A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.

Some symmetric, invariant measures of multivariate association

Evaluative criteria for psychiatric classification.

Reliability is one of the most basic concepts of science and a prerequisite for scientific work. It is also the psychometrician’s favorite concept. Without reliable measurements, not even scientific empirical psychology could exist, let alone multivariate psychology or multivariate experimental psychology.

Multivariate Reliability Theory

As measured with the Psychopathy Checklist-Revised (PCL-R), psychopathy is a dimensional construct underpinned by 4 correlated factors: Interpersonal, Affective, Lifestyle, and Antisocial. Theorists and clinicians (e.g., Karpman and Arieti) have posited 2 variants of "primary" or "true" psychopathy, both distinct from so-called "secondary" or "pseudopsychopathy." We used latent profile analysis to determine if homogeneous classes exist within a sample of 1,451 male offenders with high PCL-R scores (≥ 27). The 4 PCL-R factors were the dependent variables for clustering. A solution with 3 latent classes showed a better fit to the data than did a unitary model without latent classes. Tentative labels for the latent classes are Manipulative (Latent Class 1 [LC1]), Aggressive (Latent Class 2 [LC2]), and Sociopathic (Latent Class 3 [LC3]). The latter class represented an antisocial group that lacked the emotional detachment observed in the other 2 groups. We propose that LC1 and LC2 reflect phenotypic variations on a theme of the traditional construct of psychopathy, and that LC3 is consistent with conceptions of antisocial personality disorder and sociopathy. Replication and external classification with an independent data set of 497 adult male offenders again yielded clearly separable clusters, as well as meaningful differences or trends among latent classes on education, intelligence, symptoms of antisocial personality disorder, and self-reported psychopathic traits and negative affect. The conceptualization of psychopathy in terms of manipulative and aggressive variants is consistent with clinical theory and is empirically grounded.

Variants of psychopathy in adult male offenders: A latent profile analysis.

A Cross-Cultural Exploration Into the Meaning of Achievement

A general index of reliability, termed “canonical reliability,” is developed for use with profiles, or more generally, for use with vectors of random variables. Canonical reliability is defined as the ratio of the average squared distance among true scores to the average squared distance among observed scores. Based on Mahalonobis distances, canonical reliability is shown to be consistent with multivariate analogues of parallel form correlations, squared correlation between true and observed scores, and an analysis of variance formulation. The index of reliability based on Cronbach and Gleser'sD
2 is also derived from the general formulation. A comparison of the Mahalonobis andD
2 approaches indicates that score vectors usingD
2 distances are more reliable; however, both methods of comparing profiles are useful depending on the nature of the information that is desired. Transforming the observed variables to independent canonical variates provides a basis for comparing profiles on maximally reliable profile dimensions. For illustrative purposes, profile reliability is calculated and interpreted for the WISC subscales for a 7 1/2 year age group.

Raanan Lipshitz

Papers

Measures of reliability for profiles and test batteries.