A note on Roy's largest root

The name Roy's largest root and similar names are used in practice to label two different but functionally related statistics—one proportional to anF, and the other, a squared canonical correlation. This note presents the logic that leads to the two formulations, states which statistic some popular statistical packages use, and shows the possible source of this inconsistency in the original work of Roy (1953) and Heck (1960).     

The mixed model for multivariate repeated measures: validity conditions and an approximate test

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate correction factor is derived. The correction factor generalizes Box's and can be used to construct an adjusted MMM test.I am grateful to an anonymous referee for carefully attending to the mathematical details of this paper.     

Stability of the internal structure of the millon clinical multiaxial inventory

The Millon Clinical Multiaxial Inventory (MCMI) has become increasingly popular in clinical use. Along with this, there has been more interest in the internal structure of the 20 scales of the original 175-item MCMI-I. The literature reports some agreement on four components, although both three- and five-component solutions have been reported. The degree of similarity of these components across populations remains arguable, as none of the previous studies have used quantitative measures of component similarity. The present study reports on two new samples of psychiatric patients, one of 82 cases from a general hospital and the other of 145 inpatients from a psychiatric hospital. It also reanalyzes the data from nine samples from the literature, using Tucker's coefficient of congruence and ten Berge's analysis of principal component weights (PCW). The congruence analyses showed good agreement of the first three components across samples and notably lower agreement for the fourth. The PCW analyses showed two major types of structure matrices. In the first, there was a large and dominant first component, with three smaller ones. In the second, the variance was distributed more evenly across the four components. The results are discussed in terms of the overlapping scales of the MCMI-I.This work was supported by funds from the Departments of Psychiatry and Psychology, The University of Western Ontario.     

A general model for preferential and triadic choice in terms of centralF distribution functions

A model for preferential and triadic choice is derived in terms of weighted sums of centralF distribution functions. This model is a probabilistic generalization of Coombs' (1964) unfolding model and special cases, such as the model of Zinnes and Griggs (1974), can be derived easily from it. This new form extends previous work by Mullen and Ennis (1991) and provides more insight into the same problem that they discussed.     

Evaluation of job candidates with missing information: Effects of attribute importance and interattribute correlation

How do people make evaluations when important items of information are missing? In the context of personnel evaluations or product evaluations, researchers have proposed that decision makers may predict the missing attribute based on its assumed relationship to attributes that are present. In addition, some researchers have suggested that there is a penalty for missing information. An experiment was conducted to investigate the influence of the importance of the missing attribute on the hypothesized inference effect. Hypothetical job candidates were evaluated based on one or two attributes. The correlation between the attributes was varied between groups of subjects as was the importance of one of the attributes. The pattern of ratings of candidates with missing information varied significantly with the correlation condition when the missing attribute was very important, but did not vary as much when the missing attribute was less important. The results were generally consistent with the predictions of the Inferred Information Model (Johnson and Levin, 1985). On average, candidates with missing information were rated lower than comparable candidates with complete information and the missing attribute at an average level.     

Caveats on a process-tracing measure and a remedy

In process-tracing studies, a frequently used index describes whether information selection behavior is attribute — or alternativewise. The performance of this index is investigated under the assumption that subjects select information in a random fashion. The results show that the index may lead to inaccurate conclusions regarding the information-acquisition strategy of a subject. In addition, a Monte Carlo study is conducted that examines the sensitivity of the index to strategy changes. An alternative index is derived and a latent-class model is proposed for a parsimonious representation of individual differences in information acquisition.     

A distance-based variety of nonlinear multivariate data analysis, including weights for objects and variables

In the distance approach to nonlinear multivariate data analysis the focus is on the optimal representation of the relationships between the objects in the analysis. In this paper two methods are presented for including weights in distance-based nonlinear multivariate data analysis. In the first method, weights are assigned to the objects while the second method is concerned with differential weighting of groups of variables. When each analysis variable defines a group the latter method becomes a variable weighting method. For objects the weights are assumed to be given; for groups of variables they may be given, or estimated. These weighting schemes can also be combined and have several important applications. For example, they make it possible to perform efficient analyses of large data sets, to use the distance-based variety of nonlinear multivariate data analysis as an addition to loglinear analysis of multiway contingency tables, and to do stability studies of the solutions by applying the bootstrap on the objects or the variables in the analysis. These and other applications are discussed, and an efficient algorithm is proposed to minimize the corresponding loss function.This study is funded by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the PIONEER project Subject Oriented Multivariate Analysis to the third author.     

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.     

Spatial,non-spatial and hybrid models for scaling

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.1976 Psychometric Society Presidential Address.While people too numerous to list here have contributed ideas, inspiration, and other help, I particularly wish to acknowledge the contributions of Sandra Pruzansky, without whom this paper could not have been written. I would also like to acknowledge the past contributions of my long-time colleague Jih-Jie Chang, without whose help I probably would not have beenasked to write it.     

Factor simplicity index and transformations

A scale-invariant index of factorial simplicity is proposed as a summary statistic for principal components and factor analysis. The index ranges from zero to one, and attains its maximum when all variables are simple rather than factorially complex. A factor scale-free oblique transformation method is developed to maximize the index. In addition, a new orthogonal rotation procedure is developed. These factor transformation methods are implemented using rapidly convergent computer programs. Observed results indicate that the procedures produce meaningfully simple factor pattern solutions.This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and research grants (MH24149 and DA01070) from the U. S. Public Health Service. The assistance of Andrew L. Comrey, Henry F. Kaiser, Bonnie Barron, Marion Hee, and several anonymous reviewers is gratefully acknowledged.