Fungible parameter estimates in structural equation modeling.

Extending work by Waller (2008) on fungible regression coefficients, we propose a method for computation of fungible parameter estimates in structural equation modeling. Such estimates are defined as distinct alternative solutions for parameter estimates, where all fungible solutions yield identical model fit that is only slightly worse than the fit provided by optimal estimates. When such alternative estimates are found to be highly discrepant from optimal estimates, then substantive interpretation based on optimal estimates is called into question. We present a computational method and 3 illustrations showing the potential impact of this approach in applied research, and we discuss implications and issues for further research. (PsycINFO Database Record

Modified Distribution-Free Goodness-of-Fit Test Statistic.

Covariance structure analysis and its structural equation modeling extensions have become one of the most widely used methodologies in social sciences such as psychology, education, and economics. An important issue in such analysis is to assess the goodness of fit of a model under analysis. One of the most popular test statistics used in covariance structure analysis is the asymptotically distribution-free (ADF) test statistic introduced by Browne (Br J Math Stat Psychol 37:62-83, 1984). The ADF statistic can be used to test models without any specific distribution assumption (e.g., multivariate normal distribution) of the observed data. Despite its advantage, it has been shown in various empirical studies that unless sample sizes are extremely large, this ADF statistic could perform very poorly in practice. In this paper, we provide a theoretical explanation for this phenomenon and further propose a modified test statistic that improves the performance in samples of realistic size. The proposed statistic deals with the possible ill-conditioning of the involved large-scale covariance matrices.

Quantifying Adventitious Error in a Covariance Structure as a Random Effect.

We present an approach to quantifying errors in covariance structures in which adventitious error, identified as the process underlying the discrepancy between the population and the structured model, is explicitly modeled as a random effect with a distribution, and the dispersion parameter of this distribution to be estimated gives a measure of misspecification. Analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation. An algorithm is developed for numerical implementation of the procedure. The consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard Pitman drift assumption. Simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error. Two examples are also given as illustrations.

Random Model Discrepancy: Interpretations and Technicalities (A Rejoinder).

In this rejoinder we discuss the following aspects of our approach to model discrepancy: the interpretations of the two populations and adventitious error, the choice of inverse Wishart distribution, the perceived danger of justifying a model with bad fit, the relationship among our new approach, Chen's (J R Stat Soc Ser B, 41:235-248, 1979) approach and the existing RMSEA-based approach, and the Pitman drift assumption.

Fitting direct covariance structures by the MSTRUCT modeling language of the CALIS procedure.

This paper demonstrates the usefulness and flexibility of the general structural equation modelling (SEM) approach to fitting direct covariance patterns or structures (as opposed to fitting implied covariance structures from functional relationships among variables). In particular, the MSTRUCT modelling language (or syntax) of the CALIS procedure (SAS/STAT version 9.22 or later: SAS Institute, 2010) is used to illustrate the SEM approach. The MSTRUCT modelling language supports a direct covariance pattern specification of each covariance element. It also supports the input of additional independent and dependent parameters. Model tests, fit statistics, estimates, and their standard errors are then produced under the general SEM framework. By using numerical and computational examples, the following tests of basic covariance patterns are illustrated: sphericity, compound symmetry, and multiple-group covariance patterns. Specification and testing of two complex correlation structures, the circumplex pattern and the composite direct product models with or without composite errors and scales, are also illustrated by the MSTRUCT syntax. It is concluded that the SEM approach offers a general and flexible modelling of direct covariance and correlation patterns. In conjunction with the use of SAS macros, the MSTRUCT syntax provides an easy-to-use interface for specifying and fitting complex covariance and correlation structures, even when the number of variables or parameters becomes large.

Comment on the asymptotics of a distribution-free goodness of fit test statistic.

In a recent article Jennrich and Satorra (Psychometrika 78: 545-552, 2013) showed that a proof by Browne (British Journal of Mathematical and Statistical Psychology 37: 62-83, 1984) of the asymptotic distribution of a goodness of fit test statistic is incomplete because it fails to prove that the orthogonal component function employed is continuous. Jennrich and Satorra (Psychometrika 78: 545-552, 2013) showed how Browne's proof can be completed satisfactorily but this required the development of an extensive and mathematically sophisticated framework for continuous orthogonal component functions. This short note provides a simple proof of the asymptotic distribution of Browne's (British Journal of Mathematical and Statistical Psychology 37: 62-83, 1984) test statistic by using an equivalent form of the statistic that does not involve orthogonal component functions and consequently avoids all complicating issues associated with them.

Analytic standard errors for exploratory process factor analysis.

Exploratory process factor analysis (EPFA) is a data-driven latent variable model for multivariate time series. This article presents analytic standard errors for EPFA. Unlike standard errors for exploratory factor analysis with independent data, the analytic standard errors for EPFA take into account the time dependency in time series data. In addition, factor rotation is treated as the imposition of equality constraints on model parameters. Properties of the analytic standard errors are demonstrated using empirical and simulated data.

A structural equation modelling approach to predicting adoption of a patient-handling intervention developed for EMS providers.

Patient-handling tasks are integral to Emergency Medical Service (EMS) work as are the musculoskeletal injuries associated with these tasks. The aim of this study was to develop and test a structural equation model that describes the interactions between previously identified factors that contribute to the adoption of a specific ergonomics intervention designed for EMS work. EMS responders (n = 187), from six different organisations, participated in a 2-month longitudinal study following the introduction of a foldable patient transfer-board (slide-board) designed to assist with lateral patient transfers. Surveys administered at baseline, after 1 month and after 2 months sampled factors potentially influencing EMS responders' adoption decisions. Perceived ergonomics advantage, which was influenced by access and storage concerns and prior tool experience, contributed most strongly to intention to use at the end of the first month and to the emergence of champions, which contributed to the intention to use at the end of the second month.

Identification of factors that affect the adoption of an ergonomic intervention among Emergency Medical Service workers.

This study explored factors contributing to intervention adoption decisions among Emergency Medical Service (EMS) workers. Emergency Medical Service workers (n = 190), from six different organisations, participated in a two-month longitudinal study following the introduction of a patient transfer-board (also known as slide-board) designed to ease lateral transfers of patients to and from ambulance cots. Surveys administered at baseline, after one month and after two months sampled factors potentially influencing the EMS providers' decision process. 'Ergonomics Advantage' and 'Patient Advantage' entered into a stepwise regression model predicting 'intention to use' at the end of month one (R (2 )= 0.78). After the second month, the stepwise regression indicated only two factors were predictive of intention to use: 'Ergonomics Advantage,' and 'Endorsed by Champions' (R (2 )= 0.58). Actual use was predicted by: 'Ergonomics Advantage' and 'Previous Tool Experience.' These results relate to key concepts identified in the diffusion of innovation literature and have the potential to further ergonomics intervention adoption efforts. Practitioner Summary. This study explored factors that potentially facilitate the adoption of voluntarily used ergonomics interventions. EMS workers were provided with foldable transfer-boards (slideboards) designed to reduce the physical demands when laterally transferring patients. Factors predictive of adoption measures included perceived ergonomics advantage, the endorsement by champions, and prior tool experience.

Bootstrap Standard Error Estimates in Dynamic Factor Analysis.

Dynamic factor analysis summarizes changes in scores on a battery of manifest variables over repeated measurements in terms of a time series in a substantially smaller number of latent factors. Algebraic formulae for standard errors of parameter estimates are more difficult to obtain than in the usual intersubject factor analysis because of the interdependence of successive observations. Bootstrap methods can fill this need, however. The standard bootstrap of individual timepoints is not appropriate because it destroys their order in time and consequently gives incorrect standard error estimates. Two bootstrap procedures that are appropriate for dynamic factor analysis are described. The moving block bootstrap breaks down the original time series into blocks and draws samples of blocks instead of individual timepoints. A parametric bootstrap is essentially a Monte Carlo study in which the population parameters are taken to be estimates obtained from the available sample. These bootstrap procedures are demonstrated using 103 days of affective mood self-ratings from a pregnant woman, 90 days of personality self-ratings from a psychology freshman, and a simulation study.

Structural Equation Modeling of Multivariate Time Series.

The covariance structure of a vector autoregressive process with moving average residuals (VARMA) is derived. It differs from other available expressions for the covariance function of a stationary VARMA process and is compatible with current structural equation methodology. Structural equation modeling programs, such as LISREL, may therefore be employed to fit the model. Particular attention is given to assumptions concerning the process before the first observation. An application to a repeated time series is used to demonstrate the effect of these assumptions on the structure of the reproduced covariance matrix.

Testing differences between nested covariance structure models: Power analysis and null hypotheses.

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.

How can I connect with thee? Let me count the ways.

Two studies were conducted to examine mental representations of loneliness and social connectedness. In Study 1, young adults (N = 2,531) completed the revised UCLA Loneliness Scale (R-UCLA scale) and demographic questionnaires. An exploratory factor analysis of the R-UCLA scale on half the sample revealed a three-dimensional conceptual structure that generalized across gender. This mental representation consisted of correlated facets labeled Isolation, Relational Connectedness, and Collective Connectedness. A confirmatory factor analysis on the other half of the sample corroborated this three-factor solution. In Study 2, a population-based sample of 197 older males and females (M(age) = 57.5 years) completed the R-UCLA scale and measures of objective social circumstances. The confirmatory factor analysis supported the three-factor structure in this diverse and older adult sample. Each facet was uniquely predicted by theoretically related social circumstances. These findings suggest how humans make meaning of their social relationships in their mental representations of loneliness and connectedness.

Assessing stress in cancer patients: a second-order factor analysis model for the Perceived Stress Scale.

Using the Perceived Stress Scale (PSS), perceptions of global stress were assessed in 111 women following breast cancer surgery and at 12 and 24 months later This is the first study to factor analyze the PSS. The PSS data were factor analyzed each time using exploratory factor analysis with oblique direct quartimin rotation. Goodness-of-fit indices (root mean square error of approximation [RMSEA]), magnitude and pattern of factor loadings, and confidence interval data revealed a two-factor solution of positive versus negative stress items. The findings, replicated across time, also indicate factor stability. Hierarchical factor analyses supported a second-order factor of "perceived stress." This alternative factor model of the PSS is presented along with observations regarding the measure's use in cancer research.

Comments on the Meehl-Waller (2002) procedure for appraisal of path analysis models.

P. E. Meehl and N. G. Waller (2002) proposed an innovative method for assessing path analysis models wherein they subjected a given model, along with a set of alternatives, to risky tests using selected elements of a sample correlation matrix. Although the authors find much common ground with the perspective underlying the Meehl-Waller approach, they suggest that there are aspects of the proposed procedure that require close examination and further development. These include the selection of only one subset of correlations to estimate parameters when multiple solutions are generally available, the fact that the risky tests may test only a subset of parameters rather than the full model of interest, and the potential for different results to be obtained from analysis of equivalent models.

When fit indices and residuals are incompatible.

Standard chi-square-based fit indices for factor analysis and related models have a little known property: They are more sensitive to misfit when unique variances are small than when they are large. Consequently, very small correlation residuals indicating excellent fit can be accompanied by indications of bad fit by the fit indices when unique variances are small. An empirical example of this incompatibility between residuals and fit indices is provided. For illustrative purposes, an artificial example is provided that yields exactly the same correlation residuals as the empirical example but has larger unique variances. For this example, the fit indices indicate excellent fit. A theoretical explanation for this phenomenon is provided using relationships between unique variances and eigenvalues of the fitted correlation matrix.