Generating Nonnormal Multivariate Data Using Copulas: Applications to SEM

This article develops a procedure based on copulas to simulate multivariate nonnormal data that satisfy a prespecified variance-covariance matrix. The covariance matrix used can comply with a specific moment structure form (e.g., a factor analysis or a general structural equation model). Thus, the method is particularly useful for Monte Carlo evaluation of structural equation models within the context of nonnormal data. The new procedure for nonnormal data simulation is theoretically described and also implemented in the widely used R environment. The quality of the method is assessed by Monte Carlo simulations. A 1-sample test on the observed covariance matrix based on the copula methodology is proposed. This new test for evaluating the quality of a simulation is defined through a particular structural model specification and is robust against normality violations.     

A Third Moment Adjusted Test Statistic for Small Sample Factor Analysis

Goodness-of-fit testing in factor analysis is based on the assumption that the test statistic is asymptotically chi-square, but this property may not hold in small samples even when the factors and errors are normally distributed in the population. Robust methods such as Browne's (1984) asymptotically distribution-free method and Satorra Bentler's (1988, 1994) mean scaling statistic were developed under the presumption of nonnormality in the factors and errors. This article finds new application to the case where factors and errors are normally distributed in the population but the skewness of the obtained test statistic is still high due to sampling error in the observed indicators. An extension of Satorra Bentler's statistic is proposed that not only scales the mean but also adjusts the degrees of freedom based on the skewness of the obtained test statistic in order to improve its robustness under small samples. A simple simulation study shows that this third moment adjusted statistic asymptotically performs on par with previously proposed methods and at a very small sample size offers superior Type I error rates under a properly specified model. Data from Mardia, Kent, and Bibby's (1980) study of students tested for their ability in 5 content areas that were either open or closed book were used to illustrate the real-world performance of this statistic.     

On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes. The authors thank an associate editor and three referees for their constructive comments, which led to an improved version of the paper. This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of Notre Dame faculty research grant.     

Drug use,educational aspirations,and work force involvement: The transition from adolescence to young adulthood

The impact of high school drug use and academic potential on high school outcome (graduate or dropout) and young adult work force involvement, college involvement, and educational aspirations was examined. Frequency of drug use, grade point average, and educational plans were assessed for 479 adolescents while in high school. Four years later this same group of individuals, now in their early 20s, reported their current level of drug use, present life involvement on a variety of measures, and whether they had ever graduated from high school. Results indicate that high school graduation is predicted from a lack of cigarette and hard drug use and the presence of high academic aspirations. Using latent variable causal models, it was found that high school Academic Potential and Drug Use were significantly correlated in a negative direction. Across-time analyses indicate that high school Academic Potential significantly predicted young adult Educational Aspirations, College Involvement, and college attendance. High school Drug Use significantly predicted young adult Drug Use, a lack of College Involvement, and increased Work Force Involvement. These results do not support a psychogenic hypothesis of drug use and academic potential but rather confirm an impaired abilities or general deviance interpretation.     

Substance use and ethnicity: differential impact of peer and adult models

Modeling of a specific behavior by peers and adults is an important influence on an adolescent's choice to engage in that behavior. This study examined the impact of perceived peer and adult substance use on reported self-use within groups of black, Hispanic, Asian, and white teenagers. Effects were studied for four types of drugs: beer/wine, hard liquor, marijuana, and pills. In general, Asians and blacks reported the least self-use, whites reported the most peer models, and blacks reported the most adult models. Differential patterns of increase in use and perceived use by grade level were found across the four ethnic groups. Overall, perceived peer use of beer/wine, liquor, and marijuana was associated more with self-use than was perceived adult use. For pill use, only whites reported significantly greater influence of peers over adults on their self-use. For all substances, blacks reported the least impact of peer use on self-use relative to the three other groups. In regard to the impact of adult use on self-use, blacks and Hispanics reported the lowest for beer or wine; blacks and Asians reported the lowest for marijuana; and whites reported the highest for pills. These results are discussed in the context of differential vulnerability to modeling and vicarious learning.     

A three-stage estimation procedure for structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. A computationally efficient three-stage estimator of the thresholds and the covariance structure parameters, based on partition maximum likelihood and generalized least squares estimation, is proposed. An example is presented to illustrate the method.This research was supported in part by a research grant DA01070 from the U.S. Public Health Service. The production assistance of Julie Speckart is gratefully acknowledged.     

Problems with EM algorithms for ML factor analysis

Rubin and Thayer recently presented equations to implement maximum likelihood (ML) estimation in factor analysis via the EM algorithm. They present an example to demonstrate the efficacy of the algorithm, and propose that their recovery of multiple local maxima of the ML function “certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.” It is shown here, in contrast, that these second derivatives verify that Rubin and Thayer did not find multiple local maxima as claimed. The only known maximum remains the one found by Jöreskog over a decade earlier. The standard errors obtained from the second derivatives and the Fisher information matrix thus remain appropriate where ML assumptions are met. The advantages of the EM algorithm over other algorithms for ML factor analysis remain to be demonstrated.     

Tests for linear trend in the smallest eigenvalues of the correlation matrix

A test for linear trend among a set of eigenvalues of a correlation matrix is developed. As a technical implementation of Cattell's scree test, this is a generalization of Anderson's test for the equality of eigenvalues, and extends Bentler and Yuan's work on linear trends in eigenvalues of a covariance matrix. The power of minimumx ² and maximum likelihood ratio tests are compared. Examples show that the linear trend hypothesis is more realistic than the standard hypothesis of equality of eigenvalues, and that the hypothesis is compatible with standard decisions on the number of factors or components to retain in data analysis.This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017. The assistance of Maia Berkane and several anonymous reviewers is gratefully acknowledged.     

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.