08.Latent variable modeling06.Structural equation modeling - sporedata/researchdesigneR GitHub Wiki
Structural equation modeling (SEM) demonstrates the association between two or more latent variables as well as other variables. Under certain circumstances, SEM can represent causal relations. Common use cases include the assessment of:
- Mediation effect a variable [9]
- Interaction between constructs [8]
- The influence of one domain on another [6], [7]
- Compare the magnitude of the relationship between any two variables (single paths or total effects) with a p-value.
- Differences between groups in how domains interact - genders, races, ethnicities, religious groups, disease severity, ages, etc. [4]
- Longitudinal evolution of the relationship among different domains [10]
- Data representing clinical phenomena with at least two latent variables
Test for the equality of standardized requession coefficients in SEM using Wald tests [11]
Measurement invariance attempts to verify that the estimated factors measure the same underlying latent construct across groups or time. Invariance is often seen as an initial step before investigating larger multigroup models to ensure equivalent measurement across groups and rule out artifacts when comparing means or predictions across groups. If we have group invariance, we know the construct is being measured the same across groups. We can then compare the differences between groups in path analysis. If we have group non-invariance, we cannot compare the construct between groups. The general approach to factorial invariance tests is to use likelihood ratio tests that compare nested models in which one model in which that set of parameters (e.g., loadings) is constrained to be equal across groups to another model in which that set of parameters is allowed to be estimated freely and separately across groups [12].
Multigroup structural equation models test separate structural models in two or more groups. Such models may involve path models, comparison of indirect effects, confirmatory factor models, or full structural equation models. Multigroup models generally follow the same structure in each group and can provide separate estimates of within-group parameters (e.g., loadings, paths, and correlations). Chi-square and fit indices can be obtained for each group separately, and global fit indices joint, multigroup model [13].
path analysis can explain the causal relationships among variables by creating a path diagram. A standard function of path analysis is mediation, which assumes that a variable can influence an outcome directly and indirectly through another variable. SEM is composed of the measurement model and the structural model. A measurement model measures latent or composite variables, while the structural model tests all hypothetical dependencies based on path analysis. In other words, SEM is a combination of two statistical methods: confirmatory factor analysis and path analysis. SEM evaluation is based on the fit indices for the test of a single path coefficient (i.e., p-value and standard error) and the overall model fit (i.e., χ 2, RMSEA) [14].
mediation can be defined as “the process by which one variable transmits an effect onto another through one or more mediating variables.”¹ Let’s say we have a variable X (depression, for example) associated with a variable Y (liver disease, for example) and we want to know how the effect of X on Y is mediated by another variable M (drinking, for example). To know how M mediates the association between X and Y, we need to measure three associations at the same time: X -> Y, X -> M, and M -> Y, and the association between M and Y. Using Wright's tracing rules, we can create three paths for the effect of X on Y:
- the direct path X -> Y, known as the direct effect of X on Y;
- the indirect path X -> M -> Y, known as the indirect effect of X on Y, mediated by M;
- the full path, that is the sum of the two paths above, known as the total effect of X on Y.
These effects are calculated using the regression coefficients (beta) of each association, so they must be significant to be interpreted in a meaningful way. Wright's rules say you must multiply the path coefficients to get the path effect. So, for the first path (1) we have only the regression coefficient of X -> Y. For the effect of the second path (2), we have to multiply the regression coefficients of X -> M and M -> Y. For the effect of the third path (3), we have to sum the effects of path 1 and path 2.
Depending on the relative size of the indirect effect, we can have a partial mediation or a full mediation. According to Hair et al (2014)[15]., if (indirect effect) / (total effect) * 100 is greater than 80%, we can say we have a full mediation, and the association of X and Y is fully explained by the mediator.
A three-way interaction occurs when two moderator variables work together to influence the regression of the dependent variable on an independent variable. In other words, a regression model with a significant three-way interaction of continuous variables. For instance, to study the moderating effect of income and gender on the relationship between tolerance to workplace aggression and the prediction of workplace withdrawal behavior, one hypothesizes that the tolerance to workplace aggression on the prediction of workplace withdrawal behavior was moderated by income and gender Article Figure 1 We should test for model fitness. To be considered a good fit, the model must yield a Comparative Fit Index (CFI) with values greater than.90 and the Root Mean Square Error of Approximation (RMSEA) and Standardized Root Mean Residuals (SRMR) with values lower than .08. (Article Table 1), we must report the maximum likelihood estimates. The paths from the endogenous variables to other endogenous variables are labeled beta β. (Article Table 3). We also must report the graphic representation of the 3-way interaction. Article Figure 2 The plot shows that when income was low, tolerance to workplace aggression was more likely to influence workplace withdrawal behavior among women than among men. When income was high, women were more likely to engage in workplace withdrawal behavior than men because of their tolerance of workplace aggression. However, this impact was less pronounced than it was when their income was low. On the other hand, male workplace withdrawal behavior is less likely to be influenced by tolerance to workplace aggressiveness when income is high compared to low.
Regarding the statistics used to evaluate model fit in SEM, RMSEA is an absolute fit index that assesses how far a hypothesized model is from a perfect model. On the contrary, CFI and TLI are incremental fit indices that compare the fit of a hypothesized model with that of a baseline model (i.e., a model with the worst fit). GFI is the proportion of variance accounted for by the estimated population covariance. RNI uses the non centrality parameter as an index of lack of fit, just as the RMSEA method does. IFI adjusts the Normed Fit Index (NFI) for sample size and degrees of freedom. The last one is SRMR, representing the standardized square root of the difference between the residuals of the sample covariance matrix and the hypothesized model.
1.Reporting structural equation modeling results in Psychology and Aging: Some proposed guidelines.
2.Reporting Structural Equation Modeling and Confirmatory Factor Analysis Results: A Review
- lavaan: Latent Variable Analysis [1].
- psych: Procedures for Psychological, Psychometric, and Personality Research [2].
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Books
- Latent Variable Modeling Using R: A Step-By-Step Guide [3].
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Articles
[1] Rosseel Y. Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Journal of statistical software. 2012 Dec 19;48(2):1-36.
[2] Revelle W. psych: Procedures for psychological, psychometric, and personality research. Northwestern University, Evanston, Illinois. 2014 Aug;165:1-0.
[3] Beaujean AA. Latent Variable Modeling Using R: A Step-By-Step Guide. Routledge; 2014 May 9.
[4] Pham PN, Sharma M, Bindu KK, et al. Protective Behaviors Associated With Gender During the 2018-2020 Ebola Outbreak in Eastern Democratic Republic of the Congo.. JAMA Netw Open. 2022;5(2):e2147462. Published 2022 Feb 1.
[5] Uymaz P, Uymaz AO. Assessing acceptance of augmented reality in nursing education.. PLoS One. 2022;17(2):e0263937. Published 2022 Feb 17.
[6] Hirose J, Kotani K. How does inquisitiveness matter for generativity and happiness?.. PLoS One. 2022;17(2):e0264222. Published 2022 Feb 25.
[7] Akther T, Nur T. A model of factors influencing COVID-19 vaccine acceptance: A synthesis of the theory of reasoned action, conspiracy theory belief, awareness, perceived usefulness, and perceived ease of use. PLoS One. 2022;17(1):e0261869. Published 2022 Jan 12.
[8] Kroemeke A, Sobczyk-Kruszelnicka M. Interaction effect of coping self-efficacy and received support in daily life of hematopoietic cell transplant patient-caregiver dyads. PLoS One. 2021;16(11):e0260128. Published 2021 Nov 17.
[9] Hu S, Luo W, Darzi A, et al. Do racial and ethnic disparities in following stay-at-home orders influence COVID-19 health outcomes? A mediation analysis approach. PLoS One. 2021;16(11):e0259803. Published 2021 Nov 11.
[10] Stenlund S, Junttila N, Koivumaa-Honkanen H, et al. Longitudinal stability and interrelations between health behavior and subjective well-being in a follow-up of nine years. PLoS One. 2021;16(10):e0259280. Published 2021 Oct 29.
[11] Klopp E. A Tutorial on Testing the Equality of Standardized Regression Coefficients in Structural Equation Models using Wald Tests with lavaan. PsyArXiv. 2020;16(4):315-133.
[12] Putnick DL, Bornstein MH. Measurement invariance conventions and reporting: The state of the art and future directions for psychological research. Developmental review. 2016 Sep;41:71-90.
[13] Yuan KH, Bentler PM. A unified approach to multigroup structural equation modeling with nonstandard samples. New developments and techniques in structural equation modeling. 2001;35-56.
[14] Fan Y, Chen J, Shirkey G, John R, Wu SR, Park H, Shao C. Applications of structural equation modeling (SEM) in ecological studies: an updated review. Ecological Processes. 20l6 Nov 22;5(1):1-12.
[15] Hair JF, Hult GTM, Ringle CM, Sarstedt M. A Primer on Partial Least Squares Structural Equation Modeling. Thousand Oaks:Sage. 2014; 384 .