Effect of outliers on estimators and tests in covariance structure analysis.

A small proportion of outliers can distort the results based on classical procedures in covariance structure analysis. We look at the quantitative effect of outliers on estimators and test statistics based on normal theory maximum likelihood and the asymptotically distribution-free procedures. Even if a proposed structure is correct for the majority of the data in a sample, a small proportion of outliers leads to biased estimators and significant test statistics. An especially unfortunate consequence is that the power to reject a model can be made arbitrarily--but misleadingly--large by inclusion of outliers in an analysis.

Robust transformation with applications to structural equation modelling.

Data sets in social and behavioural sciences are seldom normal. Influential cases or outliers can lead to inappropriate solutions and problematic conclusions in structural equation modelling. By giving a proper weight to each case, the influence of outliers on a robust procedure can be minimized. We propose using a robust procedure as a transformation technique, generating a new data matrix that can be analysed by a variety of multivariate methods. Mardia's multivariate skewness and kurtosis statistics are used to measure the effect of the transformation in achieving approximate normality. Since the transformation makes the data approximately normal, applying a classical normal theory based procedure to the transformed data gives more efficient parameter estimates. Three procedures for parameter evaluation and model testing are discussed. Six examples illustrate the various aspects with the robust transformation.

On measures of explained variance in nonrecursive structural equation models.

Whereas measures of explained variance in a regression and an equation of a recursive structural equation model can be simply summarized by a standard R2 measure, this is not possible in nonrecursive models in which there are reciprocal interdependencies among variables. This article provides a general approach to defining variance explained in latent dependent variables of nonrecursive linear structural equation models. A new method of its estimation, easily implemented in EQS or LISREL and available in EQS 6, is described and illustrated.

Structural Equation Modeling with Small Samples: Test Statistics.

Structural equation modeling is a well-known technique for studying relationships among multivariate data. In practice, high dimensional nonnormal data with small to medium sample sizes are very common, and large sample theory, on which almost all modeling statistics are based, cannot be invoked for model evaluation with test statistics. The most natural method for nonnormal data, the asymptotically distribution free procedure, is not defined when the sample size is less than the number of nonduplicated elements in the sample covariance. Since normal theory maximum likelihood estimation remains defined for intermediate to small sample size, it may be invoked but with the probable consequence of distorted performance in model evaluation. This article studies the small sample behavior of several test statistics that are based on maximum likelihood estimator, but are designed to perform better with nonnormal data. We aim to identify statistics that work reasonably well for a range of small sample sizes and distribution conditions. Monte Carlo results indicate that Yuan and Bentler's recently proposed F-statistic performs satisfactorily.

Normal theory based test statistics in structural equation modelling.

Even though data sets in psychology are seldom normal, the statistics used to evaluate covariance structure models are typically based on the assumption of multivariate normality. Consequently, many conclusions based on normal theory methods are suspect. In this paper, we develop test statistics that can be correctly applied to the normal theory maximum likelihood estimator. We propose three new asymptotically distribution-free (ADF) test statistics that technically must yield improved behaviour in samples of realistic size, and use Monte Carlo methods to study their actual finite sample behaviour. Results indicate that there exists an ADF test statistic that also performs quite well in finite sample situations. Our analysis shows that various forms of ADF test statistics are sensitive to model degrees of freedom rather than to model complexity. A new index is proposed for evaluating whether a rescaled statistic will be robust. Recommendations are given regarding the application of each test statistic.

Use of structural equation modeling to test the construct validity of the SF-36 Health Survey in ten countries: results from the IQOLA Project. International Quality of Life Assessment.

A crucial prerequisite to the use of the SF-36 Health Survey in multinational studies is the reproduction of the conceptual model underlying its scoring and interpretation. Structural equation modeling (SEM) was used to test these aspects of the construct validity of the SF-36 in ten IQOLA countries: Denmark, France, Germany, Italy, the Netherlands, Norway, Spain, Sweden, the United Kingdom, and the United States. Data came from general population surveys fielded to gather normative data. Measurement and structural models developed in the United States were cross-validated in random halves of the sample in each country. SEM analyses supported the eight first-order factor model of health that underlies the scoring of SF-36 scales and two second-order factors that are the basis for summary physical and mental health measures. A single third-order factor was also observed in support of the hypothesis that all responses to the SF-36 are generated by a single, underlying construct--health. In addition, a third second-order factors, interpreted as general well-being, was shown to improve the fit of the model. This model (including eight first-order factors, three second-order factors, and one third-order factor) was cross-validated using a holdout sample within the United States and in each of the nine other countries. These results confirm the hypothesized relationships between SF-36 items and scales and justify their scoring in each country using standard algorithms. Results also suggest that SF-36 scales and summary physical and mental health measures will have similar interpretations across countries. The practical implications of a third second-order SF-36 factor (general well-being) warrant further study.

Robust mean and covariance structure analysis.

Covariance structure analysis is used to evaluate hypothesized influences among unmeasured latent and observed variables. As implemented, it is not robust to outliers and bad data. Several robust methods in model fitting and testing are proposed. These include direct estimation of M-estimators of structured parameters and a two-stage procedure based on robust M- and S-estimators of population covariances. The large sample properties of these estimators are obtained. The equivalence between a direct M-estimator and a two-stage estimator based on an M-estimator of population covariance is established when sampling from an elliptical distribution. Two test statistics are presented in judging the adequacy of a hypothesized model; both are asymptotically distribution free if using distribution free weight matrices. So these test statistics possess both finite sample and large sample robustness. The two-stage procedures can be easily adapted into standard software packages by modifying existing asymptotically distribution free procedures. To demonstrate the two-stage procedure, S-estimator and M-estimators under different weight functions are calculated for some real data sets.

Pathways Linking Treatment Intensity and Psychosocial Outcomes among Adult Survivors of Childhood Leukemia.

To determine the pathways between treatment intensity (age at diagnosis, dosage of chemotherapy [intrathecal methotrexate; IT-MTX] and cranial radiation [CRT]) and various psychosocial outcomes, review of medical records and structured interviews were carried out in 510 adult survivors of childhood leukemia. Structural equation modeling revealed that higher treatment intensity during childhood (indicated by treatment with high-dose CRT, low-dose IT-MTX, and adjusted by younger age at diagnosis) predicted more health- compromising behaviors as adults through lower educational achievement. Additionally, higher childhood treatment intensity predicted current negative mood both directly and via changes in perceived limitations. The present study's findings suggest that higher treatment intensity during childhood may serve as a risk factor for adult survivors' health-compromising behaviors through neuropsychological deficits that arise from cancer treatment.

Maximum likelihood estimation in covariance structure analysis with truncated data.

We study an estimation procedure for maximum likelihood estimation in covariance structure analysis with truncated data, and obtain the statistical properties of the estimator as well as a test of the model structure. Truncated data with and without knowledge about the number of unmeasured observations are both considered. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, which requires only first derivatives, is proposed to obtain the maximum likelihood estimates. We illustrate the statistics and parameter estimates by a fictitious example. The maximum likelihood method is compared to an alternative two-stage method.

Test of linear trend in eigenvalues of a covariance matrix with application to data analysis.

Principal component analysis and factor analysis are the most widely used tools for dimension reduction in data analysis. Both methods require some good criterion to judge the number of dimensions to be kept. The classical method focuses on testing the equality of eigenvalues. As real data hardly have this property, practitioners turn to some ad hoc criterion in judging the dimensionality of their data. One such popular method, the 'scree test' or 'scree plot' as described in many texts and statistical programs, is based on the trend in eigenvalues of sample covariance (correlation) matrix. The principal components or common factors corresponding to eigenvalues which exhibit a slow linear decrease are discarded in further data analysis. This paper develops a formal statistical test for the 'scree plot'. A special case of this test is the classical test for equality of eigenvalues which has been suggested in several texts as the criterion to decide the number of principal components to retain. Comparisons between equality of eigenvalues and the slow linear decrease in eigenvalues on some classical examples support the hypothesis of slow linear decrease. A physical background to such a phenomenon is also suggested.

Covariance structure analysis: statistical practice, theory, and directions.

Although covariance structure analysis is used increasingly to analyze nonexperimental data, important statistical requirements for its proper use are frequently ignored. Valid conclusions about the adequacy of a model as an acceptable representation of data, which are based on goodness-of-fit test statistics and standard errors of parameter estimates, rely on the model estimation procedure being appropriate for the data. Using analogies to linear regression and anova, this review examines conditions under which conclusions drawn from various estimation methods will be correct and the consequences of ignoring these conditions. A distinction is made between estimation methods that are either correctly or incorrectly specified for the distribution of data being analyzed, and it is shown that valid conclusions are possible even under misspecification. A brief example illustrates the ideas. Internet access is given to a computer code for several methods that are not available in programs such as EQS or LISREL.

Covariance Structure Analysis of Partially Additive Ipsative Data Using Restricted Maximum Likelihood Estimation.

A data matrix is said to be ipsative when the sum of the scores obtained over the variables for each subject is a constant. In this article, a general type of ipsative data known as partially additive ipsative data (PAID) is defined. Ordinary additive ipsative data (All311 is a special case. Due to the specific nature of the research design or measurement process, the observed vector is X PAID with an underlying nonipsative vector y. It is shown that if the underlying distribution of y is multivariate normal with structured covariance matrix Σ = Σ(Θ), the observed X will have a degenerate normal distribution. As a result, ordinary maximum likelihood estimation of Θ cannot be carried out directly. A transformation of X is suggested so that the transformed vector X* = BX will have a nonsingular density and restricted maximum likelihood (REML) estimation can be applied. A simulation study is conducted to investigate the effect of sample size and other model characteristics on the performance of the ML estimators and the sampling behavior of the goodness of fit statistic. It is found that REML estimates are in general close to the true parameter values, but they have larger dard errors as compared with the ordinary MLE based on y. The test statistic is well behaved when sample size is large enough. Moreover, the likelihood of obtaining a convergent solution depends on a number of factors such as sample size, number of indicators per latent factor, and degree of ipsativity. Finally, statistical decisions (reject or not reject the hypothesized model) based on X* are in general consistent with that based on y.

Application of AIC to Wald and Lagrange Multiplier Tests in Covariance Structure Analysis.

The Akaike Information Criterion (AIC) has been proposed as an alternative to the conventional χ(2) goodness-of-fit test. In this article some efficient procedures for the use of AIC in covariance structure analysis are proposed, based on the backward search via the Wald test to impose constraints and the forward search via the Lagrange Multiplier rest to release constraints. An Approximated AIC, AAIC, is developed that is considerably more efficient computationally in providing information on AIC than the conventional approach based on the likelihood ratio test. AAIC can be effectively computed with a stepwise procedure for more general and for more restricted models that do not need to be explicitly estimated. The necessity of a given restriction is shown within the AIC theory not to depend on an a-level cut off in the χ(2) distribution, but on the absolute cutoff value of 2.0. As a consequence, the AIC-based procedure did not yield the simplest model in an example examined in this study. Results also showed that the univariate increment tests, which are products of stepwise procedures in both backward and forward searches, generated the same modifications as the AIC.

A two-stage estimation of structural equation models with continuous and polytomous variables.

This paper develops a computationally efficient procedure for analysis of structural equation models with continuous and polytomous variables. A partition maximum likelihood approach is used to obtain the first stage estimates of the thresholds and the polyserial and polychoric correlations in the underlying correlation matrix. Then, based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, a generalized least squares approach is employed to estimate the structural parameters in the correlation structure. Asymptotic properties of the estimators are derived. Some simulation studies are conducted to study the empirical behaviours and robustness of the procedure, and compare it with some existing methods.

A Note on Using and Unbiased Weight Matrix in the ADF Test Statistic.

In covariance structure analysis, the asymptotically distribution-free (ADF) method fails to work satisfactorily unless the sample is extremely large. Simulation studies report that the ADF test statistics observed arc usually too large and correct models arc then over-rejected. It is known that the accuracy of the ADF test statistic depends on the estimation of the weight matrix. In existing literature and computer software, a biased estimator W is used as an estimate of the unknown weight matrix. In this article. we suggest that W, an unbiased estimate of the weight matrix, may eliminate the small or intermediate sample size bias of the ADF test statistic. Results show that the test statistics based on W and W arc highly similar. The poor performance of the ADF method was not caused by the use of a biased weight matrix in the model studied in this article.

Psychosocial correlates and predictors of AIDS risk behaviors, abortion, and drug use among a community sample, of young adult women.

Relations among latent constructs of Social Conformity, Sensation Seeking, Polydrug Use, Sexual Experience, Abortion, and Risky AIDS Behaviors were examined among a community sample of women (N = 438, mean age = 25.5 years) using confirmatory factor analysis (CFA) and predictive structural equation models (SEM). In the CFA, Risky AIDS Behavior was strongly related to more Polydrug Use and less Social Conformity and modestly related to Sexual Experience and Abortions. In SEMs, Social Conformity significantly predicted less Risky AIDS Behavior and less Polydrug Use but did not predict Abortions. Prior Sexual Experience predicted more Polydrug Use and Abortions. We conclude that the same psychological processes and predispositions that relate low social conformity to drug use and other unhealthy behaviors also influence AIDS-risk behaviors, even among a community sample of women.

Structural equation analyses of clinical subpopulation differences and comparative treatment outcomes: characterizing the daily lives of drug addicts.

The use of structural equation modeling (SEM) is illustrated for comparative treatment outcome research conducted with heterogeneous clinical subpopulations within large multimodality treatment settings. All analyses are accomplished with SEM analogs of more familiar classical multivariate techniques. The effect of the early period of treatment on the daily lives of 486 clients in two drug abuse treatment modalities (methadone maintenance and outpatient counseling) is evaluated. Structured means analysis is used to assess initial differences between modalities on the latent means of 6 latent constructs reflecting daily life. The effect of treatment modality and attrition from the program on daily life latent constructs is evaluated while initial selection differences are statistically controlled. Effect sizes are computed on the basis of SEM parameter estimates. The advantage of SEM over classic multivariate approaches for correcting for selection bias when assessing comparative outcomes is explained.

Bootstrap-corrected ADF test statistics in covariance structure analysis.

The asymptotically distribution-free (ADF) test statistic for covariance structure analysis (CSA) has been reported to perform very poorly in simulation studies, i.e. it leads to inaccurate decisions regarding the adequacy of models of psychological processes. It is shown in the present study that the poor performance of the ADF test statistic is due to inadequate estimation of the weight matrix (W = gamma -1), which is a critical quantity in the ADF theory. Bootstrap procedures based on Hall's bias reduction perspective are proposed to correct the ADF test statistic. It is shown that the bootstrap correction of additive bias on the ADF test statistic yields the desired tail behaviour as the sample size reaches 500 for a 15-variable-3-factor confirmatory factor-analytic model, even if the distribution of the observed variables is not multivariate normal and the latent factors are dependent. These results help to revive the ADF theory in CSA.

Consequences of adolescent drug use and personality factors on adult drug use.

This study examined the stability of adolescent drug use into young adulthood and explored the possible influence of personality on adolescent and adult drug use. Participants in this longitudinal study (N = 640) completed questionnaires which assessed multiple indicators for latent constructs of tobacco, alcohol, cannabis, and hard drugs, and also for the personality constructs of Socialization. In addition, the effects of obedience and extraversion were examined. Results showed that a general drug use factor in adolescence significantly predicted young adult drug use. A particular effect of adolescent obedience on adult drug use was noted. Within adolescence, obedience, extraversion, and the construct of Socialization were significant predictors of drug use. Early onset of smoking predicted adolescent drug use. The implications of these findings for early drug use education and intervention are discussed. Additional analysis explored the possibility of treating obedience as another indicator of Socialization. This model could not provide as good a fit as the original model. The measure of obedience acted as a better predictor of drug use than an overall factor of Socialization. Gender differences are discussed.

Attitudes and health behavior in diverse populations: drunk driving. Alcohol use, binge eating, marijuana use, and cigarette use.

Five different health behaviors (cigarette use, alcohol use, binge eating, illicit drug use, and drunk driving) were studied prospectively in 5 different groups of subjects. Associations between attitudes toward these behaviors and the behaviors themselves were investigated over at least 2 waves of measurement. Findings revealed that attitudes predicted behavior nonspuriously in 2 instances: alcohol use and marijuana use. Attitudes did not predict drunk driving, binge eating, or smoking behaviors. Past behavior predicted attitude in the domains of binge eating and smoking, but not in the domains of alcohol use, drunk driving, or marijuana use. The results are discussed in terms of several alternative approaches that have implications for interventions that attempt to influence health behavior through attitude change.