, Volume 65, Issue 4, pp 457–474 | Cite as

Maximum likelihood estimation of latent interaction effects with the LMS method

  • Andreas Klein
  • Helfried Moosbrugger


In the context of structural equation modeling, a general interaction model with multiple latent interaction effects is introduced. A stochastic analysis represents the nonnormal distribution of the joint indicator vector as a finite mixture of normal distributions. The Latent Moderated Structural Equations (LMS) approach is a new method developed for the analysis of the general interaction model that utilizes the mixture distribution and provides a ML estimation of model parameters by adapting the EM algorithm. The finite sample properties and the robustness of LMS are discussed. Finally, the applicability of the new method is illustrated by an empirical example.

Key words

latent interaction effects mixture distribution ML estimation structural equation modeling (SEM) EM algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitz, M., & Stegun, I. A. (1971).Handbook of mathematical functions. New York, NY: Dover Publications.Google Scholar
  2. Aiken, L. S., & West, S. G. (1991).Multiple regression: Testing and interpreting interactions. Newbury Park: SAGE Publications.Google Scholar
  3. Arbuckle, J. L. (1997).AMOS Users' Guide Version 3.6. Chicago: Small Waters Corporation.Google Scholar
  4. Bentler, P. M. (1995).EQS structural equations program manual. Encino, CA: Multivariate Software.Google Scholar
  5. Bentler, P. M., & Wu, E. J. C. (1993).EQS/Windows user's guide. Los Angeles: BMDP Statistical Software.Google Scholar
  6. Bollen, K. A. (1995). Structural equation models that are nonlinear in latent variables. In P. V. Marsden (Ed.),Sociological methodology 1995 (Vol. 25). Washington, DC: American Sociological Association.Google Scholar
  7. Bollen, K. A. (1996). An alternative two stage least squares (2SLS) estimator for latent variable equations.Psychometrika, 61, 109–121.CrossRefGoogle Scholar
  8. Brandstädter, J., & Renner, G. (1990). Tenacious goal pursuit and flexible goal adjustment: Explication and age-related analysis of assimilative and accomodative strategies of coping.Psychology and Aging, 5, 58–67.Google Scholar
  9. Cohen, J., & Cohen, P. (1975).Applied multiple regression/correlation analyses for the behavioral sciences. Hillsdale, NJ: Erlbaum.Google Scholar
  10. Degenhardt, A., & Schmidt, H. (1994). Physische Leistungsvariablen als Indikatoren für die Diagnose “Klimakterium Virile” (Physical efficiency variables as indicators for the diagnosis of ‘climacterium virile’).Sexuologie, 3, 131–141.Google Scholar
  11. Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm.Journal of the Royal Statistical Society, Series B, 39, 1–38.Google Scholar
  12. Deusinger, I. (1998).Frankfurter Körperkonzept-Skalen [Frankfurt bodily self-concept scales]. Göttingen: Hogrefe.Google Scholar
  13. Grim, J. (1982). On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions.Kybernetika, 18(3), 173–190.Google Scholar
  14. Hayduk, L.A. (1987).Structural equation modeling with LISREL. Baltimore, MD: Johns Hopkins University Press.Google Scholar
  15. Isaacson, E., & Keller, H.B. (1966).Analysis of numerical methods. New York, NY: Wiley.Google Scholar
  16. Jaccard, J., Turrisi, R., & Wan, C.K. (1990).Interaction effects in multiple regression. Newbury Park, CA: Sage Publications.Google Scholar
  17. Jöreskog, K.G., & Sörbom, D. (1989).LISREL 7: A guide to the program and applications (2nd ed.). Chicago, IL: SPSS.Google Scholar
  18. Jöreskog, K.G., & Sörbom, D. (1993).New features in LISREL 8. Chicago, IL: Scientific Software.Google Scholar
  19. Jöreskog, K. G., & Sörbom, D. (1996).PRELIS 2: User's guide. Chicago: Scientific Software.Google Scholar
  20. Jöreskog, K.G., & Yang, F. (1996). Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In G. Markoulides & R. Schumacker (Eds.),Advanced structural equation modeling (pp. 57–87). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  21. Jöreskog, K. G., & Yang, F. (1997). Estimation of interaction models using the augmented moment matrix: Comparison of asymptotic standard errors. In W. Bandilla & F. Faulbaum (Eds.),SoftStat '97. Advances in statistical software 6 (pp. 467–478). Stuttgart: Lucius & Lucius.Google Scholar
  22. Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effects of latent variables.Psychological Bulletin, 96, 201–210.Google Scholar
  23. Klein, A., Moosbrugger, H., Schermelleh-Engel, K., & Frank, D. (1997). A new approach to the estimation of latent interaction effects in structural equation models. In W. Bandilla & F. Faulbaum (Eds.),SoftStat '97. Advances in statistical software 6 (pp. 479–486). Stuttgart: Lucius & Lucius.Google Scholar
  24. Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications.Biometrika, 57, 519–530.Google Scholar
  25. Mardia, K. V. (1974). Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies.Sankhya, Series B, 36, 115–128.Google Scholar
  26. Moosbrugger, H., Frank, D., & Schermelleh-Engel, K. (1991). Zur Überprüfung von latenten Moderatoreffekten mit linearen Strukturgleichungsmodellen [Estimating latent interaction effects in structural equation models].Zeitschrift für Differentielle und Diagnostische Psychologie, 12, 245–255.Google Scholar
  27. Moosbrugger, H., Schermelleh-Engel, K., & Klein, A. (1997). Methodological problems of estimating latent interaction effects.Methods of Psychological Research Online, 2, 95–111.Google Scholar
  28. Ping, R. A. (1996a). Latent variable and quadratic effect estimation: A two-step technique using structural equation analysis.Psychological Bulletin, 119, 166–175.CrossRefGoogle Scholar
  29. Ping, R. A. (1996b). Latent variable regression: A technique for estimating interaction and quadratic coefficients.Multivariate Behavioral Research, 31, 95–120.Google Scholar
  30. Redner, R. A., & Walker, H. F. (1984). Mixture densities, maximum likelihood and the EM algorithm.SIAM Review, 26, 195–239.CrossRefGoogle Scholar
  31. Schermelleh-Engel, K., Klein, A., & Moosbrugger, H. (1998). Estimating nonlinear effects using a Latent Moderated Structural Equations Approach. In R. E. Schumacker & G. A. Marcoulides (Eds.),Interaction and nonlinear effects in structural equation modeling (pp. 203–238). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  32. Schmitt, M. (1990).Konsistenz als Persönlichkeitseigenschaft? Moderatorvariablen in der Persönlichkeits- und Einstellungsforschung [Consistency as a personality trait? Moderator variables in personality and attitude research]. Berlin: Springer.Google Scholar
  33. Schwarz, H. R. (1993).Numerische Mathematik [Numerical mathematics]. Stuttgart: Teubner.Google Scholar
  34. Thiele, A. (1998).Verlust körperlicher Leistungsfähigkeit: Bewältigung des Alterns bei Männern im mittleren Lebensalter [Loss of bodily efficacy: The coping of aging for men of medium age]. Idstein, Germany: Schulz-Kirchner-Verlag.Google Scholar
  35. Yang Jonsson, F. (1997).Nonlinear structural equation models: Simulation studies of the Kenny-Judd model. Uppsala: University of Uppsala.Google Scholar

Copyright information

© The Psychometric Society 2000

Authors and Affiliations

  1. 1.Johann Wolfgang Goethe-UniversityFrankfurt Am MainGermany

Personalised recommendations