4712 ✦ Chapter 56: <strong>The</strong> <strong>MIXED</strong> <strong>Procedure</strong> Ghosh, M. (1992), Discussion of Schervish, M., “Bayesian Analysis of Linear Models,” Bayesian Statistics 4, eds. J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith, Oxford: University Press, 432–433. Giesbrecht, F. G. (1989), “A General Structure for the Class of Mixed Linear Models,” Applications of Mixed Models in Agriculture and Related Disciplines, Southern Cooperative Series Bulletin No. 343, Louisiana Agricultural Experiment Station, Baton Rouge, 183–201. Giesbrecht, F. G. and Burns, J. C. (1985), “Two-Stage Analysis Based on a Mixed Model: Largesample Asymptotic <strong>The</strong>ory and Small-Sample Simulation Results,” Biometrics, 41, 477–486. Golub, G. H. and Van Loan, C. F. (1989), Matrix Computations, Second Edition, Baltimore: Johns Hopkins University Press. Goodnight, J. H. (1978), <strong>SAS</strong> Technical Report R-101, Tests of Hypotheses in Fixed-Effects Linear Models, Cary, NC: <strong>SAS</strong> Institute Inc. Goodnight, J. H. (1979), “A Tutorial on the Sweep Operator,” American Statistician, 33, 149–158. Goodnight, J. H. and Hemmerle, W. J. (1979), “A Simplified Algorithm for the W-Transformation in Variance Component Estimation,” Technometrics, 21, 265–268. Gotway, C. A. and Stroup, W. W. (1997), “A Generalized Linear Model Approach to Spatial Data and Prediction,” Journal of Agricultural, Biological, and Environmental Statistics, 2, 157–187. Greenhouse, S. W. and Geisser, S. (1959), “On Methods in the Analysis of Profile Data,” Psychometrika, 32, 95–112. Gregoire, T. G., Schabenberger, O., and Barrett, J. P. (1995), “Linear Modelling of Irregularly Spaced, Unbalanced, Longitudinal Data from Permanent Plot Measurements,” Canadian Journal of Forest Research, 25, 137–156. Handcock, M. S. and Stein, M. L. (1993), “A Bayesian Analysis of Kriging,” Technometrics, 35(4), 403–410 Handcock, M. S. and Wallis, J. R. (1994), “An Approach to Statistical Spatial-Temporal Modeling of Meteorological Fields (with Discussion),” Journal of the American Statistical Association, 89, 368–390. Hanks, R.J., Sisson, D.V., Hurst, R.L, and Hubbard K.G. (1980), “Statistical Analysis of Results from Irrigation Experiments Using the Line-Source Sprinkler System,” Soil Science Society American Journal, 44, 886–888. Hannan, E.J. and Quinn, B.G. (1979), “<strong>The</strong> Determination of the Order of an Autoregression,” Journal of the Royal Statistical Society, Series B, 41, 190–195. Hartley, H. O. and Rao, J. N. K. (1967), “Maximum-Likelihood Estimation for the Mixed Analysis of Variance Model,” Biometrika, 54, 93–108. Harville, D. A. (1977), “Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems,” Journal of the American Statistical Association, 72, 320–338.
References ✦ 4713 Harville, D. A. (1988), “Mixed-Model Methodology: <strong>The</strong>oretical Justifications and Future Directions,” Proceedings of the Statistical Computing Section, American Statistical Association, New Orleans, 41–49. Harville, D. A. (1990), “BLUP (Best Linear Unbiased Prediction), and Beyond,” in Advances in Statistical Methods for Genetic Improvement of Livestock, Springer-Verlag, 239–276. Harville, D. A. and 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. Hemmerle, W. J. and Hartley, H. O. (1973), “Computing Maximum Likelihood Estimates for the Mixed AOV Model Using the W-Transformation,” Technometrics, 15, 819–831. Henderson, C. R. (1984), Applications of Linear Models in Animal Breeding, University of Guelph. Henderson, C. R. (1990), “Statistical Method in Animal Improvement: Historical Overview,” in Advances in Statistical Methods for Genetic Improvement of Livestock, New York: Springer-Verlag, 1–14. Hsu, J. C. (1992), “<strong>The</strong> Factor Analytic Approach to Simultaneous Inference in the General Linear Model,” Journal of Computational and Graphical Statistics, 1, 151–168. Huber, P. J. (1967), “<strong>The</strong> Behavior of Maximum Likelihood Estimates under Nonstandard Conditions,” Proc. Fifth Berkeley Symp. Math. Statist. Prob., 1, 221–233. Hurtado, G. I. H. (1993), Detection of Influential Observations in Linear Mixed Models, Ph.D. dissertation, Department of Statistics, North Carolina State University, Raleigh, NC. Hurvich, C. M. and Tsai, C.-L. (1989), “Regression and Time Series Model Selection in Small Samples,” Biometrika, 76, 297–307. Huynh, H. and Feldt, L. S. (1970), “Conditions Under Which Mean Square Ratios in Repeated Measurements Designs Have Exact F-Distributions,” Journal of the American Statistical Association, 65, 1582–1589. Jennrich, R. I. and Schluchter, M. D. (1986), “Unbalanced Repeated-Measures Models with Structured Covariance Matrices,” Biometrics, 42, 805–820. Johnson, D. E., Chaudhuri, U. N., and Kanemasu, E. T. (1983), “Statistical Analysis of Line- Source Sprinkler Irrigation Experiments and Other Nonrandomized Experiments Using Multivariate Methods,” Soil Science Society American Journal, 47, 309–312. Jones, R. H. and Boadi-Boateng, F. (1991), “Unequally Spaced Longitudinal Data with AR(1) Serial Correlation,” Biometrics, 47, 161–175. Kackar, R. N. and 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. Kass, R. E. and Steffey, D. (1989), “Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models),” Journal of the American Statistical Association, 84, 717–726.
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SAS/STAT ® 9.22 User’s Guide The
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Chapter 56 The MIXED Procedure Cont
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Basic Features ✦ 4515 Repeated me
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PROC MIXED Contrasted with Other SA
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proc mixed data=heights; class Fami
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Figure 56.6 Fit Statistics Fit Stat
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Figure 56.10 Dimensions and Number
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Syntax: MIXED Procedure The followi
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Table 56.1 continued Statement Desc
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PROC MIXED Statement ✦ 4529 CL< =
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INFO PROC MIXED Statement ✦ 4531
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ORD PROC MIXED Statement ✦ 4533 d
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PROC MIXED Statement ✦ 4535 RESID
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PROC MIXED Statement ✦ 4537 RANDO
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CLASS Statement ✦ 4539 Because so
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contrast 'A narrow' A 1 -1 0 A*B .5
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ESTIMATE Statement ESTIMATE ’labe
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ID Statement ID variables ; ID Stat
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ADJUST=BON proc mixed; class A; mod
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LSMEANS Statement ✦ 4549 BYLEVEL
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SLICE= fixed-effect LSMESTIMATE Sta
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Table 56.6 Summary of Important MOD
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MODEL Statement ✦ 4555 DDFM=KENWA
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E E1 E2 E3 MODEL Statement ✦ 4557
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Table 56.8 continued Description Su
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SIZE=n MODEL Statement ✦ 4561 If
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MODEL Statement ✦ 4563 Table 56.1
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The estimated prediction variance i
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PARMS Statement PARMS (value-list)
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OLS PRIOR Statement ✦ 4569 By def
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PRIOR Statement ✦ 4571 JEFFREYS s
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RANDOM Statement ✦ 4573 TDATA= en
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RANDOM Statement ✦ 4575 GCORR dis
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RANDOM Statement ✦ 4577 component
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Table 56.12 continued Option Descri
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proc mixed; class a; model y = a x;
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REPEATED Statement ✦ 4583 assumed
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REPEATED Statement ✦ 4585 functio
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REPEATED Statement ✦ 4587 The fol
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TYPE=SP(EXPGA)(c1 c2) TYPE=SP(GAUGA
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TYPE=UN@AR(1) TYPE=UN@CS REPEATED S
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WEIGHT Statement WEIGHT variable ;
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Example: Growth Curve with Compound
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Mixed Models Theory ✦ 4597 Such a
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2 6 Z D 6 4 2 6 G D 6 4 1 1 1 1 1 1
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Estimating Fixed and Random Effects
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Inference and Test Statistics Mixed
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Mixed Models Theory ✦ 4605 (2002,
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Intercept Parameterization of Mixed
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Table 56.18 continued Data I A B(A)
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Residuals and Influence Diagnostics
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Scaled Residuals Residuals and Infl
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Predicted Values, PRESS Residual, a
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When ITER=0 and 2 is profiled, then
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Using results in Cook and Weisberg
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Default Output ✦ 4621 The Evaluat
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Null Model Likelihood Ratio Test De
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Table 56.22 continued ODS Table Nam
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ODS Table Names ✦ 4627 CAUTION: T
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ODS Graphics ✦ 4629 Some of the v
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Figure 56.16 Panel of the Condition
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Figure 56.18 Deletion Estimates ODS
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Computational Issues Computational
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Memory Computational Issues ✦ 463
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Examples: MIXED Procedure Examples:
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Example 56.1: Split-Plot Design ✦
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Example 56.1: Split-Plot Design ✦
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4 F 23.5 24.5 25.0 26.5 5 F 21.5 23
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Output 56.2.4 Repeated Measures Ana
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Output 56.2.7 Repeated Measures Ana
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Example 56.2: Repeated Measures ✦
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Output 56.2.14 Analysis with Compou
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Example 56.2: Repeated Measures ✦
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Example 56.3: Plotting the Likeliho
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Output 56.3.3 continued Number of O
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