Psychometrika

, Volume 81, Issue 4, pp 1046–1068 | Cite as

Pairwise Likelihood Ratio Tests and Model Selection Criteria for Structural Equation Models with Ordinal Variables

Article
First Online:
Received:
Revised:

Abstract

Correlated multivariate ordinal data can be analysed with structural equation models. Parameter estimation has been tackled in the literature using limited-information methods including three-stage least squares and pseudo-likelihood estimation methods such as pairwise maximum likelihood estimation. In this paper, two likelihood ratio test statistics and their asymptotic distributions are derived for testing overall goodness-of-fit and nested models, respectively, under the estimation framework of pairwise maximum likelihood estimation. Simulation results show a satisfactory performance of type I error and power for the proposed test statistics and also suggest that the performance of the proposed test statistics is similar to that of the test statistics derived under the three-stage diagonally weighted and unweighted least squares. Furthermore, the corresponding, under the pairwise framework, model selection criteria, AIC and BIC, show satisfactory results in selecting the right model in our simulation examples. The derivation of the likelihood ratio test statistics and model selection criteria under the pairwise framework together with pairwise estimation provide a flexible framework for fitting and testing structural equation models for ordinal as well as for other types of data. The test statistics derived and the model selection criteria are used on data on ‘trust in the police’ selected from the 2010 European Social Survey. The proposed test statistics and the model selection criteria have been implemented in the R package lavaan.

Keywords

latent variable modelling composite likelihood underlying variable approach 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We thank Professor Yves Rosseel, the developer of the R package lavaan, for adopting our R code related to PML methodology and incorporating into lavaan. This project has been supported by ESRC Grant ES/L009838/1.

Supplementary material

11336_2016_9523_MOESM1_ESM.pdf (263 kb)
Supplementary material 1 (pdf 262 KB)

References

  1. Agresti, A. (2010). Analysis of ordinal categorical data (2nd ed.). New York: Wiley.CrossRefGoogle Scholar
  2. Ansari, A., & Jedidi, K. (2000). Bayesian factor analysis for multilevel binary observations. Psychometrika, 65(4), 475–496.CrossRefGoogle Scholar
  3. Ansari, A., & Jedidi, K. (2002). Heterogeneous factor analysis models: A Bayesian approach. Psychometrika, 67(1), 49–78.CrossRefGoogle Scholar
  4. Arminger, G., & Küsters, U. (1988). Latent trait models with indicators of mixed measurement level. In I. R. Langeheine & J. Rost (Eds.), Latent trait and latent class models. New York: Plenum.Google Scholar
  5. Asparouhov, T., & Muthén, B. (2006). Robust chi-square difference testing with mean and variance adjusted test statistics. Mplus Web Notes: No. 10.Google Scholar
  6. Asparouhov, T., & Muthén, B. (2010). Simple second order chi-square correction.Google Scholar
  7. Bartholomew, D., Knott, M., & Moustaki, I. (2011). Latent variable models and factor analysis: A unified approach (3rd ed.)., John Wiley series in probability and statistics New York: Wiley.CrossRefGoogle Scholar
  8. Bellio, R., & Varin, C. (2005). A pairwise likelihood approach to generalized linear models with crossed random effects. Statistical Modelling, 5, 217–227.CrossRefGoogle Scholar
  9. Bentler, P. M. (2006). EQS 6 Structural Equations Program Manual.Google Scholar
  10. Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of Royal Statistical Society Series B, 36, 192–236.Google Scholar
  11. Bhat, C. R., Varin, C., & Ferdous, N. (2010). Maximum simulated likelihood methods and applications. In R. C. H. William Greene (Ed.), Advances in Econometrics (Vol. 26, pp. 65–106). Bingley: Emerald Group Publishing.Google Scholar
  12. Bollen, K., & Curran, P. J. (2006). Latent curve models: A structural equation perspective., Wiley series in probability and mathematical statistics New York: Wiley.Google Scholar
  13. De Leon, A. R. (2005). Pairwise likelihood approach to grouped continuous model and its extension. Statistics & Probability Letters, 75, 49–57.CrossRefGoogle Scholar
  14. Efron, B., & Hinkley, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika, 65(3), 457–487.CrossRefGoogle Scholar
  15. ESS Round 5: European Social Survey: ESS-5 Documentation Report. Edition 3.2 (2014). Bergen, European Social Survey Data Archive, Norwegian Social Science Data Services.Google Scholar
  16. ESS Round 5: European Social Survey Round 5 Data. Data file edition 3.2 (2010). Norwegian Social Science Data Services. Norway, Data Archive and distributor of ESS data.Google Scholar
  17. Fan, W., & Hancock, G. R. (2012). Robust means modeling: An alternative for hypothesis testing of independent means under variance heterogeneity and nonnormality. Journal of Educational and Behavioral Statistics, 37, 137–156.CrossRefGoogle Scholar
  18. Feddag, M.-L., & Bacci, S. (2009). Pairwise likelihood for the longitudinal mixed Rasch model. Computational Statistics and Data Analysis, 53, 1027–1037.CrossRefGoogle Scholar
  19. Fieuws, S., & Verbeke, G. (2006). Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles. Biometrics, 62, 424–431.CrossRefPubMedGoogle Scholar
  20. Gao, X., & Song, P. X. (2010). Composite likelihood Bayesian information criteria for model selection in high dimensional data. Journal of the American Statistical Association, 105(492), 1531–1540.CrossRefGoogle Scholar
  21. Heagerty, P. J., & Lele, S. (1998). A composite likelihood approach to binary spatial data. Journal of the American Statistical Association, 93, 1099–1111.CrossRefGoogle Scholar
  22. Jackson, J., Hough, M., Bradford, B., Hohl, K. & Kuha, J. (2012). Policing by consent: Topline results (UK) from Round 5 of the European social survey. ESS Country Specific Topline Results Series 1.Google Scholar
  23. Joe, H., & Lee, Y. (2009). On weighting of bivariate margins in pairwise likelihood. Journal of Multivariate Analysis, 100, 670–685.CrossRefGoogle Scholar
  24. Jöreskog, K., & Yang, F. (1996). Nonlinear structural equation models: The Kenny–Judd model with interaction effects. In G. Marcoulides & R. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 57–88). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  25. Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202.CrossRefGoogle Scholar
  26. Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409–426.CrossRefGoogle Scholar
  27. Jöreskog, K. G. (1990). New developments in LISREL: Analysis of ordinal variables using polychoric correlations and weighted least squares. Quality and Quantity, 24, 387–404.CrossRefGoogle Scholar
  28. Jöreskog, K. G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59, 381–389.CrossRefGoogle Scholar
  29. Jöreskog, K. G. (2002). Structural equation modeling with ordinal variables using LISREL.Google Scholar
  30. Jöreskog, K. G., & Moustaki, I. (2001). Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36, 347–387.CrossRefPubMedGoogle Scholar
  31. Jöreskog, K. G. Sörbom, D. (1996). LISREL 8 User’s Reference Guide.Google Scholar
  32. Katsikatsou, M. (2013). Composite Likelihood estimation for latent variable models with ordinal and continuous or ranking variables. Unpublished doctoral dissertation, Uppsala University, Uppsala.Google Scholar
  33. Katsikatsou, M., Moustaki, I., Yang-Wallentin, F., & Jöreskog, K. G. (2012). Pairwise likelihood estimation for factor analysis models with ordinal data. Computational Statistics and Data Analysis, 56, 4243–4258.CrossRefGoogle Scholar
  34. Kenward, M. G., & Molenberghs, G. (1998). Likelihood based frequentist inference when data are missing at random. Statistical Science, 13(3), 236–247.CrossRefGoogle Scholar
  35. Lee, S.-Y. (2007). Structural equation modeling: A Bayesian approach., Wiley series in probability and statistics New York: Wiley.CrossRefGoogle Scholar
  36. Lee, S.-Y., Poon, W.-Y., & Bentler, P. (1990a). Full maximum likelihood analysis of structural equation models with polytomous variables. Statistics and Probability Letters, 9, 91–97.CrossRefGoogle Scholar
  37. Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1990b). A three-stage estimation procedure for structural equation models with polytomous variables. Psychometrika, 55, 45–51.CrossRefGoogle Scholar
  38. Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57, 89–105.CrossRefGoogle Scholar
  39. Lele, S. R. (2006). Sampling variability and estimates of density dependence: A composite likelihood approach. Ecology, 87, 189–202.CrossRefPubMedGoogle Scholar
  40. Lele, S. R., & Taper, M. L. (2002). A composite likelihood approach to (co)variance components estimation. Journal of Statistical Planning and Inference, 103, 117–135.CrossRefGoogle Scholar
  41. Lindsay, B. (1988). Composite likelihood methods. Contemporary Mathematics, 80, 221–239.CrossRefGoogle Scholar
  42. Liu, J. (2007). Multivariate ordinal data analysis with pairwise likelihood and its extension to SEM. Unpublished doctoral dissertation, University of California, Los Angeles.Google Scholar
  43. Magnus, J. (1978). The moments of products of quadratic forms in normal variables (Technical Repot No. AE4/78). Institute of Actuarial Science and Econometrics, Amsterdam University.Google Scholar
  44. Maydeu-Olivares, A., & Joe, H. (2005). Limited- and full-information estimation and goodness-of-fit testing in \(2^n\) contingency tables: A unified approach. Journal of the American Statistical Association, 100, 1009–1020.CrossRefGoogle Scholar
  45. Maydeu-Olivares, A., & Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71(4), 713–732.CrossRefGoogle Scholar
  46. Millsap, E., & Yun-Tein, J. (2004). Assessing factorial invariance in ordered-categorical measures. Multivariate Behavioral Research, 39(3), 479–515.CrossRefGoogle Scholar
  47. Muthén, B. (1984). A general structural equation model with dichotomous, ordered, categorical, and continuous latent variables indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
  48. Muthén, B. (1989). Multi-group structural modelling with non-normal continuous variables. British Journal of Mathematical and Statistical Psychology, 42, 55–62.CrossRefGoogle Scholar
  49. Muthén, B. (1993). Goodness of fit with categorical and other nonnormal variables. In K. Bollen & J. Long (Eds.), Testing structural equation models (pp. 205–234). Newbury Park: Sage Publications.Google Scholar
  50. Muthén, B., & Asparouhov, T. (2002). Latent variable analysis with categorical outcomes: Multiple-group and growth modeling in Mplus. Mplus Web Notes 4.Google Scholar
  51. Muthén, B., du Toit, S., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes.Google Scholar
  52. Muthén, L. K., & Muthén, B. O. (2010). Mplus 6. Los Angeles.Google Scholar
  53. Pace, L., Salvan, A., & Sartori, N. (2011). Adjusting composite likelihood ratio statistics. Statistica Sinica, 21, 129–148.Google Scholar
  54. Palomo, J., Dunson, D. B., & Bollen, K. (2007). Handbook of computing and statistics with applications. In S.-Y. Lee (Ed.), Handbook of latent variable and related models (Vol. 1). Amsterdam: Elsevier.CrossRefGoogle Scholar
  55. Poon, W.-Y., & Lee, S.-Y. (1987). Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients. Psychometrika, 52, 409–430.CrossRefGoogle Scholar
  56. R Development Core Team. (2008). R: A language and environment for statistical computing. Vienna, Austria.Google Scholar
  57. Raftery, A. (1993). Bayesian model selection in structural equation models. In K. Bollen & J. Long (Eds.), Testing Structural Equation Models. Newbury Park, CA: Sage.Google Scholar
  58. Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36.CrossRefGoogle Scholar
  59. Rosseel, Y., Oberski, D., Byrnes, J., Vanbrabant, L., Savalei, V. & Merkle, E. (2012, September). Package lavaan.Google Scholar
  60. Satorra, A. (2000). Scaled and adjusted restricted tests in multi-sample analysis of moment structures. In R. D. H. Heijmans, D. S. G. Pollock, & A. Satorra (Eds.), Innovations in multivariate statistical analysis (pp. 233–247)., A Festschrift for Heinz Neudecker London: Kluwer Academic Publishers.CrossRefGoogle Scholar
  61. Satorra, A., & Bentler, P. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. In Proceedings of the Business and Economic Statistics Section of the American Statistical Association (pp. 308–313).Google Scholar
  62. Satorra, A., & Bentler, P. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. Clogg (Eds.), Latent variable analysis: Applications to developmental research (pp. 399–419). Thousand Oaks, CA: Sage Publications.Google Scholar
  63. Satorra, A., & Bentler, P. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66(4), 507–514.CrossRefGoogle Scholar
  64. Satorra, A., & Bentler, P. (2010). Ensuring positiveness of the scaled difference chi-square test statistic. Psychometrika, 75(2), 243–248.CrossRefPubMedPubMedCentralGoogle Scholar
  65. Savalei, V., & Kolenikov, S. (2008). Constrained vs. unconstrained estimation in structural equation modeling. Psychological Methods, 13, 150–170.CrossRefPubMedGoogle Scholar
  66. Savalei, V., & Rhemtulla, M. (2013). The performance of robust test statistics with categorical data. British Journal of Mathematical and Statistical Psychology, 66(2), 201–223.CrossRefPubMedGoogle Scholar
  67. Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel. Longitudinal and structural equation models.Google Scholar
  68. Varin, C. (2008). On composite marginal likelihoods. Advances in Statistical Analysis, 92, 1–28.CrossRefGoogle Scholar
  69. Varin, C., Høst, G., & Øivind, S. (2005). Pairwise likelihood inference in spatial generalized linear mixed models. Computational Statistics and Data Analysis, 49, 1173–1191.CrossRefGoogle Scholar
  70. Varin, C., Reid, N., & Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 1–41.Google Scholar
  71. Varin, C., & Vidoni, P. (2005). A note on composite likelihood inference and model selection. Biometrika, 92, 519–528.CrossRefGoogle Scholar
  72. Varin, C., & Vidoni, P. (2006). Pairwise likelihood inference for ordinal categorical time series. Computational Statistics and Data Analysis, 51, 2365–2373.CrossRefGoogle Scholar
  73. Vasdekis, V., Cagnone, S., & Moustaki, I. (2012). A composite likelihood inference in latent variable models for ordinal longitudinal responses. Psychometrika, 77, 425–441.CrossRefPubMedGoogle Scholar
  74. Wall, M., & Amemiya, Y. (2000). Estimation of polynomial structural equation models. Journal of the American Statistical Association, 95, 929–940.CrossRefGoogle Scholar
  75. Xi, N. (2011). A composite likelihood approach for factor analyzing ordinal data. Unpublished doctoral dissertation. Columbus: The Ohio State University.Google Scholar
  76. Zhao, Y., & Joe, H. (2005). Composite likelihood estimation in multivariate data analysis. The Canadian Journal of Statistics, 33, 335–356.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2016

Authors and Affiliations

  1. 1.Department of StatisticsLondon School of EconomicsLondonUK

Personalised recommendations