Behavior Research Methods

, Volume 39, Issue 4, pp 695–708 | Cite as

A Newton procedure for conditionally linear mixed-effects models

  • Shelley A. BlozisEmail author


This article reviews Newton procedures for the analysis of mean and covariance structures that may be functions of parameters that enter a model nonlinearly. The kind of model considered is a mixed-effects model that is conditionally linear with regard to its parameters. This means parameters entering the model nonlinearly must be fixed, whereas those entering linearly may vary across individuals. This framework encompasses several models, including hierarchical linear models, linear and nonlinear factor analysis models, and nonlinear latent curve models. A full maximum-likelihood estimation procedure is described. Mx, a statistical software package often used to estimate structural equation models, is considered for estimation of these models. An example with Mx syntax is provided.


Gradient Vector Hierarchical Linear Model Reading Score Factor Analysis Model Exponential Growth Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Amemiya, Y. (1993). Instrumental variable estimation for nonlinear factor analysis. In C. M. Cuadras & C. R. Rao (Eds.),Multivariate analysis: Future directions 2 (pp. 113–129). Amsterdam: Elsevier.Google Scholar
  2. Blozis, S. A. (2004). Structured latent curve models for the study of change in multivariate repeated measures.Psychological Methods,9, 334–353.CrossRefPubMedGoogle Scholar
  3. Blozis, S. A., &Cudeck, R. (1999). Conditionally linear mixed-effects models with latent variable covariates.Journal of Educational & Behavioral Statistics,24, 245–270.Google Scholar
  4. Browne, M. W. (1993). Structured latent curve models. In C. M. Cuadras & C. R. Rao (Eds.),Multivariate analysis: Future directions 2 (pp. 171–197). Amsterdam: Elsevier.Google Scholar
  5. Browne, M. W., &Du Toit, S. H. C. (1991). Models for learning data. In L. M. Collins & J. L. Horn (Eds.),Best methods for the analysis of change (pp. 47–68). Washington: American Psychological Association.Google Scholar
  6. Collins, L. M., &Graham, J. W. (2002). The effect of the timing and spacing of observations in longitudinal studies of tobacco and other drug use: Temporal design considerations.Drug & Alcohol Dependence,68, S85-S96.CrossRefGoogle Scholar
  7. Curran, P. J. (1997, April). The bridging of quantitative methodology and applied developmental research. In P. J. Curran (Chair),Comparing three modern approaches to longitudinal data analysis: An examination of a single developmental sample. Symposium conducted at the 1997 meeting of the Society for Research on Child Development, Washington, DC.Google Scholar
  8. Davidian, M., &Giltinan, D. M. (1995).Nonlinear models for repeated measurement data. New York: Chapman & Hall.Google Scholar
  9. Dennis, J. E., &Schnabel, R. B. (1986).Numerical methods for unconstrained optimization and nonlinear equations. Englewood Cliffs, NJ: Prentice Hall.Google Scholar
  10. Etezadi-Amoli, J., &McDonald, R. P. (1983). A second generation nonlinear factor analysis.Psychometrika,48, 315–342.CrossRefGoogle Scholar
  11. Finkbeiner, C. (1979). Estimation for the multiple factor model when data are missing.Psychometrika,44, 409–420.CrossRefGoogle Scholar
  12. Francis, D. J., Shaywitz, S. E., Stuebing, K. K., Shaywitz, B. A., &Fletcher, J. M. (1996). Developmental lag versus deficit models of reading disability: A longitudinal, individual growth curves analysis.Journal of Educational Psychology,88, 3–17.CrossRefGoogle Scholar
  13. Goldstein, H., Rasbash, J., Plewis, I., Draper, D., Browne, W., Yang, M., et al. (1998).A user’s guide to MLwiN (Tech. Rep.). London: University of London, Institute of Education.Google Scholar
  14. Hedeker, D., &Gibbons, R. D. (1997). Application of random-effects pattern-mixture models for missing data in longitudinal studies.Psychological Methods,2, 64–78.CrossRefGoogle Scholar
  15. Hox, J. J. (2002).Multilevel analysis: Techniques and applications. Mahwah, NJ: Erlbaum.Google Scholar
  16. Jamshidian, M., &Jennrich, R. I. (1997). Acceleration of the EM algorithm using quasi-Newton methods.Journal of the Royal Statistical Society: Series B,59, 569–587.CrossRefGoogle Scholar
  17. Jennrich, R. I., &Schluchter, M. D. (1986). Unbalanced repeated measures models with structured covariance matrices.Biometrics,42, 805–820.CrossRefPubMedGoogle Scholar
  18. Jöreskog, K. G., Sörbom, D., Du Toit, S., &Du Toit, M. (2001).LISREL 8: New statistical features. Lincolnwood, IL: Scientific Software International.Google Scholar
  19. Kreft, I., &de Leeuw, J. (1998).Introducing multilevel modeling. Thousand Oaks, CA: Sage.Google Scholar
  20. Laird, N. M. (1988). Missing data in longitudinal studies.Statistics in Medicine,7, 305–315.CrossRefPubMedGoogle Scholar
  21. Lawley, D. N., &Maxwell, A. E. (1971).Factor analysis as a statistical method. New York: Elsevier.Google Scholar
  22. Lindstrom, M. J., &Bates, D. M. (1988). Newton—Raphson and EM algorithms for linear mixed-effects models for repeated-measures data.Journal of the American Statistical Association,83, 1014–1022.CrossRefGoogle Scholar
  23. Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D., &Schabenberger, O. (2006).SAS for mixed models (2nd ed.). Cary, NC: SAS Institute.Google Scholar
  24. Little, R. J. A. (1995). Modeling the drop-out mechanism in repeatedmeasures studies.Journal of the American Statistical Association,90, 1112–1121.CrossRefGoogle Scholar
  25. Little, R. J. A., &Rubin, D. B. (2002).Statistical analysis with missing data (2nd ed.). New York: Wiley.Google Scholar
  26. Longford, N. [T.] (1990).VARCL: Software for variance component analysis of data with nested random effects (maximum likelihood). Princeton, NJ: Educational Testing Service.Google Scholar
  27. Longford, N. T. (1993).Random coefficient models. Oxford: Oxford University Press.Google Scholar
  28. MacCallum, R. C., Kim, C., Malarkey, W. B., &Kiecolt-Glaser, J. K. (1997). Studying multivariate change using multilevel models and latent curve models.Multivariate Behavioral Research,32, 215–253.CrossRefGoogle Scholar
  29. MathSoft, Inc. (1997). S-PLUS user’s guide. Seattle: Author.Google Scholar
  30. McDonald, R. P. (1962). A general approach to nonlinear factor analysis.Psychometrika,27, 397–415.CrossRefGoogle Scholar
  31. McDonald, R. P. (1965). Difficulty factors and nonlinear factor analysis.British Journal of Mathematical & Statistical Psychology,18, 11–23.Google Scholar
  32. McDonald, R. P. (1967a). Factor interaction in nonlinear factor analysis.British Journal of Mathematical & Statistical Psychology,20, 205–215.Google Scholar
  33. McDonald, R. P. (1967b). Numerical methods for polynomial models in nonlinear factor analysis.Psychometrika,32, 77–112.CrossRefGoogle Scholar
  34. Meredith, W., & Tisak, J. (1984).Tuckerizing curves. Paper presented at the annual meeting of the Psychometric Society, Santa Barbara, CA.Google Scholar
  35. Meredith, W., &Tisak, J. (1990). Latent curve analysis.Psychometrika,55, 107–122.CrossRefGoogle Scholar
  36. Muthén, L. K., &Muthén, B. O. (1998–2004).Mplus user’s guide (3rd ed.). Los Angeles: Authors.Google Scholar
  37. Neale, M. C., Boker, S. M., Xie, G., &Maes, H. H. (2003).Mx: Statistical modeling (6th ed.). Richmond, VA: Virginia Commonwealth University, Department of Psychiatry.Google Scholar
  38. Rao, C. R. (1973).Linear statistical inference and its applications (2nd ed.). New York: Wiley.CrossRefGoogle Scholar
  39. Raudenbush, S. W., &Bryk, A. S. (2002).Hierarchical linear models. Thousand Oaks, CA: Sage.Google Scholar
  40. Raudenbush, S. W., Bryk, A. S., Cheong, Y. F., &Congdon, R. (2004).HLM 6: Hierarchical linear and nonlinear modeling. Lincolnwood, IL: Scientific Software International.Google Scholar
  41. Rubin, D. B. (1976). Inference and missing data.Biometrika,63, 581–592.CrossRefGoogle Scholar
  42. Shah, B. V., Barnwell, B. G., &Bieler, G. S. (1997).SUDAAN user’s manual (Version 7.5). Research Triangle Park, NC: Research Triangle Institute.Google Scholar
  43. Singer, J. D., &Willett, J. B. (2003).Applied longitudinal data analysis. Oxford: Oxford University Press.CrossRefGoogle Scholar
  44. Snijders, T., &Bosker, R. (1999).Multilevel analysis. London: Sage.Google Scholar
  45. SPSS for Windows, Rel. 11.0.1. (2001). Chicago: SPSS.Google Scholar
  46. STATACorporation (1997).STATA reference manual. College Station, TX: Stata Press.Google Scholar
  47. Wolfinger, R. (1999). Fitting nonlinear mixed models with the new NLMIXED procedure. InProceedings of the twenty-fourth annual SAS Users Group International Conference (paper 287), 1–10. ( Scholar
  48. Yalcin, I., &Amemiya, Y. (2001). Nonlinear factor analysis as a statistical method.Statistical Science,16, 275–294.CrossRefGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 2007

Authors and Affiliations

  1. 1.Psychology DepartmentUniversity of CaliforniaDavis

Personalised recommendations