Fitting Nonlinear Structural Equation Models in R with Package nlsem

Structural equation mixture modeling (SEMM) has become a standard procedure in latent variable modeling over the last two decades (Jedidi, Jagpal, and DeSarbo'97b; Muthen and Shedden'99; Muthen 2001, 2004; Muthen and Asparouhov 2009). SEMM was proposed as a technique for the approximation of nonlinear latent variable relationships by finite mixtures of linear relationships (Bauer 2005, 2007; Bauer, Baldasaro, and Gottfredson 2012). In addition to this semiparametric approach to nonlinear latent variable modeling, there are numerous parametric nonlinear approaches for normally distributed variables (e.g., LMS in Mplus; Klein and Moosbrugger 2000). Recently, an additional semiparametric nonlinear structural equation mixture modeling (NSEMM) approach was proposed by Kelava, Nagengast, and Brandt (2014) that is capable of dealing with nonnormal predictors. In the nlsem package presented here, the SEMM, two distribution analytic (QML and LMS) and NSEMM approaches can be specified and estimated. We provide examples of how to use the package in the context of nonlinear latent variable modeling.

[1]  Daniel J Bauer,et al.  The integration of continuous and discrete latent variable models: potential problems and promising opportunities. , 2004, Psychological methods.

[2]  A. Genz,et al.  Computation of Multivariate Normal and t Probabilities , 2009 .

[3]  Yasuo Amemiya,et al.  Estimation for Polynomial Structural Equation Models , 2000 .

[4]  Holger Brandt,et al.  A Nonlinear Structural Equation Mixture Modeling Approach for Nonnormally Distributed Latent Predictor Variables , 2014 .

[5]  Daniel J. Bauer.  Observations on the Use of Growth Mixture Models in Psychological Research , 2007 .

[6]  Yasuo Amemiya,et al.  A method of moments technique for fitting interaction effects in structural equation models. , 2003, The British journal of mathematical and statistical psychology.

[7]  Xin-Yuan Song,et al.  Structure detection of semiparametric structural equation models with Bayesian adaptive group lasso , 2015, Statistics in medicine.

[8]  B. Muthén,et al.  Finite Mixture Modeling with Mixture Outcomes Using the EM Algorithm , 1999, Biometrics.

[9]  D. A. Kenny,et al.  Estimating the nonlinear and interactive effects of latent variables. , 1984 .

[10]  Andrew Thomas,et al.  WinBUGS - A Bayesian modelling framework: Concepts, structure, and extensibility , 2000, Stat. Comput..

[11]  T. Hothorn,et al.  Multivariate Normal and t Distributions , 2016 .

[12]  Gerhard Arminger,et al.  Mixtures of conditional mean- and covariance-structure models , 1999 .

[13]  B. Nagengast,et al.  A Bayesian Model For The Estimation Of Latent Interaction And Quadratic Effects When Latent Variables Are Non-Normally Distributed , 2012, Multivariate behavioral research.

[14]  Snigdhansu Chatterjee,et al.  Structural Equation Modeling, A Bayesian Approach , 2008, Technometrics.

[15]  James Jaccard,et al.  Measurement error in the analysis of interaction effects between continuous predictors using multiple regression: Multiple indicator and structural equation approaches. , 1995 .

[16]  Holger Brandt,et al.  A general non-linear multilevel structural equation mixture model , 2014, Front. Psychol..

[17]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[18]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[19]  Daniel J. Bauer.  A Semiparametric Approach to Modeling Nonlinear Relations Among Latent Variables , 2005 .

[20]  Dieter Zapf,et al.  Advanced Nonlinear Latent Variable Modeling: Distribution Analytic LMS and QML Estimators of Interaction and Quadratic Effects , 2011 .

[21]  Bengt Muthén,et al.  Latent Variable Analysis: Growth Mixture Modeling and Related Techniques for Longitudinal Data , 2004 .

[22]  James Algina,et al.  Comparison of Methods for Estimating and Testing Latent Variable Interactions , 2002 .

[23]  Gerhard Arminger,et al.  Finite Mixtures of Covariance Structure Models with Regressors , 1997 .

[24]  A. Kelava,et al.  Estimation of nonlinear latent structural equation models using the extended unconstrained approach , 2009 .

[25]  Bengt O. Muthén,et al.  Quasi-Maximum Likelihood Estimation of Structural Equation Models With Multiple Interaction and Quadratic Effects , 2007 .

[26]  Yasuo Amemiya,et al.  Generalized Appended Product Indicator Procedure for Nonlinear Structural Equation Analysis , 2001 .

[27]  Ulrich Trautwein,et al.  Who took the "x" out of expectancy-value theory? A psychological mystery, a substantive-methodological synergy, and a cross-national generalization. , 2011, Psychological science.

[28]  John Fox,et al.  OpenMx: An Open Source Extended Structural Equation Modeling Framework , 2011, Psychometrika.

[29]  Han L. J. van der Maas,et al.  Fitting multivariage normal finite mixtures subject to structural equation modeling , 1998 .

[30]  B. Muthén,et al.  Growth mixture modeling , 2008 .

[31]  H. Marsh,et al.  Structural Equation Models of Latent Interactions: An Appropriate Standardized Solution and Its Scale-Free Properties , 2010 .

[32]  H. Marsh,et al.  A Comparison of Strategies for Forming Product Indicators for Unequal Numbers of Items in Structural Equation Models of Latent Interactions , 2013 .

[33]  P. Bentler,et al.  An Alternative Approach for Nonlinear Latent Variable Models , 2010 .

[34]  Daniel J Bauer,et al.  Estimating and Visualizing Nonlinear Relations Among Latent Variables: A Semiparametric Approach , 2009, Multivariate behavioral research.

[35]  Yves Rosseel,et al.  lavaan: An R Package for Structural Equation Modeling , 2012 .

[36]  B. Muthén,et al.  A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the metropolis-hastings algorithm , 1998 .

[37]  Kenneth A. Bollen,et al.  STRUCTURAL EQUATION MODELS THAT ARE NONLINEAR IN LATENT VARIABLES: A LEAST- SQUARES ESTIMATOR , 1995 .

[38]  Holger Brandt,et al.  The Standardization of Linear and Nonlinear Effects in Direct and Indirect Applications of Structural Equation Mixture Models for Normal and Nonnormal Data , 2015, Front. Psychol..

[39]  R. Ping.  A Parsimonious Estimating Technique for Interaction and Quadratic Latent Variables , 1995 .

[40]  Nisha C. Gottfredson,et al.  Diagnostic Procedures for Detecting Nonlinear Relationships Between Latent Variables , 2012 .

[41]  Peter M. Bentler,et al.  EQS : structural equations program manual , 1989 .

[42]  Kamel Jedidi,et al.  STEMM: A General Finite Mixture Structural Equation Model , 1997 .

[43]  Bengt Muthén,et al.  Second-generation structural equation modeling with a combination of categorical and continuous latent variables: New opportunities for latent class–latent growth modeling. , 2001 .

[44]  H. Marsh,et al.  Structural equation models of latent interactions: evaluation of alternative estimation strategies and indicator construction. , 2004, Psychological methods.

[45]  James A. Bovaird,et al.  On the Merits of Orthogonalizing Powered and Product Terms: Implications for Modeling Interactions Among Latent Variables , 2006 .

[46]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[47]  W. DeSarbo,et al.  Finite-Mixture Structural Equation Models for Response-Based Segmentation and Unobserved Heterogeneity , 1997 .

[48]  L. Hayduk,et al.  Latent Variable Interaction and Quadratic Effect Estimation: A Two-Step Technique Using Structural Equation Analysis , 1996 .

[49]  Helfried Moosbrugger,et al.  Maximum likelihood estimation of latent interaction effects with the LMS method , 2000 .

[50]  H. Marsh,et al.  Structural equation models of latent interaction and quadratic effects. , 2013 .

[51]  K. Jöreskog,et al.  LISREL 8: New Statistical Features , 1999 .

[52]  W. Meredith.  Measurement invariance, factor analysis and factorial invariance , 1993 .

[53]  B. Efron.  Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods , 1981 .

[54]  Fan Yang,et al.  Nonlinear structural equation models: The Kenny-Judd model with Interaction effects , 1996 .

[55]  R. P. McDonald,et al.  Structural Equations with Latent Variables , 1989 .

[56]  Daniel J. Bauer,et al.  Confidence Intervals for a Semiparametric Approach to Modeling Nonlinear Relations among Latent Variables , 2011 .

[57]  Xinyuan Song,et al.  Semiparametric Latent Variable Models With Bayesian P-Splines , 2010 .

[58]  Holger Brandt,et al.  A Simulation Study Comparing Recent Approaches for the Estimation of Nonlinear Effects in SEM Under the Condition of Nonnormality , 2014 .

[59]  Andrew Thomas,et al.  The BUGS project: Evolution, critique and future directions , 2009, Statistics in medicine.

[60]  John Fox,et al.  TEACHER'S CORNER: Structural Equation Modeling With the sem Package in R , 2006 .