On equivariance and invariance of standard errors in three exploratory factor models

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standardz-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect onz-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.We gratefully acknowledge the help of the Associate Editor and three referees whose constructive comments lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.     

On the greatest lower bound to reliability

Certain ambiguities in a recent paper on the computation and statistics of the greatest lower bound are clarified.     

An 8-year study of multiple influences on drug use and drug use consequences

Multiple long-term influences on young adult drug use and abuse were tested within an interactionist perspective using latent-variable causal models. Intrapersonal influences included early drug use and social conformity. Proximal interpersonal influences were captured by perceived peer drug use, perceived adult drug use, and family disruption. More distal influences included perceptions of community approval or disapproval for drug use. Outcome measures included not only use of drugs but also disruptive drug use (getting high at work or school) and self-admitted problems with drugs. Data were obtained from 654 participants at three equally spaced time points during an 8-year longitudinal study that began when the subjects were in junior high school. Confirmatory factor analyses were used to test the adequacy of the hypothesized measurement model. Subsequently, a structural equation, or path model, was used to examine the across-time relations among the latent and manifest variables. Stability effects were found for all repeatedly measured latent variables across the 4-year spans. Social conformity strongly influenced other latent variables across time. Early drug use and perceived adult drug use were prominent predictors of young adult drug use, whereas prior drug use predicted disruptive drug use and a lack of social conformity predicted problems with drug use. Early adult alcohol use predicted later disruptive drug use and problems with drug use. Perceived adult drug use generally influenced later alcohol use, whereas peer drug use influenced later cannabis and hard-drug use. Implications for prevention and treatment using a multicausal interactionist perspective are discussed.     

Cigarette Smoking,Academic Lifestyle,and Social Impact Efficacy: An Eight-Year Study from Early Adolescence to Young Adulthood1

Two hundred and twenty-one males and 518 females were followed for 8 years from early adolescence. A causal model was proposed that related five constructs measured in adolescence to four corresponding constructs measured 8 years later. The predictor constructs included smoking involvement, academic lifestyle orientation, emotional well-being, social impact efficacy, and peer smoking behavior. Academic lifestyle orientation was found to be a central organizing construct, with the strongest within-time and across-time correlations with other constructs. The relatively weak causal influences of teenage smoking and teenage social impact efficacy suggested an explanation for the limited impact of past drug abuse prevention programs. Policy implications are discussed.     

Can test statistics in covariance structure analysis be trusted?

Covariance structure analysis uses chi 2 goodness-of-fit test statistics whose adequacy is not known. Scientific conclusions based on models may be distorted when researchers violate sample size, variate independence, and distributional assumptions. The behavior of 6 test statistics is evaluated with a Monte Carlo confirmatory factor analysis study. The tests performed dramatically differently under 7 distributional conditions at 6 sample sizes. Two normal-theory tests worked well under some conditions but completely broke down under other conditions. A test that permits homogeneous nonzero kurtoses performed variably. A test that permits heterogeneous marginal kurtoses performed better. A distribution-free test performed spectacularly badly in all conditions at all but the largest sample sizes. The Satorra-Bentler scaled test statistic performed best overall.     

The effect of agency and communality on self-esteem: Gender differences in longitudinal data

Prior studies have noted a positive relationship between masculinity (M) and self-esteem (S-E) for both males and females. However, during adolescence, males and females have been described as experiencing a divergence into two domains: agency for males and communality for females. The present study hypothesized that, contrary to the cross-sectional positive correlational relationship between M and S-E for both males and females, an agentic orientation during adolescence would predict heightened S-E at young adulthood for males only. Conversely, a communal orientation was hypothesized to predict heightened S-E for females only at young adulthood. Using structural equation models with latent variables to analyze data from a community sample of 654 subjects (M=192, F=462; White=432; Black=97; Hispanic=64; Asian=61), these hypotheses were confirmed. In addition, as found in typical masculinity-femininity (M-F) studies, both males and females evidenced a positive relationship between S-E and M, as measured on the Personal Attributes Questionnaire and essentially no relationship between F and S-E. Possible reasons for this complexity and implications for future research are discussed.     

Linear structural equations with latent variables

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.     

Statistical aspects of a three-mode factor analysis model

A special case of Bloxom's version of Tucker's three-mode model is developed statistically. A distinction is made between modes in terms of whether they are fixed or random. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. The identification problem is discussed, and unknown parameters of the model are estimated by a weighted least squares method based upon a Gauss-Newton algorithm. A goodness-of-fit statistic is presented. An example based upon self-report and peer-report measures of personality shows that the model is applicable to real data. The model represents a generalization of Thurstonian factor analysis; weighted least squares estimators and maximum likelihood estimators of the factor model can be obtained using the proposed theory.This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and a research grant (DA01070) from the U. S. Public Health Service. The very helpful comments of several anonymous reviewers are gratefully acknowledged.     

Function invariant and parameter scale-free transformation methods

The parameter matrices of factor analysis and principal component analysis are arbitrary with respect to the scale of the factors or components; typically, the scale is fixed so that the factors have unit variance. Oblique transformations to optimize an objective statement of a principle such as simple structure or factor simplicity yield arbitrary solutions, unless the criterion function is invariant with respect to the scale of the factors, or the parameter matrix is scale free with respect to the factors. Criterion functions that are factor scale-free have a number of invariance characteristics, such as being equally applicable to primary pattern or reference structure matrices. A scale-invariant simple structure function of previously studied function components is defined. First and second partial derivatives are obtained, and Newton-Raphson iterations are utilized. The resulting solutions are locally optimal and subjectively pleasing.Aspects of this paper were presented at the 1970 and 1974 annual meetings, Society of Multivariate Experimental Psychology, and the 1975 annual meeting, Psychometric Society. 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 Bonnie Barron, Sik-Yum Lee, and several extremely helpful reviewers is gratefully acknowledged.     

Alpha,Dimension-Free,and Model-Based Internal Consistency Reliability

As pointed out by Sijtsma (in press), coefficient alpha is inappropriate as a single summary of the internal consistency of a composite score. Better estimators of internal consistency are available. In addition to those mentioned by Sijtsma, an old dimension-free coefficient and structural equation model based coefficients are proposed to improve the routine reporting of psychometric internal consistency. The various ways to measure internal consistency are also shown to be appropriate to binary and polytomous items. Research supported in part by grants DA00017 and DA01070 from the National Institute on Drug Abuse. This paper is based in part on Bentler (2003).