Abstract
There are two versions of efficient partial least squares (PLSe) methodology, PLSe1 and PLSe2. PLSe2 utilizes the generalized least squares (GLS) covariance structure estimation methodology, and its performance has been verified under different normality and non-normality conditions. Based on this methodology, there must be no independent observed variables in the model. However, there are many instances where researchers would like to estimate models which contain independent observed variables. To address this issue, we propose the methodology of representing the independent observed variables with dummy factors. We take two model-implied covariance matrices from two studies on a nonstandard model and a simple mediation model, use them to generate random samples for our simulations under normality and non-normality conditions, and validate the proposed methodology. We also compare our results across PLSe2 and maximum likelihood (ML) and provide evidence for the estimates’ statistical properties being maintained when artificial variables (i.e., dummy factors) are included in the model. Our results show that while the proposed methodology works well due to the comparability of the estimates and the root mean square error (RMSE) statistics across PLSe2 and ML, the Satorra–Bentler methodology should be considered when PLSe2 is used to estimate models involving dummy factors using both multivariate normal and non-normal data. Last, we provide an illustrative application to demonstrate our simple, practical, and remedial approach in EQS.
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The sample covariance matrices and simulation files have been provided in appendices.
Notes
“Appendix A1” shows the *.eqs file to run the simulation based on the nonstandard model (Method = PLSe2) under non-normality condition (cases = 500, replications = 500). Notably, the population values shown next to the asterisks (e.g., 0.696 in the equation for V1) are the start values used to speed up estimation in each replication.
“Appendix A2” shows the *.eqs file to run the simulation based on the mediation model (Method = PLSe2) under non-normality condition (cases = 500, replications = 500). Notably, the population values shown next to the asterisks (e.g., 0.745 in the equation for V2) are the start values used to speed up estimation in each replication.
Since the sample covariance matrix has been supplied in the model files in “Appendix A3 to A5”, the parameter estimates will be identical to the results reported in Tables 7 and 8. However, the standard errors (Tables 7 and 8) and fit indices (Table 9) will be slightly different from the reported statistics because they are generated using the supplied sample covariance matrices (and not the raw data).
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Acknowledgements
The author is grateful to Zeynab Khodaei and Ilia Ghasemy and dedicates the paper to his late mother, Zahra Soltan Zamani, and his father, Ali Naghi Ghasemy, for their unconditional care and love. This paper is based on a research project related to the theory and practice of servant leadership in academic institutions led by Majid Ghasemy with co-researcher Hazri Jamil.
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This study was funded by the Universiti Sains Malaysia (Grant # 304/CIPPTN/6315200).
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Appendices
Appendices
1.1 Appendix A1
The *.eqs file to run the simulation based on the nonstandard model under non-normality condition (Method = PLSe2, Cases = 500, Replications = 500)
1.2 Appendix A2
The *.eqs file to run the simulation based on the mediation model under non-normality condition (Method = PLSe2, Cases = 500, Replications = 500)
1.3 Appendix A3
ML estimation of the bivariate regression CFA model with gender as a covariate
1.4 Appendix A4
ML estimation of the bivariate regression CFA model with gender as a covariate replaced with a dummy factor
1.5 Appendix A5
PLSe2 estimation of the bivariate regression CFA model with gender as a covariate replaced with a dummy factor
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Ghasemy, M. Estimating models with independent observed variables based on the PLSe2 methodology: a Monte Carlo simulation study. Qual Quant 56, 4129–4159 (2022). https://doi.org/10.1007/s11135-021-01297-2
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DOI: https://doi.org/10.1007/s11135-021-01297-2