You are previewing SAS for Mixed Models, Second Edition, 2nd Edition.
O'Reilly logo
SAS for Mixed Models, Second Edition, 2nd Edition

Book Description

The indispensable, up-to-date guide to mixed models using SAS. Discover the latest capabilities available for a variety of applications featuring the MIXED, GLIMMIX, and NLMIXED procedures in this valuable edition of the comprehensive mixed models guide for data analysis, completely revised and updated for SAS®9. The theory underlying the models, the forms of the models for various applications, and a wealth of examples from different fields of study are integrated in the discussions of these models:

random effect only and random coefficients models

split-plot, multilocation, and repeated measures models

hierarchical models with nested random effects

analysis of covariance models

spatial correlation models

generalized linear mixed models

nonlinear mixed models

Table of Contents

  1. Copyright
  2. Praise from the Experts
  3. Preface
  4. 1. Introduction
    1. 1.1. Types of Models That Produce Data
    2. 1.2. Statistical Models
    3. 1.3. Fixed and Random Effects
    4. 1.4. Mixed Models
    5. 1.5. Typical Studies and the Modeling Issues They Raise
      1. 1.5.1. Random Effects Model
      2. 1.5.2. Multi-location Example
      3. 1.5.3. Repeated Measures and Split-Plot Experiments
      4. 1.5.4. Fixed Treatment, Random Block, Non-normal (Binomial) Data Example
      5. 1.5.5. Repeated Measures with Non-normal (Count) Data
      6. 1.5.6. Repeated Measures and Split Plots with Effects Modeled by Nonlinear Regression Model
    6. 1.6. A Typology for Mixed Models
    7. 1.7. Flowcharts to Select SAS Software to Run Various Mixed Models
  5. 2. Randomized Block Designs
    1. 2.1. Introduction
    2. 2.2. Mixed Model for a Randomized Complete Blocks Design
      1. 2.2.1. Means and Variances from Randomized Blocks Design
      2. 2.2.2. The Traditional Method: Analysis of Variance
      3. 2.2.3. Using Expected Mean Squares
      4. 2.2.4. Example: A Randomized Complete Blocks Design
    3. 2.3. Using PROC MIXED to Analyze RCBD Data
      1. 2.3.1. Basic PROC MIXED Analysis Based on Sums of Squares
        1. Program
        2. Results
        3. Interpretation
        4. Results
        5. Interpretation
      2. 2.3.2. Basic PROC MIXED Analysis Based on Likelihood
        1. Program
        2. Results
      3. 2.3.3. Estimating and Comparing Means: LSMEANS, ESTIMATE, and CONTRAST Statements
        1. Program
        2. Results
        3. Interpretation
        4. Interpretation
      4. 2.3.4. Multiple Comparisons and Multiple Tests about Means
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Results
        6. Interpretation
      5. 2.3.4. Confidence Intervals for Variance Components
        1. Program
        2. Result
        3. Interpretation
      6. 2.3.5. Comparison of PROC MIXED with PROC GLM for the RCBD Data
        1. Program
        2. Results
        3. Interpretation
    4. 2.4. Introduction to Theory of Mixed Models
      1. 2.4.1. Review of Regression Model in Matrix Notation
      2. 2.4.2. The RCBD Model in Matrix Notation
    5. 2.5. Example of an Unbalanced Two-Way Mixed Model: Incomplete Block Design
      1. Model
      2. 2.5.1. The Usual Intra-block Analysis of PBIB Data Using PROC GLM
        1. Program
        2. Results
        3. Interpretation
      3. 2.5.2. The Combined Intra- and Inter-block Analysis of PBIB Data Using PROC MIXED
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Results
        6. Interpretation
        7. Program
        8. Interpretation
    6. 2.6. Summary
  6. 3. Random Effects Models
    1. 3.1. Introduction: Descriptions of Random Effects Models
      1. 3.1.1. Using PROC MIXED to Estimate the Variance Components
      2. 3.1.2. The Mixed Model Equations
      3. 3.1.3. Method of Moments Estimators Using PROC MIXED
    2. 3.2. Example: One-Way Random Effects Treatment Structure
      1. 3.2.1. Using the MIXED Procedure
        1. Results
        2. Program
        3. Results
        4. Interpretation
        5. Program
        6. Results
        7. Interpretation
      2. 3.2.2. Using the GLM Procedure
        1. Program
        2. Results
        3. Interpretation
      3. 3.2.3. Confidence Intervals about the Variance Components
        1. Program
        2. Results
        3. Interpretation
    3. 3.3. Example: A Simple Conditional Hierarchical Linear Model
      1. 3.3.1. Using PROC MIXED to Analyze the Data
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Interpretation
      2. 3.3.2. PROC GLM Part of the Analysis
    4. 3.4. Example: Three-Level Nested Design Structure
      1. 3.4.1. Three-Level Nested Linear Model, an Unconditional Hierarchical Nested Linear Model
      2. 3.4.2. Data Analysis Using the PROC MIXED METHOD=REML to Estimate the Variance Components
        1. Results
        2. Interpretation
      3. 3.4.3. Using the PROC MIXED METHOD=TYPE1 to Estimate the Variance Components
        1. Program
        2. Results
        3. Interpretation
      4. 3.4.4. Conditional Hierarchical Linear Model or Mixed Model
      5. 3.4.5. Using the MIXED Procedure
        1. Results
        2. Interpretation
      6. 3.4.6. Unequal Variance Model
        1. Program
        2. Results
        3. Interpretation
    5. 3.5. Example: A Two-Way Random Effects Treatment Structure to Estimate Heritability
      1. 3.5.1. Using PROC MIXED METHOD=REML
        1. Results
        2. Interpretation
      2. 3.5.2. Using PROC MIXED METHOD=TYPE3
        1. Program
        2. Results
        3. Interpretation
    6. 3.6. Summary
  7. 4. Multi-factor Treatment Designs with Multiple Error Terms
    1. 4.1. Introduction
    2. 4.2. Treatment and Experiment Structure and Associated Models
      1. 4.2.1. Possible Layouts of Factorial and Split-Plot Experiments
      2. 4.2.2. Determining the Appropriate Mixed Model for a Given Layout
    3. 4.3. Inference with Mixed Models for Factorial Treatment Designs
      1. 4.3.1. Effects of Interest in Factorial Treatment Designs
        1. 4.3.1.1. Parameters of Interest
      2. 4.3.2. More on Effects of Interest—Estimability and Use of ESTIMATE, CONTRAST, and LSMEANS Statements
        1. 4.3.2.1. Simple Effects
        2. 4.3.2.2. Main Effects
        3. 4.3.2.3. Interactions and Contrasts
      3. 4.3.3. Standard Errors
        1. 4.3.3.1. Variance of Treatment Mean and Difference Estimates
        2. 4.3.3.2. Generalization: Variance of Estimates in Matrix Form
        3. 4.3.3.3. Completing the Standard Error: Variance Component Estimates and Degrees of Freedom
    4. 4.4. Example: A Split-Plot Semiconductor Experiment
      1. 4.4.1. Tests of Interest in the Semiconductor Experiment
      2. 4.4.2. Matrix Generalization of Mixed Model F-tests
      3. 4.4.3. PROC MIXED Analysis of Semiconductor Data
        1. 4.4.3.1. Fitting the Mixed Model
          1. Interpretation
        2. 4.4.3.2. Treatment Means
        3. 4.4.3.3. Degrees of Freedom
          1. Default Degrees of Freedom in PROC MIXED
          2. Overriding the Default
          3. Interpretation
        4. 4.4.3.4. Differences among Means
          1. Main Effect Means
          2. Interpretation
          3. Defining a Specific Treatment as a Control
          4. Simple Effects and ET × POSITION Means
          5. Interpretation
          6. Using a Control for Simple Effects
          7. Interpretation
      4. 4.4.4. Alternative Mean Comparisons Using PROC GLIMMIX
    5. 4.5. Comparison with PROC GLM
      1. Comments
      2. Least-Squares Means
      3. Comments
      4. Estimates and Contrasts
      5. Comments
    6. 4.6. Example: Type × Dose Response
      1. 4.6.1. PROC MIXED Analysis of DOSE × TYPE Main Effects and Interactions
        1. Interpretation
      2. 4.6.2. Regression Analysis over DOSE by TYPE
        1. Orthogonal Polynomial Analysis
        2. Interpretation
        3. Direct Regression Model
        4. Interpretation
        5. Interpretation
        6. Extensions
        7. The Final Fit
    7. 4.7. Example: Variance Component Estimates Equal to Zero
      1. 4.7.1. Default Analysis Using PROC MIXED
        1. Comments
      2. 4.7.2. Recommended Alternative: Analysis with NOBOUND or METHOD=TYPE3 Options
        1. Comments
      3. 4.7.3. Conceptual Alternative: Negative Variance or Correlation?
        1. Interpretation
    8. 4.8. More on PROC GLM Compared to PROC MIXED: Incomplete Blocks, Missing Data, and Estimability
    9. 4.9. Summary
  8. 5. Analysis of Repeated Measures Data
    1. 5.1. Introduction
      1. 5.1.1. Basic Concepts of Repeated Measures
      2. 5.1.2. Types of Repeated Measures Analyses
      3. 5.1.3. A Statistical Model for Repeated Measures
        1. Model
    2. 5.2. Example: Mixed Model Analysis of Data from Basic Repeated Measures Design
      1. 5.2.1. Using the REPEATED Statement in PROC MIXED
        1. Program
        2. Interpretation
        3. Interpretation
        4. Interpretation
        5. Interpretation
        6. Interpretation
      2. 5.2.2. Comparing Results from Two Covariance Structures
        1. Interpretation
        2. Interpretation
        3. Interpretation
    3. 5.3. Modeling Covariance Structure
      1. 5.3.1. Some Candidate Covariance Structures
      2. 5.3.2. Selecting an Appropriate Covariance Model
        1. 5.3.2.1. Graphical Methods
          1. Interpretation
          2. Interpretation
          3. Interpretation
        2. 5.3.2.2. Information Criteria for Comparing Covariance Structures
          1. Interpretation
          2. Interpretation
      3. 5.3.3. Reassessing the Covariance Structure with a Means Model Accounting for Baseline Measurement
      4. 5.3.4. PROC MIXED Analysis of FEV1 Data
      5. 5.3.6. Inference on Treatment and Time Effects of FEV1 Data Using PROC MIXED
        1. 5.3.6.1. Comparisons of DRUG × HOUR Means
        2. 5.3.6.2. Comparisons Using Regression
    4. 5.4. Example: Unequally Spaced Repeated Measures
      1. Model
      2. Program
      3. Results
      4. Interpretation
    5. 5.5. Summary
  9. 6. Best Linear Unbiased Prediction
    1. 6.1. Introduction
    2. 6.2. Examples of BLUP
      1. 6.2.1. Random Effects Model
      2. 6.2.2. Two-Way Mixed Model
      3. 6.2.3. A Random Coefficient Regression Model
      4. 6.2.4. A Mixed Model with Multiple Error Terms
    3. 6.3. Basic Concepts of BLUP
    4. 6.4. Example: Obtaining BLUPs in a Random Effects Model
      1. 6.4.1. Program Using the MIXED Procedure
        1. Program
        2. Results
        3. Interpretation
      2. 6.4.2. Comparison of PROC MIXED and PROC GLM Results
        1. Analysis with PROC GLM
        2. Results
        3. PROC MIXED Program to Duplicate PROC GLM Results
        4. Program
        5. Results
        6. Interpretation
      3. 6.4.3. Relationship between Sire Means and BLUPs
    5. 6.5. Example: Two-Factor Mixed Model
      1. 6.5.1. Model
      2. 6.5.2. Program to Obtain Estimates and Predictors
        1. Program
        2. Results
        3. Interpretation
      3. 6.5.3. Intermediate Inference
      4. 6.5.4. Broad Space BLUP
      5. 6.5.5. Comparison of PROC MIXED with PROC GLM
        1. Program
        2. Results
        3. Interpretation
        4. Interpretation
    6. 6.6. A Multilocation Example
      1. 6.6.1. Model Issues for Multicenter Trial
      2. 6.6.2. Analysis with Fixed Location Effects
        1. Program
        2. Interpretation
        3. Comments on Location × Treatment LS Means
      3. 6.6.3. Analysis with Random Location Effects
        1. 6.6.3.1. SAS Program: Basic Analysis and Population-Averaged Inference
          1. Program
          2. Results
          3. Interpretation
        2. 6.6.3.2. Narrow Inference Space Treatment Means
          1. Interpretation
    7. 6.7. Location-Specific Inference in Multicenter Example
      1. Obtaining BLUPs
      2. Implementation Using PROC MIXED
      3. Interpretation
      4. Interpretation
    8. 6.8. Summary
  10. 7. Analysis of Covariance
    1. 7.1. Introduction
    2. 7.2. One-Way Fixed Effects Treatment Structure with Simple Linear Regression Models
      1. 7.2.1. Comparing the Regression Models
      2. 7.2.2. Summary of an Analysis of Covariance Strategy
      3. 7.2.3. Extensions to More Complex Regression Models
    3. 7.3. Example: One-Way Treatment Structure in a Randomized Complete Block Design Structure—Equal Slopes Model
      1. 7.3.1. Step 1: Fit the Model to Test the Slopes-Equal-to-Zero Hypothesis
        1. Model
        2. Program
        3. Results
        4. Interpretation
      2. 7.3.2. Step 2: Determine If a Common Slope Model Is Adequate to Describe the Data
        1. Model
        2. Program
        3. Results
      3. 7.3.3. Step 3: Fit a Common Slope Model
        1. Model
        2. Program
        3. Results
        4. Interpretation
      4. 7.3.4. Mixed Model Estimator: A Combined Estimator
        1. Intra-block Information
        2. Program
        3. Results
        4. Interpretation
        5. Inter-block Information
        6. Program
        7. Results
        8. Interpretation
        9. Combined Estimate of the Common Slope
        10. Program
        11. Results
        12. Interpretation
    4. 7.4. Example: One-Way Treatment Structure in an Incomplete Block Design Structure—Time to Boil Water
      1. Model
      2. 7.4.1. Step 1: Fit the Model to Test the Equal Slopes Hypothesis
        1. Program
        2. Results
        3. Interpretation
      3. 7.4.2. Step 2: Fit the Unequal Slopes Model
        1. Program
        2. Results
        3. Interpretation
      4. 7.4.3. Step 3: Test for Lack of Fit of the Simple Linear Regression Model
        1. Program
        2. Results
        3. Interpretation
      5. 7.4.4. Step 4: Testing the Equality of the Models at a Preselected Value of the Covariate
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Results
        6. Interpretation
    5. 7.5. Example: One-Way Treatment Structure in a Balanced Incomplete Block Design Structure
      1. 7.5.1. Step 1: Fit the Unequal Slopes Model
        1. Model
        2. Program
        3. Results
        4. Interpretation
      2. 7.5.2. Step 2: Test the Equal Slopes Hypothesis
        1. Program
        2. Results
        3. Interpretation
      3. 7.5.3. Step 3: Fit Model with Unequal Slopes for Each Level of GRP
        1. Program
        2. Results
        3. Interpretation
    6. 7.6. Example: One-Way Treatment Structure in an Unbalanced Incomplete Block Design Structure
      1. 7.6.1. Step 1: Fit the Unequal Slopes Model and Test the Slopes-Equal-to-Zero Hypothesis
        1. Program
        2. Results
        3. Interpretation
      2. 7.6.2. Step 2: Test the Equal Slopes Hypothesis
        1. Program
        2. Results
        3. Interpretation
      3. 7.6.3. Step 3: Fit a Common Slope Model
        1. Program
        2. Results
        3. Interpretation
    7. 7.7. Example: Split-Plot Design with the Covariate Measured on the Large-Size Experimental Unit or Whole Plot
      1. 7.7.1. Step 1: Fit Unequal Slopes Model and Test Slopes-Equal-to-Zero Hypothesis
        1. Model
        2. Program
        3. Results
      2. 7.7.2. Step 2: Fit Factorial Effects Model for Intercepts and Slopes
        1. Program
        2. Results
        3. Interpretation
      3. 7.7.3. Step 3: Fit Model with Slopes Expressed as Main Effects of the Two Factors
        1. Model
        2. Program
        3. Results
        4. Interpretation
      4. 7.7.4. Step 4: Fit Model with Unequal Slopes for Each Gender
        1. Model
        2. Program
        3. Results
        4. Intepretation
        5. Program
        6. Results
        7. Interpretation
    8. 7.8. Example: Split-Plot Design with the Covariate Measured on the Small-Size Experimental Unit or Subplot
      1. Model
      2. 7.8.1. Step 1: Fit Model with Factorial Effects for Both Intercepts and Slopes
        1. Program
        2. Results
        3. Interpretation
      3. 7.8.2. Step 2: Fit the Factorial Effects for Intercepts and a Main Effects Representation for the Slopes
        1. Program
        2. Results
        3. Interpretation
      4. 7.8.3. Step 3: Fit the Factorial Effects Model for Intercepts and the Slopes as a Function of Teaching Method
        1. Program
        2. Results
        3. Interpretation
      5. 7.8.4. Step 4: Fit a Common Slope Model
        1. Model
        2. Program
        3. Results
        4. Interpretation
        5. Program
        6. Results
        7. Interpretation
        8. Interpretation
        9. Interpretation
      6. 7.8.5. Comparison with PROC GLM
        1. Program
        2. Results
        3. Interpretation
    9. 7.9. Example: Complex Strip-Plot Design with the Covariate Measured on an Intermediate-Size Experimental Unit
      1. Model
      2. Program
      3. Results
      4. Interpretation
      5. Interpretation
    10. 7.10. Summary
  11. 8. Random Coefficient Models
    1. 8.1. Introduction
    2. 8.2. Example: One-Way Random Effects Treatment Structure in a Completely Randomized Design Structure
      1. Model
      2. Program
      3. Results
      4. Interpretation
      5. Program
      6. Results
      7. Interpretation
      8. Program
      9. Results
      10. Interpretation
    3. 8.3. Example: Random Student Effects
      1. Model
      2. Program
      3. Results
      4. Interpretation
      5. Program
      6. Results
      7. Interpretation
    4. 8.4. Example: Repeated Measures Growth Study
      1. 8.4.1. Repeated Measures Analysis
        1. Program
        2. Results
        3. Interpretation
      2. 8.4.2. Random Coefficient Analysis
        1. Program
        2. Results
        3. Interpretation
        4. Results
        5. Interpretation
      3. 8.4.3. Test of Lack of Fit
        1. Program
        2. Results
        3. Interpretation
    5. 8.5. Summary
  12. 9. Heterogeneous Variance Models
    1. 9.1. Introduction
    2. 9.2. Example: Two-Way Analysis of Variance with Unequal Variances
      1. 9.2.1. Testing for Homogeneity of Variances
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Results
        6. Interpretation
        7. Program
        8. Interpretation
        9. Program
        10. Results
        11. Interpretation
        12. Program
        13. Results
        14. Interpretation
      2. 9.2.2. Fitting a Model with Heterogeneous Variances
        1. Program
        2. Results
        3. Interpretation
      3. 9.2.3. Fitting a Model with Reduced Heterogeneous Variance Structure
        1. Program
        2. Results
        3. Interpretation
    3. 9.3. Example: Simple Linear Regression Model with Unequal Variances
      1. 9.3.1. Fit Power-of-X Model Using REML
        1. Program
        2. Results
        3. Interpretation
      2. 9.3.2. Fit Power-of-X Model Using ML
        1. Program
        2. Results
        3. Interpretation
      3. 9.3.3. Fit Power-of-X Model Using ML from PROC NLMIXED
        1. Program
        2. Results
        3. Interpretation
      4. 9.3.4. Fit Power-of-the-Mean Model Using REML
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Results
        6. Interpretation
        7. Results
        8. Interpretation
      5. 9.3.5. Fit Power-of-the-Mean Model Using ML from PROC MIXED
        1. Program
        2. Results
        3. Interpretation
      6. 9.3.6. Fit Power-of-the-Mean Model Using ML from PROC NLMIXED
        1. Program
        2. Results
        3. Interpretation
      7. 9.3.7. Comparison of Power-of-X and Power-of-the-Mean Model Covariance Structures
        1. Program
        2. Results
        3. Interpretation
    4. 9.4. Example: Nested Model with Unequal Variances for a Random Effect
      1. 9.4.1. Model with Equal Variances
        1. Program
        2. Results
        3. Interpretation
      2. 9.4.2. Testing for Equal District Variances across Regions
        1. Program
        2. Results
        3. Interpretation
        4. Program
        5. Results
        6. Interpretation
      3. 9.4.3. Test for Equal Residual Variances
        1. Program
        2. Results
        3. Interpretation
      4. 9.4.4. Fitting Model with Unequal Random Effect Variances
        1. Program
        2. Results
        3. Interpretation
    5. 9.5. Example: Within-Subject Variability
      1. Program
      2. 9.5.1. Basic Unstructured Covariance Model
        1. Model
        2. Program
        3. Results
        4. Interpretation
      3. 9.5.2. Other Covariance Structures
      4. 9.5.3. Selecting a Covariance Structure
        1. Program
        2. Results
        3. Interpretation
      5. 9.5.4. Power-of-the-Mean Model
        1. Model
        2. Program
        3. Results
        4. Interpretation
        5. Program
        6. Results
        7. Interpretation
    6. 9.6. Example: Combining Between- and Within-Subject Heterogeneity
      1. Model
      2. 9.6.1. Basic Unstructured Covariance Model
        1. Program
        2. Results
        3. Interpretation
      3. 9.6.2. Incorporating Between-Subject Heterogeneity
        1. Program
        2. Results
        3. Interpretation
      4. 9.6.3. Heterogeneous Random Coefficients
        1. Program
        2. Results
        3. Program
        4. Results
        5. Interpretation
    7. 9.7. Example: Log-Linear Variance Models
      1. 9.7.1. Initial Model
        1. Program
        2. Results
        3. Interpretation
      2. 9.7.2. Full Model with Power-of-X Dispersion Effects
        1. Program
        2. Results
        3. Interpretation
      3. 9.7.3. Reduced Model with Power-of-X Dispersion Effects
        1. Program
        2. Results
        3. Interpretation
      4. 9.7.4. Model with UN Repeated Measures Error Structure
        1. Program
        2. Results
        3. Interpretation
    8. 9.8. Summary
  13. 10. Mixed Model Diagnostics
    1. 10.1. Introduction
    2. 10.2. From Linear to Linear Mixed Models
      1. 10.2.1. Residuals in the Linear Model
        1. 10.2.1.1. Studentization
        2. 10.2.1.2. Error Recovery
      2. 10.2.2. Influence Measures in the Linear Model
      3. 10.2.3. Random Effects and the Consequences of Generalized Least Squares
        1. 10.2.3.1. Marginal and Conditional Residuals
        2. 10.2.3.2. Example: A Split-Plot Design
          1. Program
          2. Results
          3. Interpretation
          4. Program
          5. Results
          6. Interpretation
        3. 10.2.3.3. Generalized Least Squares and Influence Diagnostics
    3. 10.3. The Influence Diagnostics
      1. 10.3.1. Overall Influence
      2. 10.3.2. Influence on the Parameter Estimates
      3. 10.3.3. Influence on the Precision of Estimates
      4. 10.3.4. Iterative or Non-iterative Analysis
    4. 10.4. Example: Unequally Spaced Repeated Measures
      1. 10.4.1. No Temporal Effects in Mean Structure
        1. Program
        2. Results
        3. Interpretation
      2. 10.4.2. With Temporal Effects in Mean Structure
        1. Program
        2. Program
        3. Interpretation
    5. 10.5. Summary
  14. 11. Spatial Variability
    1. 11.1. Introduction
    2. 11.2. Description
    3. 11.3. Spatial Correlation Models
      1. 11.3.1. Spatial Correlation Models Used in PROC MIXED
      2. 11.3.2. Models with a Nugget Effect
    4. 11.4. Spatial Variability and Mixed Models
      1. 11.4.1. Integrating Spatial Variability into Mixed Models
        1. Model with Nugget (G-side)
        2. Model without Nugget (R-side)
        3. Model with Nugget (R-side)
      2. 11.4.2. Geostatistics Related to PROC MIXED
    5. 11.5. Example: Estimating Spatial Covariance
      1. 11.5.1. Estimating the No-Nugget Model
        1. Program
        2. Results
        3. Interpretation
      2. 11.5.2. Likelihood Ratio Test of Spatial Covariance
      3. 11.5.3. Comparing Spatial Covariance Models
        1. Program
        2. Results
        3. Interpretation
      4. 11.5.4. Comparing Spatial Models to Nonspatial Models
        1. Program
        2. Results
        3. Interpretation
      5. 11.5.5. Estimating the Model with a Nugget Effect
        1. Program
        2. Results
        3. Interpretation
        4. Interpretation
    6. 11.6. Using Spatial Covariance for Adjustment: Part 1, Regression
      1. 11.6.1. A Regression Example
        1. Spatial Covariance Model
        2. Program
        3. Results
        4. Interpretation
      2. 11.6.2. Comparison with Independent Errors Model
        1. Different Covariance Models
    7. 11.7. Using Spatial Covariance for Adjustment: Part 2, Analysis of Variance
      1. 11.7.1. Possible Models for Wheat Yield Trial
      2. 11.7.2. Analysis Using REML Estimation of Covariance Components
        1. Results
        2. Interpretation
      3. 11.7.3. Alternative Estimate of Covariance Parameters Based on Semivariogram
      4. 11.7.4. Impact of Spatial Adjustment on Treatment Effect Inference
        1. LSMEANS of RCB Analysis Compared with Spatial Covariance Model
        2. Interpretation
    8. 11.8. Example: Spatial Prediction—Kriging
      1. 11.8.1. Ordinary Kriging
        1. Program
        2. Interpretation
        3. Program
        4. Interpretation
        5. Interpretation
      2. 11.8.2. Universal Kriging
        1. Program
        2. Interpretation
        3. Interpretation
    9. 11.9. Summary
  15. 12. Power Calculations for Mixed Models
    1. 12.1. Introduction
    2. 12.2. Power Analysis of a Pilot Study
      1. Results
      2. Results
      3. Interpretation
      4. Program
      5. Results
      6. Interpretation
      7. Program
      8. Results
      9. Interpretation
    3. 12.3. Constructing Power Curves
      1. Program
      2. Results
      3. Interpretation
    4. 12.4. Comparing Spatial Designs
      1. Program
      2. Results
      3. Interpretation
    5. 12.5. Power via Simulation
      1. Program
      2. Program
      3. Results
      4. Interpretation
      5. Program
      6. Results
      7. Interpretation
    6. 12.6. Summary
  16. 13. Some Bayesian Approaches to Mixed Models
    1. 13.1. Introduction and Background
    2. 13.2. P-Values and Some Alternatives
    3. 13.3. Bayes Factors and Posterior Probabilities of Null Hypotheses
      1. Program
      2. Results
    4. 13.4. Example: Teaching Methods
      1. Program
      2. Results
      3. Interpretation
    5. 13.5. Generating a Sample from the Posterior Distribution with the PRIOR Statement
    6. 13.6. Example: Beetle Fecundity
      1. Program
      2. Results
      3. Interpretation
      4. Program
      5. Results
      6. Interpretation
      7. Program
      8. Results
      9. Program
      10. Results
      11. Interpretation
      12. Program
      13. Results
      14. Interpretation
      15. Program
      16. Results
    7. 13.7. Summary
  17. 14. Generalized Linear Mixed Models
    1. 14.1. Introduction
    2. 14.2. Two Examples to Illustrate When Generalized Linear Mixed Models Are Needed
      1. 14.2.1. Binomial Data in a Multi-center Clinical Trial
      2. 14.2.2. Count Data in a Split-Plot Design
    3. 14.3. Generalized Linear Model Background
      1. 14.3.1. Fixed Effects Generalized Linear Models
      2. 14.3.2. Probability Distributions
        1. Binomial
        2. Poisson
        3. Normal
        4. Common Features
        5. The Exponential Family of Distributions
      3. 14.3.3. Link Functions and Variance Structure
        1. Link Functions
        2. Variance Structure
      4. 14.3.4. Predicting Means from the Inverse Link Function
      5. 14.3.5. Deviance
        1. Goodness of Fit
        2. Hypothesis Testing
      6. 14.3.6. Inference Using Estimable Functions
        1. Using the Wald Statistic for χ2 Tests
        2. F-Tests
        3. Relation between Inference Using Generalized Linear Models versus Standard Linear Models
    4. 14.4. From GLMs to GLMMs
      1. 14.4.1. Incorporating Random Effects
      2. 14.4.2. The Pseudo-likelihood Approach
        1. Generalized Mixed Model Equations
        2. Predictable Functions
        3. Wald Statistics
        4. F-Tests
      3. 14.4.3. The Role of the Scale Parameter
      4. 14.4.4. Estimation Methods in PROC GLIMMIX
    5. 14.5. Example: Binomial Data in a Multi-center Clinical Trial
      1. 14.5.1. Analysis of Binomial Data Using Logit (Canonical) Link
        1. Model
        2. Program
        3. Results
        4. Interpretation
        5. Program
        6. Results
        7. Interpretation
        8. Example: How Estimates and Means Are Obtained
        9. Interpretation
        10. Caution: How to Correctly Determine a Treatment Difference
      2. 14.5.2. Obtaining Predicted Values
        1. Program
        2. Results
        3. Interpretation
      3. 14.5.3. Alternative Link Functions
        1. Model Using Probit Link
        2. Program for Analysis Using Probit Link
        3. Results
        4. Interpretation
        5. Results—Fixed and Random Effect Solutions for Probit Model
        6. Interpretation
    6. 14.6. Example: Count Data in a Split-Plot Design
      1. 14.6.1. Standard Analysis of Poisson GLMM
        1. Model
        2. Program to Analyze Count Data
        3. Results
        4. Interpretation
        5. Program
        6. Result
        7. Interpretation
      2. 14.6.2. Analysis of Poisson GLMM with Overdispersion
        1. Program
        2. Results
        3. Interpretation
        4. Failing to Account for Whole-Plot Error
        5. Interpretation
    7. 14.7. Summary
  18. 15. Nonlinear Mixed Models
    1. 15.1. Introduction
    2. 15.2. Background on PROC NLMIXED
    3. 15.3. Example: Logistic Growth Curve Model
      1. 15.3.1. PROC NLMIXED Analysis
        1. Results
        2. Interpretation
      2. 15.3.2. %NLINMIX Macro Analysis—Method 1
        1. Program for %NLINMIX Macro—Method 1
        2. Results for NLINMIX Macro—Method 1
        3. Interpretation for NLINMIX Macro—Method 1
      3. 15.3.3. %NLINMIX Macro Analysis—Method 2
        1. Program for %NLINMIX Macro—Method 2
      4. 15.3.4. %NLINMIX Macro Analysis—Method 3, With and Without Variance Weighting
        1. Program for %NLINMIX Macro—Method 3
        2. Program for NLINMIX Macro—Method 3 with Variance Weighting
        3. Results for %NLINMIX Macro—Method 3 with Variance Weighting
        4. Interpretation for NLINMIX Macro—Method 3 with Variance Weighting
      5. 15.3.5. Choosing the Exponent for the Weight Matrix That Leads to the Best Fit
      6. 15.3.6. Comparing the Logistic Model Fitting Methods
    4. 15.4. Example: Nested Nonlinear Random Effects Models
      1. Program
      2. Results
    5. 15.5. Example: Zero-Inflated Poisson and Hurdle Poisson Models
      1. Program
      2. Results
      3. Interpretation
      4. Program
      5. Results
      6. Interpretation
      7. Program
      8. Results
      9. Interpretation
    6. 15.6. Example: Joint Survival and Longitudinal Model
      1. Program
      2. Results
      3. Interpretation
      4. Program
      5. Results
      6. Interpretation
      7. Program
      8. Results
      9. Program
      10. Results
      11. Interpretation
    7. 15.7. Example: One-Compartment Pharmacokinetic Model
      1. Program
      2. Results
      3. Interpretation
      4. Pharmacokinetic Model
      5. Program
      6. Results
      7. Interpretation
      8. Program
      9. Results
      10. Interpretation
      11. Program
      12. Results
      13. Interpretation
      14. Program
      15. Results
      16. Interpretation
      17. Program
      18. Results
      19. Interpretation
    8. 15.8. Comparison of PROC NLMIXED and the %NLINMIX Macro
    9. 15.9. Three General Fitting Methods Available in the %NLINMIX Macro
    10. 15.10. Troubleshooting Nonlinear Mixed Model Fitting
      1. 15.10.1. General Considerations
      2. 15.10.2. PROC NLMIXED Tips
        1. Starting Values
        2. Parameterizing Variances and Covariance Matrices
        3. Other Tips to Reduce Run Time and Improve Convergence
      3. 15.10.3. %NLINMIX Macro Tips
    11. 15.11. Summary
  19. 16. Case Studies
    1. 16.1. Introduction
    2. 16.2. Response Surface Experiment in a Split-Plot Design
      1. 16.2.1. Introduction
      2. 16.2.2. Analysis Using PROC MIXED
        1. Program
        2. Results
        3. Interpretation
        4. Program—Including DDFM=SATTERTH
        5. Results
        6. Interpretation
    3. 16.3. Response Surface Experiment with Repeated Measures
      1. 16.3.1. Introduction
      2. 16.3.2. Analysis Using Compound Symmetry
        1. Program
        2. Results
        3. Interpretation
      3. 16.3.3. Analysis Using Correlated Error Model
        1. Program
        2. Results
        3. Interpretation
    4. 16.4. A Split-Plot Experiment with Correlated Whole Plots
      1. 16.4.1. Introduction
      2. 16.4.2. Analysis Using PROC GLIMMIX
        1. Program
        2. Results
        3. Interpretation
      3. 16.4.3. Comparison with Standard Split-Plot ANOVA
        1. Program
        2. Results
        3. Interpretation
    5. 16.5. A Complex Split Plot: Whole Plot Conducted as an Incomplete Latin Square
      1. 16.5.1. Introduction
      2. 16.5.2. Analysis Using PROC MIXED
        1. Program
        2. Results
        3. Interpretation
        4. Program—Evaluation of Regression over N Levels by Genotype
        5. Results
        6. Interpretation
        7. Interpretation
        8. PROC GLIMMIX Program Using ODS Graphics to Plot N × G Means
        9. Results
        10. Interpretation
        11. Program—Estimate Linear Regression for Each Genotype
        12. Results
        13. Interpretation
        14. Plot of Predicted Regression Line
        15. Program
        16. Interpretation
      3. 16.5.3. Problems Encountered with Using PROC GLM
    6. 16.6. A Complex Strip-Split-Split-Plot Example
      1. 16.6.1. Introduction
      2. 16.6.2. Analysis Using PROC GLIMMIX
        1. Program
        2. Results
        3. Interpretation
        4. Program—LSMEANS
        5. Results
        6. Interpretation
        7. Results for EC × IR × V Means
        8. Interpretation
      3. 16.6.3. Standard Errors in Complex Designs
    7. 16.7. Unreplicated Split-Plot Design
      1. 16.7.1. Introduction
      2. 16.7.2. Analysis Using PROC MIXED
        1. Program
        2. Results
        3. Interpretation
        4. Program to Estimate Model Parameters
        5. Results
        6. Interpretation
        7. Final Comment
    8. 16.8. 23 Treatment Structure in a Split-Plot Design with the Three-Way Interaction as the Whole-Plot Comparison
      1. 16.8.1. Introduction
      2. 16.8.2. Analysis Using PROC MIXED and PROC GLIMMIX
        1. Program—Analysis of Variance Model
        2. Results
        3. Interpretation
        4. Program—Regression Model
        5. Results
        6. Interpretation
      3. 16.8.3. Comparison with PROC GLM
        1. Program—Naive PROC GLM
        2. Results
        3. Interpretation
        4. Program—Modified PROC GLM to Obtain Tests of Model Effects
        5. Results
        6. Interpretation
        7. Program—Modified PROC GLM to Obtain LSMEANS
        8. Results
        9. Interpretation
    9. 16.9. 23 Treatment Structure in an Incomplete Block Design Structure with Balanced Confounding
      1. 16.9.1. Introduction
      2. 16.9.2. Analysis Using PROC GLIMMIX and PROC MIXED
        1. Program—Analysis of Variance (Means) Model
        2. Results
        3. Interpretation
        4. Program—Regression Model
        5. Results
        6. Interpretation
    10. 16.10. Product Acceptability Study with Crossover and Repeated Measures
      1. 16.10.1. Introduction
      2. 16.10.2. Variable Definitions
      3. 16.10.3. Independent Error Model
        1. Program—Including PRIORPRD
        2. Results
        3. Interpretation
        4. Program—Including Li Terms
        5. Results
        6. Interpretation
        7. Interpretation
        8. Program—ODS Graphics with PROC GLIMMIX to Plot TIME Effect by PROD
      4. 16.10.4. Autoregressive Errors for the Time Interval Part of the Model
        1. Program—Autoregressive Errors
        2. Results
        3. Interpretation
        4. Program—AR(1) Time Errors and the Li Terms
        5. Results
        6. Interpretation
      5. 16.10.5. Autoregressive Errors for the Time Interval and Period Parts of the Model
        1. PROC MIXED Program—Model with PRIORPRD Effect and AR(1) Error Structure for Time Interval and Period Effects
        2. Results
        3. Interpretation
        4. Program—Replace PRIORCND with Li’s to Estimate LSMEANS
        5. Results
        6. Interpretation
      6. 16.10.6. Conclusions
    11. 16.11. Random Coefficients Modeling of an AIDS Trial
      1. 16.11.1. Introduction
      2. 16.11.2. Analysis Using PROC MIXED
        1. Program—Model Assuming Homogeneous Residual Variance Group
        2. Results
        3. Interpretation
        4. Program—Heterogeneous Residual Variance Groups
        5. Results
        6. Interpretation
        7. Program—Fit the Final Reduced “Hockey Stick” Model
        8. Results
        9. Interpretation
        10. Program—Plot Predicted Values
    12. 16.12. Microarray Example
      1. 16.12.1. Introduction
      2. 16.12.2. Analysis with PROC GLIMMIX
        1. Program
        2. Results
        3. Interpretation
  20. 1. Linear Mixed Model Theory
    1. A1.1. Introduction
    2. A1.2 . Matrix Notation
    3. A1.3 . Formulation of the Mixed Model
      1. A1.3.1. The General Linear Mixed Model
      2. A1.3.2. Conditional and Marginal Distributions
      3. A1.3.3. Example: Growth Curve with Compound Symmetry
      4. A1.3.4. Example: Split-Plot Design
    4. A1.4. Estimating Parameters, Predicting Random Effects
      1. A1.4.1. Estimating β and Predicting u: The Mixed Model Equations
      2. A1.4.2. Random Effects, Ridging, and Shrinking
      3. A1.4.3. It’s All in the Sweep
      4. A1.4.4. Maximum Likelihood and Restricted Maximum Likelihood for Covariance Parameters
        1. Maximum Likelihood (ML)
        2. Restricted Maximum Likelihood (REML)
        3. Connecting the Dots
    5. A1.5. Statistical Properties
    6. A1.6. Model Selection
      1. A1.6.1. Model Comparisons via Likelihood Ratio Tests
      2. A1.6.2. Model Comparisons via Information Criteria
    7. A1.7. Inference and Test Statistics
      1. A1.7.1. Inference about the Covariance Parameters
      2. A1.7.2. Inference about Fixed and Random Effects
  21. 2. Data Sets
    1. A2.2. Randomized Block Designs
    2. A2.3. Random Effects Models
    3. A2.4. Analyzing Multi-level and Split-Plot Designs
    4. A2.5. Analysis of Repeated Measures Data
    5. A2.6. Best Linear Unbiased Prediction
    6. A2.7. Analysis of Covariance
    7. A2.8. Random Coefficient Models
    8. A2.9. Heterogeneous Variance Models
    9. A2.10. Mixed Model Diagnostics
    10. A2.11. Spatial Variability
    11. A2.13. Some Bayesian Approaches to Mixed Models
    12. A2.14. Generalized Linear Mixed Models
    13. A2.15. Nonlinear Mixed Models
    14. A2.16. Case Studies
  22. References
  23. Books Available from SAS Press
    1. JMP® Books