Wai Chan

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. Mardias 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.

A Two-Stage Approach to Synthesizing Covariance Matrices in Meta-Analytic Structural Equation Modeling.

Structural equation modeling (SEM) is widely used as a statistical framework to test complex models in behavioral and social sciences. When the number of publications increases, there is a need to systematically synthesize them. Methodology of synthesizing findings in the context of SEM is known as meta-analytic SEM (MASEM). Although correlation matrices are usually preferred in MASEM, there are cases in which synthesizing covariance matrices is useful, especially when the scales of the measurement are comparable. This study extends the 2-stage SEM (TSSEM) approach proposed by M. W. L. Cheung and Chan (2005b) to synthesizing covariance matrices in MASEM. A simulation study was conducted to compare the TSSEM approach with several approximate methods. An empirical example is used to illustrate the procedures and future directions for MASEM are discussed.

Structural equation modeling with heavy tailed distributions

Data in social and behavioral sciences typically possess heavy tails. Structural equation modeling is commonly used in analyzing interrelations among variables of such data. Classical methods for structural equation modeling fit a proposed model to the sample covariance matrix, which can lead to very inefficient parameter estimates. By fitting a structural model to a robust covariance matrix for data with heavy tails, one generally gets more efficient parameter estimates. Because many robust procedures are available, we propose using the empirical efficiency of a set of invariant parameter estimates in identifying an optimal robust procedure. Within the class of elliptical distributions, analytical results show that the robust procedure leading to the most efficient parameter estimates also yields a most powerful test statistic. Examples illustrate the merit of the proposed procedure. The relevance of this procedure to data analysis in a broader context is noted.

Testing Dependent Correlation Coefficients via Structural Equation Modeling

Organizational researchers are sometimes interested in testing if independent or dependent correlation coefficients are equal. Olkin and Finn and Steiger proposed several statistical procedures to test dependent correlation coefficients in a single group, whereas meta-analytic procedures can be used to test independent correlation coefficients in two or more groups. Because computer programming is usually involved, applied researchers may find these procedures hard to implement, especially in testing the dependent correlation coefficients. This article suggests using a structural equation modeling (SEM) approach as a unified framework to test independent and dependent correlational hypotheses. To demonstrate the comparability among these approaches, examples and ad hoc simulation studies are used. Advantages of the SEM approach are also discussed.

Covariance structure analysis of ordinal ipsative data

Data are ipsative if they are subject to a constant-sum constraint for each individual. In the present study, ordinal ipsative data (OID) are defined as the ordinal rankings across a vector of variables. It is assumed that OID are the manifestations of their underlying nonipsative vector y, which are difficult to observe directly. A two-stage estimation procedure is suggested for the analysis of structural equation models with OID. In the first stage, the partition maximum likelihood (PML) method and the generalized least squares (GLS) method are proposed for estimating the means and the covariance matrix of Acy, where Ac is a known contrast matrix. Based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, the generalized least squares method is used to estimate the structural parameters in the second stage. A goodness-of-fit statistic is given for testing the hypothesized covariance structure. Simulation results show that the proposed method works properly when a sufficiently large sample is available.

The covariance structure analysis of ipsative data

Data are called ipsative if they are subject to a constant-sum constraint for each observation. Usually, ipsative data are the consequence of transformation of their corresponding preipsative data. In this article, two kinds of ipsative data are defined. They are the additive ipsative data (AID) and the multiplicative ipsative data (MID). Jackson and Alwin proposed a method to analyze AID in exploratory factor analysis. However, they failed to provide the estimates of the original factor loadings. In this study, first, their method is modified in the context of covariance structure analysis. It is discovered that the original parameter estimates can usually be recovered provided the model of the preipsative data is well defined. An artificial example is used to demonstrate the suggested method. Second, the method is extended to the case of MID. A real example is considered also. Finally, some related issues, problems, and generalizations are addressed and discussed.

Assessing Structural Equation Models by Equivalence Testing With Adjusted Fit Indexes

Conventional null hypothesis testing (NHT) is a very important tool if the ultimate goal is to find a difference or to reject a model. However, the purpose of structural equation modeling (SEM) is to identify a model and use it to account for the relationship among substantive variables. With the setup of NHT, a nonsignificant test statistic does not necessarily imply that the model is correctly specified or the size of misspecification is properly controlled. To overcome this problem, this article proposes to replace NHT by equivalence testing, the goal of which is to endorse a model under a null hypothesis rather than to reject it. Differences and similarities between equivalence testing and NHT are discussed, and new “T-size” terminology is introduced to convey the goodness of the current model under equivalence testing. Adjusted cutoff values of root mean square error of approximation (RMSEA) and comparative fit index (CFI) corresponding to those conventionally used in the literature are obtained to facilitate the understanding of T-size RMSEA and CFI. The single most notable property of equivalence testing is that it allows a researcher to confidently claim that the size of misspecification in the current model is below the T-size RMSEA or CFI, which gives SEM a desirable property to be a scientific methodology. R code for conducting equivalence testing is provided in an appendix.

Classifying Correlation Matrices Into Relatively Homogeneous Subgroups: A Cluster Analytic Approach

Researchers are becoming interested in combining meta-analytic techniques and structural equation modeling to test theoretical models from a pool of studies. Most existing procedures are based on the assumption that all correlation matrices are homogeneous. Few studies have addressed what the next step should be when studies being analyzed are heterogeneous and the search for moderator variables for homogeneous subgroup analysis fails. Cluster analysis is proposed and evaluated in this article as an exploratory tool to classify studies into relatively homogeneous groups. Simulation studies indicate that using Euclidean distance on raw correlation coefficients or U-transformed scores with the complete linkage or Ward’s minimum-variance methods will provide satisfactory results.

Bootstrap Standard Error and Confidence Intervals for the Correlation Corrected for Range Restriction: A Simulation Study.

The standard Pearson correlation coefficient is a biased estimator of the true population correlation, rho, when the predictor and the criterion are range restricted. To correct the bias, the correlation corrected for range restriction, rc, has been recommended, and a standard formula based on asymptotic results for estimating its standard error is also available. In the present study, the bootstrap standard-error estimate is proposed as an alternative. Monte Carlo simulation studies involving both normal and nonnormal data were conducted to examine the empirical performance of the proposed procedure under different levels of rho, selection ratio, sample size, and truncation types. Results indicated that, with normal data, the bootstrap standard-error estimate is more accurate than the traditional estimate, particularly with small sample size. With nonnormal data, performance of both estimates depends critically on the distribution type. Furthermore, the bootstrap bias-corrected and accelerated interval consistently provided the most accurate coverage probability for rho.

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.