Jos M. F. ten Berge

Tucker's Congruence Coefficient as a Meaningful Index of Factor Similarity

When Tuckers congruence coefficient is used to assess the similarity of factor interpretations, it is desirable to have a critical congruence level less than unity that can be regarded as indicative of identity of the factors. The literature only reports rules of thumb. The present article repeats and broadens the approach used in the study by Haven and ten Berge (1977). It aims to find a critical congruence level on the basis of judgments of factor similarity by practitioners of factor analysis. Our results suggest that a value in the range .85-.94 corresponds to a fair similarity, while a value higher than .95 implies that the two factors or components compared can be considered equal.

PARAFAC2 - Part I. A direct fitting algorithm for the PARAFAC2 model

PARAFAC is a generalization of principal component analysis (PCA) to the situation where a set of data matrices is to be analysed. If each data matrix has the same row and column units, the resulting data are three‐way data and can be modelled by the PARAFAC1 model. If each data matrix has the same column units but different (numbers of) row units, the PARAFAC2 model can be used. Like the PARAFAC1 model, the PARAFAC2 model gives unique solutions under certain mild assumptions, whereas it is less severely constrained than PARAFAC1. It may therefore also be used for regular three‐way data in situations where the PARAFAC1 model is too restricted. Usually the PARAFAC2 model is fitted to a set of matrices with cross‐products between the column units. However, this model‐fitting procedure is computationally complex and inefficient. In the present paper a procedure for fitting the PARAFAC2 model directly to the set of data matrices is proposed. It is shown that this algorithm is more efficient than the indirect fitting algorithm. Moreover, it is more easily adjusted so as to allow for constraints on the parameter matrices, to handle missing data, as well as to handle generalizations to sets of three‐ and higher‐way data. Furthermore, with the direct fitting approach we also gain information on the row units, in the form of ‘factor scores’. As will be shown, this elaboration of the model in no way limits the feasibility of the method. Even though full information on the row units becomes available, the algorithm is based on the usually much smaller cross‐product matrices only. Copyright

Orthogonal procrustes rotation for two or more matrices

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gowers method of generalized Procrustes analysis are suggested.

The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality

To assess the reliability of congeneric tests, specifically designed reliability measures have been proposed. This paper emphasizes that such measures rely on a unidimensionality hypothesis, which can neither be confirmed nor rejected when there are only three test parts, and will invariably be rejected when there are more than three test parts. Jackson and Agunwambas (1977) greatest lower bound to reliability is proposed instead. Although this bound has a reputation for overestimating the population value when the sample size is small, this is no reason to prefer the unidimensionality-based reliability. Firstly, the sampling bias problem of the glb does not play a role when the number of test parts is small, as is often the case with congeneric measures. Secondly, glb and unidimensionality based reliability are often equal when there are three test parts, and when there are more test parts, their numerical values are still very similar. To the extent that the bias problem of the greatest lower bound does play a role, unidimensionality-based reliability is equally affected. Although unidimensionality and reliability are often thought of as unrelated, this paper shows that, from at least two perspectives, they act as antagonistic concepts. A measure, based on the same framework that led to the greatest lower bound, is discussed for assessing how close is a set of variables to unidimensionality. It is the percentage of common variance that can be explained by a single factor. An empirical example is given to demonstrate the main points of the paper.

On uniqueness in CANDECOMP/PARAFAC

One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP) has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of thek-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskals sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP solution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It will be shown that Kruskals condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher.

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.

A family of association coefficients for metric scales

Four types of metric scales are distinguished: the absolute scale, the ratio scale, the difference scale and the interval scale. A general coefficient of association for two variables of the same metric scale type is developed. Some properties of this general coefficient are discussed. It is shown that the matrix containing these coefficients between any number of variables is Gramian. The general coefficient reduces to specific coefficients of association for each of the four metric scales. Two of these coefficients are well known, the product-moment correlation and Tuckers congruence coefficient. Applications of the new coefficients are discussed.

Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices in two or more populations

Millsap and Meredith (1988) have developed a generalization of principal components analysis for the simultaneous analysis of a number of variables observed in several populations or on several occasions. The algorithm they provide has some disadvantages. The present paper offers two alternating least squares algorithms for their method, suitable for small and large data sets, respectively. Lower and upper bounds are given for the loss function to be minimized in the Millsap and Meredith method. These can serve to indicate whether or not a global optimum for the simultaneous components analysis problem has been attained.

Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the three-way array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal.

Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set ofn×n matrices that have a rank less thann has zero volume. Kruskal pointed out that a 2×2×2 array has rank three or less, and that the subsets of those 2×2×2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskals results to 2×n×n arrays. Incidentally, it is shown that twon ×n matrices can be diagonalized simultaneously with positive probability.