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ARMD Trial: Linear Model with Homogeneous Variance

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Linear Mixed-Effects Models Using R

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Abstract

In this chapter, we illustrate the use of the R tools, described in Sects. 5.2–5.5.We apply them to fit an LM with independent, homoscedastic residual errors to the visual acuity measurements from the ARMD dataset. Note that the model is considered for software illustration purposes only. In view of the structure of the data and of the results of the exploratory analysis presented in Sect. 3.2, the assumptions of the independence and homoscedasticity of the visual acuity measurements are not correct. More advanced LMs, which properly take into account the structure of the data and do not require these assumptions, will be presented in Chaps. 12 and 16.

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References

  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.

    Article  Google 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. http://CRAN.R-project.org/package=Matrix.

  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.

    Article  Google 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.

    Article  MathSciNet  MATH  Google Scholar 

  12. Cressie, N., & Hawkins, D. (1980). Robust estimation of the variogram: I. Mathematical Geology, 12(2), 115–125.

    Article  MathSciNet  Google 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. http://CRAN.R-project.org/package=xtable.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  19. Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2004). Applied Longitudinal Analysis. Wiley Series in Probability and Statistics. John Wiley & Sons.

    MATH  Google 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.

    Article  MATH  Google 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.

    Article  MathSciNet  Google 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.

    Article  Google 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.

    Article  MATH  Google Scholar 

  31. Laird, N., & Ware, J. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974.

    Article  MATH  Google 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.

    Article  MathSciNet  Google 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.

    MathSciNet  MATH  Google 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.

    Article  MathSciNet  Google 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. Irwin

    Google 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.

    Article  Google 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.

    Article  MathSciNet  MATH  Google Scholar 

  45. Santos Nobre, J., & da Motta Singer, J. (2007). Residual analysis for linear mixed models. Biometrical Journal, 49(6), 863–875.

    Article  MathSciNet  Google 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.

    Article  MathSciNet  MATH  Google 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.

    Article  MathSciNet  MATH  Google 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.

    Article  MATH  Google 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.

    Article  MathSciNet  MATH  Google Scholar 

  54. Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika, 72(1), 133–144.

    Article  MathSciNet  MATH  Google Scholar 

  55. Stram, D., & Lee, J. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics, 50(4), 1171–1177.

    Article  MATH  Google 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.

    Article  Google 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.

    Article  Google 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.

    Article  MathSciNet  MATH  Google 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.

    Article  Google Scholar 

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Gałecki, A., Burzykowski, T. (2013). ARMD Trial: Linear Model with Homogeneous Variance. In: Linear Mixed-Effects Models Using R. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3900-4_6

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