Norman Cliff

Orthogonal rotation to congruence

Two problems are considered. The first is that of rotating two factor solutions orthogonally to a position where corresponding factors are as similar as possible. A least-squares solution for transformations of the two factor matrices is developed. The second problem is that of rotating a factor matrix orthogonally to a specified target matrix. The solution to the second problem is related to the first. Applications are discussed.

INTERPRETATION OF CANONICAL ANALYSIS: ROTATED VS, UNROTATED SOLUTIONS

Orthogonal rotation of canonical variates is shown to preserve the major properties of the canonical solution and may increase its interpretability.

Answering Ordinal Questions with Ordinal Data Using Ordinal Statistics

It is argued that ordinal statistical methods are often more appropriate than their more common counterparts for three types of reasons: Conclusions from them will be unaffected by monotonic transformation of the variables, they are statistically more robust when used appropriately, and they often correspond more closely to the goals of the investigator. Kendalls tau (Kendall, 1970) and its counterpart delta are recommended as having wide I applicability and good statistical behavior. It is recommended that they be estimated as population parameters and their standard errors estimated form the data. Ways in which they can then substitute for Pearson correlations and mean comparisons in a number of research contexts are suggested.

Multidimensional comparisons of structures of vocally and facially expressed emotion

The cognitive structure of vocally expressed emotion was examined using a nonmetric multidimensional scaling analysis and a factor analysis of seven tone-of-voice rating scales. A canonical correlation analysis suggested that the same two dimensions clearly emerged from these analyses, pleasant-unpleasant and excitement. These two general dimensions were each highly related to single tone-of-voice rating scales, those for pleasantness and thinness, respectively. The two dimensions of pleasant-unpleasant and excitement were shown to correspond to the facial expression scales of pleasant-unpleasant and attention-rejection developed by Engen, Levy, and Schlosberg in 1958. When the stimuli for both vocally and facially expressed emotion were mapped together on a two-dimensional space, notable differences became apparent. An attempt to interpret some of these differences was made.

A theory of consistency of ordering generalizable to tailored testing

Measures of consistency and completeness of order relations derived from test-type data are proposed. The measures are generalized to apply to incomplete data such as from tailored testing. The measures are based on consideration of the items-plus-persons by items-plus-persons matrix as an adjacency matrix in which a 1 means that the row element, whether item or person, dominated the column element. From this the number of item-item and person-person dominance relations can be derived. Indices of consistency are developed which compare the number of dominance relations in a matrix to the number that would occur if it were perfectly consistent and to the number in a random matrix; relations to other indices are noted.

On the relation between undimensional judgments and multidimensional scaling

Abstract Three studies are described. They are based on the hypothesis that the individuals responses to members of a set of stimuli are based on his internalized conception of them and that this internalized conception can be revealed by multidimensional scaling analysis of his judgments of interstimulus similarity. Other responses were expected to be systematically related to the structure so revealed. In a study of facial expressions it was found that judgments of their intensity seemed to represent the expressions distance in two-dimensional space from a point representing no emotion. A study of semantic meaning showed that favorableness of adjectives was represented by a direction in 2-space and different kinds of favorableness were somewhat different directions. In a study of simulated air-raids, the degree of threat posed by a raid corresponded to a direction in the space and the action appropriate to counter it was an area of the space. The results were taken to support the validity of the hypothesis and the usefulness of multidimensional scaling.

Empirical Size, Coverage, and Power of Confidence Intervals for Spearman's Rho.

Several methods of constructing confidence intervals (CIs) for Spearmans rho were tested in a Monte Carlo investigation. A total of 2,000 samples of sizes 10, 50, and 200 were randomly drawn from bivariate normal populations with p, equal to .00, .29, .43, .58, .73, and .89. Each method for computing a 95% CI around p3 was evaluated with regard to size in the null case and power and coverage in non-null cases. Fishers z transformation of r, worked well provided N was not small and Ps was not too large. The CIs constructed using the variance estimate for product-moment correlations had coverages that were consistently too liberal. Kraemers method for establishing CIs produced coverages that were conservative. An empirical attempt to adjust the Fisher CI maintained Type I error rate near the nominal level in all cases with no loss of power. Arguments are made for the continued use of r, in behavioral research.

A generalization of the interpoint distance model

Coombs and Kao have conjectured that, given the distances between a set of points representing persons and another representing objects, factor analytic methods can be used to recover the locations of the persons and objects. In the present paper it is proved that the conjecture applies, with some qualifications, to squared distances rather than distances themselves. Possible applications to scaling and to inventory responses are discussed.

The “idealized individual” interpretation of individual differences in multidimensional scaling

It is shown that several variants of the Tucker-Messick points of view procedure are interpretable in terms of their “idealized individual” concepts. It is contended that the main function of the method is that it allows the computation ofseveral vectors of mean judgments instead of only one and that each of these may be used to stand for the actual judgments of a subgroup of subjects. Vectors that do not correspond to the judgments of any subjects are to be avoided. It is alleged that Rosss interpretation of “points of view” is quite different from that intended by Tucker and Messick.

Using the longitudinal guttman simplex as a basis for measuring growth

Many difficulties inherent in the measurement of growth stem from the use of traditional measurement methodologies. The longitudinal Guttman simplex (LGS), an alternative approach based on a model of growth, is discussed in this article. The LGS has several advantages over traditional methodology. First, interindividual differences in developmental rates are a part of the model. Second, the LGS procedure can easily handle any number of occasions of measurement. Third, the LGS is suited to nonlinear as well as linear monotonic growth. Fourth, a consistency index associated with the LGS methodology, CL, indicates the extent to which cumulative, unitary development characterizes a particular latent variable. Finally, and perhaps most important, because a model of the growth undergone by the latent variable being measured is incorporated in the LGS model the resulting instruments enjoy a high level of construct validity. The LGS is limited to cumulative, unitary development; additional measurement theories are needed for other kinds of development.