SAS for Mixed Models: Introduction and Basic Applications

✍ Scribed by Walter W. Stroup; George A. Milliken; Elizabeth A. Claassen Claassen

• Publisher: SAS Institute
• Year: 2018
• Tongue: English
• Leaves: 608
• Category: Library

✦ Synopsis

Discover the power of mixed models with SAS. Mixed models-now the mainstream vehicle for analyzing most research data-are part of the core curriculum in most master's degree programs in statistics and data science. In a single volume, this book updates both SAS(R) for Linear Models, Fourth Edition, and SAS(R) for Mixed Models, Second Edition, covering the latest capabilities for a variety of applications featuring the SAS GLIMMIX and MIXED procedures. Written for instructors of statistics, graduate students, scientists, statisticians in business or government, and other decision makers, SAS(R) for Mixed Models is the perfect entry for those with a background in two-way analysis of variance, regression, and intermediate-level use of SAS. This book expands coverage of mixed models for non-normal data and mixed-model-based precision and power analysis, including the following topics: Random-effect-only and random-coefficients models Multilevel, split-plot, multilocation, and repeated measures models Hierarchical models with nested random effects Analysis of covariance models Generalized linear mixed models This book is part of the SAS Press program.

✦ Table of Contents

Contents
About This Book
What Does This Book Cover?
What’s New in This Edition?
Is This Book for You?
What Should You Know about the Examples?
Software Used to Develop the Book's Content
Example Code and Data
Output and Figures
SAS University Edition
SAS Press Wants to Hear from You
Dedication and Acknowledgments
Chapter 1: Mixed Model Basics
1.1 Introduction
1.2 Statistical Models
1.3 Forms of Linear Predictors
1.4 Fixed and Random Effects
1.5 Mixed Models
1.6 Typical Studies and Modeling Issues That Arise
1.7 A Typology for Mixed Models
1.8 Flowcharts to Select SAS Software to Run Various Mixed Models
Chapter 2: Design Structure I: Single Random Effect
2.1 Introduction
2.2 Mixed Model for a Randomized Block Design
2.3 The MIXED and GLIMMIX Procedures to Analyze RCBD Data
2.4 Unbalanced Two-Way Mixed Model: Examples with Incomplete Block Design
2.5 Analysis with a Negative Block Variance Estimate: An Example
2.6 Introduction to Mixed Model Theory
2.7 Summary
Chapter 3: Mean Comparisons for Fixed Effects
3.1 Introduction
3.2 Comparison of Two Treatments
3.3 Comparison of Several Means: Analysis of Variance
3.4 Comparison of Quantitative Factors: Polynomial Regression
3.5 Mean Comparisons in Factorial Designs
3.6 Summary
Chapter 4: Power, Precision, and Sample Size I: Basic Concepts
4.1 Introduction
4.2 Understanding Essential Background for Mixed Model Power and Precision
4.3 Computing Precision and Power for CRD: An Example
4.4 Comparing Competing Designs I—CRD versus RCBD: An Example
4.5 Comparing Competing Designs II—Complete versus Incomplete Block Designs: An Example
4.6 Using Simulation for Precision and Power
4.7 Summary
Chapter 5: Design Structure II: Models with Multiple Random Effects
5.1 Introduction
5.2 Treatment and Experiment Structure and Associated Models
5.2.1 Essential Terminology: Components of Treatment and Design Structure
5.2.2 Possible Design Structures for 2 × 2 Factorial Treatment Design
Figure 5.1: Visualization of Treatment Codes in 2 × 2 Factorial Experiment
Completely Randomized Design (CRD)
Figure 5.2: Completely Randomized Design
Randomized Complete Block
Figure 5.3: Randomized Complete Block Design
Row and Column Designs (Latin Square)
Figure 5.4: Latin Square Design
Split-Plot, Variation 1—Whole-Plot as CRD
Figure 5.5: Split-Plot Variation 1– Whole Plot as CRD
Split-Plot Variation 2—Whole Plot Conducted as RCBD
Figure 5.6: Split-Plot Variation 2—Whole-Plot as RCBD
Table 5.1: Sources of Variation for Split-Plot with Whole Plot as RCBD
Strip-Split-Plot
Figure 5.7: Strip-Plot Design
Table 5.2: Sources of Variation for a 2 × 2 Factorial Experiment Conducted as a Strip-Split-Plot
Split-Plot with Whole Plot Conducted as a Replicated 2 × 2 Latin Square
Figure 5.8: Split-Plot Variation 3—Whole-Plot Blocked on Row and Column
Table 5.3. Sources of Variation for Split-Plot with Whole Plot Conducted as a Latin Square
5.2.3 A Final Note on the Design Structures
5.2.4 Determination of the Appropriate Mixed Model for a Given Layout
Table 5.4: Model–Design Association, by Figure Number
Table 5.5: CLASS, MODEL, and RANDOM Statements for Each Design Shown in Figures 5.2 through 5.8
5.3 Inference with Factorial Treatment Designs with Various Mixed Models
5.3.1 Standard Errors
5.3.2 Variance of Treatment Mean and Difference Estimates
5.3.3 Completing the Standard Error: Variance Component Estimates and Degrees of Freedom
Table 5.6 Analysis of Variance for Layout in Figure 5.5
5.4 A Split-Plot Semiconductor Experiment: An Example
5.4.1 Tests of Interest in the Semiconductor Experiment
Table 5.7: Analysis of Variance for Semiconductor Example
5.4.2 Matrix Generalization of Mixed Model F Tests
Table 5.8: Degrees of Freedom for Semiconductor Example
5.4.3 PROC GLIMMIX Analysis of Semiconductor Data
Program
Program 5.1
Results
Output 5.1: Basic PROC GLIMMIX Model Fitting Output for Semiconductor Data
Interpretation
Program
Program 5.2
Output 5.2: Analysis of Variance Using PROC MIXED for Semiconductor Data
Visualization of Treatment Means
Figure 5.9: Interaction Plot for Semiconductor Split Plot Data
Interpretation
Formal Decomposition of the ET × POSITION Interaction
Program
Program 5.3
Results
Output 5.3: Contrasts to Decompose ET*POSITION Interaction
Interpretation
Inference on Main Effect Means
Output 5.4: Semiconductor Data Least Squares Means Using PROC GLIMMIX Default
Degrees of Freedom
Default Degrees of Freedom
Override of the Default
Output 5.5: Least Squares Means for POS Using Specified Degrees of Freedom
Interpretation
Differences among Means
Main Effect Means
Program 5.4
Output 5.6: Comparison of POSITION Least Squares Means Using LSMEANS ( / DIFF), ESTIMATE, and CONTRAST Statements
Interpretation
Definition of a Specific Treatment as a Control
Output 5.7: Dunnett Test of Position 3 versus Other Positions in Semiconductor Data
5.5 A Brief Comment about PROC GLM
5.6 Type × Dose Response: An Example
5.6.1 PROC GLIMMIX Analysis of DOSE and TYPE Effects
Program 5.5
Figure 5.10: Interaction Plot Produced by PROC GLIMMIX MEANPLOT Option
Output 5.8 Variance Estimates, Main Effect, and Interaction Tests for Variety Evaluation
Output 5.9 Test of SLICEs for Variety Evaluation Data
5.6.2 A Closer Look at the Interaction Plot
Program 5.6
Figure 5.11: Interaction Plot: Means over DOSE by TYPE
5.6.3 Regression Analysis over DOSE by TYPE
Program
Program 5.7
Results
Output 5.10: Orthogonal Polynomial Results for DOSE Effect in Variety Evaluation Data
Interpretation
Program
Program 5.8
Results
Output 5.11: Orthogonal Polynomial Results for Log-Dose in Variety Evaluation Data
Program
Program 5.9
Results
Output 5.12: Order of Polynomial Regression Fit by TYPE
Direct Regression Model and Program to Estimate It
Program
Program 5.10
Results
Output 5.13: Regression of LOGDOSE on Y by TYPE Using Direct Regression
Interpretation
Estimate Statements to Add to Program 5.10
Results
Output 5.14: Estimates of Regression Model by TYPE
A Better Way to Estimate the Regression Equation
Program
Program 5.11
Results
Output 5.15: Direct Regression by TYPE Using Nested Model
Interpretation
Extensions
The Final Fit
Program
Program 5.12
Results
Output 5.16 Final Regression Model Parameter Estimates for Variety Evaluation Data
5.7 Variance Component Estimates Equal to Zero: An Example
5.7.1 Default Analysis Using PROC GLIMMIX
Program 5.13
Output 5.17 Standard PROC GLIMMIX Analysis of Mouse Data
Output 5.18: Analysis of Variance for Mouse Data—PROC MIXED
5.7.2 One Recommended Alternative: Override Set-to-Zero Default Using NOBOUND or METHOD=TYPE3
Output 5.19: NOBOUND and METHOD=TYPE3 Results Overriding Set-to-Zero Default
NOBOUND
5.7.3 Conceptual Alternative: Negative Variance or Correlation?
Programs
Program 5.14
Program 5.15
Results
Output 5.20: Compound Symmetry Analysis of Mouse Data
Interpretation
5.8 A Note on PROC GLM Compared to PROC GLIMMIX and PROC MIXED: Incomplete Blocks, Missing Data, and Spurious Non-Estimability
Program 5.16
Output 5.21: PROC GLM Output of Least Squares Means for Mouse Data
5.9 Summary
Chapter 6: Random Effects Models
6.1 Introduction: Descriptions of Random Effects Models
6.2 One-Way Random Effects Treatment Structure: Influent Example
6.3 A Simple Conditional Hierarchical Linear Model: An Example
6.4 Three-Level Nested Design Structure: An Example
6.5 A Two-Way Random Effects Treatment Structure to Estimate Heritability: An Example
6.6 Modern ANOVA with Variance Components
6.7 Summary
Chapter 7: Analysis of Covariance
7.1 Introduction
7.2 One-Way Fixed Effects Treatment Structure with Simple Linear Regression Models
7.3 One-Way Treatment Structure in an RCB Design Structure—Equal Slopes Model: An Example
7.4 One-Way Treatment Structure in an Incomplete Block Design Structure: An Example
7.5 One-Way Treatment Structure in a BIB Design Structure: An Example
7.6 One-Way Treatment Structure in an Unbalanced Incomplete Block Design Structure: An Example
7.7 Multilevel or Split-Plot Design with the Covariate Measured on the Large-Size Experimental Unit or Whole Plot: An Example
7.8 Summary
Chapter 8: Analysis of Repeated Measures Data
8.1 Introduction
8.2 Mixed Model Analysis of Data from Basic Repeated Measures Design: An Example
8.3 Covariance Structures
8.4 PROC GLIMMIX Analysis of FEV1 Data
8.5 Unequally Spaced Repeated Measures: An Example
8.6 Summary
Chapter 9: Best Linear Unbiased Prediction (BLUP) and Inference on Random Effects
9.1 Introduction
9.2 Examples Motivating BLUP
9.3 Obtainment of BLUPs in the Breeding Random Effects Model
9.4 Machine-Operator Two-Factor Mixed Model
9.5 A Multilocation Example
9.6 Matrix Notation for BLUP
9.7 Summary
Chapter 10: Random Coefficient Models
10.1 Introduction
10.2 One-Way Random Effects Treatment Structure in a Completely Randomized Design Structure: An Example
10.3 Random Student Effects: An Example
10.4 Repeated Measures Growth Study: An Example
10.5 Prediction of the Shelf Life of a Product
10.6 Summary
Chapter 11: Generalized Linear Mixed Models for Binomial Data
11.1 Introduction
11.2 Three Examples of Generalized Linear Mixed Models for Binomial Data
11.3 Example 1: Binomial O-Ring Data
11.4 Generalized Linear Model Background
11.5 Example 2: Binomial Data in a Multicenter Clinical Trial
11.6 Example 3: Binary Data from a Dairy Cattle Breeding Trial
11.7 Summary
Chapter 12: Generalized Linear Mixed Models for Count Data
12.1 Introduction
12.2 Three Examples Illustrating Generalized Linear Mixed Models with Count Data
12.3 Overview of Modeling Considerations for Count Data
12.4 Example 1: Completely Random Design with Count Data
12.5 Example 2: Count Data from an Incomplete Block Design
12.6 Example 3: Linear Regression with a Discrete Count Dependent Variable
12.7 Blocked Design Revisited: What to Do When Block Variance Estimate is Negative
12.8 Summary
Chapter 13: Generalized Linear Mixed Models for Multilevel and Repeated Measures Experiments
13.1 Introduction
13.2 Two Examples Illustrating Generalized Linear Mixed Models with Complex Data
13.3 Example 1: Split-Plot Experiment with Count Data
13.4 Example 2: Repeated Measures Experiment with Binomial Data
Chapter 14: Power, Precision, and Sample Size II: General Approaches
14.1 Introduction
14.2 Split Plot Example Suggesting the Need for a Follow-Up Study
Program 14.1
Output 14.1: Type III Tests of Fixed Effects, Means, and Differences from Split Plot Data
14.3 Precision and Power Analysis for Planning a Split-Plot Experiment
Programs
Program 14.2
Program 14.3
Results
Output 14.2 Precision Analysis: Simple Effects from Split Plot with 24 blocks
Interpretation
Program
Program 14.4
Results
Output 14.3: Power for Split Plot Experiment with 24 Blocks
Interpretation
Results
Output 14.4 Power for Split Plot with 62 Blocks
Interpretation
14.4 Use of Mixed Model Methods to Compare Two Proposed Designs
Programs for Two Designs
Program 14.7
Results
Output 14.5: Precision Analysis of Design 14.4.1 – Balanced Incomplete Block
Output 14.6: Precision Analysis for Design 14.4.2—Split Plot
Interpretation
14.5 Precision and Power Analysis: A Repeated Measures Example
14.5.1 Using Pilot Data to Inform Precision and Power Analysis
Program
Program 14.8
Results
Output 14.7 Information Criteria for Three Covariance Models from Pilot Repeated Measures
Interpretation
Program
Program 14.9
Results
Output 14.8: Covariance and Mean Estimates for Repeated Measures Pilot Data
Interpretation
14.5.2 Repeated Measures Precision and Power Analysis
Programs
Program 14.10
Program 14.11
Results
Output 14.9: Precision and Power Results for Study with 48 Subjects per Treatment
Interpretation
Program
Program 14.12
Results
Output 14.10 Power Analysis: Repeated Measures Experiment with 84 Subjects per Treatment
Interpretation
14.5.3 Final Thoughts on Repeated Measures Power Analysis
What to Do If You Have No Pilot Data
A Caveat
14.6 Precision and Power Analysis for Non-Gaussian Data: A Binomial Example
14.6.1 A Simplistic Approach
Program 14.13
Output 14.11: PROC POWER Listing: (Naive) Required Sample Size for Binomial Example
14.6.2 Precision and Power: Blocked Design, Binomial Response
Programs
Program 14.14
Program 14.15
Results
Output 14.12: Power for Test to Compare Binomial Probabilities, 5 Locations, 250 Subjects
Interpretation
Program
Program 14.16
Results
Output 14.13: Power for Test to Compare Binomial Probabilities, 16 Locations, 250 Subjects
Table 14.1 Tradeoff between Number of Subjects, Number of Locations and Available Power
14.7 Precision and Power: Example with Incomplete Blocks and Count Data
Programs
Program 14.17
Program 14.18
Program 14.19
Results
Output 14.14: Precision and Power Analysis for Incomplete Block Design with Count Data
Interpretation
14.8 Summary
Chapter 15: Mixed Model Troubleshooting and Diagnostics
Appendix A: Linear Mixed Model Theory
A.1 Introduction
A.2 Matrix Notation
A.3 Formulation of the Mixed Model
A.3.1 The General Linear Mixed Model
A.3.2 Conditional and Marginal Distributions
A.3.3 Example: Growth Curve with Compound Symmetry
A.3.4 Example: Split-Plot Design
A.4 Estimating Parameters, Predicting Random Effects
A.4.1 Estimating  and Predicting u: The Mixed Model Equations
A.4.2 Random Effects, Ridging, and Shrinking
A.4.3 Use of the Sweep Operation for Solutions
A.4.4 Maximum Likelihood and Restricted Maximum Likelihood for Covariance Parameters
Maximum Likelihood (ML)
Restricted Maximum Likelihood (REML)
Final Connections
A.5 Statistical Properties
A.6 Model Selection
A.6.1 Model Comparisons via Likelihood Ratio Tests
A.6.2 Model Comparisons via Information Criteria
A.7 Inference and Test Statistics
A.7.1 Inference about the Covariance Parameters
A.7.2 Inference about Fixed and Random Effects
Appendix B: Generalized Linear Mixed Model Theory
B.1 Introduction
B.2 Formulation of the Generalized Linear Model
B.2.1 Essential Background
Table B.1: Log-likelihood Parameters of Gaussian, Binomial and Poisson
B.2.2 Required Elements of the Generalized Linear Model
B.2.3 Estimating Equations for the Generalized Linear Model
B.2.4 Quasi-Likelihood
B.3 Formulation of the Generalized Linear Mixed Model
B.3.1 Pseudo-Likelihood Estimating Equations
B.3.2 Inference about Fixed and Random Effects
B.4 Conditional versus Marginal Models and Inference Space
Figure B.1: Plot of Block Effect Density Function
Figure B.2: Conditional Density Function of Observations, Given Block Effect
Figure B.3: Marginal Density Function of Observations
Interpretation
A Final Word about a Pervasive Misconception
B.5 Integral Approximation
B.5.1 Adaptive Quadrature
B.5.2 Laplace Approximation
B.5.3 Integral Approximation or Pseudo-Likelihood: Pros and Cons
Integral Approximation
Pseudo-Likelihood
References
Index
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