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

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).

Is the Factor Observed in Investigations on the Item-Position Effect Actually the Difficulty Factor?

“…The application of this method can also be useful if a structural investigation does not reveal the expected homogeneity. In larger numbers of items there is always some change that an additional factor may arise (Davis-Stober, Doignon, & Suck, 2015) since there may be inhomogeneity due to subsets of very similar items. Such inhomogeneity is likely to lead to model misfit.…”

Threshold-free estimation of entropy from a Pearson matrix

“…Arguably, this may seem a too restricting imposition [38]. But perhaps a bit surprising, it has been verified numerically that certain results (like principal component analysis and certain properties of covariance matrices) for homogeneous data do not depart very much from the case where the N i 's display a normal distribution (provided the populations have ample variability [37,38]). Therefore, it is correct to affirm that data obeying the homogeneous Guttman scaling is not ubiquitous.…”

“…A second possibility is to consider an algorithmically oriented method, not relying on any a priori qualitative information about the process. But then, admittedly it has a higher chance of not satisfying the perfect homogeneous Guttman scaling [38,43]. For the full data set, let us define a and b as the minimum and maximum values assumed by the entire collection of x (j) i 's.…”

“…A test of the entropic brain hypothesis Here we discuss an idealized type of data, following the Guttman scaling [35,36]. In the homogeneous case [37] (see below), many exact results are known for the corresponding Pearson correlation matrix R hG [37,38]. As we will show, this allows to write an analytical expression for the von Neumann entropy S hG of R hG .…”

“…Frontal Pole (3,4) Insular Cortex (5,6) Superior Frontal Gyrus (7,8) Middle Frontal Gyrus (9,10) Inferior Frontal Gyrus, pars triangularis (11,12) Inferior Frontal Gyrus, pars opercularis (13,14) Precentral Gyrus (15,16) Temporal Pole (17,18) Superior Temporal Gyrus, anterior division (19,20) Superior Temporal Gyrus, posterior division (21,22) Middle Temporal Gyrus, anterior division (23,24) Middle Temporal Gyrus, posterior division (25,26) Middle Temporal Gyrus, temporooccipital part (27,28) Inferior Temporal Gyrus, anterior division* (29,30) Inferior Temporal Gyrus, posterior division (31,32) Inferior Temporal Gyrus, temporooccipital part (33,34) Postcentral Gyrus (35,36) Superior Parietal Lobule* (37,38) Supramarginal Gyrus, anterior division (39,40) Supramarginal Gyrus, posterior division (41,42) Angular Gyrus (43,44) Lateral Occipital Cortex, superior division (45,46) Later...…”