the analysis of repeated measures (but written some 20 years after Greenhouse and Geisser), see the informative review by Huynh Huynh (1978).
It should be noted that the Greenhouse–Geisser Psychometrika paper does not include proofs for the explicit form that \(\epsilon \) should take nor for its lower bound. Formal demonstrations are given in a companion 1958 paper in The Annals of Mathematical Statistics, with the authors listed in the traditional alphabetical order (Geisser and Greenhouse, 1958). As is common for an Annals paper, the proofs are extremely cryptic and given with little or no interpolated verbal explanation or discussion.
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Commentaries on the Ten Most Highly Cited Psychometrika Articles from 1936 to the Present. Psychometrika 81, 1177–1211 (2016). https://doi.org/10.1007/s11336-016-9540-y
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DOI: https://doi.org/10.1007/s11336-016-9540-y