Introduction to Multilevel Analysis

  • Jan de Leeuw
  • Erik Meijer


Loss Function Multilevel Analysis Royal Statistical Society Full Information Maximum Likelihood Dispersion Matrix 


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  1. 1.
    L. S. Aiken and S. G. West. Multiple Regression: Testing and Interpreting Interaction. Sage Publications, Newbury Park, CA, 1991.Google Scholar
  2. 2.
    M. Aitkin and N. Longford. Statistical modelling issues in school effectiveness studies. Journal of the Royal Statistical Society, Series A, 149: 1–43, 1986. (with discussion).CrossRefGoogle Scholar
  3. 3.
    L. Anselin. Spatial econometrics. In B. H. Baltagi, editor, A Companion to Theoretical Econometrics, pages 310–330. Blackwell, Malden, MA, 2001.Google Scholar
  4. 4.
    M. Arellano. Panel Data Econometrics. Oxford University Press, Oxford, UK, 2003.MATHGoogle Scholar
  5. 5.
    T. Asparouhov. Sampling weights in latent variable modeling. Structural Equation Modeling, 12:411–434, 2005.MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Asparouhov. General multi-level modeling with sampling weights. Communications in Statistics—Theory & Methods, 35:439–460, 2006.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Bates and D. Sarkar. The lme4 Package, 2006. URL http://cran.r-project.orgGoogle Scholar
  8. 8.
    P. M. Bentler. EQS 6 Structural Equations Program Manual. Multivariate Software, Encino, CA, 2006.Google Scholar
  9. 9.
    H. M. Blalock. Contextual effects models: Theoretical and methodological issues. Annual Review of Sociology, 10:353–372, 1984.CrossRefGoogle Scholar
  10. 10.
    R. J. Bosker, T. A. B. Snijders, and H. Guldemond. PINT: Estimating Standard Errors of Regression Coefficients in Hierarchical Linear Models for Power Calculations. User’s Manual Version 1.6. University of Twente, Enschede, The Netherlands, 1999.Google Scholar
  11. 11.
    L. H. Boyd and G. R. Iversen. Contextual Analysis: Concepts and Statistical Techniques. Wadsworth, Belmont, CA, 1979.MATHGoogle Scholar
  12. 12.
    M. W. Browne. Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37:62–83, 1984.MATHMathSciNetGoogle Scholar
  13. 13.
    A. S. Bryk and S. W. Raudenbush. Hierachical Linear Models: Applications and Data Analysis Methods. Sage, Newbury Park, CA, 1992.Google Scholar
  14. 14.
    L. Burstein. The analysis of multilevel data in educational research and evaluation. Review of Research in Education, 8:158–233, 1980.CrossRefGoogle Scholar
  15. 15.
    L. Burstein, R. L. Linn, and F. J. Capell. Analyzing multilevel data in the presence of heterogeneous within-class regressions. Journal of Educational Statistics, 3:347–383, 1978.CrossRefGoogle Scholar
  16. 16.
    F. M. T. A. Busing, E. Meijer, and R. Van der Leeden. MLA: Software for MultiLevel Analysis of Data with Two Levels. User's Guide for Version 4.1. Leiden University, Department of Psychology, Leiden, 2005.Google Scholar
  17. 17.
    A. C. Cameron and P. K. Trivedi. Microeconometrics: Methods and Applications. Cambridge University Press, Cambridge, UK, 2005.Google Scholar
  18. 18.
    G. Chamberlain. Panel data. In Z. Griliches and M. D. Intriligator, editors, Handbook of Econometrics, volume 2, pages 1247–1318. North-Holland, Amsterdam, 1984.Google Scholar
  19. 19.
    G. Chamberlain and E. E. Leamer. Matrix weighted averages and posterior bounds. Journal of the Royal Statistical Society, Series B, 38:73–84, 1976.MATHMathSciNetGoogle Scholar
  20. 20.
    K. Chantala, D. Blanchette, and C. M. Suchindran. Software to compute sampling weights for multilevel analysis, 2006. URL Scholar
  21. 21.
    Y. F. Cheong, R. P. Fotiu, and S. W. Raudenbush. Efficiency and robustness of alternative estimators for two- and three-level models: The case of NAEP. Journal of Educational and Behavioral Statistics, 26:411–429, 2001.CrossRefGoogle Scholar
  22. 22.
    J. S. Coleman, E. Q. Campbell, C. J. Hobson, J. McPartland, A. M. Mood, F. D. Weinfeld, and R. L. York. Equality of Educational Opportunity. U.S. Government Printing Office, Washington, DC, 1966.Google Scholar
  23. 23.
    D. R. Cox. Interaction. International Statistical Review, 52:1–31, 1984.MATHMathSciNetGoogle Scholar
  24. 24.
    M. Davidian and D. M. Giltinan. Nonlinear Models for Repeated Mesurement Data. Chapman & Hall, London, 1995.Google Scholar
  25. 25.
    R. Davidson and J. G. MacKinnon. Estimation and Inference in Econometrics. Oxford University Press, Oxford, UK, 1993.MATHGoogle Scholar
  26. 26.
    F. R. de Hoog, T. P. Speed, and E. R. Williams. On a matrix identity associated with generalized least squares. Linear Algebra and its Applications, 127:449–456, 1990.Google Scholar
  27. 27.
    J. de Leeuw. Centering in multilevel analysis. In B. S. Everitt and D. C. Howell, editors, Encyclopedia of Statistics in Behavioral Science, volume 1, pages 247–249. Wiley, New York, 2005.Google Scholar
  28. 28.
    J. de Leeuw and I. G. G. Kreft. Random coefficient models for multilevel analysis. Journal of Educational Statistics, 11:57–85, 1986.CrossRefGoogle Scholar
  29. 29.
    J. de Leeuw and I. G. G. Kreft. Questioning multilevel models. Journal of Educational and Behavioral Statistics, 20:171–190, 1995.CrossRefGoogle Scholar
  30. 30.
    J. de Leeuw and I. G. G. Kreft. Software for multilevel analysis. In A. H. Leyland and H. Goldstein, editors, Multilevel Modelling of Health Statistics, pages 187–204. Wiley, Chichester, 2001.Google Scholar
  31. 31.
    J. de Leeuw and G. Liu. Augmentation algorithms for mixed model analysis. Preprint 115, UCLA Statistics, Los Angeles, CA, 1993.Google Scholar
  32. 32.
    G. del Pino. The unifying role of iterative generalized least squares in statistical algorithms. Statistical Science, 4:394–408, 1989. (with discussion).MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39:1–38, 1977. (with discussion).MATHMathSciNetGoogle Scholar
  34. 34.
    A. P. Dempster, D. B. Rubin, and R. K. Tsutakawa. Estimation in covariance components models. Journal of the American Statistical Association, 76: 341–353, 1981.MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    M. Du Toit and S. H. C. Du Toit. Interactive LISREL: User's Guide. Scientific Software International, Chicago, 2002.Google Scholar
  36. 36.
    W. J. Duncan. Some devices for the solution of large sets of simultaneous linear equations (with an appendix on the reciprocation of partitioned matrices). The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 7th Series, 35:660–670, 1944.MATHMathSciNetGoogle Scholar
  37. 37.
    J. P. Elhorst and A. S. Zeilstra. Labour force participation rates at the regional and national levels of the European Union: An integrated analysis. Papers in Regional Science, forthcoming.Google Scholar
  38. 38.
    T. S. Ferguson. A Course in Large Sample Theory. Chapman & Hall, London, 1996.MATHGoogle Scholar
  39. 39.
    D. A. Freedman. On the so-called “Huber sandwich estimator” and “robust standard errors”. Unpublished manuscript, 2006.Google Scholar
  40. 40.
    A. Gelman. Multilevel (hierarchical) modeling: What it can and cannot do. Technometrics, 48:432–435, 2006.MathSciNetCrossRefGoogle Scholar
  41. 41.
    A. Gelman and I. Pardoe. Bayesian measures of explained variance and pooling in multilevel (hierarchical) models. Technometrics, 48:241–251, 2006.MathSciNetCrossRefGoogle Scholar
  42. 42.
    A. Gelman, D. K. Park, S. Anselobehere, P. N. Price, and L. C. Minnete. Models, assumptions and model checking in ecological regressions. Journal of the Royal Statistical Society, Series A, 164:101–118, 2001.MATHGoogle Scholar
  43. 43.
    A. S. Goldberger. Best linear unbiased prediction in the generalized linear regression model. Journal of the American Statistical Association, 57:369–375, 1962.MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    H. Goldstein. Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika, 73:43–56, 1986.MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    H. Goldstein. Restricted unbiased iterative generalized least-squares estimation. Biometrika, 76:622–623, 1989.MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    H. Goldstein. Multilevel Statistical Models, 3rd edition. Edward Arnold, London, 2003.MATHGoogle Scholar
  47. 47.
    H. Goldstein and J. Rasbash. Efficient computational procedures for the estimation of parameters in multilevel models based on iterative generalised least squares. Computational Statistics & Data Analysis, 13:63–71, 1992.Google Scholar
  48. 48.
    P. J. Green. Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. Journal of the Royal Statistical Society, Series B, 46:149–192, 1984. (with discussion).MATHGoogle Scholar
  49. 49.
    L. Grilli and M. Pratesi. Weighted estimation in multilevel ordinal and binary models in the presence of informative sampling designs. Survey Methodology, 30:93–103, 2004.Google Scholar
  50. 50.
    E. A. Hanushek. Efficient estimates for regressing regression coefficients. American Statistician, 28:66–67, 1974.MathSciNetCrossRefGoogle Scholar
  51. 51.
    H. O. Hartley and J. N. K. Rao. Maximum likelihood estimation for the mixed analysis of variance model. Biometrika, 54:93–108, 1967.MathSciNetGoogle Scholar
  52. 52.
    D. A. Harville. Baysian inference for variance components using only error contrasts. Biometrika, 61:383–385, 1974.MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    D. A. Harville. Matrix Algebra From a Statistician’s Perspective. Springer, New York, 1997.MATHGoogle Scholar
  54. 54.
    T. Hastie and R. Tibshirani. Varying-coefficient models. Journal of the Royal Statistical Society, Series B, 55:757–796, 1993. (with discussion).MATHMathSciNetGoogle Scholar
  55. 55.
    D. Hedeker. MIXNO: A computer program for mixed-effects nominal logistic regression. Journal of Statistical Software, 4(5):1–92, 1999.MathSciNetGoogle Scholar
  56. 56.
    D. Hedeker and R. D. Gibbons. MIXOR: A computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine, 49:157–176, 1996.Google Scholar
  57. 57.
    D. Hedeker and R. D. Gibbons. MIXREG: A computer program for mixed-effects regression analysis with autocorrelated errors. Computer Methods and Programs in Biomedicine, 49:229–252, 1997.CrossRefGoogle Scholar
  58. 58.
    J. Hemelrijk. Underlining random variables. Statistica Neerlandica, 20:1–7, 1966.MathSciNetCrossRefGoogle Scholar
  59. 59.
    J. J. Hox. Multilevel Analysis: Techniques and Applications. Erlbaum, Mahwah, NJ, 2002.Google Scholar
  60. 60.
    C. Hsiao. Analysis of Panel Data, 2nd edition. Cambridge University Press, Cambridge, UK, 2003.Google Scholar
  61. 61.
    J. Z. Huang, C. O. Wu, and L. Zhou. Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika, 89:111–128, 2002.MATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    C. Jencks, M. Smith, H. Acland, M. J. Bane, D. Cohen, H. Gintis, B. Heyns, and S. Michelson. Inequality: A Reassessment of the Effect of Family and Schooling in America. Basic Books, New York, 1972.Google Scholar
  63. 63.
    R. I. Jennrich and M. D. Schluchter. Unbalanced repeated-measures models with structured covariance matrices. Biometrics, 42:805–820, 1986.MATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    K. G. Jöreskog, D. Sörbom, S. H. C. Du Toit, and M. Du Toit. LISREL 8: New Statistical Features. Scientific Software International, Chicago, 2001. (3rd printing with revisions).Google Scholar
  65. 65.
    J. Kim and E. W. Frees. Multilevel modeling with correlated effects. Psychometrika, forthcoming.Google Scholar
  66. 66.
    M. S. Kovačević and S. N. Rai. A pseudo maximum likelihood approach to multilevel modelling of survey data. Communications in Statistics—Theory and Methods, 32:103–121, 2003.MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    I. G. G. Kreft and J. de Leeuw. Introducing Multilevel Modeling. Sage, London, 1998.Google Scholar
  68. 68.
    I. G. G. Kreft, J. de Leeuw, and L. S. Aiken. The effect of different forms of centering in hierarchical linear models. Multivariate Behavioral Research, 30:1–21, 1995.CrossRefGoogle Scholar
  69. 69.
    N. M. Laird, N. Lange, and D. Stram. Maximum likelihood computations with repeated measures: Application of the EM algorithm. Journal of the American Statistical Association, 82:97–105, 1987.MATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    N. M. Laird and J. H. Ware. Random-effects models for longitudinal data. Biometrics, 38:963–974, 1982.MATHCrossRefGoogle Scholar
  71. 71.
    L. I. Langbein. Schools or students: Aggregation problems in the study of student achievement. Evaluation Studies Review Annual, 2:270–298, 1977.Google Scholar
  72. 72.
    D. V. Lindley and A. F. M. Smith. Bayes estimates for the linear model. Journal of the Royal Statistical Society, Series B, 34:1–41, 1972.Google Scholar
  73. 73.
    M. J. Lindstrom and D. M. Bates. Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. Journal of the American Statistical Association, 83:1014–1022, 1988.MATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    N. T. Longford. A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects. Biometrika, 74:817–827, 1987.MATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    N. T. Longford. VARCL. Software for Variance Component Analysis of Data with Nested Random Effects (Maximum Likelihood). Educational Testing Service, Princeton, NJ, 1990.Google Scholar
  76. 76.
    N. T. Longford. Random Coefficient Models. Oxford University Press, Oxford, UK, 1993.MATHGoogle Scholar
  77. 77.
    K. Löwner. Über monotone Matrixfunktionen. Mathematische Zeitschrift, 38:177–216, 1934.MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    C. J. M. Maas and J. J. Hox. The influence of violations of assumptions on multilevel parameter estimates and their standard errors. Computational Statistics & Data Analysis, 46:427–440, 2004.MathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    J. R. Magnus and H. Neudecker. Symmetry, 0–1 matrices and Jacobians: A review. Econometric Theory, 2:157–190, 1986.Google Scholar
  80. 80.
    J. R. Magnus and H. Neudecker. Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, Chichester, 1988.MATHGoogle Scholar
  81. 81.
    W. M. Mason, G. Y. Wong, and B. Entwisle. Contextual analysis through the multilevel linear model. Sociological Methodology, 14:72–103, 1983.CrossRefGoogle Scholar
  82. 82.
    P. McCullagh and J. A. Nelder. Generalized Linear Models, 2nd edition. Chapman & Hall, London, 1989.MATHGoogle Scholar
  83. 83.
    E. Meijer and J. Rouwendal. Measuring welfare effects in models with random coefficients. Journal of Applied Econometrics, 21:227–244, 2006.MathSciNetCrossRefGoogle Scholar
  84. 84.
    B. O. Muthén and A. Satorra. Complex sample data in structural equation modeling. Sociological Methodology, 25:267–316, 1995.CrossRefGoogle Scholar
  85. 85.
    L. K. Muthén and B. O. Muthén. Mplus User's Guide, 4th edition. Muthén & Muthén, Los Angeles, 1998–2006.Google Scholar
  86. 86.
    J. Nocedal and S. J. Wright. Numerical Optimization. Springer, New York, 1999.MATHGoogle Scholar
  87. 87.
    D. Pfeffermann. The role of sampling weights when modeling survey data. International Statistical Review, 61:317–337, 1993.MATHCrossRefGoogle Scholar
  88. 88.
    D. Pfeffermann, C. J. Skinner, D. J. Holmes, H. Goldstein, and J. Rasbash. Weighting for unequal selection probabilities in multilevel models. Journal of the Royal Statistical Society, Series B, 60:23–56, 1998. (with discussion).Google Scholar
  89. 89.
    J. C. Pinheiro and D. M. Bates. Mixed-Effects Models in S and S-PLUS. Springer, New York, 2000.MATHGoogle Scholar
  90. 90.
    J. C. Pinheiro, D. M. Bates, S. DebRoy, and D. Sarkar. The nlme Package, 2006. URL http://cran.r-project.orgGoogle Scholar
  91. 91.
    R. F. Potthoff and S. N. Roy. A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51:313–326, 1964.MATHMathSciNetGoogle Scholar
  92. 92.
    R. F. Potthoff, M. A. Woodbury, and K. G. Manton. “Equivalent sample size” and “equivalent degrees of freedom” refinements for inference using survey weights under superpopulation models. Journal of the American Statistical Association, 87:383–396, 1992.MATHMathSciNetCrossRefGoogle Scholar
  93. 93.
    R Development Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2006. URL http://www.r-project.orgGoogle Scholar
  94. 94.
    S. Rabe-Hesketh and A. Skrondal. Multilevel modelling of complex survey data. Journal of the Royal Statistical Society, Series A, 169:805–827, 2006.MathSciNetGoogle Scholar
  95. 95.
    S. Rabe-Hesketh, A. Skrondal, and A. Pickles. GLLAMM manual. Working Paper 160, U.C. Berkeley Division of Biostatistics, Berkeley, CA, 2004. (Downloadable from Scholar
  96. 96.
    C. R. Rao. Linear Statistical Inference and its Applications, 2nd edition. Wiley, New York, 1973.MATHGoogle Scholar
  97. 97.
    J. Rasbash, F. Steele, W. J. Browne, and B. Prosser. A User’s Guide to MLwiN. Version 2.0. Centre for Multilevel Modelling, University of Bristol, Bristol, UK, 2005.Google Scholar
  98. 98.
    S. W. Raudenbush. Reexamining, reaffirming, and improving application of hierarchical models. Journal of Educational and Behavioral Statistics, 20:210–220, 1995.CrossRefGoogle Scholar
  99. 99.
    S. W. Raudenbush and A. S. Bryk. Empirical Bayes meta-analysis. Journal of Educational Statistics, 10:75–98, 1985.CrossRefGoogle Scholar
  100. 100.
    S. W. Raudenbush and A. S. Bryk. A hierarchical model for studying school effects. Sociology of Education, 59:1–17, 1986.CrossRefGoogle Scholar
  101. 101.
    S. W. Raudenbush and A. S. Bryk. Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd edition. Sage, Thousand Oaks, CA, 2002.Google Scholar
  102. 102.
    S. W. Raudenbush, A. S. Bryk, Y. F. Cheong, and R. Congdon. HLM 6: Hierarchical Linear and Nonlinear Modeling. Scientific Software International, Chicago, 2004.Google Scholar
  103. 103.
    G. K. Robinson. That BLUP is a good thing: the estimation of random effects. Statistical Science, 6:15–51, 1991. (with discussion).Google Scholar
  104. 104.
    W. S. Robinson. Ecological correlations and the behavior of individuals. Sociological Review, 15:351–357, 1950.CrossRefGoogle Scholar
  105. 105.
    J. Rouwendal and E. Meijer. Preferences for housing, jobs, and commuting: A mixed logit analysis. Journal of Regional Science, 41:475–505, 2001.CrossRefGoogle Scholar
  106. 106.
    SAS/Stat. SAS/Stat User's Guide, version 9.1. SAS Institute, Cary, NC, 2004.Google Scholar
  107. 107.
    M. D. Schluchter. BMDP5V – Unbalanced repeated measures models with structured covariance matrices. Technical Report 86, BMDP Statistical Software, Los Angeles, 1988.Google Scholar
  108. 108.
    C. J. Skinner. Domain means, regression and multivariate analysis. In C. J. Skinner, D. Holt, and T. M. F. Smith, editors, Analysis of Complex Surveys, pages 59–87. Wiley, New York, 1989.Google Scholar
  109. 109.
    C. J. Skinner, D. Holt, and T. M. F. Smith, editors. Analysis of Complex Surveys. Wiley, New York, 1989.MATHGoogle Scholar
  110. 110.
    A. Skrondal and S. Rabe-Hesketh. Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models. Chapman & Hall/CRC, Boca Raton, FL, 2004.MATHGoogle Scholar
  111. 111.
    T. A. B. Snijders and R. J. Bosker. Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling. Sage, London, 1999.MATHGoogle Scholar
  112. 112.
    D. J. Spiegelhalter, N. G. Best, B. P. Carlin, and A. van der Linde. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, 64:583–639, 2002. (with discussion).MATHCrossRefGoogle Scholar
  113. 113.
    D. J. Spiegelhalter, A. Thomas, N. G. Best, and D. Lunn. WinBUGS User Manual, Version 1.4. MRC Biostatistics Unit, Cambridge, UK, 2003.Google Scholar
  114. 114.
    SPSS. SPSS Advanced Models™ 15.0 Manual. SPSS, Chicago, 2006.Google Scholar
  115. 115.
    StataCorp. Stata Statistical Software: Release 9. Stata Corporation, College Station, TX, 2005.Google Scholar
  116. 116.
    J. L. F. Strenio, H. I. Weisberg, and A. S. Bryk. Empirical Bayes estimation of individual growth curve parameters and their relationship to covariates. Biometrics, 39:71–86, 1983.MATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    P. A. V. B. Swamy. Statistical Inference in a Random Coefficient Model. Springer, New York, 1971.Google Scholar
  118. 118.
    R. L. Tate and Y. Wongbundhit. Random versus nonrandom coefficient models for multilevel analysis. Journal of Educational Statistics, 8:103–120, 1983.CrossRefGoogle Scholar
  119. 119.
    G. Van Landeghem, P. Onghena, and J. Van Damme. The effect of different forms of centering in hierarchical linear models re-examined. Technical Report 2001-04, Catholic University of Leuven, University Centre for Statistics, Leuven, Belgium, 2001.Google Scholar
  120. 120.
    G. Verbeke and E. Lesaffre. Large sample properties of the maximum likelihood estimators in linear mixed models with misspecified random-effects distributions. Technical Report 1996.1, Catholic University of Leuven, Biostatistical Centre for Clinical Trials, Leuven, 1996.Google Scholar
  121. 121.
    G. Verbeke and E. Lesaffre. The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data. Computational Statistics & Data Analysis, 23:541–556, 1997.MATHMathSciNetCrossRefGoogle Scholar
  122. 122.
    G. Verbeke and G. Molenberghs. Linear Mixed Models for Longitudinal Data. Springer, New York, 2000.MATHGoogle Scholar
  123. 123.
    T. Wansbeek and E. Meijer. Measurement Error and Latent Variables in Econometrics. North-Holland, Amsterdam, 2000.MATHGoogle Scholar
  124. 124.
    J. M. Wooldridge. Asymptotic properties of weighted M-estimators for variable probability samples. Econometrica, 67:1385–1406, 1999.MATHMathSciNetCrossRefGoogle Scholar
  125. 125.
    J. M. Wooldridge. Asymptotic properties of weighted M-estimators for standard stratified samples. Econometric Theory, 17:451–470, 2001.MATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    J. M. Wooldridge. Econometric Analysis of Cross Section and Panel Data. MIT Press, Cambridge, MA, 2002.Google Scholar
  127. 127.
    J. M. Wooldridge. Inverse probability weighted M-estimators for sample selection, attrition, and stratification. Portuguese Economic Journal, 1:117–139, 2002.Google Scholar
  128. 128.
    R. Xu. Measuring explained variation in linear mixed effects models. Statistics in Medicine, 22:3527–3541, 2003.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Jan de Leeuw
    • 1
  • Erik Meijer
    • 2
  1. 1.Department of StatisticsUniversity of California at Los AngelesLos AngelesUSA
  2. 2.Faculty of Economics and RAND CorporationUniversity of GroningenSanta MonicaUSA

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