Ehri Ryu

Level-Specific Evaluation of Model Fit in Multilevel Structural Equation Modeling

In multilevel structural equation modeling, the “standard” approach to evaluating the goodness of model fit has a potential limitation in detecting the lack of fit at the higher level. Level-specific model fit evaluation can address this limitation and is more informative in locating the source of lack of model fit. We proposed level-specific test statistics for the test of overall model fit, comparative fit index, and root mean squared error of approximation using partially saturated models, and we also considered another level-specific approach proposed by Yuan and Bentler (2007). A simulation study showed that the standard approach failed to detect the lack of fit at the group level. The fit indexes produced by the level-specific approaches both successfully detected the lack of model fit at each level. There were only minor differences in the performance of the 2 level-specific approaches.

Multilevel Modeling: Current and Future Applications in Personality Research

Traditional statistical analyses can be compromised when data are collected from groups or multiple observations are collected from individuals. We present an introduction to multilevel models designed to address dependency in data. We review current use of multilevel modeling in 3 personality journals showing use concentrated in the 2 areas of experience sampling and longitudinal growth. Using an empirical example, we illustrate specification and interpretation of the results of series of models as predictor variables are introduced at Levels 1 and 2. Attention is given to possible trends and cycles in longitudinal data and to different forms of centering. We consider issues that may arise in estimation, model comparison, model evaluation, and data evaluation (outliers), highlighting similarities to and differences from standard regression approaches. Finally, we consider newer developments, including 3-level models, cross-classified models, nonstandard (limited) dependent variables, multilevel structural equation modeling, and nonlinear growth. Multilevel approaches both address traditional problems of dependency in data and provide personality researchers with the opportunity to ask new questions of their data.

Effects of skewness and kurtosis on normal-theory based maximum likelihood test statistic in multilevel structural equation modeling

A simulation study investigated the effects of skewness and kurtosis on level-specific maximum likelihood (ML) test statistics based on normal theory in multilevel structural equation models. The levels of skewness and kurtosis at each level were manipulated in multilevel data, and the effects of skewness and kurtosis on level-specific ML test statistics were examined. When the assumption of multivariate normality was violated, the level-specific ML test statistics were inflated, resulting in Type I error rates that were higher than the nominal level for the correctly specified model. Q-Q plots of the test statistics against a theoretical chi-square distribution showed that skewness led to a thicker upper tail and kurtosis led to a longer upper tail of the observed distribution of the level-specific ML test statistic for the correctly specified model.

Mediation and Moderation: Testing Relationships between Symptom Status, Functional Health, and Quality of Life in HIV Patients.

We extended Wilson and Clearys (1995) health-related quality of life model to examine the relationships among symptom status (Symptoms), functional health (Disability), and quality of life (QOL). Using a community sample (N = 956) of male HIV positive patients, we tested a mediation model in which the relationship between Symptoms and QOL is partially mediated by Disability. Common and unique ideas from 3 approaches to examining moderation of effects in mediational models (Edwards & Lambert, 2007; MacKinnon, 2008; Preacher, Rucker, & Hayes, 2007) were used to test whether (a) the direct relationship of Symptoms to QOL and (b) the relationship of Disability to QOL are moderated by age. In the mediation model, both the direct and the indirect (mediated) effects were significant. The direct relationship of Symptoms to QOL was significantly moderated by age, but the relationship of Disability to QOL was not. High Symptoms were associated with lower QOL at all ages, but this relationship became stronger at older ages. We compare the 3 approaches and consider their advantages over traditional approaches to combining mediation and moderation.

Factorial invariance in multilevel confirmatory factor analysis

This paper presents a procedure to test factorial invariance in multilevel confirmatory factor analysis. When the group membership is at level 2, multilevel factorial invariance can be tested by a simple extension of the standard procedure. However level-1 group membership raises problems which cannot be appropriately handled by the standard procedure, because the dependency between members of different level-1 groups is not appropriately taken into account. The procedure presented in this article provides a solution to this problem. This paper also shows Muthéns maximum likelihood (MUML) estimation for testing multilevel factorial invariance across level-1 groups as a viable alternative to maximum likelihood estimation. Testing multilevel factorial invariance across level-2 groups and testing multilevel factorial invariance across level-1 groups are illustrated using empirical examples. SAS macro and Mplus syntax are provided.

Confirmation of the validity of the HAQ-DI in two populations living with chronic illnesses.

The assessment of functional health in chronic illnesses such as HIV/AIDS and rheumatoid arthritis is central to the measurement of health-related quality of life. The purpose of this article is to report the testing and comparison of the measurement structure of the Health Assessment Questionnaire-Disability Index (HAQ-DI), a measure of functional health, in 917 persons living with HIV/AIDS and 901 individuals with rheumatoid arthritis. The samples come from data collected as part of the Arthritis, Rheumatism, and Aging Medical Information System (ARAMIS) and AIDS Time-Oriented Health Outcome Study (ATHOS) projects. Using confirmatory factor analysis (CFA), the hypothesized structure represented by a general factor (functional health) and eight measured items was tested separately. Based on the fit indexes, the model fit the ATHOS data (χ2 = 36.933, p < .0117; CFI = 1.000; SRMR = 0.025). After correlating the error terms for two of the measured items, the model also fit the ARAMIS data (χ2 = 302.34, p = .0000; CFI = 0.937; SRMR = 0.041). This analysis provides further support of the construct validity of the HAQ-DI for persons living with HIV/AIDS or rheumatoid arthritis.

The Role of Centering for Interaction of Level 1 Variables in Multilevel Structural Equation Models

This article examined the role of centering in estimating interaction effects in multilevel structural equation models. Interactions are typically represented by product term of 2 variables that are hypothesized to interact. In multilevel structural equation modeling (MSEM), the product term involving Level 1 variables is decomposed into within-cluster and between-cluster random components. The choice of centering affects the decomposition of the product term, and therefore affects the sample variance and covariance associated with the product term used in the maximum likelihood fitting function. The simulation study showed that for an interaction between a Level 1 variable and a Level 2 variable, the product term of uncentered variables or the product term of grand mean centered variables produced unbiased estimates in both Level 1 and Level 2 models. The product term of cluster mean centered variables produced biased estimates in the Level 1 model. For an interaction between 2 Level 1 variables, the product term of cluster mean centered variables produced unbiased estimates in the Level 1 model, whereas the product term of grand mean centered variables produced unbiased estimates for the Level 1 model. Recommendations for researchers who wish to estimate interactions in MSEM are provided.

Distinguishing Between-Person and Within-Person Relationships in Longitudinal Health Research: Arthritis and Quality of Life

BackgroundMany health measures (e.g., blood pressure, quality of life) have meaningful fluctuation over time around a relatively stable mean level for each person.PurposeThis didactic paper describes two closely related statistical models for examining between-person and within-person relationships between two or more sets of measures collected over time: the latent intercept model with correlated residuals (LI) in structural equation modeling framework and the multivariate multilevel model (MVML) in multilevel modeling framework.ResultsWe illustrated that the basic LI model and the MVML model are equivalent. We presented an illustrative example using a national arthritis data resource to examine between-person and within-person relationships of symptom status, functional health, and quality of life in arthritis patients.DiscussionAdditional design and modeling issues for the treatment of missing data are considered. We discuss contexts in which one of the two models may be preferred. Mplus and SAS syntax are available.

Multiple Group Analysis in Multilevel Structural Equation Model Across Level 1 Groups.

This article introduces and evaluates a procedure for conducting multiple group analysis in multilevel structural equation model across Level 1 groups (MG1-MSEM; Ryu, 2014). When group membership is at Level 1, multiple group analysis raises two issues that cannot be solved by a simple extension of the standard multiple group analysis in single-level structural equation model. First, the Level 2 data are not independent between Level 1 groups. Second, the standard procedure fails to take into account the dependency between members of different Level 1 groups within the same cluster. The MG1-MSEM approach provides solutions to these problems. In MG1-MSEM, the Level 1 mean structure is necessary to represent the differences between Level 1 groups within clusters. The Level 2 model is the same regardless of Level 1 group membership. A simulation study examined the performance of MUML (Muthéns maximum likelihood) estimation in MG1-MSEM. The MG1-MSEM approach is illustrated for both a multilevel path model and a multilevel factor model using empirical data sets.

Multilevel Factorial Invariance in n-Level Structural Equation Modeling (nSEM)

We present a multigroup multilevel confirmatory factor analysis (CFA) model and a procedure for testing multilevel factorial invariance in n-level structural equation modeling (nSEM). Multigroup multilevel CFA introduces a complexity when the group membership at the lower level intersects the clustered structure, because the observations in different groups but in the same cluster are not independent of one another. nSEM provides a framework in which the multigroup multilevel data structure is represented with the dependency between groups at the lower level properly taken into account. The procedure for testing multilevel factorial invariance is illustrated with an empirical example using an R package xxm2.