The MIXED Procedure

Akaike, H. (1974). “A New Look at the Statistical Model Identification.” IEEE Transactions 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. New York: John Wiley & Sons.

Beckman, R. J., Nachtsheim, C. J., and Cook, R. D. (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. 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. (1993). Statistics for Spatial Data. Rev. ed. 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.” Journal of the Royal Statistical Society, Series D 44:113–135. http://www.jstor.org/stable/2348622.

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 Bayesian Statistics, vol. 4, edited by J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, 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. 2nd ed. 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.” 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. Alexandria, VA: American Statistical Association.

Harville, D. A. (1990). “BLUP (Best Linear Unbiased Prediction), and Beyond.” In Advances in Statistical Methods for Genetic Improvement of Livestock, edited by D. Gianola, and K. Hammond, 239–276. Vol. 18 of Advanced Series in Agricultural Sciences. Berlin: Springer-Verlag.

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. 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.” Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1:221–233.

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

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.” Journal of the Royal Statistical Society, Series C 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.

Kenward, M. G., and 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.

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—Simulation and Computation 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. T., and Stram, D. O. (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. 2nd ed. Cary, NC: SAS Institute Inc.

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. 2nd ed. New York: Springer-Verlag.

McKeon, J. J. (1974). “F Approximations to the Distribution of Hotelling’s .” Biometrika 61:381–383.

McLean, R. A., and Sanders, W. L. (1988). “Approximating Degrees of Freedom for Standard Errors in Mixed Linear Models.” In Proceedings of the Statistical Computing Section, 50–59. Alexandria, VA: American Statistical Association.

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

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

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

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. 2nd ed. Belmont, CA: PWS-Kent.

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

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. S., and Samson, P., Jr. (1959). “On Hotelling’s Generalization of .” 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 Analysis of Linear Models.” In Bayesian Statistics, vol. 4, edited by J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, 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.” American Statistician 46:84–88.

Snedecor, G. W., and Cochran, W. G. (1980). Statistical Methods. 7th ed. Ames: Iowa State University Press.

Steel, R. G. D., Torrie, J. H., and Dickey, D. A. (1997). Principles and Procedures of Statistics: A Biometrical Approach. 3rd ed. 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.” Applications of Mixed Models in Agriculture and Related Disciplines 39–48. Southern Cooperative Series Bulletin No. 343, Louisiana Agricultural Experiment Station, Baton Rouge.

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 104–122. Southern Cooperative Series Bulletin No. 343, Louisiana Agricultural Experiment Station, Baton Rouge.

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 22:1701–1762.

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. H., and Young, S. S. (1993). Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment. 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. 2nd ed. New York: McGraw-Hill.

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

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 7:481–500.

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, 1172–1182. Cary, NC: SAS Institute Inc. http://www.sascommunity.org/sugi/SUGI95/Sugi-95-198%20Wolfinger%20Chang.pdf.

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, 1380–1388. Cary, NC: SAS Institute Inc. http://www.sascommunity.org/sugi/SUGI91/Sugi-91-249%20Wolfinger%20Tobias%20Sall.pdf.

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, S. P. (1994). Adjusted F Tests for Repeated Measures with the MIXED Procedure. Knoxville: 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.