Sensitivity analysis in factor analysis: Difference between using covariance and correlation matrices

Influence curves of some parameters under various methods of factor analysis have been given in the literature. These influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis. The differences between the influence curves based on the covariance and the correlation matrices are derived in this paper. Simple formulas for the differences of the influence curves, based on the two matrices, for the unique variance matrix, factor loadings and some other parameter are obtained under scale-invariant estimation methods, though the influence curves themselves are in complex forms.The authors are most grateful to the referees, the Associate Editor, the Editor and Raymond Lam for helpful suggestions for improving the clarity of the paper.     

Quantitative analysis of qualitative data

This paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach, Alternating Least Squares and Optimal Scaling. Using these two principles, my colleagues and I have extended a variety of analysis procedures originally proposed for quantitative (interval or ratio) data to qualitative (nominal or ordinal) data, including additivity analysis and analysis of variance; multiple and canonical regression; principal components; common factor and three mode factor analysis; and multidimensional scaling. The approach has two advantages: (a) If a least squares procedure is known for analyzing quantitative data, it can be extended to qualitative data; and (b) the resulting algorithm will be convergent. Three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented.Presented as the Presidential Address to the Psychometric Society's Annual meeting, May, 1981. I wish to express my deep appreciation to Jan de Leeuw and Yoshio Takane. Our team effort was essential for the developments reported in this paper. Without this effort the present paper would not exist. Portions of this paper appear in Lantermann, E. D. & Feger, H. (Eds.)Similarity and Choice, Hans Huber, Vienna, 1980. The present paper benefits greatly from a set of detailed comments made by Joseph Kruskal on the earlier paper.     

Influence in canonical correlation analysis

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.We thank the Editor and the anonymous reviewers for their helpful comments. This research was carried out with the financial support of the Italian Ministry of the University and the National Research Council.     

Redundancy analysis an alternative for canonical correlation analysis

A component method is presented maximizing Stewart and Love's redundancy index. Relationships with multiple correlation and principal component analysis are pointed out and a rotational procedure for obtaining bi-orthogonal variates is given. An elaborate example comparing canonical correlation analysis and redundancy analysis on artificial data is presented.A Fortran IV program for the method of redundancy analysis described in this paper can be obtained from the author upon request.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

Part and bipartial canonical correlation analysis

Extending the definitions of part and bipartial correlation to sets of variates, the notion of part and bipartial canonical correlation analysis are developed and illustrated.     

The role of correlation in analysis of variance

A test of the significance of a row or column agent in an analysis of variance may be expressed in the form of correlation between the agent and the variate. A test of the significance of interaction variance may be expressed in the form of correlation between the agents. These expressions are principally of theoretical interest in that the degree of significance in anF test or the value of a correlation coefficient may be controlled at will, or inadvertently, within certain limits.This study was made possible by the Bureau of Psychological Services, Institute for Human Adjustment, Horace H. Rackham School of Graduate Studies, University of Michigan.     

EMG waveform duration: A validation method for the surface electromyogram

The electromyogram (EMG) recorded with surface electrodes was examined using a triggered oscilloscope technique. When all but the high-amplitude waveforms in a given segment of activity were excluded from analysis, it was found that the duration of the remaining waveforms was specific to the muscle that produced them. That is, waveforms of a given duration were recorded whenever conventional indices-observable contraction, joint movement. needle recordings-showed activity in a specific muscle. This led to the suggestion that waveform duration might be used to validate the surface EMG. The suggestion. which is limited by the need to control posture and electrode placement, both of which may influence waveform duration, was illustrated in application to psychological research.     

Quantitative measurement of observed workload in the analysis of aircrew performance

A methodology for the quantitative evaluation of observed workload was proposed. The model was designed to provide point estimations of observed workload at any time during the performance of a set of tasks. The model was also designed to provide information about the task-scheduling strategies used to complete a set of tasks. The proposed model was then tested with data from a full mission flight simulation. The model predictions correlated significantly with expert ratings of workload management made during the flight simulation. The model was also able to distinguish between low- and high-performing aircrews when performance was based on the number of errors made during flight simulations.     

Generalizations of the partial,part and bipartial canonical correlation analysis

Theg ₁- andg ₂-bipartial canonical correlation analyses are developed as generalizations of the partial, part, and bipartial canonical correlation analysis. Illustrative examples are provided.     

An oscillatory correlation model of visual motion analysis

We describe and evaluate a model of motion perception based on the integration of information from two parallel pathways: a motion pathway and a luminance pathway. The motion pathway has two stages. The first stage measures and pools local motion across the input animation sequence and assigns reliability indices to these pooled measurements. The second stage groups locations on the basis of these measurements. In the luminance pathway, the input scene is segmented into regions on the basis of similarities in luminance. In a subsequent integration stage, motion and luminance segments are combined to obtain the final estimates of object motion. The neural network architecture we employ is based on LEGION (locally excitatory globally inhibitory oscillator networks), a scheme for feature binding and region labeling based on oscillatory correlation. Many aspects of the model are implemented at the neural network level, whereas others are implemented at a more abstract level. We apply this model to the computation of moving, uniformly illuminated, two-dimensional surfaces that are either opaque or transparent. Model performance replicates a number of distinctive features of human motion perception.     

Quantitative analysis of human movement synergies: constructive pattern analysis for gait

To record three-dimensional coordinates of the joints from normal human subjects during locomotion, we used a digital motion analysis system (ELITE). Recordings were obtained under several different conditions, which included normal walking and stepping over obstacles. Principal component analysis was used to analyze coordinate data after conversion of the data to segmental angles. This technique gave a stable summary of the redundancy in gait kinematic data in the form of reduced variables (principal components). By modeling the shapes of the phase plots of reduced variables (distortion analysis) and using a limited number of model parameters, good resolution was obtained between subtly different conditions. Hence, it was possible to accurately resolve small distributed changes in gait patterns within subjects. These methods seem particularly suited to longitudinal studies in which relevant movement features are not known a priori. Assumptions and neurophysiological applications are discussed.     

A non-parametric test of correlation using rank orders within subgroups

Kendall's rank order test for association between two variables is generalized to the case where the total sample is made up of several subgroups and the data on one or both variables consist of the rank order within each subgroup. The test involves no assumptions concerning scales of measurement, shapes of distributions, or relative level of excellence or amount of variability of the different subgroups. Two empirical examples indicate that the normal approximation to the exact test of significance can be considered adequate for most practical situations. Special consideration is given to the case of tied ranks. If ties occur in but one variable within any given subgroup, only a slight modification in procedure is needed. Extensive ties in both variables within subgroups lead to difficulties in determining the appropriate correction for continuity.     

Methodological and computational considerations for multiple correlation analysis

The squared multiple correlation coefficient has been widely employed to assess the goodness-of-fit of linear regression models in many applications. Although there are numerous published sources that present inferential issues and computing algorithms for multinormal correlation models, the statistical procedure for testing substantive significance by specifying the nonzero-effect null hypothesis has received little attention. This article emphasizes the importance of determining whether the squared multiple correlation coefficient is small or large in comparison with some prescribed standard and develops corresponding Excel worksheets that facilitate the implementation of various aspects of the suggested significance tests. In view of the extensive accessibility of Microsoft Excel software and the ultimate convenience of general-purpose statistical packages, the associated computer routines for interval estimation, power calculation, a nd samplesize determination are alsoprovided for completeness. The statistical methods and available programs of multiple correlation analysis described in this article purport to enhance pedagogical presentation in academic curricula and practical application in psychological research.     

Multi-attribute utility analysis using the promes approach

Traditional utility analyses have focused on the economic value of increased job performance. While providing a great deal of information to decision makers, such approaches fail to account for other potentially critical outcomes of implementing a selection system (e.g. legal exposure). This paper proposes a new method for incorporating multiple outcomes in utility analysis. An example shows how to incorporate economic gains due to job performance, legal exposure, effect on protected groups, and organizational image in the decision to implement a new selection system. Integrating these four outcomes into utility analysis captures managerial policy that will allow examination of the process of using utility information to aid decision making. In addition, the method is amenable to use of sensitivity, breakeven, and Monte Carlo analyses advocated by utility researchers (Boudreau, 1991).