Have a personal or library account? Click to login
Estimation, Testing, and Prediction Regions of the Fixed and Random Effects by Solving the Henderson’s Mixed Model Equations Cover

Estimation, Testing, and Prediction Regions of the Fixed and Random Effects by Solving the Henderson’s Mixed Model Equations

Measurement Science Review
Volume 12 (2012): Issue 6 (December 2012)
By: Viktor Witkovský  
Open Access
|Dec 2012

References

  1. [1] Alnosaier, W.S. (2007). Kenward-Roger Approximate F Test for Fixed Effects in Mixed Linear Models. Dissertation Thesis submitted to Oregon State University, April 25, 2007. http://ir.library.oregonstate.edu
    Search in Google Scholar
  2. [2] Arendacká, B. (2007). Fiducial generalized pivots for a variance component vs. an approximate confidence Interval. Measurement Science Review 7 (6), 55 - 63.
    Search in Google Scholar
  3. [3] Arendacká, B. (2012). Approximate interval for the between group variance under heteroscedasticity. Journal of Statistical Computation & Simulation 82 (2), 209 - 218.10.1080/00949655.2011.606548
    Search in Google Scholar
  4. [4] Arendacká, B. (2012). A note on fiducial generalized pivots for σ2 A in one-way heteroscedastic ANOVA with random effects. Statistics 46 (4), 489 - 504.
    Search in Google Scholar
  5. [5] Caliński, T., Kageyama, S. (2008). On the analysis of experiments in affine resolvable designs. Journal of Statistical Planning and Inference 138, 3350 - 3356.10.1016/j.jspi.2007.04.036
    Search in Google Scholar
  6. [6] Chvosteková, M., Witkovský, V. (2009). Exact likelihood ratio test for the parameters of the linear regression model with normal errors. Measurement Science Review 9 (1), 1 - 8.10.2478/v10048-009-0003-9
    Search in Google Scholar
  7. [7] Cui, X., Churchill, G.A. (2003). How many mice and how many arrays? Replication of cDNA microarray experiments. In Lin, S.M. and Allred, E.T. (eds), Methods of Microarray DataAnalysis III, New York: Kluwer.
    Search in Google Scholar
  8. [8] Cui, X., Hwang, J.T.G, Qiu, J., Blades, N.J., Churchill, G.A. (2005). Improved statistical tests for differential gene expression by shrinking variance components estimates. Biostatistics 6 (1), 59 - 75.10.1093/biostatistics/kxh01815618528
    Search in Google Scholar
  9. [9] Cui, X., Affourtit, J., Shockley, K.R., Woo, Y., Churchill, G.A. (2006). Inheritance patterns of transcript levels in F1 hybrid mice. Genetics 174, 627 - 637.10.1534/genetics.106.060251160207716888332
    Search in Google Scholar
  10. [10] Das, K., Jiang, J., Rao, J.N.K. (2004). Mean squared error of empirical predictor. Annals of Statistics 32, 818 -- 840.10.1214/009053604000000201
    Search in Google Scholar
  11. [11] Domotor, Z. (2012). Algebraic frameworks for measurement in the natural sciences. Measurement Science Review 12 (6), 213 - 233.10.2478/v10048-012-0032-7
    Search in Google Scholar
  12. [12] Elston, D.A. (1998). Estimation of denominator degrees of freedom of F-distributions for assessing Wald statistics for fixed-effect factors in unbalanced mixed models. Biometrics 54 (3), 1085 - 1096.10.2307/2533859
    Search in Google Scholar
  13. [13] Fai, A.H.T., Cornelius, P.L. (1996). Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments. Journal of Statistical Computing and Simulation 54, 363 - 378.
    Search in Google Scholar
  14. [14] Fonseca, M., Mathew, T., Mexia, J.T., Zmyślony, R. (2007). Tolerance intervals in a two-way nested model with mixed or random effects. Statistics 41 (4), 289 - 300.10.1080/02331880701270648
    Search in Google Scholar
  15. [15] Gelman, A. (2005). Analysis of variance: Why it is more important than ever. The Annals of Statistics 33 (1), 1 - 31.10.1214/009053604000001048
    Search in Google Scholar
  16. [16] Giesbrecht, F.G., Burns, J.C. (1985). Two-stage analysis based on a mixed model: Large sample asymptotic theory and smallsample simulation results. Biometrics 41, 477 - 486.10.2307/2530872
    Search in Google Scholar
  17. [17] Hartley, H.O., Rao, J.N.K. (1967). Maximum-likelihood estimation for the mixed analysis of variance model. Biometrika 54, 93 - 108.10.1093/biomet/54.1-2.93
    Search in Google Scholar
  18. [18] Hartung, J., Knapp, G., Sinha, B.K. (2008). Statistical Meta-Analysis with Applications, New York: Wiley.10.1002/9780470386347
    Search in Google Scholar
  19. [19] Harville, D.A. (1977). Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association 72, 320 - 38.10.1080/01621459.1977.10480998
    Search in Google Scholar
  20. [20] Harville, D.A. (1985). Decomposition of prediction error. Journal of the American Statistical Association 80, 132 - 138.10.1080/01621459.1985.10477152
    Search in Google Scholar
  21. [21] Harville, D.A., Jeske, D.R. (1992). Mean squared error of estimation or prediction under a general linear model. Journal of the American Statistical Association 87, 724 - 731.10.1080/01621459.1992.10475274
    Search in Google Scholar
  22. [22] Harville, D.A. (2008). Accounting for the estimation of variances and covariances in prediction under general linear model: An overview. Tatra Mountains Mathematical Publications 39, 1 - 15.
    Search in Google Scholar
  23. [23] Henderson, C.R. (1953). Estimation of variance and covariance components Biometrics 9 (2), 226 - 252.10.2307/3001853
    Search in Google Scholar
  24. [24] Herdahl, M. (2008). Linear mixed model for compressor head and flow data with an application. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Norway, June 2008. http://ntnu.diva-portal.org.
    Search in Google Scholar
  25. [25] Jeske, D.R., Harville, D.A., (1988). Prediction-interval procedures and (fixed-effects) confidence-interval procedures for mixed linear models. Communications in Statistics - Theory and Methods 17, 1053 - 1087.10.1080/03610928808829672
    Search in Google Scholar
  26. [26] Jiang, J. (1999). On unbiasedness of the empirical BLUE and BLUP. Statistical and Probability Letters 41 (1999), 19 - 24.
    Search in Google Scholar
  27. [27] Kackar, R.N., Harville, D.A. (1981). Unbiasedness of two-stage estimation and prediction procedures for mixed linear models, Communications in Statistics - Theory and Methods 10, 1249 - 1261.10.1080/03610928108828108
    Search in Google Scholar
  28. [28] Kackar, R.N., Harville, D.A. (1984). Approximations for standard errors of estimators of fixed and random effects in mixed linear models. Journal of the American Statistical Association 79, 853 - 862.
    Search in Google Scholar
  29. [29] Kenward, M.G., Roger, J.H. (1997). Small sample inference for fixed effects from restricted maximum likelihood, Biometrics 53, 983 - 997.10.2307/2533558
    Search in Google Scholar
  30. [30] Kenward, M.G., Roger, J.H. (2009). An improved approximation to the precision of fixed effects from restricted maximum likelihood. Computational Statistics and Data Analysis 53 2583 - 2595.10.1016/j.csda.2008.12.013
    Search in Google Scholar
  31. [31] Krishnamoorthy, K., Mathew, T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation, New York: Wiley.10.1002/9780470473900
    Search in Google Scholar
  32. [32] Laird, N.M., Lange, N., Stram, D. (1987). Maximum likelihood computations with repeated measures: Application of the EM algorithm. Journal of the American Statistical Association 82, 97 - 105.10.1080/01621459.1987.10478395
    Search in Google Scholar
  33. [33] La Motte, L.R. (1973). Quadratic estimation of variance components. Biometrics 29, 311 - 330.10.2307/2529395
    Search in Google Scholar
  34. [34] 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.
    Search in Google Scholar
  35. [35] Littell, R.C., Milliken, G.A., Stroup, W.W., Wolfinger, R.D., Schabenberger, O. (2006). SAS for Mixed Models, Second Edition, Cary, NC: SAS Institute Inc.
    Search in Google Scholar
  36. [36] McCulloch, C.E., Searle, S.R. (2001). Generalized, Linear, and Mixed Models, New York: Wiley.10.1002/9780470057339.vag009
    Search in Google Scholar
  37. [37] McLean, R.A., Sanders, W.L. (1988). Approximating degrees of freedom for standard errors in mixed linear models. In: Proceedings of the Statistical Computing Section, Alexandria, VA: American Statistical Association, 50 - 59.
    Search in Google Scholar
  38. [38] McLean, R.A., Sanders, W.L., Stroup, W.W. (1991). A unified approach to mixed linear models. The American Statistician 45, 54 - 64.
    Search in Google Scholar
  39. [39] Patterson, H.D., Thompson, R. (1971). Recovery of interblock information when block sizes are unequal. Biometrika 58, 545 - 554.10.1093/biomet/58.3.545
    Search in Google Scholar
  40. [40] Pinheiro, J.C., Bates, D.M. (2000): Mixed-Effects Models in Sand S-PLUS. Springer-Verlag, New York.10.1007/978-1-4419-0318-1
    Search in Google Scholar
  41. [41] Prasad, N.G.N., Rao, J.N.K. (1990). The estimation of the mean squared error of small area estimators. Journal of the American Statistical Association 85, 163 - 171.10.1080/01621459.1990.10475320
    Search in Google Scholar
  42. [42] Rao, C.R. (1972). Estimation of variance and covariance components in linear models. Journal of the American Statistical Association 67, 112 -- 115.10.1080/01621459.1972.10481212
    Search in Google Scholar
  43. [43] Rao, C.R., Kleffe, J. (1988). Estimation of Variance Components and Applications. North-Holland Publishing Company, Amsterdam.
    Search in Google Scholar
  44. [44] Robinson, G.K. (1991). That BLUP is a good thing: The estimation of random effects. Statistical Science 6, 15 - 51.
    Search in Google Scholar
  45. [45] SAS Institute, Inc. (2012), SAS/STAT 9.2 User’s Guide The MIXED Procedure. Online Help, Cary, NC: SAS Institute.
    Search in Google Scholar
  46. [46] Satterthwaite, F.E. (1941). Synthesis of variance. Pyschometrika 6, 309 - 316.10.1007/BF02288586
    Search in Google Scholar
  47. [47] Satterthwaite, F.E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin 2 (6), 110 -114.10.2307/3002019
    Search in Google Scholar
  48. [48] Savin, A., Wimmer, G., Witkovský, V. (2003). On Kenward- Roger confidence intervals for common mean in interlaboratory trials. Measurement Science Review 3, 53 - 56.
    Search in Google Scholar
  49. [49] Searle, S.R., Casella, G. McCulloch, C.E. (1992). Variance Components. John Wiley & Sons, New York.10.1002/9780470316856
    Search in Google Scholar
  50. [50] Schaalje, B.G., McBride, J.B., Fellingham, G.W. (2001). Approximations to distributions of test statistics in complex mixed linear models using SAS Proc MIXED. In: SUGI 26 Proceedings,Statistics, Data Analysis and Data Mining Section, Paper262, 1 - 5.
    Search in Google Scholar
  51. [51] Schulz, E., Tiemann, L., Witkovský, V., Schmidt, P., Ploner, M. (2012). Gamma oscillations are involved in the sensorimotor transformation of pain. Journal of Neurophysiology 108, 1025 - 1031.10.1152/jn.00186.201222623490
    Search in Google Scholar
  52. [52] Širková, L.,Witkovský, V. (2001). On testing variance components in unbalanced mixed linear model. Applications of Mathematics 46 (3), 191 - 213.10.1023/A:1013739907119
    Search in Google Scholar
  53. [53] Štulajter, F. (2002). The MSE of the BLUP in a finite discrete spectrum LRM. Tatra Mountains Mathematical Publications 26, 125 -- 131.
    Search in Google Scholar
  54. [54] Štulajter, F. (2002). Predictions in Time Series Using Regression Models, Springer-Verlag, New York.10.1007/978-1-4757-3629-8
    Search in Google Scholar
  55. [55] Tuček, P., Tučková, M., Fišerová, E., Tuček, J., Kubáček, L. (2012). Design of experiment for measurement of Langevin function. Measurement Science Review 12 (4), 121 - 127.10.2478/v10048-012-0019-4
    Search in Google Scholar
  56. [56] Volaufová J. (2005). Statistical methods in biomedical research and measurement science. Measurement Science Review 5, Section 1, 1 - 10.
    Search in Google Scholar
  57. [57] Volaufová J., Witkovský, V. (1992). Estimation of variance components in mixed linear models. Applications of Mathematics 37 (2), 139 - 148.10.21136/AM.1992.104497
    Search in Google Scholar
  58. [58] Volaufová J., Witkovský, V. (1992). Least-squares and minimum MSE estimators of variance components in mixed linear models. Biometrical Journal 33 (8), 923 - 936.
    Search in Google Scholar
  59. [59] Volaufová J., Witkovský, V. (2012). On exact inference in linear models with two variance-covariance components. TatraMountains Mathematical Publications 51, 173 - 181.10.2478/v10127-012-0017-9
    Search in Google Scholar
  60. [60] Wolfinger, R.D., Tobias, R.D., Sall, J. (1994). Computing gaussian likelihoods and their derivatives for general linear mixed models. SIAM Journal on Scientific Computing 15 (6), 1294 -1310.10.1137/0915079
    Search in Google Scholar
  61. [61] Wimmer G., Witkovský, V. (2003). Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model. Journal of Statistical Computation and Simulation 73 (5), 333 - 346.10.1080/0094965021000038940
    Search in Google Scholar
  62. [62] Wimmer G.,Witkovský, V. (2007). Linear comparative calibration with correlated measurements. Kybernetika 43 (4), 443 -452.
    Search in Google Scholar
  63. [63] Wimmer G., Witkovský, V. (2007). Univariate linear calibration via replicated errors-in-variables model. Journal of Statistical Computation and Simulation 77 (3), 213 - 227.10.1080/10629360600679433
    Search in Google Scholar
  64. [64] Wimmer G.,Witkovský, V. (2011). Note on a calibration problem: Selected results and extensions of professor Kubáˇcek’s research. Acta Universitatis Palackianae Olomucensis, Facul-tas Rerum Naturalium, Mathematica 50 (2), 123 - 130.
    Search in Google Scholar
  65. [65] Witkovský, V. (1996). On variance-covariance components estimation in linear models with AR(1). Acta Mathematica UniviversitatisComenianae 65 (1), 129 - 139.
    Search in Google Scholar
  66. [66] Witkovský, V. (1998). Estimation of variance components with constraints. Journal of Statistical Planning and Inference 69 (1), 81 - 87.10.1016/S0378-3758(97)00127-4
    Search in Google Scholar
  67. [67] Witkovský, V. (1998). Modified minimax quadratic estimation of variance components. Kybernetika 34 (5), 535 - 543.
    Search in Google Scholar
  68. [68] Witkovský, V. (2002). MATLAB algorithm mixed.m for solving Henderson’s mixed model equations. Technical Report, Institute of Measurement Science, Slovak Academy of Sciences, Bratislava, 2002. http://www.mathworks.com/matlabcentral/fileexchange/200-mixed.
    Search in Google Scholar
  69. [69] Witkovský, V. (2005). Comparison of some exact and approximate interval estimators for common mean. Measurement ScienceReview 5 (1), 19 - 22.
    Search in Google Scholar
  70. [70] Witkovský, Savin, A., Wimmer G. (2003). On small sample inference for common mean in heteroscedastic one-way model. Discussiones Mathematicae Probability and Statistics 23 (2), 123 - 145.
    Search in Google Scholar
  71. [71] Witkovský, V., Wimmer G. (2001). On statistical models for consensus values. Measurement Science Review 1 (1), 33 - 36.
    Search in Google Scholar
  72. [72] Witkovský, V.,Wimmer G. (2003). Consensus mean and interval estimators for the common mean. Tatra Mountains MathematicalPublications 26 (1), 183 - 194.
    Search in Google Scholar
  73. [73] Witkovský, V., Wimmer G. (2007). Confidence interval for common mean in interlaboratory comparisons with systematic laboratory biases. Measurement Science Review 7 (6), 64 - 73.
    Search in Google Scholar
  74. [74] Wu, H., Yang, H., Churchill, G.A. (2012). R/MAANOVA: Anextensive R environment for the analysis of microarray experiments. http://www.bioconductor.org
    Search in Google Scholar
DOI: https://doi.org/10.2478/v10048-012-0033-6 | Journal eISSN: 1335-8871
Journal RSS Feed
Language: English
Page range: 234 - 248
Published on: Dec 15, 2012
Published by: Slovak Academy of Sciences, Institute of Measurement Science
In partnership with: Paradigm Publishing Services
Publication frequency: Volume open
Keywords:
Linear mixed model,
mixed model equations,
fixed effects,
random effects,
REML,
BLUP,
EBLUP,
MSE,
Satterthwaite approximation,
Fai-Cornelius approximation,
Harville-Jeske and Prasad-Rao approximation,
Kenward-Roger approximation
Related subjects:
Engineering,
Electrical engineering,
Control engineering, metrology and testing

© 2012 Viktor Witkovský, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons License.

Previous articleVolume 12 (2012): Issue 6 (December 2012)Next article