The MIXED Procedure

References

  • Akaike, H. (1974), “A New Look at the Statistical Model Identification,” IEEE Transaction on Automatic Control, AC-19, 716–723.

  • Akritas, M. G., Arnold, S. F., and Brunner, E. (1997), “Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs,” Journal of the American Statistical Association, 92, 258–265.

  • Allen, D. M. (1974), “The Relationship between Variable Selection and Data Augmentation and a Method of Prediction,” Technometrics, 16, 125–127.

  • Bates, D. M. and Watts, D. G. (1988), Nonlinear Regression Analysis and Its Applications, John Wiley & Sons.

  • Beckman, R. J., Nachtsheim, C. J., and Cook, D. R. (1987), “Diagnostics for Mixed-Model Analysis of Variance,” Technometrics, 29, 413–426.

  • Belsley, D. A., Kuh, E., and Welsch, R. E. (1980), Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, New York: John Wiley & Sons.

  • Box, G. E. P. and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, Wiley Classics Library Edition, published 1992, New York: John Wiley & Sons.

  • Bozdogan, H. (1987), “Model Selection and Akaike’s Information Criterion (AIC): The General Theory and Its Analytical Extensions,” Psychometrika, 52, 345–370.

  • Brown, H. and Prescott, R. (1999), Applied Mixed Models in Medicine, New York: John Wiley & Sons.

  • Brownie, C., Bowman, D. T., and Burton, J. W. (1993), “Estimating Spatial Variation in Analysis of Data from Yield Trials: A Comparison of Methods,” Agronomy Journal, 85, 1244–1253.

  • Brownie, C. and Gumpertz, M. L. (1997), “Validity of Spatial Analysis of Large Field Trials,” Journal of Agricultural, Biological, and Environmental Statistics, 2, 1–23.

  • Brunner, E., Dette, H., and Munk, A. (1997), “Box-Type Approximations in Nonparametric Factorial Designs,” Journal of the American Statistical Association, 92, 1494–1502.

  • Brunner, E., Domhof, S., and Langer, F. (2002), Nonparametric Analysis of Longitudinal Data in Factorial Experiments, New York: John Wiley & Sons.

  • Burdick, R. K. and Graybill, F. A. (1992), Confidence Intervals on Variance Components, New York: Marcel Dekker.

  • Burnham, K. P. and Anderson, D. R. (1998), Model Selection and Inference: A Practical Information-Theoretic Approach, New York: Springer-Verlag.

  • Carlin, B. P. and Louis, T. A. (1996), Bayes and Empirical Bayes Methods for Data Analysis, London: Chapman & Hall.

  • Carroll, R. J. and Ruppert, D. (1988), Transformation and Weighting in Regression, London: Chapman & Hall.

  • Chilès, J. P. and Delfiner, P. (1999), Geostatistics-Modeling Spatial Uncertainty, New York: John Wiley & Sons.

  • Christensen, R., Pearson, L. M., and Johnson, W. (1992), “Case-Deletion Diagnostics for Mixed Models,” Technometrics, 34, 38–45.

  • Cook, R. D. (1977), “Detection of Influential Observations in Linear Regression,” Technometrics, 19, 15–18.

  • Cook, R. D. (1979), “Influential Observations in Linear Regression,” Journal of the American Statistical Association, 74, 169–174.

  • Cook, R. D. and Weisberg, S. (1982), Residuals and Influence in Regression, New York: Chapman & Hall.

  • Cressie, N. A. C. (1993), Statistics for Spatial Data, New York: John Wiley & Sons.

  • Crowder, M. J. and Hand, D. J. (1990), Analysis of Repeated Measures, New York: Chapman & Hall.

  • Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society, Series B, 39, 1–38.

  • Diggle, P. J. (1988), “An Approach to the Analysis of Repeated Measurements,” Biometrics, 44, 959–971.

  • Diggle, P. J., Liang, K.-Y., and Zeger, S. L. (1994), Analysis of Longitudinal Data, Oxford: Clarendon Press.

  • Dunnett, C. W. (1980), “Pairwise Multiple Comparisons in the Unequal Variance Case,” Journal of the American Statistical Association, 75, 796–800.

  • Edwards, D. and Berry, J. J. (1987), “The Efficiency of Simulation-Based Multiple Comparisons,” Biometrics, 43, 913–928.

  • Everitt, B. S. (1995), “The Analysis of Repeated Measures: A Practical Review with Examples,” The Statistician, 44, 113–135.

  • Fai, A. H. T. and 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 Computation and Simulation, 54, 363–378.

  • Federer, W. T. and Wolfinger, R. D. (1998), “SAS Code for Recovering Intereffect Information in Experiments with Incomplete Block and Lattice Rectangle Designs,” Agronomy Journal, 90, 545–551.

  • Fuller, W. A. (1976), Introduction to Statistical Time Series, New York: John Wiley & Sons.

  • Fuller, W. A. and Battese, G. E. (1973), “Transformations for Estimation of Linear Models with Nested Error Structure,” Journal of the American Statistical Association, 68, 626–632.

  • Galecki, A. T. (1994), “General Class of Covariance Structures for Two or More Repeated Factors in Longitudinal Data Analysis,” Communications in Statistics—Theory and Methods, 23, 3105–3109.

  • Games, P. A. and Howell, J. F. (1976), “Pairwise Multiple Comparison Procedures with Unequal n’s and/or Variances: A Monte Carlo Study,” Journal of Educational Statistics, 1, 113–125.

  • Gelfand, A. E., Hills, S. E., Racine-Poon, A., and Smith, A. F. M. (1990), “Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling,” Journal of the American Statistical Association, 85, 972–985.

  • Ghosh, M. (1992), “Discussion of Schervish, M., 'Bayesian Analysis of Linear Models',” in J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds., Bayesian Statistics 4, 432–433, Oxford: Oxford University Press.

  • Giesbrecht, F. G. (1989), A General Structure for the Class of Mixed Linear Models, Southern Cooperative Series Bulletin 343, Louisiana Agricultural Experiment Station, Baton Rouge.

  • Giesbrecht, F. G. and Burns, J. C. (1985), “Two-Stage Analysis Based on a Mixed Model: Large-Sample Asymptotic Theory 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), Tests of Hypotheses in Fixed-Effects Linear Models, Technical Report R-101, SAS Institute Inc, Cary, NC.

  • Goodnight, J. H. (1979), “A Tutorial on the Sweep Operator,” The 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, 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), “The 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.

  • Harville, D. A. (1988), “Mixed-Model Methodology: Theoretical Justifications and Future Directions,” in Proceedings of the Statistical Computing Section, 41–49, New Orleans: American Statistical Association.

  • 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.

  • Harville, J. A. (1990), “BLUP (Best Linear Unbiased Prediction), and Beyond,” in Advances in Statistical Methods for Genetic Improvement of Livestock, 239–276, Springer-Verlag.

  • 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, Guelph, ON: University of Guelph.

  • Henderson, C. R. (1990), “Statistical Method in Animal Improvement: Historical Overview,” in Advances in Statistical Methods for Genetic Improvement of Livestock, 1–14, New York: Springer-Verlag.

  • Hirotsu, C. and Srivastava, M. (2000), “Simultaneous Confidence Intervals Based on One-Sided Max t Test,” Statistics and Probability Letters, 49, 25–37.

  • Hsu, J. C. (1992), “The Factor Analytic Approach to Simultaneous Inference in the General Linear Model,” Journal of Computational and Graphical Statistics, 1, 151–168.

  • Huber, P. J. (1967), “The Behavior of Maximum Likelihood Estimates under Nonstandard Conditions,” Proc. Fifth Berkeley Symp. Math. Statist. Prob., 1, 221–233.

  • Hurtado, G. I. (1993), Detection of Influential Observations in Linear Mixed Models, Ph.d. dissertation, Department of Statistics, North Carolina State University, 1993.

  • 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.

  • Kenward, M. G. (1987), “A Method for Comparing Profiles of Repeated Measurements,” Applied Statistics, 36, 296–308.

  • Kenward, M. G. and Roger, J. H. (1997), “Small Sample Inference for Fixed Effects from Restricted Maximum Likelihood,” Biometrics, 53, 983–997.

  • Keselman, H. J., Algina, J., Kowalchuk, R. K., and Wolfinger, R. D. (1998), “A Comparison of Two Approaches for Selecting Covariance Structures in the Analysis of Repeated Measures,” Communications in Statistics—Computation and Simulation, 27, 591–604.

  • Keselman, H. J., Algina, J., Kowalchuk, R. K., and Wolfinger, R. D. (1999), “A Comparison of Recent Approaches to the Analysis of Repeated Measurements,” British Journal of Mathematical and Statistical Psychology, 52, 63–78.

  • Kramer, C. Y. (1956), “Extension of Multiple Range Tests to Group Means with Unequal Numbers of Replications,” Biometrics, 12, 307–310.

  • Laird, N. M., Lange, N., and Stram, D. (1987), “Maximum Likelihood Computations with Repeated Measures: Application of the EM Algorithm,” Journal of the American Statistical Association, 82, 97–105.

  • Laird, N. M. and Ware, J. H. (1982), “Random-Effects Models for Longitudinal Data,” Biometrics, 38, 963–974.

  • LaMotte, L. R. (1973), “Quadratic Estimation of Variance Components,” Biometrics, 29, 311–330.

  • Liang, K.-Y. and Zeger, S. L. (1986), “Longitudinal Data Analysis Using Generalized Linear Models,” Biometrika, 73, 13–22.

  • Lindsey, J. K. (1993), Models for Repeated Measurements, Oxford: Clarendon Press.

  • Lindstrom, M. J. and 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.

  • Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D., and Schabenberger, O. (2006), SAS for Mixed Models, Second Edition, Cary, NC: SAS Press.

  • Little, R. J. A. (1995), “Modeling the Drop-Out Mechanism in Repeated-Measures Studies,” Journal of the American Statistical Association, 90, 1112–1121.

  • Louis, T. A. (1988), “General Methods for Analyzing Repeated Measures,” Statistics in Medicine, 7, 29–45.

  • Macchiavelli, R. E. and Arnold, S. F. (1994), “Variable Order Ante-dependence Models,” Communications in Statistics—Theory and Methods, 23, 2683–2699.

  • Marx, D. and Thompson, K. (1987), Practical Aspects of Agricultural Kriging, Bulletin 903, Arkansas Agricultural Experiment Station, Fayetteville, NC.

  • Matérn, B. (1986), Spatial Variation, Second Edition, New York: Springer-Verlag.

  • McKeon, J. J. (1974), “F Approximations to the Distribution of Hotelling’s $T_0^2$,” Biometrika, 61, 381–383.

  • McLean, R. A. and Sanders, W. L. (1988), “Approximating Degrees of Freedom for Standard Errors in Mixed Linear Models,” Proceedings of the Statistical Computing Section.

  • McLean, R. A., Sanders, W. L., and Stroup, W. W. (1991), “A Unified Approach to Mixed Linear Models,” The American Statistician, 45, 54–64.

  • Milliken, G. A. and Johnson, D. E. (1992), Analysis of Messy Data, Volume 1: Designed Experiments, New York: Chapman & Hall.

  • Moriguchi, S., ed. (1976), Statistical Method for Quality Control, (in Japanese), Tokyo: Japan Standards Association.

  • Murray, D. M. (1998), Design and Analysis of Group-Randomized Trials, New York: Oxford University Press.

  • Myers, R. H. (1990), Classical and Modern Regression with Applications, Second Edition, Belmont, CA: PWS-Kent.

  • Obenchain, R. L. (1990), STABLSIM.EXE, Version 9010, Indianapolis: Eli Lilly, unpublished C code.

  • Patel, H. I. (1991), “Analysis of Incomplete Data from a Clinical Trial with Repeated Measurements,” Biometrika, 78, 609–619.

  • Patterson, H. D. and Thompson, R. (1971), “Recovery of Inter-Block Information When Block Sizes Are Unequal,” Biometrika, 58, 545–554.

  • Pillai, K. C. and Samson, P. (1959), “On Hotelling’s Generalization of $T^2$,” Biometrika, 46, 160–168.

  • Pothoff, R. F. and Roy, S. N. (1964), “A Generalized Multivariate Analysis of Variance Model Useful Especially for Growth Curve Problems,” Biometrika, 51, 313–326.

  • Prasad, N. G. N. and Rao, J. N. K. (1990), “The Estimation of Mean Squared Error of Small-Area Estimators,” Journal of the American Statistical Association, 85, 163–171.

  • Pringle, R. M. and Rayner, A. A. (1971), Generalized Inverse Matrices with Applications to Statistics, New York: Hafner Publishing.

  • Rao, C. R. (1972), “Estimation of Variance and Covariance Components in Linear Models,” Journal of the American Statistical Association, 67, 112–115.

  • Ripley, B. D. (1987), Stochastic Simulation, New York: John Wiley & Sons.

  • Robinson, G. K. (1991), “That BLUP Is a Good Thing: The Estimation of Random Effects,” Statistical Science, 6, 15–51.

  • Rubin, D. B. (1976), “Inference and Missing Data,” Biometrika, 63, 581–592.

  • Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989), “Design and Analysis of Computer Experiments,” Statistical Science, 4, 409–435.

  • Schabenberger, O. and Gotway, C. A. (2005), Statistical Methods for Spatial Data Analysis, Boca Raton, FL: Chapman & Hall/CRC.

  • Schervish, M. J. (1992), Bayesian Statistics 4, chapter Bayesian Analysis of Linear Models, 419–434, Oxford: Oxford University Press.

  • Schluchter, M. D. and Elashoff, J. D. (1990), “Small-Sample Adjustments to Tests with Unbalanced Repeated Measures Assuming Several Covariance Structures,” Journal of Statistical Computation and Simulation, 37, 69–87.

  • Schwarz, G. (1978), “Estimating the Dimension of a Model,” Annals of Statistics, 6, 461–464.

  • Searle, S. R. (1971), Linear Models, New York: John Wiley & Sons.

  • Searle, S. R. (1982), Matrix Algebra Useful for Statisticians, New York: John Wiley & Sons.

  • Searle, S. R. (1988), “Mixed Models and Unbalanced Data: Wherefrom, Whereat, and Whereto?” Communications in Statistics—Theory and Methods, 17, 935–968.

  • Searle, S. R., Casella, G., and McCulloch, C. E. (1992), Variance Components, New York: John Wiley & Sons.

  • Self, S. G. and Liang, K.-Y. (1987), “Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions,” Journal of the American Statistical Association, 82, 605–610.

  • Serfling, R. J. (1980), Approximation Theorems of Mathematical Statistics, New York: John Wiley & Sons.

  • Singer, J. D. (1998), “Using SAS PROC MIXED to Fit Multilevel Models, Hierarchical Models, and Individual Growth Models,” Journal of Educational and Behavioral Statistics, 23, 323–355.

  • Smith, A. F. M. and Gelfand, A. E. (1992), “Bayesian Statistics without Tears: a Sampling-Resampling Perspective,” The American Statistician, 46, 84–88.

  • Snedecor, G. W. and Cochran, W. G. (1976), Statistical Methods, Sixth Edition, Ames: Iowa State University Press.

  • Snedecor, G. W. and Cochran, W. G. (1980), Statistical Methods, Seventh Edition, Ames: Iowa State University Press.

  • Steel, R. G. D., Torrie, J. H., and Dickey, D. (1997), Principles and Procedures of Statistics: A Biometrical Approach, Third Edition, New York: McGraw-Hill.

  • Stram, D. O. and Lee, J. W. (1994), “Variance Components Testing in the Longitudinal Mixed Effects Model,” Biometrics, 50, 1171–1177.

  • Stroup, W. W. (1989a), Predictable Functions and Prediction Space in the Mixed Model Procedure, Southern Cooperative Series Bulletin 343, Louisiana Agricultural Experiment Station, Baton Rouge, LA.

  • Stroup, W. W. (1989b), “Use of Mixed Model Procedure to Analyze Spatially Correlated Data: An Example Applied to a Line-Source Sprinkler Irrigation Experiment,” Applications of Mixed Models in Agriculture and Related Disciplines.

  • Stroup, W. W., Baenziger, P. S., and Mulitze, D. K. (1994), “Removing Spatial Variation from Wheat Yield Trials: A Comparison of Methods,” Crop Science, 86, 62–66.

  • Sullivan, L. M., Dukes, K. A., and Losina, E. (1999), “An Introduction to Hierarchical Linear Modelling,” Statistics in Medicine, 18, 855–888.

  • Swallow, W. H. and Monahan, J. F. (1984), “Monte Carlo Comparison of ANOVA, MIVQUE, REML, and ML Estimators of Variance Components,” Technometrics, 28, 47–57.

  • Tamhane, A. C. (1979), “A Comparison of Procedures for Multiple Comparisons of Means with Unequal Variances,” Journal of the American Statistical Association, 74, 471–480.

  • Tierney, L. (1994), “Markov Chains for Exploring Posterior Distributions,” Annals of Statistics, 22(4), 1701–1762.

  • Verbeke, G. and Molenberghs, G., eds. (1997), Linear Mixed Models in Practice: A SAS-Oriented Approach, New York: Springer.

  • Verbeke, G. and Molenberghs, G. (2000), Linear Mixed Models for Longitudinal Data, New York: Springer.

  • Westfall, P. H., Tobias, R. D., Rom, D., Wolfinger, R. D., and Hochberg, Y. (1999), Multiple Comparisons and Multiple Tests Using the SAS System, Cary, NC: SAS Institute Inc.

  • Westfall, P. J. and Young, S. S. (1993), Resampling-Based Multiple Testing, New York: John Wiley & Sons.

  • White, H. (1980), “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity,” Econometrica, 48, 817–838.

  • Whittle, P. (1954), “On Stationary Processes in the Plane,” Biometrika, 41, 434–449.

  • Winer, B. J. (1971), Statistical Principles in Experimental Design, Second Edition, New York: McGraw-Hill.

  • Wolfinger, R. D. (1993), “Covariance Structure Selection in General Mixed Models,” Communications in Statistics, Simulation, and Computation, 22(4), 1079–1106.

  • Wolfinger, R. D. (1996), “Heterogeneous Variance-Covariance Structures for Repeated Measures,” Journal of Agricultural, Biological, and Environmental Statistics, 1, 205–230.

  • Wolfinger, R. D. (1997), “An Example of Using Mixed Models and PROC MIXED for Longitudinal Data,” Journal of Biopharmaceutical Statistics, 7(4), 481–500.

  • Wolfinger, R. D. and Chang, M. (1995), “Comparing the SAS GLM and MIXED Procedures for Repeated Measures,” in Proceedings of the Twentieth Annual SAS Users Group Conference, Cary, NC: SAS Institute Inc.

  • Wolfinger, R. D., Tobias, R. D., and Sall, J. (1991), “Mixed Models: A Future Direction,” in Proceedings of the Sixteenth Annual SAS Users Group Conference, Cary, NC: SAS Institute Inc.

  • Wolfinger, R. D., Tobias, R. D., and Sall, J. (1994), “Computing Gaussian Likelihoods and Their Derivatives for General Linear Mixed Models,” SIAM Journal on Scientific Computing, 15, 1294–1310.

  • Wright, P. S. (1994), Adjusted F Tests for Repeated Measures with the MIXED Procedure, 328 SMC-Statistics Department, University of Tennessee.

  • Zimmerman, D. L. and Harville, D. A. (1991), “A Random Field Approach to the Analysis of Field-Plot Experiments and Other Spatial Experiments,” Biometrics, 47, 223–239.