{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:15:31Z","timestamp":1775643331488,"version":"3.50.1"},"reference-count":79,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T00:00:00Z","timestamp":1681689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Structural equation models (SEM) are widely used in the social sciences. They model the relationships between latent variables in structural models, while defining the latent variables by observed variables in measurement models. Frequently, it is of interest to compare particular parameters in an SEM as a function of a discrete grouping variable. Multiple-group SEM is employed to compare structural relationships between groups. In this article, estimation approaches for the multiple-group are reviewed. We focus on comparing different estimation strategies in the presence of local model misspecifications (i.e., model errors). In detail, maximum likelihood and weighted least-squares estimation approaches are compared with a newly proposed robust Lp loss function and regularized maximum likelihood estimation. The latter methods are referred to as model-robust estimators because they show some resistance to model errors. In particular, we focus on the performance of the different estimators in the presence of unmodelled residual error correlations and measurement noninvariance (i.e., group-specific item intercepts). The performance of the different estimators is compared in two simulation studies and an empirical example. It turned out that the robust loss function approach is computationally much less demanding than regularized maximum likelihood estimation but resulted in similar statistical performance.<\/jats:p>","DOI":"10.3390\/a16040210","type":"journal-article","created":{"date-parts":[[2023,4,18]],"date-time":"2023-04-18T01:36:45Z","timestamp":1681781805000},"page":"210","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Model-Robust Estimation of Multiple-Group Structural Equation Models"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8226-3132","authenticated-orcid":false,"given":"Alexander","family":"Robitzsch","sequence":"first","affiliation":[{"name":"IPN\u2014Leibniz Institute for Science and Mathematics Education, Olshausenstra\u00dfe 62, 24118 Kiel, Germany"},{"name":"Centre for International Student Assessment (ZIB), Olshausenstra\u00dfe 62, 24118 Kiel, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bartholomew, D.J., Knott, M., and Moustaki, I. (2011). Latent Variable Models and Factor Analysis: A Unified Approach, Wiley.","DOI":"10.1002\/9781119970583"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bollen, K.A. (1989). Structural Equations with Latent Variables, John Wiley & Sons.","DOI":"10.1002\/9781118619179"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Arminger, G., Clogg, C.C., and Sobel, M.E. (1995). Handbook of Statistical Modeling for the Social and Behavioral Sciences, Springer.","DOI":"10.1007\/978-1-4899-1292-3"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"J\u00f6reskog, K.G., Olsson, U.H., and Wallentin, F.Y. (2016). Multivariate Analysis with LISREL, Springer.","DOI":"10.1007\/978-3-319-33153-9"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Mulaik, S.A. (2009). Foundations of Factor Analysis, CRC Press.","DOI":"10.1201\/b15851"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Edwards, M.C., and MacCallum, R.C. (2012). Current Topics in the Theory and Application of Latent Variable Models, Routledge.","DOI":"10.4324\/9780203813409"},{"key":"ref_7","first-page":"297","article-title":"Structural equation modeling","volume":"Volume 26","author":"Rao","year":"2007","journal-title":"Handbook of Statistics, Vol. 26: Psychometrics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"631","DOI":"10.3390\/stats5030039","article-title":"Comparing the robustness of the structural after measurement (SAM) approach to structural equation modeling (SEM) against local model misspecifications with alternative estimation approaches","volume":"5","author":"Robitzsch","year":"2022","journal-title":"Stats"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1007\/s11336-015-9451-3","article-title":"Quantifying adventitious error in a covariance structure as a random effect","volume":"80","author":"Wu","year":"2015","journal-title":"Psychometrika"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Huber, P.J., and Ronchetti, E.M. (2009). Robust Statistics, Wiley.","DOI":"10.1002\/9780470434697"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Maronna, R.A., Martin, R.D., and Yohai, V.J. (2006). Robust Statistics: Theory and Methods, Wiley.","DOI":"10.1002\/0470010940"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s40300-020-00185-3","article-title":"The main contributions of robust statistics to statistical science and a new challenge","volume":"79","author":"Ronchetti","year":"2021","journal-title":"Metron"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Robitzsch, A. (2022). Estimation methods of the multiple-group one-dimensional factor model: Implied identification constraints in the violation of measurement invariance. Axioms, 11.","DOI":"10.3390\/axioms11030119"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"102805","DOI":"10.1016\/j.ssresearch.2022.102805","article-title":"Measurement invariance in the social sciences: Historical development, methodological challenges, state of the art, and future perspectives","volume":"110","author":"Seddig","year":"2023","journal-title":"Soc. Sci. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/BF02294825","article-title":"Measurement invariance, factor analysis and factorial invariance","volume":"58","author":"Meredith","year":"1993","journal-title":"Psychometrika"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Millsap, R.E. (2011). Statistical Approaches to Measurement Invariance, Routledge.","DOI":"10.4324\/9780203821961"},{"key":"ref_17","unstructured":"Holland, P.W., and Wainer, H. (1993). Differential Item Functioning: Theory and Practice, Lawrence Erlbaum."},{"key":"ref_18","unstructured":"Rao, C.R., and Sinharay, S. (2007). Handbook of Statistics, Vol. 26: Psychometrics, Elsevier."},{"key":"ref_19","unstructured":"Chen, Y., Li, C., and Xu, G. (2021). DIF statistical inference and detection without knowing anchoring items. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"192","DOI":"10.3390\/stats6010012","article-title":"Comparing robust linking and regularized estimation for linking two groups in the 1PL and 2PL models in the presence of sparse uniform differential item functioning","volume":"6","author":"Robitzsch","year":"2023","journal-title":"Stats"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"666","DOI":"10.3102\/10769986221109208","article-title":"Testing differential item functioning without predefined anchor items using robust regression","volume":"47","author":"Wang","year":"2022","journal-title":"J. Educ. Behav. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Boos, D.D., and Stefanski, L.A. (2013). Essential Statistical Inference, Springer.","DOI":"10.1007\/978-1-4614-4818-1"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1111\/j.1467-9531.2011.01236.x","article-title":"Biases of parameter estimates in misspecified structural equation models","volume":"41","author":"Kolenikov","year":"2011","journal-title":"Sociol. Methodol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1912526","article-title":"Maximum likelihood estimation of misspecified models","volume":"50","author":"White","year":"1982","journal-title":"Econometrica"},{"key":"ref_25","first-page":"1","article-title":"Generalized least squares estimators in the analysis of covariance structures","volume":"8","author":"Browne","year":"1974","journal-title":"S. Afr. Stat. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1080\/10705511.2013.824793","article-title":"Understanding robust corrections in structural equation modeling","volume":"21","author":"Savalei","year":"2014","journal-title":"Struct. Equ. Model."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Cudeck, R., and MacCallum, R.C. (2007). Factor Analysis at 100, Lawrence Erlbaum.","DOI":"10.4324\/9780203936764"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1177\/0049124107301946","article-title":"Least absolute deviation estimation in structural equation modeling","volume":"36","author":"Siemsen","year":"2007","journal-title":"Sociol. Methods Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1080\/10705511.2021.1971527","article-title":"Flexible extensions to structural equation models using computation graphs","volume":"29","author":"Oberski","year":"2022","journal-title":"Struct. Equ. Model."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"246","DOI":"10.3390\/stats3030019","article-title":"Lp loss functions in invariance alignment and Haberman linking with few or many groups","volume":"3","author":"Robitzsch","year":"2020","journal-title":"Stats"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kolen, M.J., and Brennan, R.L. (2014). Test Equating, Scaling, and Linking, Springer.","DOI":"10.1007\/978-1-4939-0317-7"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1080\/10705511.2014.919210","article-title":"Multiple-group factor analysis alignment","volume":"21","author":"Asparouhov","year":"2014","journal-title":"Struct. Equ. Model."},{"key":"ref_33","first-page":"978","article-title":"IRT studies of many groups: The alignment method","volume":"5","author":"Asparouhov","year":"2014","journal-title":"Front. Psychol."},{"key":"ref_34","first-page":"303","article-title":"An extension of the invariance alignment method for scale linking","volume":"62","author":"Pokropek","year":"2020","journal-title":"Psych. Test Assess. Model."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1080\/00273171.2019.1681252","article-title":"Regularized estimation of the nominal response model","volume":"55","author":"Battauz","year":"2020","journal-title":"Multivar. Behav. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s11634-015-0205-y","article-title":"A uniform framework for the combination of penalties in generalized structured models","volume":"11","author":"Oelker","year":"2017","journal-title":"Adv. Data Anal. Classif."},{"key":"ref_37","unstructured":"Lee, S.Y. (2007). Handbook of Latent Variable and Related Models, Elsevier."},{"key":"ref_38","unstructured":"Fox, J., and Weisberg, S. (2023, March 27). Robust Regression in R: An Appendix to an R Companion to Applied Regression. Available online: https:\/\/bit.ly\/3canwcw."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1080\/03610927708827533","article-title":"Robust regression using iteratively reweighted least-squares","volume":"6","author":"Holland","year":"1977","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1080\/03610929708831988","article-title":"Robust regression: A weighted least squares approach","volume":"26","author":"Chatterjee","year":"1997","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1080\/00031305.2012.687494","article-title":"Who invented the delta method?","volume":"66","year":"2012","journal-title":"Am. Stat."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1177\/1536867X1001000201","article-title":"Resampling variance estimation for complex survey data","volume":"10","author":"Kolenikov","year":"2010","journal-title":"Stata J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1080\/10705511.2022.2039660","article-title":"Partially confirmatory approach to factor analysis with Bayesian learning: A LAWBL tutorial","volume":"22","author":"Chen","year":"2022","journal-title":"Struct. Equ. Model."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s11336-021-09751-8","article-title":"Single- and multiple-group penalized factor analysis: A trust-region algorithm approach with integrated automatic multiple tuning parameter selection","volume":"86","author":"Geminiani","year":"2021","journal-title":"Psychometrika"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Hirose, K., and Terada, Y. (2022). Sparse and simple structure estimation via prenet penalization. Psychometrika.","DOI":"10.1007\/s11336-022-09868-4"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s11336-017-9566-9","article-title":"A penalized likelihood method for structural equation modeling","volume":"82","author":"Huang","year":"2017","journal-title":"Psychometrika"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v093.i07","article-title":"lslx: Semi-confirmatory structural equation modeling via penalized likelihood","volume":"93","author":"Huang","year":"2020","journal-title":"J. Stat. Softw."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1080\/10705511.2016.1154793","article-title":"Regularized structural equation modeling","volume":"23","author":"Jacobucci","year":"2016","journal-title":"Struct. Equ. Model."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"579","DOI":"10.3390\/psych3040038","article-title":"Tutorial on the use of the regsem package in R","volume":"3","author":"Li","year":"2021","journal-title":"Psych"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1080\/10705511.2018.1558060","article-title":"Should regularization replace simple structure rotation in exploratory factor analysis?","volume":"26","author":"Scharf","year":"2019","journal-title":"Struct. Equ. Model."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Fan, J., Li, R., Zhang, C.H., and Zou, H. (2020). Statistical Foundations of Data Science, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429096280"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., and Wainwright, M. (2015). Statistical Learning with Sparsity: The Lasso and Generalizations, CRC Press.","DOI":"10.1201\/b18401"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1177\/09622802221133557","article-title":"Regularization approaches in clinical biostatistics: A review of methods and their applications","volume":"32","author":"Friedrich","year":"2023","journal-title":"Stat. Methods Med. Res."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","article-title":"Variable selection via nonconcave penalized likelihood and its oracle properties","volume":"96","author":"Fan","year":"2001","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1080\/01621459.2014.934827","article-title":"Statistical analysis of Q-matrix based diagnostic classification models","volume":"110","author":"Chen","year":"2015","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"5755","DOI":"10.1038\/s41598-020-62473-2","article-title":"Meta-analysis based on nonconvex regularization","volume":"10","author":"Zhang","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1177\/1471082X16642560","article-title":"Regularized regression for categorical data","volume":"16","author":"Tutz","year":"2016","journal-title":"Stat. Model."},{"key":"ref_58","first-page":"369","article-title":"Penalized regression, standard errors, and Bayesian lassos","volume":"5","author":"Kyung","year":"2010","journal-title":"Bayesian Anal."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1198\/jasa.2011.tm10382","article-title":"A perturbation method for inference on regularized regression estimates","volume":"106","author":"Minnier","year":"2011","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_60","unstructured":"R Core Team (2022). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_61","unstructured":"Robitzsch, A. (2023, April 02). sirt: Supplementary Item Response Theory Models; R Package Version 3.13-128. Available online: https:\/\/github.com\/alexanderrobitzsch\/sirt."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R.J. (1994). An Introduction to the Bootstrap, CRC Press.","DOI":"10.1201\/9780429246593"},{"key":"ref_63","first-page":"91","article-title":"Do we have to combine values in the Schwartz\u2019 human values scale? A comment on the Davidov studies","volume":"3","author":"Knoppen","year":"2009","journal-title":"Surv. Res. Methods"},{"key":"ref_64","first-page":"25","article-title":"Testing the discriminant validity of Schwartz\u2019 portrait value questionnaire items\u2014A replication and extension of Knoppen and Saris (2009)","volume":"6","author":"Beierlein","year":"2012","journal-title":"Surv. Res. Methods"},{"key":"ref_65","unstructured":"Muth\u00e9n, B., and Asparouhov, T. (2023, March 04). New Methods for the Study of Measurement Invariance with Many Groups. Technical Report. Available online: https:\/\/bit.ly\/3nBbr5M."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1177\/0049124117701488","article-title":"Recent methods for the study of measurement invariance with many groups: Alignment and random effects","volume":"47","author":"Asparouhov","year":"2018","journal-title":"Sociol. Methods Res."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF02294210","article-title":"A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators","volume":"49","year":"1984","journal-title":"Psychometrika"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Robitzsch, A. (2022). On the bias in confirmatory factor analysis when treating discrete variables as ordinal instead of continuous. Axioms, 11.","DOI":"10.31234\/osf.io\/xfrca"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/BF02295644","article-title":"Structural equation modeling with heavy tailed distributions","volume":"69","author":"Yuan","year":"2004","journal-title":"Psychometrika"},{"key":"ref_70","unstructured":"Lee, S.Y. (2007). Handbook of Latent Variable and Related Models, Elsevier."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1080\/10705511.2018.1561293","article-title":"A Monte Carlo simulation study to assess the appropriateness of traditional and newer approaches to test for measurement invariance","volume":"26","author":"Pokropek","year":"2019","journal-title":"Struct. Equ. Model."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1177\/0013164413498257","article-title":"Assessing the hypothesis of measurement invariance in the context of large-scale international surveys","volume":"74","author":"Rutkowski","year":"2014","journal-title":"Educ. Psychol. Meas."},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Liu, X., Wallin, G., Chen, Y., and Moustaki, I. (2023). Rotation to sparse loadings using Lp losses and related inference problems. Psychometrika.","DOI":"10.1007\/s11336-023-09911-y"},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Asparouhov, T., and Muth\u00e9n, B. (2023, March 04). Penalized Structural Equation Models. Technical Report. Available online: https:\/\/bit.ly\/3TlbxdC.","DOI":"10.1080\/10705511.2023.2263913"},{"key":"ref_75","first-page":"19","article-title":"Some thoughts about the design of loss functions","volume":"5","author":"Hennig","year":"2007","journal-title":"Revstat Stat. J."},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Robitzsch, A., and L\u00fcdtke, O. (2023). Why full, partial, or approximate measurement invariance are not a prerequisite for meaningful and valid group comparisons. Struct. Equ. Model.","DOI":"10.1080\/10705511.2023.2191292"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1198\/000313002753631330","article-title":"The calculus of M-estimation","volume":"56","author":"Stefanski","year":"2002","journal-title":"Am. Stat."},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Schader, M., Gaul, W., and Vichi, M. (2003). Between Data Science and Applied Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization, Springer.","DOI":"10.1007\/978-3-642-18991-3"},{"key":"ref_79","first-page":"79","article-title":"When all models are wrong","volume":"30","author":"Saltelli","year":"2014","journal-title":"Issues Sci. Technol."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/4\/210\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:17:32Z","timestamp":1760123852000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/4\/210"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,17]]},"references-count":79,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["a16040210"],"URL":"https:\/\/doi.org\/10.3390\/a16040210","relation":{"has-preprint":[{"id-type":"doi","id":"10.31234\/osf.io\/25md9","asserted-by":"object"}]},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,17]]}}}