Some extensions of image analysis

A general model for describing the interrelations of common scores is derived. Guttman's image analysis is a special case. In addition, a new factor model based upon estimation of the person product-moment matrix is described for the case in which the number of variables exceeds the number of persons.     

Tests of homogeneity of means and covariance matrices for multivariate incomplete data

Existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns. The likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects. A generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples. Three factors were varied for a simulation: number of variables, percent missing completely at random, and sample size. One thousand data sets were simulated for each condition. The generalized least squares test of homogeneity of means performed close to an ideal Type I error rate for most of the conditions. The generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also.Preliminary results on this research were presented at the 1999 Western Psychological Association convention, Irvine, CA, and in the UCLA Statistics Preprint No. 265 (http://www.stat.ucla.edu). The assistance of Ke-Hai Yuan and several anonymous reviewers is gratefully acknowledged.     

On robusiness of the normal-theory based asymptotic distributions of three reliability coefficient estimates

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

On the relations among regular, equal unique variances, and image factor analysis models

We investigate under what conditions the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. We show that the two models will give similar matrices of factor loadings if Schneeweiss' condition, that the difference between the largest and the smallest value of unique variances is small relative to the sizes of the column sums of squared factor loadings, holds. Furthermore, we generalize our results and discus the conditions under which the matrix of factor loadings from the regular factor analysis model will be well approximated by the matrix of factor loadings from Jöreskog's image factor analysis model. Especially, we discuss Guttman's condition (i.e., the number of variables increases without limit) for the two models to agree, in relation with the condition we have shown, and conclude that Schneeweiss' condition is a generalization of Guttman's condition. Some implications for practice are discussed.Kentaro Hayashi is a visiting Assistant Professor, Department of Mathematics, Bucknell University, Lewisburg PA 17837, and Peter M. Bentler is Professor, Departments of Psychology and Statistics, University of California, Los Angeles CA 90095-1563. (Emails: Khayashi@bucknell.edu, bentler@ucla.edu) Parts of this paper were discussed in a session on Factor Analysis (J. ten Berge, Chair) at the IFCS-98 International Conference, Rome, July, 1998. This work was supported by National Institute on Drug Abuse grant DA 01070. The authors thank Professors Hans Schneeweiss and Ke-Hai Yuan, and four anonymous referees, for their invaluable comments which led to an improved version of this paper.     

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.