SII Project: Modeling Gains in Mathematics Achievement-Scores

  • Andrzej Gałecki
  • Tomasz Burzykowski
Part of the Springer Texts in Statistics book series (STS)


The SII Project was described in Sect.2.4. In Sect.3.4, an exploratory analysis of the data was presented. The data have a hierarchical structure, with pupils grouped in classes which, in turn, are grouped in schools. Thus, we deal with two levels of grouping in the data or, equivalently, with a three-level data hierarchy. In this chapter, we use LMMs to analyze the change in mathematics achievement-scores for pupils, MATHGAIN. In particular, we use models, which include random intercepts for schools and classes to account for the data hierarchy.


Orthogonal Polynomial Data Frame Random Intercept Math Score Function Lmer 
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  1. .
    Baayen, R., Davidson, D., & Bates, D. (2008). Mixed-effects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59(4), 390–412.CrossRefGoogle Scholar
  2. .
    Bates, D. (2012). Computational Methods for Mixed Models. R Foundation for Statistical Computing.Google Scholar
  3. .
    Bates, D., & Maechler, M. (2012). Matrix: Sparse and Dense Matrix Classes and Methods. R package version 1.0–10. Scholar
  4. .
    Bates, D., Maechler, M., & Bolker, B. (2012). Fitting linear mixed-effects models using lme4. Journal of Statistical Software (forthcoming).Google Scholar
  5. .
    Box, G. E. P., Jenkins, G. M., & Reinsel, G. C. (1994). Time Series Analysis. Prentice Hall Inc., third ed. Forecasting and control.Google Scholar
  6. .
    Cantrell, C. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press.Google Scholar
  7. .
    Carroll, R., & Ruppert, D. (1988). Transformation and Weighting in Regression. Chapman & Hall/CRC.Google Scholar
  8. .
    Chambers, J., & Hastie, T. (1992). Statistical Models in S. Wadsworth & Brooks/Cole Advanced Books & Software.Google Scholar
  9. .
    Chatterjee, S., Hadi, A., & Price, B. (2000). The Use of Regression Analysis by Example. John Wiley & Sons.Google Scholar
  10. .
    Claflin, D.R., Larkin, L.M., Cederna, P.S., Horowitz, J.F., Alexander, N.B., Cole, N.M., Galecki, A.T., Chen, S., Nyquist, L.V., Carlson, B.M., Faulkner, J.A., & Ashton-Miller, J.A. (2011) Effects of high- and low-velocity resistance training on the contractile properties of skeletal muscle fibers from young and older humans. Journal of Applied Physiology, 111, 1021–1030.CrossRefGoogle Scholar
  11. .
    Crainiceanu, C., & Ruppert, D. (2004). Likelihood ratio tests in linear mixed models with one variance component. Journal of the Royal Statistical Society: Series B, 66, 165–185.MathSciNetMATHCrossRefGoogle Scholar
  12. .
    Cressie, N., & Hawkins, D. (1980). Robust estimation of the variogram: I. Mathematical Geology, 12(2), 115–125.MathSciNetCrossRefGoogle Scholar
  13. .
    Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons.Google Scholar
  14. .
    Dahl, D. B. (2009). xtable: Export tables to LaTeX or HTML. R package version 1.5-6. Scholar
  15. .
    Dalgaard, P. (2008). Introductory Statistics with R. Springer Verlag.Google Scholar
  16. .
    Davidian, M., & Giltinan, D. (1995). Nonlinear Models for Repeated Measurement Data. Chapman & Hall.Google Scholar
  17. .
    Demidenko, E. (2004). Mixed Models: Theory and Applications. Wiley-Interscience, first ed.Google Scholar
  18. .
    Fai, A., & Cornelius, P. (1996). Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments. Journal of Statistical Computation and Simulation, 54(4), 363–378.MathSciNetMATHCrossRefGoogle Scholar
  19. .
    Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2004). Applied Longitudinal Analysis. Wiley Series in Probability and Statistics. John Wiley & Sons.MATHGoogle Scholar
  20. .
    Galecki, A. (1994). General class of covariance structures for two or more repeated factors in longitudinal data analysis. Communications in Statistics-Theory and Methods, 23(11), 3105–3119.MATHCrossRefGoogle Scholar
  21. .
    Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (1995). Bayesian Data Analysis. Texts in Statistical Science Series. Chapman & Hall.Google Scholar
  22. .
    Golub, G. H., & Van Loan, C. F. (1989). Matrix Computations, vol. 3 of Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, second ed.Google Scholar
  23. .
    Gurka, M. (2006). Selecting the best linear mixed model under REML. The American Statistician, 60(1), 19–26.MathSciNetCrossRefGoogle Scholar
  24. .
    Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer.Google Scholar
  25. .
    Helms, R. (1992). Intentionally incomplete longitudinal designs: I. Methodology and comparison of some full span designs. Statistics in Medicine, 11(14-15), 1889–1913.Google Scholar
  26. .
    Henderson, C. (1984). Applications of Linear Models in Animal Breeding. University of Guelph.Google Scholar
  27. .
    Hill, H., Rowan, B., & Ball, D. (2005). Effect of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371– 406.CrossRefGoogle Scholar
  28. .
    Janssen, R., Tuerlinckx, F., Meulders, M., & De Boeck, P. (2000). A hierarchical IRT model for criterion-referenced measurement. Journal of Educational and Behavioral Statistics, 25(3), 285.Google Scholar
  29. .
    Jones, R. (1993). Longitudinal Data with Serial Correlation: A State-space Approach. Chapman & Hall/CRC.Google Scholar
  30. .
    Kenward, M., & Roger, J. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53(3), 983–997.MATHCrossRefGoogle Scholar
  31. .
    Laird, N., & Ware, J. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974.MATHCrossRefGoogle Scholar
  32. .
    Leisch, F. (2002). Sweave: Dynamic generation of statistical reports using literate data analysis. In W. Härdle, & B. Rönz (eds.) Compstat 2002 — Proceedings in Computational Statistics, (pp. 575–580). Physica Verlag, Heidelberg.Google Scholar
  33. .
    Lenth, R. (2001). Some practical guidelines for effective sample size determination. The American Statistician, 55(3), 187–193.MathSciNetCrossRefGoogle Scholar
  34. .
    Liang, K.-Y., & Self, S. G. (1996). On the asymptotic behaviour of the pseudolikelihood ratio test statistic. Journal of the Royal Statistical Society: Series B, 58(4), 785–796.MathSciNetMATHGoogle Scholar
  35. .
    Litell, R., Milliken, G., Stroup, W., Wolfinger, R., & Schabenberger, O. (2006). SAS for Mixed Models. SAS Publishing.Google Scholar
  36. .
    Molenberghs, G., & Verbeke, G. (2007). Likelihood ratio, score, and Wald tests in a constrained parameter space. The American Statistician, 61(1), 22–27.MathSciNetCrossRefGoogle Scholar
  37. .
    Murrell, P. (2005). R Graphics. Chapman & Hall/CRC.Google Scholar
  38. .
    Murrell, P., & Ripley, B. (2006). Non-standard fonts in postscript and pdf graphics. The Newsletter of the R Project, 6(2), 41.Google Scholar
  39. .
    Neter, J., Wasserman, W., & Kutner, M. (1990). Applied Linear Statistical Models. IrwinGoogle Scholar
  40. .
    Pharmacological Therapy for Macular Degeneration Study Group (1997). Interferon α-IIA is ineffective for patients with choroidal neovascularization secondary to age-related macular degeneration. Results of a prospective randomized placebo-controlled clinical trial. Archives of Ophthalmology, 115, 865–872.Google Scholar
  41. .
    Pinheiro, J., & Bates, D. (1996). Unconstrained parametrizations for variance-covariance matrices. Statistics and Computing, 6(3), 289–296.CrossRefGoogle Scholar
  42. .
    Pinheiro, J., & Bates, D. (2000). Mixed-effects Models in S and S-PLUS. Springer.Google Scholar
  43. .
    R Development Core Team (2010). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing.Google Scholar
  44. .
    Rothenberg, T. (1984). Approximate normality of generalized least squares estimates. Econometrica, 52(4), 811–825.MathSciNetMATHCrossRefGoogle Scholar
  45. .
    Santos Nobre, J., & da Motta Singer, J. (2007). Residual analysis for linear mixed models. Biometrical Journal, 49(6), 863–875.MathSciNetCrossRefGoogle Scholar
  46. .
    Sarkar, D. (2008). Lattice: Multivariate Data Visualization with R. Springer Verlag.Google Scholar
  47. .
    Satterthwaite, F. E. (1941). Synthesis of variance. Psychometrika, 6, 309–316.MathSciNetMATHCrossRefGoogle Scholar
  48. .
    Schabenberger, O. (2004). Mixed model influence diagnostics in proceedings of the twenty-ninth annual sas users group international conference. Proceedings of the Twenty-Ninth Annual SAS Users Group International Conference, 189, 29.Google Scholar
  49. .
    Schabenberger, O., & Gotway, C. (2005). Statistical Methods for Spatial Data Analysis, vol. 65. Chapman & Hall.Google Scholar
  50. .
    Scheipl, F., Greven, S., & Kuechenhoff, H. (2008). Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models. Computational Statistics & Data Analysis, 52(7), 3283–3299.MathSciNetMATHCrossRefGoogle Scholar
  51. .
    Schluchter, M., & Elashoff, J. (1990). Small-sample adjustments to tests with unbalanced repeated measures assuming several covariance structures. Journal of Statistical Computation and Simulation, 37(1-2), 69–87.MATHCrossRefGoogle Scholar
  52. .
    Searle, S., Casella, G., & McCulloch, C. (1992). Variance Components. John Wiley & Sons.Google Scholar
  53. .
    Self, S. G., & Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82(398), 605–610.MathSciNetMATHCrossRefGoogle Scholar
  54. .
    Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika, 72(1), 133–144.MathSciNetMATHCrossRefGoogle Scholar
  55. .
    Stram, D., & Lee, J. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics, 50(4), 1171–1177.MATHCrossRefGoogle Scholar
  56. .
    Tibaldi, F., Verbeke, G., Molenberghs, G., Renard, D., Van den Noortgate, W., & De Boeck, P. (2007). Conditional mixed models with crossed random effects. British Journal of Mathematical and Statistical Psychology, 60(2), 351–365.CrossRefGoogle Scholar
  57. .
    Van den Noortgate, W., De Boeck, P., & Meulders, M. (2003). Cross-classification multilevel logistic models in psychometrics. Journal of Educational and Behavioral Statistics, 28(4), 369.CrossRefGoogle Scholar
  58. .
    Venables, W., & Ripley, B. (2010). Modern Applied Statistics with S. Springer.Google Scholar
  59. .
    Verbeke, G., & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer Verlag.Google Scholar
  60. .
    Verbeke, G., & Molenberghs, G. (2003). The use of score tests for inference on variance components. Biometrics, 59(2), 254–262.MathSciNetMATHCrossRefGoogle Scholar
  61. .
    Vonesh, E., & Chinchilli, V. (1997). Linear and Nonlinear Models for the Analysis of Repeated Measurements. CRC.Google Scholar
  62. .
    West, B. T., Welch, K. B., & Gałecki, A. T. (2007). Linear Mixed Models: A Practical Guide Using Statistical Software. Chapman and Hall/CRC.Google Scholar
  63. .
    Wickham, H. (2007). Reshaping data with the reshape package. Journal of Statistical Software, 21(12).Google Scholar
  64. .
    Wilkinson, G., & Rogers, C. (1973). Symbolic description of factorial models for analysis of variance. Applied Statistics, 22, 392–399.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Andrzej Gałecki
    • 1
  • Tomasz Burzykowski
    • 2
  1. 1.University of MichiganAnn ArborUSA
  2. 2.Center for StatisticsHasselt UniversityDiepenbeekBelgium

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