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Tables of Contents for Methods and Applications of Linear Models
Chapter/Section Title
Page #
Page Count
Data Tables and Associated Numerical Examples
xxi
PART I INTRODUCTION AND BASIC THEORY
Introduction to Linear Models
3
36
Introduction and Motivation
3
2
Definition of the Linear Model
5
7
The Cell Means Model
12
2
Regression Models
14
8
The Single-Variable Regression Model
14
5
Regression Models with Several Inputs
19
3
Analysis of Variance Models
22
11
Fixed Effects Models
23
6
Constraints on the Cell Means
29
2
Mixed Models
31
2
Additional Topics
33
2
Designed Experiments and Observational Studies
33
1
Discrete Response Variables
34
1
Multivariate Linear Models
34
1
Concluding Comments
35
4
Exercises
36
3
The Distribution of Linear and Quadratic Forms
39
26
Linear and Quadratic Forms
39
3
Properties of the Multivariate Normal Distribution
42
3
The Distribution of Quadratic Forms
45
12
The Noncentral Chi-Squared Distribution
45
4
Fundamental Theorems
49
8
Other Noncentral Distributions
57
8
The Noncentral t Distribution
57
1
The Noncentral F Distribution
58
1
Extensions
59
1
Exercises
60
5
Estimation and Inference in Simple Linear Models
65
38
Estimation of Parameters
65
17
Estimation for the Unconstrained Model
66
4
Estimation for the Constrained Model
70
7
Best Linear Unbiased Estimation (BLUE) of &theta
77
1
A Partitioned Form of the Model
78
2
Reparameterized Models
80
2
Tests of Linear Hypotheses on θ
82
13
The Unconstrained Model
81
12
The Constrained Model
93
2
Confidence Regions and Intervals
95
8
Confidence Regions
95
1
Confidence Intervals
96
2
Exercises
98
5
Simultaneous Inference: Tests and Confidence Intervals
103
28
Simultaneous Tests
103
21
Simultaneous Tests: General Methods
104
13
Simultaneous Tests: Cell Means Models
117
7
Simultaneous Confidence Intervals
124
7
Exercises
126
5
PART II REGRESSION MODELS
Regression on Functions of One Variable
131
52
The Simple Linear Regression Model
131
5
Estimation, Inference, and Prediction
136
14
Estimation
136
5
Hypothesis Testing and Confidence Intervals
141
6
Prediction and Prediction Intervals
147
3
Examining the Data and the Model
150
15
Residuals
151
4
Outliers, Extreme Points, and Influence
155
6
Normality, Independence, and Variance Homogeneity
161
4
A Test for Lack of Fit
165
3
Polynomial Regression Models
168
15
The Quadratic Model
169
5
Higher-Ordered Polynomial Models
174
2
Orthogonal Polynomials
176
1
Regression through the Origin
177
1
Exercises
178
5
Transforming the Data and Miscellaneous Topics
183
24
The Need for Transformations
183
1
Weighted Least Squares
184
3
Variance Stabilizing Transformations
187
1
Transformations to Achieve a Linear Model
188
7
Transforming the Dependent Variable
189
5
Trnasfroming the Predictors
194
1
Analysis of the Trnasformed Model
195
1
Transformations with Forbes Data
196
3
Other Topics
199
8
Random Inputs
200
1
Errors in the Inputs
201
1
Calibration
202
1
Exercises
203
4
Regression on Functions of Several Variables
207
50
The Multiple Linear Regression Model
207
1
Preliminary Data Analysis
208
5
Fitting the Multiple Linear Regression Model
213
11
Fitting the Original Data
213
7
Transforming the Response
220
4
Partial Correlation and Added-Variable Plots
224
8
Partial Correlation
224
2
Added-Variable Plots
226
4
Simple and Partial Correlations
230
2
Variable Selection
232
12
The Case of Orthogonal Predictors
232
4
Nonorthogonal Predictors
236
4
Selection Methods
240
4
Model Specification
244
13
Exercises
252
5
Collinearity in Multiple Linear Regression
257
42
The Collinearity Problem
257
8
Introduction
257
1
A Simple Example
258
4
The Picket Fence
262
2
Rotation of Coordinates
264
1
A Typical Example
265
7
Preliminary Data Analysis
266
2
Initial Regression Analysis
268
4
Collinearity Diagnostics
272
12
Variance Inflation Factors
272
4
Eigenvalues, Eigenvectors, and Principal Component Plots
276
8
Remedial Solutions: Biased Estimators
284
15
Variable Deletion
285
2
Regression on Principal Components
287
9
Ridge Regression
296
1
Exercises
296
3
Influential Observations in Multiple Linear Regression
299
38
The Influential Data Problem
299
2
The Hat Matrix
301
5
The Centered and Uncentered Hat Matrices
301
1
Properties of the Hat Matrices
302
4
The effects of Deleting Observations
306
4
Numerical Measures of Influence
310
5
The Diagonal Elements of the Hat Matrix
310
1
Residuals
310
2
The Mean Square Ratio
312
1
Cook's Distance
313
1
Other Indicators of Influential Data
314
1
The Dilemma Data
315
5
Principal Component Plots for Identifying Unusual Cases
320
9
The Projection Ellipse and Principal Component Plots
320
3
The Augmented Hat Matrix
323
3
Multiple Extremes: The Masking Problem
326
3
Robust/Resistant Methods in Regression Analysis
329
8
M-Estimation
329
1
Interative, Reweighted Least Squares
330
2
Regression with Bounded Influence
332
1
Exercises
333
4
Polynomial Models and Models with Qualitative Predictors
337
44
Polynomial Models
337
6
The Quadratic Model with Two Predictors
337
2
A Discussion of Quadratic Surfaces
339
4
The Analysis of Response Surfaces
343
5
Analysis with First-Order Models
343
3
Analysis with Second-Order Models
346
2
Models with Qualitative Predictors
348
33
Indicator Variables to Identify Groups of Data
349
17
Indicator Variables to Fit Segmented Polynomials
366
8
Exercises
374
7
Related Topics
381
20
Introduction
381
1
Nonlinear Regression Models
382
6
Some Linearizeable Functions
383
2
The Modified Gauss-Newton Method
385
3
Nonparametric Model-Fitting Methods
388
5
Locally Weighted-Average Predictors
388
3
Projection Pursuit Regression
391
2
Logistic Regression
393
8
Exercises
396
5
PART III ANALYSIS OF VARIANCE MODELS
Fixed Effects Models: I. Single-Factor Classification of Means
401
32
Concepts and Terminology
401
3
The One-Way Classification Model: Balanced Data
404
18
Parameter Estimation
405
1
The Hypothesis of Equal Means
406
4
Simultaneous Inferences about the Population Means
410
3
Simultaneous Acceptance and Confidence Ellipses
413
4
Orthogonal Contrasts
417
2
Reparameterizations of the One-Way Model
419
3
The One-Way Classification Model: Unbalanced Data
422
4
The Analysis of Covariance
426
7
Exercises
428
5
Fixed Effects Models: II. Two-Way Cross-Classification
433
62
The Unconstrained Model: Balanced Data
433
23
Parameter Estimation
435
1
Tests of Hypotheses
436
9
Simultaneous Inference
445
2
Reparameterization of the Two-Factor Model
447
6
Experiments with One Observation per Cell: A Test for Interaction
453
3
The Unconstrained Model: Unbalanced Data
456
11
Discussion in Terms of the Cell Means Model
456
3
The Reparameterized Model
459
8
The No-Inrteraction Model: Balanced Data
467
7
Parameter Estimation
468
1
Tests of Hypotheses
469
3
Simultaneous Inference
472
1
Reparameterization in the No-Interaction Model
472
2
The No-Interaction Model: Unbalanced Data
474
9
Missing Cells: Estimation
474
5
Missing Cells: Testing Hypotheses
479
3
Connected Designs
482
1
Nonhomogeneous Experimental Units: The Concept of Blocking
483
12
A Model for the Randomized, Complete Block Design
484
1
Inferences on Parameters
485
3
Exercises
488
7
Fixed Effects Models: III. Nested Factors and General Structure
495
30
The Two-Fold Nested Model
495
7
The Analysis with Balanced Data
495
4
The Analysis with Unbalanced Data
499
3
The Three-Factor Cross-Classified Model
502
5
A Three-Factor Nested-Factorial Model
507
4
The Analysis with Balanced Data
507
3
The Analysis with Unbalanced Data
510
1
A General Structure for Balanced, Fixed-Effects Models
511
14
The General Factorial Model
511
4
Nested-Factorial Models
515
4
Exercises
519
6
Mixed Effects Models: I. The AOV with Balanced Data
525
52
Introduction
525
2
Examples of Model Formulation
527
14
A General Structure for Balanced Mixed Models
541
2
The Random Model
542
1
The Mixed Model
542
1
The Mixed Model Analysis
543
12
Estimation of Variance Components: The AOV Method
543
5
Estimation of the Fixed Effects Parameters
548
1
Properties of the Estimators and Inferential Results
549
6
Numerical Examples
555
11
Alternative Developments of Mixed Models
566
11
The Scheffe Mixed Model
570
1
A Randomization Theory
571
2
Exercises
573
4
Mixed Effects Models: II. The AVE Method
577
42
Introduction
577
1
An Introduction to the AVE Method
578
15
The AVE Method for the Two-Factor Cross-Classification Model
578
8
Numerical Examples for the Two-Factor Model
586
7
The General AVE Method
593
7
The AVE Method: Factorial Models
593
3
The Ave Method: Nested and Nested-Factorial Models
596
2
The AVE Method: Computational Form of the VCE Table
598
1
The AVE Method: For General Mixed Effects Models
599
1
Numerical Examples
600
11
Properties of AVE Estimates
611
8
Diagnostic Analysis for the Two-Way Classification Model
612
2
Confidence Intervals
614
1
Exercises
614
5
Mixed Effects Models: III. Unbalanced Data
619
48
Introduction
619
2
Parameter Estimation: Likelihood Methods
621
9
Maximum Likelihood Estimation (ML)
621
3
Restricted Maximum Likelihood Estimation (REML)
624
4
A Numerical Illustration of the Methods
628
2
ML and REML Estimates with Balanced Data
630
4
ML Estimation with Balanced Data
630
3
REML Estimation with Balanced Data
633
1
An EM Algorithm for REML Estimation
634
15
A Review of the EM Algorithm
634
1
The EM Algorithm Applied to REML Estimation
635
6
Estimation of Fixed Effects
641
1
Inferences on Variance Components and Fixed Effects
642
4
Numerical Examples that Illustrate the EM Algorithm
646
3
The EM Algorithm Applied to the AVE Method
649
9
The Computational Form of the EM-AVE Method
650
1
Numerical Examples of the EM-AVE Method
651
7
Models with Covariates
658
5
Development of the Analysis
659
1
A Numerical Example
660
3
Summary
663
4
Exercises
663
4
Appendix A Mathematical Facts
667
20
A.I Matrix Algebra
667
16
A.II Optimization
683
4
Appendix B Statistical Facts
687
4
B.I Estimation
687
2
B.II Tests of Hypotheses and Confidence Regions
689
2
Appendix C Statistical Tables
691
6
Appendix D Data for Examples and Exercises
697
16
D.I Downloading Data Set Files from ftp Server
697
1
D.II Listing of Data Set Files
698
15
References
713
14
Index
727