Covariance Structure Models Research Articles

Asymptotic robustness of the normal theory likelihood ratio statistic for two-level covariance structure models

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures have been developed to analyze such data while taking into account the resulting dependence of observations. Most of these developments require a multivariate normality distribution assumption. It is important to know whether normal theory-based inference can still be valid when applied to nonnormal hierarchical data sets. Using an analytical approach for balanced data and numerical illustrations for unbalanced data, this paper shows that the likelihood ratio statistic based on the normality assumption is asymptotically robust for many nonnormal distributions. The result extends the scope of asymptotic robustness theory that has been established in different contexts.

Point and Interval Estimation of Reliability for Multiple-Component Measuring Instruments via Linear Constraint Covariance Structure Modeling

A widely and readily applicable covariance structure modeling approach is outlined that allows point and interval estimation of scale reliability with fixed components. The procedure employs only linear constraints introduced in a congeneric model, which after reparameterization permit expression of composite reliability as a function of appropriate parameters. Unlike the popular Cronbach's coefficient alpha that already at the population level is generally a misestimator of scale reliability, this method is based on the formal definition of the reliability coefficient. The proposed approach is illustrated by means of a numerical example.

Do “bad boys” really get the girls? Delinquency as a cause and consequence of dating behavior among adolescents*

Drawing on Zahavi's (1975) handicap principle, we suggest that delinquency and other risk-taking behavior might be seen as evidence of positive, adaptive qualities such as nerve and bravery. Drawing on Akers' (1998) social learning theory, we also suggest that potential romantic partners might be attracted to such traits and that this romantic attention might reinforce delinquency and other risk-taking behavior. Using data from the first and third waves of the National Youth Survey, we test these assertions with a series of longitudinal OLS and contemporaneous-effects covariance structure models. Results suggest that delinquency serves to increase romantic involvement and that romantic involvement may provide vicarious, but not necessarily direct, reinforcement for delinquency among both male and female adolescents.

Estimation of maximal reliability: A note on a covariance structure modelling approach

A one-step covariance structure analysis procedure for estimation of maximal reliability of linear composites with congeneric measures is outlined. The approach is readily employed within a single modelling session using popular covariance structure analysis software, and permits simultaneous estimation of the optimal measure weights with standard errors. The method is illustrated by a numerical example.

Robustness of Generalized Estimating Equation (GEE) Tests of Significance against Misspecification of the Error Structure Model

AbstractGeneralized linear model analyses of repeated measurements typically rely on simplifying mathematical models of the error covariance structure for testing the significance of differences in patterns of change across time. The robustness of the tests of significance depends, not only on the degree of agreement between the specified mathematical model and the actual population data structure, but also on the precision and robustness of the computational criteria for fitting the specified covariance structure to the data. Generalized estimating equation (GEE) solutions utilizing the robust empirical sandwich estimator for modeling of the error structure were compared with general linear mixed model (GLMM) solutions that utilized the commonly employed restricted maximum likelihood (REML) procedure. Under the conditions considered, the GEE and GLMM procedures were identical in assuming that the data are normally distributed and that the variance‐covariance structure of the data is the one specified by the user.The question addressed in this article concerns relative sensitivity of tests of significance for treatment effects to varying degrees of misspecification of the error covariance structure model when fitted by the alternative procedures. Simulated data that were subjected to monte carlo evaluation of actual Type I error and power of tests of the equal slopes hypothesis conformed to assumptions of ordinary linear model ANOVA for repeated measures except for autoregressive covariance structures and missing data due to dropouts. The actual within‐groups correlation structures of the simulated repeated measurements ranged from AR(1) to compound symmetry in graded steps, whereas the GEE and GLMM formulations restricted the respective error structure models to be either AR(1), compound symmetry (CS), or unstructured (UN). The GEE‐based tests utilizing empirical sandwich estimator criteria were documented to be relatively insensitive to misspecification of the covariance structure models, whereas GLMM tests which relied on restricted maximum likelihood (REML) were highly sensitive to relatively modest misspecification of the error correlation structure even though normality, variance homogeneity, and linearity were not an issue in the simulated data.Goodness‐of‐fit statistics were of little utility in identifying cases in which relatively minor misspecification of the GLMM error structure model resulted in inadequate alpha protection for tests of the equal slopes hypothesis. Both GEE and GLMM formulations that relied on unstructured (UN) error model specification produced nonconservative results regardless of the actual correlation structure of the repeated measurements. A random coefficients model produced robust tests with competitive power across all conditions examined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Mixed Model Approach to Discriminant Analysis with Longitudinal Data

AbstractIn medical diagnostics data are often characterized by repeated observations on the same subject (patient), particularly in form of time series, dose‐response curves etc. If the aim is an individual allocation of patients into one of different groups (diagnoses), a discriminant analysis has to be applied. In a first attempt, Tomasko et al., 1999 [5] modified the well‐known linear discriminant analysis using the mixed model MANOVA for the estimation of fixed effects and for a determination of various structures of covariance matrices, including unstructured, compound symmetry, and autoregressive of order 1. We extended that approach by a random subject effect (Azzalani, 1995 [1]), validated error estimations, and cross‐validated feature selection (Wernecke, 2002 [9]). In this paper, we complete the investigations of the new method with a cross‐validated selection of the adequate model (covariance structure) and extensive Monte‐Carlo simulations. An application to contrast‐enhanced breast MRI signals in a multicenter study is also given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A simulation study to investigate the use of cutoff values for assessing model fit in covariance structure models

In this paper, we used simulations to investigate the effect of sample size, number of indicators, factor loadings, and factor correlations on frequencies of the acceptance/rejection of models (true and misspecified) when selected goodness-of-fit indices were compared with prespecified cutoff values. We found the percent of true models accepted when a goodness-of-fit index was compared with a prespecified cutoff value was affected by the interaction of the sample size and the total number of indicators. In addition, for the Tucker-Lewis index (TLI) and the relative noncentrality index (RNI), model acceptance percentages were affected by the interaction of sample size and size of factor loadings. For misspecified models, model acceptance percentages were affected by the interaction of the number of indicators and the degree of model misspecification. This suggests that researchers should use caution in using cutoff values for evaluating model fit. However, the study suggests that researchers who prefer to use prespecified cutoff values should use TLI, RNI, NNCP, and root-mean-square-error-of-approximation (RMSEA) to assess model fit. The use of GFI should be discouraged.

Investigating Individual Change and Development: The Multilevel Model for Change and the Method of Latent Growth Modeling

Although controversy has always surrounded the measurement of change, it is now known that change can indeed be measured well if a longitudinal perspective is adopted. A sample of individuals must be followed over time and measured repeatedly at sensibly spaced intervals, and an explicit multilevel model for change must be postulated during data analysis. The latter contains two submodels: (a) an individual growth model (Level 1) and (b) a model for systematic interindividual differences in change (Level 2). Recently, methodologists have shown how the multilevel model for change can be mapped onto the general covariance structure model (as implemented in LISREL, say), thereby empowering the use of covariance structure analysis in the measurement of change-a strategy now known as latent growth modeling. Although the statistical model is identical under both approaches, and parameter estimates are generally very close, latent growth modeling offers advantages. It permits the modeling of change simultaneously in several domains, the modeling of change in a construct, the modeling of change as part of a network of relations, and the testing of explicit hypotheses about error covariance structure. In this article, I use longitudinal data on the early reading scores of 1740 children from the Early Childhood Longitudinal Study (National Center for Education Statistics, 2002) to introduce latent growth modeling.

LISREL 8.54: A program for structural equation modelling with latent variables

LISREL 8.54 is a computer program originally designed for estimation of a general structural equation model developed by Karl Joreskog in the early 1970s (Joreskog, 1973). Its current version, LISREL 8.54, is an advanced statistical program for multivariate analysis of linear systems with latent or unobserved variables, econometric structural equation systems, and a variety of factor analytic methods and covariance structure models. Recent developments also include powerful procedures for analysis with non-metric variables including latent variable models with ordinal indicators. The core code of the program is written in FORTRAN, but over the last decade its current vendor, Scientific Software International Inc. (SSI), developed an impressive interactive visual interface in C/C CC that made LISREL one of the most user-friendly multivariate analysis packages available. While preserving its raw power and flexibility, this multi-language architecture, however, contributed to instability problems associated with the 8.50/8.51 versions. Several subsequent upgrades have resulted in the 8.54 release that now more closely resembles the power and reliability of the earlier versions of this program. The LISREL 8.54 program (LISWIN32.EXE) is a 32-bit Windows application which interfaces with several modules, the main two of which are LISREL (LISREL85.EXE) and PRELIS (PRELIS25.EXE).1 PRELIS was developed later than LISREL with the main purpose of providing a pre-processor and data transformation front-end to LISREL.

A Cautionary Note on the Use of Information Fit Indexes in Covariance Structure Modeling With Means

Information fit indexes such as Akaike Information Criterion, Consistent Akaike Information Criterion, Bayesian Information Criterion, and the expected cross validation index can be valuable in assessing the relative fit of structural equation models that differ regarding restrictiveness. In cases in which models without mean restrictions (i.e., saturated mean structure) are compared to models with restricted (i.e., modeled) means, one should take account of the presence of means, even if the model is saturated with respect to the means. The failure to do this can result in an incorrect rank order of models in terms of the information fit indexes. We demonstrate this point by an analysis of measurement invariance in a multigroup confirmatory factor model.

A Covariance Structure Model for Factors Related to Social Activities of the Elderly Living at Home

A Covariance Structure Model for Factors Related to Social Activities of the Elderly Living at Home

A New Measure of Misfit for Covariance Structure Models

Various fit indices exist in structural equation models. Most of these indices are related to the noncentrality parameter (NCP) of the chi-square distribution that the involved test statistic is implicitly assumed to follow. Existing literature suggests that few statistics can be well approximated by chi-square distributions. The meaning of the NCP is not clear when the behavior of the statistic cannot be described by a chi-square distribution. In this paper we define a new measure of model misfit (MMM) as the difference between the expected values of a statistic under the alternative and null hypotheses. This definition does not need to assume that the population covariance matrix is in the vicinity of the proposed model, nor does it need for the test statistic to follow any distribution of a known form. The MMM does not necessarily equal the discrepancy between the model and the population covariance matrix as has been assumed in existing literature. Bootstrap approaches to estimating the MMM and a related quantity are developed. An algorithm for obtaining bootstrap confidence intervals of the MMM is constructed. Examples with practical data sets contrast several measures of model misfit. The quantile-quantile plot is used to illustrate the unrealistic nature of chi-square distribution assumptions under either the null or an alternative hypothesis in practice.

Recall and recognition in mild hypoxia: using covariance structural modeling to test competing theories of explicit memory

To test theories of explicit memory in amnesia, we examined the effect of hypoxia on memory performance in a group of 56 survivors of sudden cardiac arrest. Structural equation modeling revealed that a single-factor explanation of recall and recognition was insufficient to account for performance, thus contradicting single-process models of explicit memory. A dual-process model of recall in which two processes (e.g., declarative memory and controlled search) contribute to recall performance, whereas only one process (e.g., declarative memory) underlies recognition performance, also failed to explain the results adequately. In contrast, a dual-process model of recognition provided an acceptable account of the data. In this model, two processes—recollection and familiarity—underlie recognition memory, whereas only the recollection process contributes to free recall. The best-fitting model was one in which hypoxia and aging led to deficits in recollection, but left familiarity unaffected. Moreover, a controlled search process was correlated with recollection, but was not associated with familiarity or the severity of hypoxia. The results support models of explicit memory in which recollection depends on the hippocampus and frontal lobes, whereas familiarity-based recognition relies on other brain regions.

An application of Mean–Covariance Structure Models for the analysis of group lending behavior

Understanding the influence of group dynamics (such as group homogeneity and the domino effect) on loan repayment is key to strengthening microfinance institutions in developing countries that employ the group lending methodology. Empirically, these conceptual variables have no perfect proxies and are suited for latent variable models. Mean and Covariance Structure Models (MECOSA) are a useful methodology for the incorporation of latent variables with metric, censored metric, dichotomous and ordinal indicators. Data from 140 groups from a group lending program in Burkina Faso were used to demonstrate the application and interpretation of MECOSA.

Covariance Structure Models for Gene Expression Microarray Data

Covariance structure models are applied to gene expression data using a factor model, a path model, and their combination. The factor model is based on a few factors that capture most of the expression information. A common factor of a group of genes may represent a common protein factor for the transcript of the co-expressed genes, and hence, it has a biological interpretation. With simultaneous regressions among variables, a path analysis model is used to specify a structure based on biological pathways. Path models are applied to a simple cell cycle and the tricarboxylic acid cycle from yeast gene expression data. Finally, combining factor and path models, a covariance structure model produces a more complicated pathway structure of the yeast cell cycle involving genes and their underlying factors. All the models are estimated by maximum likelihood using the EQS software package.

Testing Equality of Proportions Explained Latent Variance in Structural Equation Models

A covariance structure modeling method to test equality in proportions explained variance in studied unobserved dimensions by means of latent predictors is outlined. The procedure is applicable with multiple-indicator, structural equation models where of interest is to compare the predictive power of sets of latent independent variables for given constructs. The approach accounts for measurement error in all manifest variables and is illustrated with data from a cognitive intervention study.

Remarks on the identifiability of thurstonian paired comparison models under multiple judgment

It is well-known that the representations of the Thurstonian models for difference judgment data are not unique. It has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the Thurstonian ranking models. In this paper, we examine the equivalence relations between Thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models.

More on the effectiveness of public spending on health care and education: a covariance structure model

AbstractUsing data for a sample of developing countries and transition economies, this paper estimates the relationship between government spending on health care and education and selected social indicators. Unlike previous studies, where social indicators are used as proxies for the unobservable health and education status of the population, this paper estimates a latent variable model. The findings suggest that public spending is an important determinant of social outcomes, particularly in the education sector. Overall, the latent variable approach yields better estimates of a social production function than the traditional approach, with higher elasticities of social indicators with respect to income and spending, therefore providing stronger evidence that increases in public spending do have a positive impact on social outcomes. Copyright © 2003 John Wiley & Sons, Ltd.

Alcohol expectancies:: Integrating cognitive science and psychometric approaches

Aims: To demonstrate an integrative methodology to explore psychological constructs, we used multiple methods in the examination of alcohol expectancies—a psychological construct that is generally measured using survey methodology. We then used the methodology in order to assess the relationship of alcohol expectancy dimensions to drinking-related outcomes. Design: We developed alcohol expectancy models using a cognitive paradigm designed to maximize cognitive activation of expectancies, in order to delineate dimensions of alcohol expectancies. Next, using multidimensional scaling (MDS), we defined heuristic models representing domains for arousing, sedating, positive, and negative alcohol expectancies. We then assessed these models in a separate sample using confirmatory factor analysis (CFA) and covariance structure modeling. Setting: The research was conducted at a large public university in the southeastern US. Participants: A total of 927 male and female college students ranging in age from 17 to 35 years participated. Measurements: Measures included the Alcohol Expectancy Inventory, the Quantity Frequency Index (QFI), and the Drinking Styles Questionnaire. Findings: Single indicator models representing expectancies for arousing and positive effects of alcohol did not differ significantly in the prediction of drinking, and both accounted for significantly more variance in self-reported drinking than expectancies for sedating and negative effects of alcohol consumption. Expectations for the sedating effects of alcohol accounted for significantly more variance than those for negative effects. Expectations for sedating effects added unique variance in the prediction of drinking when all predictors were simultaneously modeled. Conclusions: The multiple methods integrated here can be used in the development and testing of alcohol expectancy models. This integrative methodology warrants further development and validation.

Bootstrap approach to inference and power analysis based on three test statistics for covariance structure models.

We study several aspects of bootstrap inference for covariance structure models based on three test statistics, including Type I error, power and sample-size determination. Specifically, we discuss conditions for a test statistic to achieve a more accurate level of Type I error, both in theory and in practice. Details on power analysis and sample-size determination are given. For data sets with heavy tails, we propose applying a bootstrap methodology to a transformed sample by a downweighting procedure. One of the key conditions for safe bootstrap inference is generally satisfied by the transformed sample but may not be satisfied by the original sample with heavy tails. Several data sets illustrate that, by combining downweighting and bootstrapping, a researcher may find a nearly optimal procedure for evaluating various aspects of covariance structure models. A rule for handling non-convergence problems in bootstrap replications is proposed.