Limits...
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 Rasch data 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
getmorefigures.php?uid=PMC4664651&req=5

Figure 1: Each plot compares the eigenvalues obtained from Equation (3) to those obtained from simulated Rasch data 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 items using the 5000 simulated responses, i.e., we calculated the sample covariance of the 5000 responses. We then numerically calculated the eigenvalues of this covariance matrix. Figure 1 compares the eigenvalues obtained from the simulated data to the eigenvalues obtained from Equation (3), for each condition. It is interesting to note that the largest eigenvalue for the simulated data is always larger than the maximal eigenvalue obtained via Equation (3), this is similar to results obtained by Zwick (1987) within the context of the Guttman correlation matrix. In general, moving to a probabilistic response model (the Rasch model) and sampling the θ values from a normal distribution appears to yield covariance eigenvalues that greatly differ from those obtained in Equation (3). As we show in the next study, improving item discrimination will yield different results.


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 Rasch data 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 1: Each plot compares the eigenvalues obtained from Equation (3) to those obtained from simulated Rasch data 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 items using the 5000 simulated responses, i.e., we calculated the sample covariance of the 5000 responses. We then numerically calculated the eigenvalues of this covariance matrix. Figure 1 compares the eigenvalues obtained from the simulated data to the eigenvalues obtained from Equation (3), for each condition. It is interesting to note that the largest eigenvalue for the simulated data is always larger than the maximal eigenvalue obtained via Equation (3), this is similar to results obtained by Zwick (1987) within the context of the Guttman correlation matrix. In general, moving to a probabilistic response model (the Rasch model) and sampling the θ values from a normal distribution appears to yield covariance eigenvalues that greatly differ from those obtained in Equation (3). As we show in the next study, improving item discrimination will yield different results.

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