N-Level Structural Equation Models (nSEM): The Effect of Sample Size on the Parameter Estimation in Latent Random-Intercept Model
Abstract
Multilevel Structural Equation Modeling (MSEM) is claimed to address hierarchical data structures and latent response variables, but it becomes unstable with an increasing number of levels. N-Level SEM (nSEM) is an SEM framework designed to handle a growing number of levels in the model. The nSEM framework uses the Maximum Likelihood Estimation (MLE) method for parameter estimation, which requires a large sample size and correct model specification. Therefore, it is essential to consider the necessary minimal sample size to ensure accurate and efficient parameter estimation in the nSEM model. This study examined how sample size affects the performance of parameter estimators in nSEM models. We propose a method to evaluate the effect of many environments to estimate the results of factor loadings and environmental variance produced by the model. In addition, we also assess the impact of environment size on the estimation results of factor loadings and individual variance. The results were then applied to actual data on student mathematics learning motivation in Depok. The findings show that neither the number of environments nor the size of the environment affects the performance of fixed parameter estimation in the nSEM model. nSEM indicates excellent performance in estimating environmental variance at level 2 when the number of environments increases. Conversely, increasing the size of the environment worsens the performance of estimating individual variance parameters. Overall, the nSEM framework for the latent random-intercept (LatenRI) model performs well with increasing sample sizes. The application data on LatenRI models show almost similar estimation results.
Keywords: hierarchical data; latent random intercept model; multilevel structural equation modeling; n-level structural equation modeling.
Abstrak
Multilevel Structural Equation Modeling (MSEM) diklaim dapat mengatasi struktur data hierarki dan variabel respons laten, namun menjadi tidak stabil dengan bertambahnya jumlah level. N-Level SEM (nSEM) adalah kerangka kerja SEM yang dirancang untuk menangani semakin banyak level dalam model. Kerangka kerja nSEM menggunakan metode Maximum Likelihood Estimation (MLE) untuk estimasi parameter, yang memerlukan ukuran sampel yang besar dan spesifikasi model yang benar. Oleh karena itu, penting untuk mempertimbangkan ukuran sampel minimal yang diperlukan untuk memastikan estimasi parameter yang akurat dan efisien dalam model nSEM. Studi ini menguji bagaimana ukuran sampel mempengaruhi kinerja penduga parameter dalam model nSEM. Kami mengusulkan metode untuk mengevaluasi pengaruh banyak lingkungan dalam memperkirakan hasil factor loadings dan varians lingkungan yang dihasilkan oleh model. Selain itu, kami juga menilai dampak ukuran lingkungan terhadap hasil estimasi factor loadings dan varians individu. Hasilnya kemudian diterapkan pada data aktual motivasi belajar matematika siswa di Depok. Hasil menunjukkan bahwa baik jumlah lingkungan maupun ukuran lingkungan tidak mempengaruhi kinerja estimasi parameter tetap pada model nSEM. nSEM menunjukkan kinerja yang sangat baik dalam memperkirakan varians lingkungan pada level 2 ketika jumlah lingkungan meningkat. Sebaliknya, peningkatan ukuran lingkungan akan memperburuk kinerja pendugaan parameter varians individu. Secara keseluruhan, kerangka nSEM untuk model intersepsi acak laten (LatenRI) bekerja dengan baik dengan meningkatnya ukuran sampel. Data penerapan model LatenRI menunjukkan hasil estimasi yang hampir serupa.
Kata Kunci: data hirarki; model intersep acak laten; model persamaan structural multilevel; model persamaan structural n-level.
2020MSC: 62D99
Keywords
References
M. D. M. Rueda, B. Cobo, and A. Arcos, “Regression Models in Complex Survey Sampling for Sensitive Quantitative Variables,” Mathematics, vol. 9, no. 6, pp. 1–13, Mar. 2021, doi: 10.3390/math9060609.
D. Holt, A. J. Scott, and P. D. Ewings, “Chi-Squared Tests with Survey Data,” Journal of the Royal Statistical Society Series A, vol. 143, no. 3, pp. 303–320, 1980.
K. J. Preacher, M. J. Zyphur, and Z. Zhang, “Multilevel Structural Equation Models for Assessing Moderation Within and Across Levels of Analysis,” Psychol Methods, vol. 21, no. 2, pp. 189–205, 2016, doi: 10.1037/met0000052.supp.
R. D. Goddard, M. Tschannen-Moran, and W. K. Hoy, “A Multilevel Examination of the Distribution and Effects of Teacher Trust in Students and Parents in Urban Elementary Schools,” Elem Sch J, vol. 102, no. 1, pp. 3–17, 2001.
S. De Iaco and S. Maggio, “Using Multilevel Models to Evaluate the Attitude of Separate Waste Collection in Young People,” Metron, vol. 80, pp. 77–95, Apr. 2022, doi: 10.1007/s40300-020-00194-2.
B. O. Muthén, “Mean and Covariance Structure Analysis of Hierarchical Data,” in The Psychometric Society Meeting, in UCLA Statistics. Princeton: UCLA, 1990, pp. 1–57.
I. Tsaousis, G. D. Sideridis, and K. Al-Harbi, “Examining Differences in Within-and Between-Person Simple Structures of an Engineering Qualification Test Using Multilevel MIMIC Structural Equation Modeling,” Front Appl Math Stat, vol. 4, no. 3, pp. 1–14, 2018, doi: 10.3389/fams.2018.00003.
A. H. Leyland and P. P. Groenewegen, “Multilevel Data Structures,” in Multilevel Modelling for Public Health and Health Services Research, Springer International Publishing, 2020, pp. 49–68. doi: 10.1007/978-3-030-34801-4_4.
B. O. Muthen, “Multilevel Covariance Structure Analysis,” Sociol Methods Res, vol. 22, no. 3, pp. 376–398, 1994, doi: 10.1177/0049124194022003006.
P. Mehta, “n-Level Structural Equation Modeling,” in Applied Quantitative Analysis in Education and the Social Sciences, New York: Routledge, 2013, pp. 329–361.
L. Branum-Martin, “Multilevel Modeling: Practical Examples to Illustrate a Special Case of SEM,” in Applied Quantitative Analysis in Education and the Social Sciences, 2013, pp. 95–124.
O. Kwok, M. H. Lai, F. Tong, and R. Lara-alecio, “Analyzing Complex Longitudinal Data in Educational Research : A Demonstration With Project English Language and Literacy Acquisition ( ELLA ) Data Using xxM,” Frontiers in Psycology, vol. 9, no. June, pp. 1–12, 2018, doi: 10.3389/fpsyg.2018.00790.
E. Ryu, “Multiple Group Analysis in Multilevel Structural Equation Model Across Level 1 Groups,” Multivariate Behav Res, vol. 50, no. 3, pp. 300–315, 2015, doi: 10.1080/00273171.2014.1003769.
I. Devlieger and Y. Rosseel, “Multilevel Factor Score Regression,” Multivariate Behavioral Research, vol. 55, no. 4. Routledge, pp. 600–624, Jul. 03, 2019. doi: 10.1080/00273171.2019.1661817.
B. Kelcey, K. Cox, and N. Dong, “Croon’s Bias-Corrected Factor Score Path Analysis for Small- to Moderate-Sample Multilevel Structural Equation Models,” Organ Res Methods, vol. 24, no. 1, pp. 55–77, Jan. 2021, doi: 10.1177/1094428119879758.
A. Sagan, “Sample Size in Multilevel Structural Equation Modeling – the Monte Carlo Approach,” Econometrics, vol. 23, no. 4, pp. 63–79, 2019, doi: 10.15611/eada.2019.4.05.
T. A. B. Snijders, “Power and Sample Size in Multilevel Modeling,” Encyclopedia of Statistics in Behavioral Science, vol. 3, pp. 1570–1573, 2005, [Online]. Available: https://www.researchgate.net/publication/228079507
M. T. Barendse and Y. Rosseel, “Multilevel SEM with Random Slopes in Discrete Data Using the Pairwise Maximum Likelihood,” British Journal of Mathematical and Statistical Psychology, vol. 76, no. 2, pp. 327–352, May 2023, doi: 10.1111/bmsp.12294.
E. Ryu, “Model Fit Evaluation in Multilevel Structural Equation Models,” Front Psychol, vol. 5, no. 81, pp. 1–9, 2014, doi: 10.3389/fpsyg.2014.00081.
Y. Petscher and C. Schatschneider, “Using n-Level Structural Equation Models for Causal Modeling in Fully Nested, Partially Nested, and Cross-Classified Randomized Controlled Trials,” Educ Psychol Meas, vol. 79, no. 6, pp. 1075–1102, 2019, doi: 10.1177/0013164419840071.
H. Y. Hsu, J. H. Lin, O. M. Kwok, S. Acosta, and V. Willson, “The Impact of Intraclass Correlation on the Effectiveness of Level-Specific Fit Indices in Multilevel Structural Equation Modeling: A Monte Carlo Study,” Educ Psychol Meas, vol. 77, no. 1, pp. 5–31, Jan. 2017, doi: 10.1177/0013164416642823.
D. J. Barr, R. Levy, C. Scheepers, and H. J. Tily, “Random Effects Structure for Confirmatory Hypothesis Testing: Keep it Maximal,” J Mem Lang, vol. 68, no. 3, pp. 255–278, 2013.
H. Matuschek, R. Kliegl, S. Vasishth, H. Baayen, and D. Bates, “Balancing Type I Error and Power in Linear Mixed Models,” J Mem Lang, vol. 94, pp. 305–315, Jun. 2017, doi: 10.1016/j.jml.2017.01.001.
E. S. W. Ng, K. Diaz-Ordaz, R. Grieve, R. M. Nixon, S. G. Thompson, and J. R. Carpenter, “Multilevel Models for Cost-Effectiveness Analyses that Use Cluster Randomised Trial Data: An Approach to Model Choice,” Stat Methods Med Res, vol. 25, no. 5, pp. 2036–2052, Oct. 2016, doi: 10.1177/0962280213511719.
J. N. Pritikin, M. D. Hunter, T. von Oertzen, T. R. Brick, and S. M. Boker, “Many-Level Multilevel Structural Equation Modeling: An Efficient Evaluation Strategy,” Structural Equation Modeling, vol. 24, no. 5, pp. 684–698, Sep. 2017, doi: 10.1080/10705511.2017.1293542.
M. J. Brewer, A. Butler, and S. L. Cooksley, “The Relative Performance of AIC, AICC and BIC in the Presence of Unobserved Heterogeneity,” Methods Ecol Evol, vol. 7, no. 6, pp. 679–692, Jun. 2016, doi: 10.1111/2041-210X.12541.
V. Eminita, A. Saefuddin, K. Sadik, and U. D. Syafitri, “Analyzing Multilevel Model of Educational Data: Teachers’ Ability Effect on Students’ Mathematical Learning Motivation,” Journal on Mathematics Education, vol. 15, no. 2, pp. 431–450, 2024, doi: 10.22342/jme.v13i1.xxxx.
T. D. Little, Longitudinal Structural Equation Modeling. New York: The Gilford Press, 2013.
J. J. Hox and C. J. M. Maas, “The Accuracy of Multilevel Structural Equation Modeling with Pseudobalanced Groups and Small Samples,” Structural Equation Modeling, vol. 8, no. 2, pp. 157–174, 2001.
B. Meuleman and J. Billiet, “A Monte Carlo Sample Size Study: How Many Countries are Needed for Accurate Multilevel SEM?,” Surv Res Methods, vol. 3, no. 1, pp. 45–58, 2009, [Online]. Available: http://www.surveymethods.org
DOI: 10.15408/inprime.v6i1.38914
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