Peter H. Schönemann

Some new results on factor indeterminacy

Some relations between maximum likelihood factor analysis and factor indeterminacy are discussed. Bounds are derived for the minimum average correlation between equivalent sets of correlated factors which depend on the latent roots of the factor intercorrelation matrix ψ. Empirical examples are presented to illustrate some of the theory and indicate the extent to which it can be expected to be relevant in practice.

The minimum average correlation between equivalent sets of uncorrelated factors

A simplified proof of a lemma by Ledermann [1938], which lies at the core of the factor indeterminacy issue, is presented. It leads to a representation of an orthogonal matrixT, relating equivalent factor solutions, which is different from Ledermanns [1938] and Guttmans [1955].T is used to evaluate bounds on the average correlation between equivalent sets of uncorrelated factors. It is found that the minimum average correlation is independent of the data.

An individual difference model for the multidimensional analysis of preference data

A model for the analysis of paired comparison data is presented which combines features of the BTL-model with features of the Unfolding model. The model is metric, mathematically tractable, and has an exact algebraic solution. Since it is multidimensional and allows for individual differences, it is thought to be more realistic for some choice situations than either the Thurstone model or the BTL-model. No claim is made that the present model will be appropriate for all conceivable choice situations. Rather, it is argued that the fact that it is explicitly falsifiable is a point in its favor.

An algebraic solution for a class of subjective metrics models

It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carroll and Chang. In addition to being very simple, this algebraic solution also permits testing the constraints of these models explicitly. A numerical example is given.

Alternative measures of fit for the Schönemann-carroll matrix fitting algorithm

In connection with a least-squares solution for fitting one matrix,A, to another,B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure,e, and a refined similarity measure,es, which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix,B,e.g.,e(A,kB) ≠e(A,B) fork ≠ 1. Therefore, both measures are useless for answering questions of the type: “DoesA fitB better thanA fitsC?”. In this note two new measures of fit are suggested which do not depend upon the norms ofA andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.

Power Tables for Analysis of Variance.

In numerous applications of the analysis of variance it is necessary to compute the power of F tests having numerically high alpha (significance) levels. This article tabulates the power of F tests for numerator degrees of freedom, df = 1, 2, 3, 6, 9, 12; denominator df = 3, 6, 9, 12, 15, 20, 40, 60, 120; alternative hypotheses, phi = 2.(.2)3.0; and significance levels, alpha = .05, .10(.10),.50. The use of these tables is illustrated with a brief numerical example.

On models and muddles of heritability

One reason for the astonishing persistence of the IQ myth in the face of overwhelming prior and posterior odds against it may be the unbroken chain of excessive heritability claims for ‘intelligence’, which IQ tests are supposed to ‘measure’. However, if, as some critics insist, ‘intelligence’ is undefined, and Spearmans g is beset with numerous problems, not the least of which is universal rejection of Spearmans model by the data, then how can the heritability of ‘intelligence’ exceed that of milk production of cows and egg production of hens?The thesis of the present review paper is that the answer to this riddle has two parts: (a) the technical basis of heritability claims for human behavior is just as shaky as that of Spearmans g. For example, a once widely used ‘heritability estimate’ turns out to be mathematically invalid, while another such estimate, though mathematically valid, never fits any data; and (b) valid technical criticisms of flawed heritability claims typically are met with stubborn editorial resistance in the main stream journals, which tends to calcify such misinformation.

On the validity of indeterminate factor scores

A partition of the vector space of all deviation score vectors for fixed sample size N is used to show that the (indeterminate) factors of the factor model can always be constructed so as to predict any criterion perfectly, including all those that are entirely uncorrected with the observed variables.

A Solution to the Weighted Procrustes Problem in Which the Transformation is in Agreement with the Loss Function.

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.

Similarity of rectangles

Abstract Krantz and Tversky found that neither (log-) height ( y ) and width ( x ), nor area ( x + y ) and shape ( x − y ) qualify as “subjective dimensions of rectangles” because both pairs violate the decomposability condition for their dissimilarity data. However, the data suggest a nonlinear transformation of x , y into a pair of subjective dimensions u ( x , y ), v ( x , y ) for which decomposability should be approximately satisfied. An explicit statement of this mapping is given.