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A Note on the Eigensystem of the Covariance Matrix of Dichotomous Guttman Items.

Davis-Stober CP, Doignon JP, Suck R - Front Psychol (2015)

Bottom Line: In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items.Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions.We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

View Article: PubMed Central - PubMed

Affiliation: Department of Psychological Sciences, University of Missouri Columbia, MO, USA.

ABSTRACT
We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions. We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

No MeSH data available.


Related in: MedlinePlus

Each plot compares the eigenvalues obtained from Equation (3) to those obtained from simulated 2PL data with high item discrimination under the assumption that θ ~ N(0, 1) under n = 4, 6, 8, 16, 32, and 64 items.
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Figure 2: Each plot compares the eigenvalues obtained from Equation (3) to those obtained from simulated 2PL data with high item discrimination under the assumption that θ ~ N(0, 1) under n = 4, 6, 8, 16, 32, and 64 items.

Mentions: For each condition, we computed the covariance matrix of the 5000 simulated responses and numerically calculated the eigenvalues of the generated covariance matrix for each condition. Figure 2 compares the eigenvalues from these simulated data to the eigenvalues obtained via Equation (3), for each condition. It is interesting to note that there is a much closer correspondence between the two sets of eigenvalues under these conditions. Further, this relationship becomes stronger as the number of equally spaced items increases, yielding nearly a perfect match to the maximal eigenvalue as the number of items reaches 32 and 64.


A Note on the Eigensystem of the Covariance Matrix of Dichotomous Guttman Items.

Davis-Stober CP, Doignon JP, Suck R - Front Psychol (2015)

Each plot compares the eigenvalues obtained from Equation (3) to those obtained from simulated 2PL data with high item discrimination under the assumption that θ ~ N(0, 1) under n = 4, 6, 8, 16, 32, and 64 items.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4664651&req=5

Figure 2: Each plot compares the eigenvalues obtained from Equation (3) to those obtained from simulated 2PL data with high item discrimination under the assumption that θ ~ N(0, 1) under n = 4, 6, 8, 16, 32, and 64 items.
Mentions: For each condition, we computed the covariance matrix of the 5000 simulated responses and numerically calculated the eigenvalues of the generated covariance matrix for each condition. Figure 2 compares the eigenvalues from these simulated data to the eigenvalues obtained via Equation (3), for each condition. It is interesting to note that there is a much closer correspondence between the two sets of eigenvalues under these conditions. Further, this relationship becomes stronger as the number of equally spaced items increases, yielding nearly a perfect match to the maximal eigenvalue as the number of items reaches 32 and 64.

Bottom Line: In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items.Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions.We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

View Article: PubMed Central - PubMed

Affiliation: Department of Psychological Sciences, University of Missouri Columbia, MO, USA.

ABSTRACT
We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions. We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

No MeSH data available.


Related in: MedlinePlus