Normal theory based test statistics in structural equation modelling.

The asymptotically distribution free (ADF) method is often used to estimate parameters or test models without a normal distribution assumption on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance versus correlation structure analysis. The behaviors of 3 test statistics frequently used to evaluate structural equation models with nonnormally distributed variables, χ2 test TAGLS and its small-sample variants TYB and TF(AGLS) were compared. Results showed that the ADF method in correlation structure analysis with test statistic TAGLS performs much better at small sample sizes than the corresponding test for covariance structures. In contrast, test statistics TYB and TF(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. It is proposed that excessively large and variable condition numbers of weight matrices are a cause of poor behavior of ADF test statistics in small samples, and results showed that these condition numbers are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.

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.

In manufacturing processes, a vector of multiple responses is often monitored to assess if the process is in-control. A multivariate Shewhart control chart is one method of monitoring the mean vector. If the chi-square statistic exceeds an upper control limit (UCL), then an out-of-control signal occurs. The average run length (ARL) is used to determine the UCL value. ARLs have been estimated under an assumption of a multivariate normal (MVN) assumption. This research explores the sensitivity of ARLs when the MVN assumption is incorrect. ARLs for data from Multivariate t, lognormal, uniform, and beta distributions are estimated and compared to ARLs under the MVN assumption.

When the multivariate normality assumption is violated in structural equation modeling, a leading remedy involves estimation via normal theory maximum likelihood with robust corrections to standard errors. We propose that this approach might not be best for forming confidence intervals for quantities with sampling distributions that are slow to approach normality, or for functions of model parameters. We implement and study a robust analog to likelihood-based confidence intervals based on inverting the robust chi-square difference test of Satorra (2000). We compare robust standard errors and the robust likelihood-based approach versus resampling methods in confirmatory factor analysis (Studies 1 & 2) and mediation analysis models (Study 3) for both single parameters and functions of model parameters, and under a variety of nonnormal data generation conditions. The percentile bootstrap emerged as the method with the best calibrated coverage rates and should be preferred if resampling is possible, followed by the robust likelihood-based approach.

Structural equation modeling (SEM) relies on normal theory methods – such as maximum likelihood method (ML) and generalized least squares (GLS) when estimating model and testing model goodness of fit. This limits its applicability since most of the observed variables in social sciences are almost never normally distributed. In this paper, we relax this restriction and conduct a simulation experiment with the simplest observed variable SEM model to see the extent and nature of effects of nonnormality on SEM. We incorporate that the distribution of error of the model belongs to g -and- k family of distributions.

When the assumption of multivariate normality is violated or when a discrepancy function other than (normal theory) maximum likelihood is used in structural equation models, the null distribution of the test statistic may not be χ2 distributed. Most existing methods to approximate this distribution only match up to 2 moments. In this article, we propose 2 additional approximation methods: a scaled F distribution that matches 3 moments simultaneously and a direct Monte Carlo–based weighted sum of i.i.d. χ2 variates. We also conduct comprehensive simulation studies to compare the new and existing methods for both maximum likelihood and nonmaximum likelihood discrepancy functions and to separately evaluate the effect of sampling uncertainty in the estimated weights of the weighted sum on the performance of the approximation methods.

Existing methods for structural equation modeling involve fitting the ordinary sample covariance matrix by a proposed structural model. Since a sample covariance is easily influenced by a few outlying cases, the standard practice of modeling sample covariances can lead to inefficient estimates as well as inflated fit indices. By giving a proper weight to each individual case, a robust covariance will have a bounded influence function as well as a nonzero breakdown point. These robust properties of the covariance estimators will be carried over to the parameter estimators in the structural model if a technically appropriate procedure is used. We study such a procedure in which robust covariances replace ordinary sample covariances in the context of the Wishart likelihood function. This procedure is easy to implement in practice. Statistical properties of this procedure are investigated. A fit index is given based on sampling from an elliptical distribution. An estimating equation approach is used to develop a variety of robust covariances, and consistent covariances of these robust estimators, needed for standard errors and test statistics, follow from this approach. Examples illustrate the inflated statistics and distorted parameter estimates obtained by using sample covariances when compared with those obtained by using robust covariances. The merits of each method and its relevance to specific types of data are discussed.

Approximations to the distributions of goodness-of-fit indexes in structural equation modeling are derived with the assumption of multivariate normality and slight misspecification of models. The fit indexes considered in this article are Joreskog and Sorbom's goodness-of-fit index (GFI) and the adjusted GFI, McDonald's absolute GFI, Steiger and Lind's root mean squared error of approximation, Steiger's Γ1 and Γ2, Bentler and Bonett's normed fit index, Bollen's incremental fit index and ρ1, Tucker and Lewis's index ρ2, and Bentler's fit index (McDonald and Marsh's relative noncentrality index). An approximation to the asymptotic covariance matrix for the fit indexes is derived by using the delta method. Furthermore, approximations to the densities of the fit indexes are obtained from the transformations of the asymptotically noncentral chi-square distributed variable. A simulation is carried out to confirm the accuracy of the approximations.

It is difficult to detect interactions between continuous variables in field research using moderated multiple regression (MMR). One reason is that multivariate normality, which occurs in field research but not experimental research, suppresses the residual variance of interaction terms. SPSS, SAS, and Matlab programs that expose the extent of this problem for any given data set are provided. Ironically, when significant interactions are found in field research using MMR, this is because the assumption of multivariate normality has been violated, thus rendering the significance tests potentially invalid. The programs presented in this article compute more trustworthy significance levels via data randomization.

The paper develops a two-stage robust procedure for structural equation modeling (SEM) and an R package rsem to facilitate the use of the procedure by applied researchers. In the first stage, M-estimates of the saturated mean vector and covariance matrix of all variables are obtained. Those corresponding to the substantive variables are then fitted to the structural model in the second stage. A sandwich-type covariance matrix is used to obtain consistent standard errors (SE) of the structural parameter estimates. Rescaled, adjusted as well as corrected and F-statistics are proposed for overall model evaluation. Using R and EQS, the R package rsem combines the two stages and generates all the test statistics and consistent SEs. Following the robust analysis, multiple model fit indices and standardized solutions are provided in the corresponding output of EQS. An example with open/closed book examination data illustrates the proper use of the package. The method is further applied to the analysis of a data set from the National Longitudinal Survey of Youth 1997 cohort, and results show that the developed procedure not only gives a better endorsement of the substantive models but also yields estimates with uniformly smaller standard errors than the normal-distribution-based maximum likelihood.

Hypothesis tests

Monitoring and improving manufacturing processes involves identifying, investigating and eliminating problems responsible for inefficiencies in production operations. While statistical process control tools, such as control charts, are available for process monitoring at the operational level, methods for evaluating system performance from more strategic and tactical levels are limited. The traditional control charts that monitor a single process parameter at a time may not be appropriate in situations where interrelationships among various system measures exist. Although multivariate process control techniques allow for simultaneous monitoring of several process parameters, they require assumptions of independence and multivariate normality of data. In addition, their application has mostly been at an operational level. In order to assist managers in monitoring and improving manufacturing system performance, this paper proposes an individual control chart that monitors an integrated performance index generated from a non-parametric method, which effectively considers multiple performance measures and the relationships between them. The primary advantages of this method are that a single integrated measure can be monitored, does not require assumptions of independence and multivariate normality of data, and allows for the integration of decision-maker's input when the system measures that are monitored have unequal importance.

One of the major challenges for epidemiologists is to understand causal relationships between risk factors and health outcomes by analysing data from observational studies. Epidemiologists know too well that correlations, such as those in regression analysis, rarely mean causation, and it would be very desirable if there is a methodology for observational studies, analogous to randomization in experimental studies, which can discover causes and effects amongst variables or at least confirm or refute the proposed causal relationships. Randomized controlled trials (RCTs) are certainly the ‘gold standard’ of establishing causes and effects, but quite often it is either unethical or unfeasible to conduct RCTs to test causal relations in epidemiological research. Epidemiologists need a methodology, which is sort of a combination of the directed acyclic graphs (DAGs) for conceptual construction of causal models and regression analysis for testing those models. It is therefore surprising that structural equation modelling (SEM) has not been so frequently used in epidemiology as in the social sciences, given that both epidemiologists and social scientists want to delineate causes and effects from observational data. The difference between DAGs and path diagrams in SEM is almost trivial to epidemiologists: the path between two variables can only have one direction in DAGs. 1,2 An individual path in SEM is tested in the same way the regression coefficient is in regression analysis, and model fit indices provided by SEM software packages help the analysts to assess the adequacy of the proposed causal model compared with the observed associations in the sample data. 3,4 Why then is SEM still under-utilized in epidemiology? This is a question posed by a commentary 5 in this journal a few years ago. The answers cited included unfamiliar terminology (SEM theory is formulated in Greek), restriction in the assumptions of variables (outcome variables need to be continuous), difficulties in testing interaction and non-linear relationship (it can be quite tedious to set up SEM models to do these), and equivalent models (two different causal models imply the same correlation structure and consequently, it is impossible to tell which is better). 6 Recent advances in SEM theory and software development has resolved some of these issues: new estimation methods do not require the strict assumption of multivariate normality, 7 and the outcome variables can be binary, ordinal or counts. 7–9 The basic rationale behind SEM is rather simple: multiple linear equations are used to specify causal relationship between variables some of which are manifest variables (i.e. observed and collected by the researchers), whilst others are latent variables (i.e. derived from the observed variables by specifying their relations using equations), such as those in factor analysis. The multiple equations in each causal model entail a certain structure of correlational relationships between observed variables which is usually given as a correlation (or covariance) matrix . The estimation procedure is to minimize the difference between and the observed correlation/covariance matrix S formulated by a likelihood function. The 2 -test is then used to evaluate the difference between these two matrices by taking into account the number of the estimated parameters in the proposed model. When the 2 -value is large (i.e. the difference between the two matrices is large) relative to the model’s degree of freedom, the proposed model is rejected, i.e. something in the causal relationships specified by the proposed model is not quite right and require a second thought. When the 2 -value is small, we fail to reject the model or tentatively accept the model as adequate. Because of the possibility of equivalent models, our model may still be wrong in terms of the causal relations amongst variables but happen to entail the same correlation structure (the same S) of the ‘true’ model. So SEM seems to be an endeavour to search for the truth by approximation, and in this aspect, doing SEM is quite similar to the process of ‘Conjectures and Refutations’ as described by the great philosopher

This article proposes a new general methodology for constructing nonparametric and semiparametric Asymptotically Distribution-Free (ADF) tests for semiparametric hypotheses in regression models for possibly dependent data coming from a strictly stationary process. Classical tests based on the difference between the estimated distributions of the restricted and unrestricted regression errors are not ADF. In this article, we introduce a novel transformation of this difference that leads to ADF tests with well-known critical values. The general methodology is illustrated with applications to testing for parametric models against nonparametric or semiparametric alternatives, and semiparametric constrained mean–variance models. Several Monte Carlo studies and an empirical application show that the finite sample performance of the proposed tests is satisfactory in moderate sample sizes.

ABSTRACTIn this study, our aim was to investigate the changes of different data structures and different sample sizes on the structural equation modeling and the influence of these factors on the model fit measures. Examining the created structural equation modeling under different data structures and sample sizes, the evaluation of model fit measures were performed with a simulation study. As a result of the simulation study, optimization and negative variance estimation problems have been encountered depending on the sample size and changing correlations. It was observed that these problems disappeared either by increasing the sample size or the correlations between the variables in factor. For upcoming studies, the choice of RMSEA and IFI model fit measures can be suggested in all sample sizes and the correlation values for data sets are ensured the multivariate normal distribution assumption.