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Purpose The purpose of this paper is to discuss the pros and cons of partial least squares approach to structural equation modeling (PLS-SEM). The topics include bias, consistency, maximization of R2, reliability and model validation. Design/methodology/approach The approach in this study is descriptive, and the method consists of logical arguments and analysis that are supported by results in references. Findings Several optimal properties of the PLS-SEM methodology are clarified. A proposal for transforming PLS-SEM mode A to mode B is highlighted, and the transformed mode possesses the desired properties of both modes A and B. Issues with the application of regression analysis using composite scores are also discussed. The strength of PLS-SEM is also compared against that of covariance-based SEM. Research limitations/implications Additional studies on PLS-SEM are needed when the population structure contains cross-loadings and/or correlated errors. Practical implications PLS-SEM may have inflated type I errors and R2 values even with normally distributed data. Originality/value The content of this paper is new, and there does not exist such an in-depth discussion of the pros and cons of PLS-SEM methodology in the literature.

The impact of missing data on statistical inference varies depending on several factors such as the proportion of missingness, missing-data mechanism, and method employed to handle missing values. While these topics have been extensively studied, most recommendations have been made assuming that all missing values are from the same missing-data mechanism. In reality, it is very likely that a mixture of missing-data mechanisms is responsible for missing values in a dataset and even within the same pattern of missingness. Although a mixture of missing-data mechanisms and causes within a dataset is a likely scenario, the performance of popular missing-data methods under these circumstances is unknown. This study provides a realistic evaluation of methods for handling missing data in this setting using Monte Carlo simulation in the context of regression. This study also seeks to identify acceptable proportions of missing values that violate the missing-data mechanism assumed by the method used to handle missing values. Results indicate that multiple imputation (MI) performs better than other principled or ad-hoc methods. Different missing-data methods are also compared via the analysis of a real dataset in which mixtures of missingness mechanisms are created. Recommendations are provided for the use of different methods in practice.

Structural equation modeling (SEM) and path analysis using composite-scores are distinct classes of methods for modeling the relationship of theoretical constructs. The two classes of methods are integrated in the partial-least-squares approach to structural equation modeling (PLS-SEM), which systematically generates weighted composites and uses them to conduct path analysis of the structural model via the least-squares method. However, the goodness of PLS-SEM depends on the statistical properties of the composites, which are further determined by the formulations of the weights. This article studies how the formulations of PLS-SEM composites are affected by model specification, with focus on the sensitivity of the weights to common specification errors. Results indicate that the weights under PLS-SEM mode A are not affected by within-block error-covariances but those under mode B are. While between-block error-covariances and cross-loadings only affect the weights of the involved items under both PLS-SEM modes A and B, the weights under mode B are much more sensitive than those under mode A. In contrast, the weights under a recently proposed transformed mode (denoted as BA) are a compromise between those of modes A and B. The findings not only advance the understanding of the PLS-SEM methodology but also facilitate model diagnostics. Empirical applications of the results are illustrated via the analysis of a real dataset.

Structural equation modeling (SEM) has been deemed as a proper method when variables contain measurement errors. In contrast, path analysis with composite scores is preferred for prediction and diagnosis of individuals. While path analysis with composite scores has been criticized for yielding biased parameter estimates, recent literature pointed out that the population values of parameters in a latent-variable model depend on artificially assigned scales. Consequently, bias in parameter estimates is not a well-grounded concept for models involving latent constructs. This article compares path analysis with composite scores against SEM with respect to effect size and statistical power in testing the significance of the path coefficients, via the z- or t-statistics. The data come from many sources with various models that are substantively determined. Results show that SEM is not as powerful as path analysis even with equally weighted composites. However, path analysis with Bartlett-factor scores and the partial least-squares approach to SEM perform the best with respect to effect size and power.

This article proposes a two-level moderated mediation (2moME) model with single level data, and develops measures to quantify the moderated mediation (moME) effect sizes for both the conventional moME model and the 2moME model. A Bayesian approach is developed to estimate and test moME effects and the corresponding effect sizes (ES). Monte Carlo results indicate that (1) the 2moME model yields more accurate estimates of the parameters than the conventional moME model; (2) the 95% credibility interval following the 2moME model covers the moME effects and the ESs more accurately than that following the conventional moME model; and (3) statistical tests for the existence of the moME effects with the 2moME model are more reliable in controlling type I errors than those with the conventional moME model, especially under heteroscedasticity conditions. In addition, the developed measures of ES are more interpretable, and directly answer the questions regarding the extent to which a moderator can account for the change of the mediation effect between the predictor and the outcome variable through the mediator variable. An empirical example illustrates the application of the 2moME model and the ES measures.

Partial-least-squares approach to structural equation modeling (PLS-SEM) uses proxies of latent variables to conduct regression analysis, which directly addresses the needs of prediction and classi...

Chi-square type test statistics are widely used in assessing the goodness-of-fit of a theoretical model. The exact distributions of such statistics can be quite different from the nominal chi-squar...

Multilevel structural equation models (MSEM) are typically evaluated on the basis of goodness of fit indices. A problem with these indices is that they pertain to the entire model, reflecting simul...

Data-driven model modification plays an important role for a statistical methodology to advance the understanding of subjective matters. However, when the sample size is not sufficiently large mode...

In exploratory factor analysis (EFA), cross-loadings frequently occur in empirical research, but its effects on determining the number of factors to retain are seldom known. In this paper, we analy...

With single-level data, Yuan, Cheng and Maxwell developed a two-level regression model for more accurate moderation analysis. This article extends the two-level regression model to a two-level moderated latent variable (2MLV) model, and uses a Bayesian approach to estimate and test the moderation effects. Monte Carlo results indicate that: 1) the new method yields more accurate estimate of the interaction effect than those via the product-indicator (PI) approach and latent variable interaction (LVI) with single-level model, both are also estimated via Bayesian method; 2) the coverage rates of the credibility interval following the 2MLV model are closer to the nominal 95% than those following the other methods; 3) the test for the existence of the moderation effect is more reliable in controlling Type I errors than both PI and LVI, especially under heteroscedasticity conditions. Moreover, a more interpretable measure of effect size is developed based on the 2MLV model, which directly answers the question as to what extent a moderator can account for the change of the coefficient between the predictor and the outcome variable. A real data example illustrates the application of the new method.

Compared to the conventional covariance-based SEM (CB-SEM), partial-least-squares SEM (PLS-SEM) has an advantage in computation, which obtains parameter estimates by repeated least squares regressi...

Data in social and behavioral sciences are routinely collected using questionnaires, and each domain of interest is tapped by multiple indicators. Structural equation modeling (SEM) is one of the m...

issing values that are missing not at random (MNAR) can result from a variety of missingness processes. However, two fundamental subtypes of MNAR values can be obtained from the definition of the MNAR mechanism itself. The distinction between them deserves consideration because they have characteristic differences in how they distort relationships in the data. This has implications for the validity of statistical results and generalizability of methodological findings that are based on data (empirical or generated) with MNAR values. However, these MNAR subtypes have largely gone unnoticed by the literature. As few studies have considered both subtypes, their relevance to methodological and substantive research has been overlooked. This article systematically introduces the two MNAR subtypes and gives them descriptive names. A case study demonstrates they are mechanically distinct from each other and from other missing-data mechanisms. Applied examples are given to help researchers conceptually identify MNAR subtypes in real data. Methods are provided to generate missing values from both subtypes in simulation studies. Simulation studies for regression and growth curve modeling contexts show MNAR subtypes consistently differ in the severity of their impact on statistical inference. This behavior is examined in light of how relationships in the data become characteristically distorted. The contents of this article are intended to provide a foundation and tools for organized consideration of MNAR subtypes. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

This article proposes a two-level moderated mediation (2moME) model with single level data, and develops measures to quantify the moderated mediation (moME) effect sizes for both the conventional moME model and the 2moME model. A Bayesian approach is developed to estimate and test moME effects and the corresponding effect sizes (ES). Monte Carlo results indicate that (1) the 2moME model yields more accurate estimates of the parameters than the conventional moME model; (2) the 95% credibility interval following the 2moME model covers the moME effects and the ESs more accurately than that following the conventional moME model; and (3) statistical tests for the existence of the moME effects with the 2moME model are more reliable in controlling type I errors than those with the conventional moME model, especially under heteroscedasticity conditions. In addition, the developed measures of ES are more interpretable, and directly answer the questions regarding the extent to which a moderator can account for the change of the mediation effect between the predictor and the outcome variable through the mediator variable. An empirical example illustrates the application of the 2moME model and the ES measures.

Chi-square type test statistics are widely used in assessing the goodness-of-fit of a theoretical model. The exact distributions of such statistics can be quite different from the nominal chi-square distribution due to violation of conditions encountered with real data. In such instances, the bootstrap or Monte Carlo methodology might be used to approximate the distribution of the statistic. However, the sample quantile may be a poor estimate of the population counterpart when either the sample size is small or the number of different values of the replicated statistic is limited. Using statistical learning, this article develops a method that yields more accurate quantiles for chi-square type test statistics. Formulas for smoothing the quantiles of chi-square type statistics are obtained. Combined with the bootstrap methodology, the smoothed quantiles are further used to conduct equivalence testing in mean and covariance structure analysis. Two real data examples illustrate the applications of the developed formulas in quantifying the size of model misspecification under equivalence testing. The idea developed in the article can also be used to develop formulas for smoothing the quantiles of other types of test statistics or parameter estimates.

Cronbach’s coefficient alpha remains very important as a measure of internal consistency. The well-known Spearman-Brown formula indicates that as the number of items (i.e., the dimension) goes to infinity, the coefficient alpha eventually approaches one. In this work, we show that under the assumption of a one-factor model, not necessarily with parallel items, the phenomenon of the coefficient alpha approaching one is closely related to four different phenomena: (1) the closeness between factor-analysis (FA) loadings and principal-component-analysis (PCA) loadings, (2) the inverse of the population covariance matrix of the manifest variables becoming a diagonal matrix, (3) the communalities between FA and PCA approaching each other, and (4) the factor score and the principal component agreeing with each other. These results allow us to characterize the relationship between FA and PCA with respect to the coefficient alpha.

Data in social sciences are typically non-normally distributed and characterized by heavy tails. However, most widely used methods in social sciences are still based on the analyses of sample means and sample covariances. While these conventional methods continue to be used to address new substantive issues, conclusions reached can be inaccurate or misleading. Although there is no 'best method' in practice, robust methods that consider the distribution of the data can perform substantially better than the conventional methods. This article gives an overview of robust procedures, emphasizing a few that have been repeatedly shown to work well for models that are widely used in social and behavioural sciences. Real data examples show how to use the robust methods for latent variable models and for moderated mediation analysis when a regression model contains categorical covariates and product terms. Results and logical analyses indicate that robust methods yield more efficient parameter estimates, more reliable model evaluation, more reliable model/data diagnostics, and more trustworthy conclusions when conducting replication studies. R and SAS programs are provided for routine applications of the recommended robust method.

Effect size is crucial for quantifying differences and a key concept behind Type I errors and power, but measures of effect size are seldom studied in structural equation modeling (SEM). While fit ...

Moderation analysis is useful for addressing interesting research questions in social sciences and behavioural research. In practice, moderated multiple regression (MMR) models have been most widely used. However, missing data pose a challenge, mainly because the interaction term is a product of two or more variables and thus is a non-linear function of the involved variables. Normal-distribution-based maximum likelihood (NML) has been proposed and applied for estimating MMR models with incomplete data. When data are missing completely at random, moderation effect estimates are consistent. However, simulation results have found that when data in the predictor are missing at random (MAR), NML can yield inaccurate estimates of moderation effects when the moderation effects are non-null. Simulation studies are subject to the limitation of confounding systematic bias with sampling errors. Thus, the purpose of this paper is to analytically derive asymptotic bias of NML estimates of moderation effects with MAR data. Results show that when the moderation effect is zero, there is no asymptotic bias in moderation effect estimates with either normal or non-normal data. When the moderation effect is non-zero, however, asymptotic bias may exist and is determined by factors such as the moderation effect size, missing-data proportion, and type of missingness dependence. Our analytical results suggest that researchers should apply NML to MMR models with caution when missing data exist. Suggestions are given regarding moderation analysis with missing data.

Meta-analysis plays a key role in combining studies to obtain more reliable results. In social, behavioral, and health sciences, measurement units are typically not well defined. More meaningful results can be obtained by standardizing the variables and via the analysis of the correlation matrix. Structural equation modeling (SEM) with the combined correlations, called meta-analytical SEM (MASEM), is a powerful tool for examining the relationship among latent constructs as well as those between the latent constructs and the manifest variables. Three classes of methods have been proposed for MASEM: (1) generalized least squares (GLS) in combining correlations and in estimating the structural model, (2) normal-distribution-based maximum likelihood (ML) in combining the correlations and then GLS in estimating the structural model (ML-GLS), and (3) ML in combining correlations and in estimating the structural model (ML). The current article shows that these three methods are equivalent. In particular, (a) the GLS method for combining correlation matrices in meta-analysis is asymptotically equivalent to ML, (b) the three methods (GLS, ML-GLS, ML) for MASEM with correlation matrices are asymptotically equivalent, (c) they also perform equally well empirically, and (d) the GLS method for SEM with the sample correlation matrix in a single study is asymptotically equivalent to ML, which has being discussed extensively in the SEM literature regarding whether the analysis of a correlation matrix yields consistent standard errors and asymptotically valid test statistics. The results and analysis suggest that a sample-size weighted GLS method is preferred for combining correlations and for MASEM.

Ridge generalized least squares (RGLS) is a recently proposed estimation procedure for structural equation modeling. In the formulation of RGLS, there is a key element, ridge tuning parameter, whos...

Unless data are missing completely at random (MCAR), proper methodology is crucial for the analysis of incomplete data. Consequently, methods for effectively testing the MCAR mechanism become important, and procedures were developed via testing the homogeneity of means and variances–covariances across the observed patterns (e.g., Kim & Bentler in Psychometrika 67:609–624, 2002; Little in J Am Stat Assoc 83:1198–1202, 1988). The current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data, and the normal-distribution-based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random, although the values may not equal those of the underlying population distribution. The results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the MCAR mechanism even when the population distribution is multivariate normal.

Among test statistics for assessing overall model fit in structural equation modeling (SEM), the Satorra–Bentler rescaled statistic is most widely used when the normality assumption is violated. Ho...

Mean and mean-and-variance corrections are the 2 major principles to develop test statistics with violation of conditions. In structural equation modeling (SEM), mean-rescaled and mean-and-variance...

When the assumption of multivariate normality is violated and the sample sizes are relatively small, existing test statistics such as the likelihood ratio statistic and Satorra–Bentler’s rescaled and adjusted statistics often fail to provide reliable assessment of overall model fit. This article proposes four new corrected statistics, aiming for better model evaluation with nonnormally distributed data at small sample sizes. A Monte Carlo study is conducted to compare the performances of the four corrected statistics against those of existing statistics regarding Type I error rate. Results show that the performances of the four new statistics are relatively stable compared with those of existing statistics. In particular, Type I error rates of a new statistic are close to the nominal level across all sample sizes under a condition of asymptotic robustness. Other new statistics also exhibit improved Type I error control, especially with nonnormally distributed data at small sample sizes.

The normal-distribution-based likelihood ratio statistic is widely used for power analysis in structural Equation modeling (SEM). In such an analysis, power and sample size are computed by assuming that follows a central chi-square distribution under and a noncentral chi-square distribution under . However, with either violation of normality or not a large enough sample size, both empirical and analytical results indicate that the chi-square distribution assumptions are not realistic and consequently methods of power analysis based on such assumptions are not valid. This article describes a Monte Carlo (MC) method for power analysis. A measure of effect size for characterizing the power property of different rescaled statistics is also provided. Robust methods are proposed to increase the power of and other statistics. Simulation results show that the MC method reliably controls Type I errors and robust estimation methods effectively increase the power, and their combination is thus recommended for conduc...

Many aspects of multivariate analysis involve obtaining the precision matrix, i.e., the inverse of the covariance matrix. When the dimension is larger than the sample size, the sample covariance matrix is no longer positive definite, and the inverse does not exist. Under the sparsity assumption on the elements of the precision matrix, the problem can be solved by fitting a Gaussian graphical model with lasso penalty. However, in high-dimensional settings in behavioral sciences, the sparsity assumption does not necessarily hold. The dimensions are often greater than the sample sizes, while they are likely to be comparable in size. Under such circumstances, introducing some covariance structures might solve the issue of estimating the precision matrix. Factor analysis is employed for modeling the covariance structure and the Woodbury identity to find the precision matrix. Different methods are compared such as unweighted least squares and factor analysis with equal unique variances (i.e., the probabilistic principal component analysis), as well as ridge factor analysis with small ridge parameters. Results indicate that they all give relatively small mean squared errors even when the dimensions are larger than the sample size.