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SAS for Linear Models, Fourth Edition

Book Description

This clear and comprehensive guide provides everything you need for powerful linear model analysis. Using a tutorial approach and plenty of examples, authors Ramon Littell, Walter Stroup, and Rudolf Freund lead you through methods related to analysis of variance with fixed and random effects. You will learn to use the appropriate SAS procedure for most experiment designs (including completely random, randomized blocks, and split plot) as well as factorial treatment designs and repeated measures. SAS for Linear Models, Fourth Edition, also includes analysis of covariance, multivariate linear models, and generalized linear models for non-normal data. Find inside: regression models; balanced ANOVA with both fixed- and random-effects models; unbalanced data with both fixed- and random-effects models; covariance models; generalized linear models; multivariate models; and repeated measures. New in this edition: MIXED and GENMOD procedures, updated examples, new software-related features, and other new material. This book is part of the SAS Press program.

Table of Contents

  1. Acknowledgments
  2. Chapter 1 Introduction
    1. 1.1 About This Book
    2. 1.2 Statistical Topics and SAS Procedures
  3. Chapter 2 Regression
    1. 2.1 Introduction
    2. 2.2 The REG Procedure
      1. 2.2.1 Using the REG Procedure to Fit a Model with One Independent Variable
      2. 2.2.2 The P, CLM, and CLI Options: Predicted Values and Confidence Limits
      3. 2.2.3 A Model with Several Independent Variables
      4. 2.2.4 The SS1 and SS2 Options: Two Types of Sums of Squares
      5. 2.2.5 Tests of Subsets and Linear Combinations of Coefficients
      6. 2.2.6 Fitting Restricted Models: The RESTRICT Statement and NOINT Option
      7. 2.2.7 Exact Linear Dependency
    3. 2.3 The GLM Procedure
      1. 2.3.1 Using the GLM Procedure to Fit a Linear Regression Model
      2. 2.3.2 Using the CONTRAST Statement to Test Hypotheses about Regression Parameters
      3. 2.3.3 Using the ESTIMATE Statement to Estimate Linear Combinations of Parameters 26
    4. 2.4 Statistical Background
      1. 2.4.1 Terminology and Notation
      2. 2.4.2 Partitioning the Sums of Squares
      3. 2.4.3 Hypothesis Tests and Confidence Intervals
      4. 2.4.4 Using the Generalized Inverse
  4. Chapter 3 Analysis of Variance for Balanced Data
    1. 3.1 Introduction
    2. 3.2 One- and Two-Sample Tests and Statistics
      1. 3.2.1 One-Sample Statistics
      2. 3.2.2 Two Related Samples
      3. 3.2.3 Two Independent Samples
    3. 3.3 The Comparison of Several Means: Analysis of Variance
      1. 3.3.1 Terminology and Notation
        1. 3.3.1.1 Crossed Classification and Interaction Sum of Squares
        2. 3.3.1.2 Nested Effects and Nested Sum of Squares
      2. 3.3.2 Using the ANOVA and GLM Procedures
      3. 3.3.3 Multiple Comparisons and Preplanned Comparisons
    4. 3.4 The Analysis of One-Way Classification of Data
      1. 3.4.1 Computing the ANOVA Table
      2. 3.4.2 Computing Means, Multiple Comparisons of Means, and Confidence Intervals
      3. 3.4.3 Planned Comparisons for One-Way Classification: The CONTRAST Statement
      4. 3.4.4 Linear Combinations of Model Parameters
      5. 3.4.5 Testing Several Contrasts Simultaneously
      6. 3.4.6 Orthogonal Contrasts
      7. 3.4.7 Estimating Linear Combinations of Parameters: The ESTIMATE Statement
    5. 3.5 Randomized-Blocks Designs
      1. 3.5.1 Analysis of Variance for Randomized-Blocks Design
      2. 3.5.2 Additional Multiple Comparison Methods
      3. 3.5.3 Dunnett’s Test to Compare Each Treatment to a Control
    6. 3.6 A Latin Square Design with Two Response Variables
    7. 3.7 A Two-Way Factorial Experiment
      1. 3.7.1 ANOVA for a Two-Way Factorial Experiment
      2. 3.7.2 Multiple Comparisons for a Factorial Experiment
      3. 3.7.3 Multiple Comparisons of METHOD Means by VARIETY
      4. 3.7.4 Planned Comparisons in a Two-Way Factorial Experiment
      5. 3.7.5 Simple Effect Comparisons
      6. 3.7.6 Main Effect Comparisons
      7. 3.7.7 Simultaneous Contrasts in Two-Way Classifications
      8. 3.7.8 Comparing Levels of One Factor within Subgroups of Levels of Another Factor
      9. 3.7.9 An Easier Way to Set Up CONTRAST and ESTIMATE Statements
  5. Chapter 4 Analyzing Data with Random Effects
    1. 4.1 Introduction
    2. 4.2 Nested Classifications
      1. 4.2.1 Analysis of Variance for Nested Classifications
      2. 4.2.2 Computing Variances of Means from Nested Classifications and Deriving Optimum Sampling Plans
      3. 4.2.3 Analysis of Variance for Nested Classifications: Using Expected Mean Squares to Obtain Valid Tests of Hypotheses
      4. 4.2.4 Variance Component Estimation for Nested Classifications: Analysis Using PROC MIXED
      5. 4.2.5 Additional Analysis of Nested Classifications Using PROC MIXED: Overall Mean and Best Linear Unbiased Prediction
    3. 4.3 Blocked Designs with Random Blocks
      1. 4.3.1 Random-Blocks Analysis Using PROC MIXED
      2. 4.3.2 Differences between GLM and MIXED Randomized-Complete-Blocks Analysis: Fixed versus Random Blocks
        1. 4.3.2.1 Treatment Means
        2. 4.3.2.2 Treatment Differences
    4. 4.4 The Two-Way Mixed Model
      1. 4.4.1 Analysis of Variance for the Two-Way Mixed Model: Working with Expected Mean Squares to Obtain Valid Tests
      2. 4.4.2 Standard Errors for the Two-Way Mixed Model: GLM versus MIXED
      3. 4.4.3 More on Expected Mean Squares: Determining Quadratic Forms and Null Hypotheses for Fixed Effects
    5. 4.5 A Classification with Both Crossed and Nested Effects
      1. 4.5.1 Analysis of Variance for Crossed-Nested Classification
      2. 4.5.2 Using Expected Mean Squares to Set Up Several Tests of Hypotheses for Crossed-Nested Classification
      3. 4.5.3 Satterthwaite's Formula for Approximate Degrees of Freedom
      4. 4.5.4 PROC MIXED Analysis of Crossed-Nested Classification
    6. 4.6 Split-Plot Experiments
      1. 4.6.1 A Standard Split-Plot Experiment
        1. 4.6.1.1 Analysis of Variance Using PROC GLM
        2. 4.6.1.2 Analysis with PROC MIXED
  6. Chapter 5 Unbalanced Data Analysis: Basic Methods
    1. 5.1 Introduction
    2. 5.2 Applied Concepts of Analyzing Unbalanced Data
      1. 5.2.1 ANOVA for Unbalanced Data
      2. 5.2.2 Using the CONTRAST and ESTIMATE Statements with Unbalanced Data
      3. 5.2.3 The LSMEANS Statement
      4. 5.2.4 More on Comparing Means: Other Hypotheses and Types of Sums of Squares
    3. 5.3 Issues Associated with Empty Cells
      1. 5.3.1 The Effect of Empty Cells on Types of Sums of Squares
      2. 5.3.2 The Effect of Empty Cells on CONTRAST, ESTIMATE, and LSMEANS Results
    4. 5.4 Some Problems with Unbalanced Mixed-Model Data
    5. 5.5 Using the GLM Procedure to Analyze Unbalanced Mixed-Model Data
      1. 5.5.1 Approximate F-Statistics from ANOVA Mean Squares with Unbalanced Mixed-Model Data
      2. 5.5.2 Using the CONTRAST, ESTIMATE, and LSMEANS Statements in GLM with Unbalanced Mixed-Model Data
    6. 5.6 Using the MIXED Procedure to Analyze Unbalanced Mixed-Model Data
    7. 5.7 Using the GLM and MIXED Procedures to Analyze Mixed-Model Data with Empty Cells
    8. 5.8 Summary and Conclusions about Using the GLM and MIXED Procedures to Analyze Unbalanced Mixed-Model Data
  7. Chapter 6 Understanding Linear Models Concepts
    1. 6.1 Introduction
    2. 6.2 The Dummy-Variable Model
      1. 6.2.1 The Simplest Case: A One-Way Classification
      2. 6.2.2 Parameter Estimates for a One-Way Classification
      3. 6.2.3 Using PROC GLM for Analysis of Variance
      4. 6.2.4 Estimable Functions in a One-Way Classification
    3. 6.3 Two-Way Classification: Unbalanced Data
      1. 6.3.1 General Considerations
      2. 6.3.2 Sums of Squares Computed by PROC GLM
      3. 6.3.3 Interpreting Sums of Squares in Reduction Notation
      4. 6.3.4 Interpreting Sums of Squares in -Model Notation
      5. 6.3.5 An Example of Unbalanced Two-Way Classification
      6. 6.3.6 The MEANS, LSMEANS, CONTRAST, and ESTIMATE Statements in a Two-Way Layout
      7. 6.3.7 Estimable Functions for a Two-Way Classification
        1. 6.3.7.1 The General Form of Estimable Functions
        2. 6.3.7.2 Interpreting Sums of Squares Using Estimable Functions
        3. 6.3.7.3 Estimating Estimable Functions
        4. 6.3.7.4 Interpreting LSMEANS, CONTRAST, and ESTIMATE Results Using Estimable Functions
      8. 6.3.8 Empty Cells
    4. 6.4 Mixed-Model Issues
      1. 6.4.1 Proper Error Terms
      2. 6.4.2 More on Expected Mean Squares
      3. 6.4.3 An Issue of Model Formulation Related to Expected Mean Squares
    5. 6.5 ANOVA Issues for Unbalanced Mixed Models
      1. 6.5.1 Using Expected Mean Squares to Construct Approximate F-Tests for Fixed Effects
    6. 6.6 GLS and Likelihood Methodology Mixed Model
      1. 6.6.1 An Overview of Generalized Least Squares Methodology
      2. 6.6.2 Some Practical Issues about Generalized Least Squares Methodology
  8. Chapter 7 Analysis of Covariance
    1. 7.1 Introduction
    2. 7.2 A One-Way Structure
      1. 7.2.1 Covariance Model
      2. 7.2.2 Means and Least-Squares Means
      3. 7.2.3 Contrasts
      4. 7.2.4 Multiple Covariates
    3. 7.3 Unequal Slopes
      1. 7.3.1 Testing the Heterogeneity of Slopes
      2. 7.3.2 Estimating Different Slopes
      3. 7.3.3 Testing Treatment Differences with Unequal Slopes
    4. 7.4 A Two-Way Structure without Interaction
    5. 7.5 A Two-Way Structure with Interaction
    6. 7.6 Orthogonal Polynomials and Covariance Methods
      1. 7.6.1 A 2×3 Example
      2. 7.6.2 Use of the IML ORPOL Function to Obtain Orthogonal Polynomial Contrast Coefficients
      3. 7.6.3 Use of Analysis of Covariance to Compute ANOVA and Fit Regression
  9. Chapter 8 Repeated-Measures Analysis
    1. 8.1 Introduction
    2. 8.2 The Univariate ANOVA Method for Analyzing Repeated Measures
      1. 8.2.1 Using GLM to Perform Univariate ANOVA of Repeated-Measures Data
      2. 8.2.2 The CONTRAST, ESTIMATE, and LSMEANS Statements in Univariate ANOVA of Repeated-Measures Data
    3. 8.3 Multivariate and Univariate Methods Based on Contrasts of the Repeated Measures
      1. 8.3.1 Univariate ANOVA of Repeated Measures at Each Time
      2. 8.3.2 Using the REPEATED Statement in PROC GLM to Perform Multivariate Analysis of Repeated-Measures Data
      3. 8.3.3 Univariate ANOVA of Contrasts of Repeated Measures
    4. 8.4 Mixed-Model Analysis of Repeated Measures
      1. 8.4.1 The Fixed-Effects Model and Related Considerations
      2. 8.4.2 Selecting an Appropriate Covariance Model
      3. 8.4.3 Reassessing the Covariance Structure with a Means Model Accounting for Baseline Measurement
      4. 8.4.4 Information Criteria to Compare Covariance Models
      5. 8.4.5 PROC MIXED Analysis of FEV1 Data
      6. 8.4.6 Inference on the Treatment and Time Effects of FEV1 Data Using PROC MIXED
        1. 8.4.6.1 Comparisons of DRUG*HOUR Means
        2. 8.4.6.2 Comparisons Using Regression
  10. Chapter 9 Multivariate Linear Models
    1. 9.1 Introduction
    2. 9.2 A One-Way Multivariate Analysis of Variance
    3. 9.3 Hotelling’s T2 Test
    4. 9.4 A Two-Factor Factorial
    5. 9.5 Multivariate Analysis of Covariance
    6. 9.6 Contrasts in Multivariate Analyses
    7. 9.7 Statistical Background
  11. Chapter 10 Generalized Linear Models
    1. 10.1 Introduction
    2. 10.2 The Logistic and Probit Regression Models
      1. 10.2.1 Logistic Regression: The Challenger Shuttle O-Ring Data Example
      2. 10.2.2 Using the Inverse Link to Get the Predicted Probability
      3. 10.2.3 Alternative Logistic Regression Analysis Using 0-1 Data
      4. 10.2.4 An Alternative Link: Probit Regression
    3. 10.3 Binomial Models for Analysis of Variance and Analysis of Covariance
      1. 10.3.1 Logistic ANOVA
      2. 10.3.2 The Analysis-of-Variance Model with a Probit Link
      3. 10.3.3 Logistic Analysis of Covariance
    4. 10.4 Count Data and Overdispersion
      1. 10.4.1 An Insect Count Example
      2. 10.4.2 Model Checking
      3. 10.4.3 Correction for Overdispersion
      4. 10.4.4 Fitting a Negative Binomial Model
      5. 10.4.5 Using PROC GENMOD to Fit the Negative Binomial with a Log Link
      6. 10.4.6 Fitting the Negative Binomial with a Canonical Link
      7. 10.4.7 Advanced Application: A User-Supplied Program to Fit the Negative Binomial with a Canonical Link
    5. 10.5 Generalized Linear Models with Repeated Measures—Generalized Estimating Equations
      1. 10.5.1 A Poisson Repeated-Measures Example
      2. 10.5.2 Using PROC GENMOD to Compute a GEE Analysis of Repeated Measures
    6. 10.6 Background Theory
      1. 10.6.1 The Generalized Linear Model Defined
      2. 10.6.2 How the GzLM’s Parameters Are Estimated
      3. 10.6.3 Standard Errors and Test Statistics
      4. 10.6.4 Quasi-Likelihood
      5. 10.6.5 Repeated Measures and Generalized Estimating Equations
  12. Chapter 11 Examples of Special Applications
    1. 11.1 Introduction
    2. 11.2 Confounding in a Factorial Experiment
      1. 11.2.1 Confounding with Blocks
      2. 11.2.2 A Fractional Factorial Example
    3. 11.3 A Balanced Incomplete-Blocks Design
    4. 11.4 A Crossover Design with Residual Effects
    5. 11.5 Models for Experiments with Qualitative and Quantitative Variables
    6. 11.6 A Lack-of-Fit Analysis
    7. 11.7 An Unbalanced Nested Structure
    8. 11.8 An Analysis of Multi-Location Data
      1. 11.8.1 An Analysis Assuming No Location?Treatment Interaction
      2. 11.8.2 A Fixed-Location Analysis with an Interaction
      3. 11.8.3 A Random-Location Analysis
      4. 11.8.4 Further Analysis of a Location?Treatment Interaction Using a Location Index
    9. 11.9 Absorbing Nesting Effects
  13. References
  14. Index