A Comparison of Statistics for Assessing Model Invariance in Latent Class Analysis


Latent class analysis (LCA) is a widely used statistical technique for identifying subgroups in the population based upon multiple indicator variables. It has a number of advantages over other unsupervised grouping procedures such as cluster analysis, including stronger theoretical underpinnings, more clearly defined measures of model fit, and the ability to conduct confirmatory analyses. In addition, it is possible to ascertain whether an LCA solution is equally applicable to multiple known groups, using invariance assessment techniques. This study compared the effectiveness of multiple statistics for detecting group LCA invariance, including a chi-square difference test, a bootstrap likelihood ratio test, and several information indices. Results of the simulation study found that the bootstrap likelihood ratio test was the optimal invariance assessment statistic. In addition to the simulation, LCA group invariance assessment was demonstrated in an application with the Youth Risk Behavior Survey (YRBS). Implications of the simulation results for practice are discussed.

Share and Cite:

Finch, H. (2015) A Comparison of Statistics for Assessing Model Invariance in Latent Class Analysis. Open Journal of Statistics, 5, 191-210. doi: 10.4236/ojs.2015.53022.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Hoijtink, H. (2001) Confirmatory Latent Class Analysis: Model Selection Using Bayes Factors and (Pseudo) Likelihood Ratio Statistics. Multivariate Behavioral Research, 36, 563-588.
[2] McCutcheon, A.L. (2002) Basic Concepts and Procedures in Single- and Multiple-Group Latent Class Analysis. In: Hagenaars, J.A. and McCutcheon, A.L., Eds., Applied Latent Class Analysis, Cambridge University Press, Cambridge, 56-86.
[3] Laudy, O., Boom, J. and Hoijtink, H. (2005) Bayesian Computational Methods for Inequality Constrained Latent Class Analysis. In: van der Ark, L.A., Croon, M.A. and Sijtsma, K., Eds., New Developments in Categorical Data Analysis for the Social and Behavioral Sciences, Lawrence Erlbaum Associates Publishers, Mahwah, 63-82.
[4] Finch, W.H. and Bronk, K.C. (2011) Conducting Confirmatory Latent Class Analysis Using Mplus. Structural Equation Modeling, 18, 132-151.
[5] Collins, L.M. and Lanza, S.T. (2010) Latent Class and Latent Transition Analysis: With Applications in the Social, Behavioral and Health Sciences. Wiley, New York.
[6] Nylund, K.L., Asparouhov, T. and Muthén, B.O. (2007) Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study. Structural Equation Modeling, 14, 535-569.
[7] French, B.F. and Finch, W.H. (2006) Confirmatory Factor Analytic Procedures for the Determination of Measurement Invariance. Structural Equation Modeling, 13, 378-402.
[8] Lubke, G.H. and Muthén, B.O. (2004) Applying Multigroup Confirmatory Factor Models for Continuous Outcomes to Likert Scale Data Complicates Meaningful Group Comparisons. Structural Equation Modeling, 11, 514-534.
[9] Cheung, G.W. and Rensvold, R.B. (2002) Evaluating Goodness-of-Fit Indexes for Testing Measurement Invariance. Structural Equation Modeling, 9, 233-255.
[10] Akaike, H. (1987) Factor Analysis and AIC. Psychometrika, 52, 317-332.
[11] Bozdogan, H. (1987) Model Selection and Akaike’s Information Criterion (AIC): The General Theory and Its Analytical Extensions. Psychometrika, 52, 345-370.
[12] McLachlan, G. and Peel, D. (2000) Finite Mixture Models. Wiley, New York.
[13] Tofighi, D. and Enders, C.K. (2007) Identifying the Correct Number of Classes in a Growth Mixture Model. In: Hancock, G.R., Ed., Advances in Latent Variable Mixture Models, Information Age, Greenwich, 317-341.
[14] Yang, C. (2006) Evaluating Latent Class Analyses in Qualitative Phenotype Identification. Computational Statistics & Data Analysis, 50, 1090-1104.
[15] Celeux, G. and Soromenho, G. (1996) An Entropy Criterion for Assessing the Number of Clusters in a Mixture Model. Journal of Classification, 13, 195-212.
[16] Jedidi, K., Jagpal, H. and DeSarbo, W.S. (1997) Finite-Mixture Structural Equation Models for Response-Based Segmentation and Unobserved Heterogeneity. Marketing Science, 16, 39-59.
[17] Read, T.R.C. and Cressie, N.A.C. (1988) Goodness of Fit Statistics for Discrete Multivariate Data. Springer, New York.
[18] Koehler, K.J. and Larntz, K. (1980) An Empirical Investigation of Goodness of Fit Statistics for Sparse Multinomials. Journal of the American Statistical Association, 75, 336-344.
[19] Koehler, K.J. (1986) Goodness-of-Fit Tests for Log-Linear Models in Sparse Contingency Tables. Journal of the American Statistical Association, 81, 483-493.
[20] Meade, A.W. and Lautenschlager, G.J. (2004) A Monte-Carlo Study of Confirmatory Factor Analytic Tests of Measurement Equivalence/Invariance. Structural Equation Modeling, 11, 60-72.
[21] Burnham, K. and Anderson, D. (2003) Model Selection and Multimodel Inference: A Practical-Theoretic Approach. Springer-Verlag, New York.
[22] Vrieze, S.I. (2012) Model Selection and Psychological Theory: A Discussion of the Differences between the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). Psychological Methods, 17, 228-243.
[23] Shao, J. (1997) An Asymptotic Theory for Linear Model Selection (with Discussion). Statistica Sinica, 7, 221-242.
[24] Wicherts, J.M. and Dolan, C.V. (2004) A Cautionary Note on the Use of Information Fit Indexes in Covariance Structure Modeling with Mean. Structural Equation Modeling, 11, 45-50.
[25] Dziak, J.J., Lanza, S.T. and Xu, S. (2011) SimulateLcaDataset SAS Macro Users Guide, Version 1.1.0. The Methodology Center, Pennsylvania State University, University Park.
[26] Lanza, S.T., Dziak, J.J., Huang, L., Xu, S. and Collins, L.M. (2011) PROC LCA & PROC LTA Users’ Guide (Version 1.2.3). The Methodology Center, Pennsylvania State University, University Park.
[27] SAS Institute (2009) SAS Version 9.1. SAS Institute, Cary.
[28] Arnett, J. (1995) The Young and the Reckless: Adolescent Reckless Behavior. Current Directions in Psychological Science, 4, 67-70.
[29] Poulin, C., Boudreau, B. and Asbridge, M. (2007) Adolescent Passengers of Drunk Drivers: A Multi-Level Exploration into the Inequities of Risk and Safety. Addiction, 102, 51-61.
[30] McCartt, A.T. and Northrup, V.S. (2004) Factors Related to Seat Belt Use among Fatally Injured Teenage Drivers. Journal of Safety Research, 35, 29-38.
[31] Bianco, A., Trani, F., Santoro, G. and Angelillo, I.F. (2005) Adolescents’ Attitudes and Behavior towards Motorcycle Helmet Use in Italy. European Journal of Pediatrics, 164, 207-211.
[32] Everett, S., Miyamoto, J., Saraiya, M. and Berkowitz, Z. (2012) Trends in Sunscreen Use among U.S. High School Students: 1999-2009. The Journal of Adolescent Health, 50, 304-307.
[33] Klein, K., Thompson, D., Scheidt, P., Overpeck, M. and Gross, L., HBSC International Investigators (2005) Factors Associated with Bicycle Helmet Use among Young Adolescents in a Multinational Sample. Injury Prevention, 11, 288-293.
[34] Lowry, R., Powell, K.E., Kann, L., Collins, J.L. and Kolbe, L.J. (1998) Weapon-Carrying, Physical Fighting, and Fight Related Injury among U.S. Adolescents. American Journal of Preventive Medicine, 14, 122-129.
[35] Hirschberger, G., Florian, V., Mikulinger, M., Goldenberg, J.L. and Pyszczynski, T. (2002) Gender Differences in the Willingness to Engage in Risky Behavior: A Terror Management Perspective. Death Studies, 26, 117-141.
[36] Byrnes, J.P., Miller, D.C. and Schafer, W.D. (1999) Gender Differences in Risk Taking: A Meta-Analysis. Psychological Bulletin, 125, 367-383.
[37] Sclove, L. (1987) Application of Model-Selection Criteria to Some Problems in Multivariate Analysis. Psychometrika, 52, 333-343.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.