A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population

We consider a p-dimensional, centered normal population such that all variables have a positive variance σ2 and any correlation coefficient between different variables is a given nonnegative constant ρ < 1. Suppose that both the sample size n and population dimension p tend to infinity with p/n →...

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Main Authors: Akama, Yohji, Husnaqilati, Atina
格式: Article PeerReviewed
語言:English
出版: The Indonesian Mathematical Society 2022
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https://repository.ugm.ac.id/278613/
https://jims-a.org/index.php/jimsa/article/view/115
https://doi.org/10.22342/jims.28.3.1158.272-303
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spelling id-ugm-repo.2786132023-11-02T02:02:30Z https://repository.ugm.ac.id/278613/ A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population Akama, Yohji Husnaqilati, Atina Mathematics and Applied Sciences We consider a p-dimensional, centered normal population such that all variables have a positive variance σ2 and any correlation coefficient between different variables is a given nonnegative constant ρ < 1. Suppose that both the sample size n and population dimension p tend to infinity with p/n → c > 0. We prove that the limiting spectral distribution of a sample correlation matrix is Marˇcenko- Pastur distribution of index c and scale parameter 1 − ρ. By the limiting spectral distributions, we rigorously show the limiting behavior of widespread stopping rules Guttman-Kaiser criterion and cumulative-percentage-of-variation rule in PCA and EFA. As a result, we establish the following dichotomous behavior of Guttman-Kaiser criterion when both n and p are large, but p/n is small: (1) the criterion retains a small number of variables for ρ > 0, as suggested by Kaiser, Humphreys, and Tucker [Kaiser, H. F. (1992). On Cliff’s formula, the Kaiser-Guttman rule and the number of factors. Percept. Mot. Ski. 74]; and (2) the criterion retains p/2 variables for ρ = 0, as in a simulation study [Yeomans, K. A. and Golder, P. A. (1982). The Guttman-Kaiser criterion as a predictor of the number of common factors. J. Royal Stat. Soc. Series D. 31(3)]. The Indonesian Mathematical Society 2022 Article PeerReviewed application/pdf en https://repository.ugm.ac.id/278613/1/Husnaqilati_MA.pdf Akama, Yohji and Husnaqilati, Atina (2022) A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population. J. Indones. Math. Soc., 28 (3). pp. 272-303. ISSN 2460-0245 https://jims-a.org/index.php/jimsa/article/view/115 https://doi.org/10.22342/jims.28.3.1158.272-303
institution Universitas Gadjah Mada
building UGM Library
continent Asia
country Indonesia
Indonesia
content_provider UGM Library
collection Repository Civitas UGM
language English
topic Mathematics and Applied Sciences
spellingShingle Mathematics and Applied Sciences
Akama, Yohji
Husnaqilati, Atina
A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population
description We consider a p-dimensional, centered normal population such that all variables have a positive variance σ2 and any correlation coefficient between different variables is a given nonnegative constant ρ < 1. Suppose that both the sample size n and population dimension p tend to infinity with p/n → c > 0. We prove that the limiting spectral distribution of a sample correlation matrix is Marˇcenko- Pastur distribution of index c and scale parameter 1 − ρ. By the limiting spectral distributions, we rigorously show the limiting behavior of widespread stopping rules Guttman-Kaiser criterion and cumulative-percentage-of-variation rule in PCA and EFA. As a result, we establish the following dichotomous behavior of Guttman-Kaiser criterion when both n and p are large, but p/n is small: (1) the criterion retains a small number of variables for ρ > 0, as suggested by Kaiser, Humphreys, and Tucker [Kaiser, H. F. (1992). On Cliff’s formula, the Kaiser-Guttman rule and the number of factors. Percept. Mot. Ski. 74]; and (2) the criterion retains p/2 variables for ρ = 0, as in a simulation study [Yeomans, K. A. and Golder, P. A. (1982). The Guttman-Kaiser criterion as a predictor of the number of common factors. J. Royal Stat. Soc. Series D. 31(3)].
format Article
PeerReviewed
author Akama, Yohji
Husnaqilati, Atina
author_facet Akama, Yohji
Husnaqilati, Atina
author_sort Akama, Yohji
title A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population
title_short A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population
title_full A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population
title_fullStr A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population
title_full_unstemmed A Dichotomous Behavior of Guttman-Kaiser Criterion From Equi-Correlated Normal Population
title_sort dichotomous behavior of guttman-kaiser criterion from equi-correlated normal population
publisher The Indonesian Mathematical Society
publishDate 2022
url https://repository.ugm.ac.id/278613/1/Husnaqilati_MA.pdf
https://repository.ugm.ac.id/278613/
https://jims-a.org/index.php/jimsa/article/view/115
https://doi.org/10.22342/jims.28.3.1158.272-303
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