Contents
preface PART FOUR
DESIGN AND ANALYSIS OF
SINGLE-FACTOR STUDIES 641
Chapter 15
Introduction to the Design of
Experimental and Observational
Studies 642
15.1 Experimental Studies, Observational
Studies, and Causation 643
Experimental Studies 643
Observational Studies 644
Mixed Experimental and Observational
Studies 646
15.2 Experimental Studies: Basic
Concepts 647
xx Contents
Factors 647
Crossed and Nested Factors 648
Treatments 649
Choice of Treatments 649
Experimental Units 652
Sample Size and Replication 652
Randomization 653
Constrained Randomization:
Blocking 655
Measurements 658
15.3 An Overview of Standard Experimental
Designs 658
Completely Randomized Design 659
Factorial Experiments 660
Randomized Complete Block
Designs 661
Nested Designs 662
Repeated Measures Designs 663
Incomplete Block Designs 664
Two-Level Factorial and Fractional
Factorial Experiments 665
Response Surface Experiments 666
15.4 Design of Observational Studies 666
Cross-Sectional Studies 666
Prospective Studies 667
Retrospective Studies 667
Matching 668
15.5 Case Study: Paired-Comparison
Experiment 669
15.6 Concluding Remarks 672
Cited References 672
Problems 672
Exercise 676
Chapter 16
Single-Factor Studies 677
16.1 Single-Factor Experimental and
Observational Studies 677
16.2 Relation between Regression and
Analysis of Variance 679
Illustrations 679
Choice between Two Types of Models 680
16.3 Single-Factor ANOVA Model 681
Basic Ideas 681
Cell Means Model 681
Important Features of Model 682
The ANOVA Model Is a Linear
Model 683
Interpretation of Factor Level Means 684
Distinction between ANOVA Models I
and II 685
16.4 Fitting of ANOVA Model 685
Notation 686
Least Squares and Maximum Likelihood
Estimators 687
Residuals 689
16.5 Analysis of Variance 690
Partitioning of SSTO 690
Breakdown of Degrees of Freedom 693
Mean Squares 693
Analysis of Variance Table 694
Expected Mean Squares 694
16.6 F Test for Equality of Factor Level
Means 698
Test Statistic 698
Distribution of F* 699
Construction of Decision Rule 699
16.7 Alternative Formulation of Model 701
Factor Effects Model 701
Definition of µ. 702
Test for Equality of Factor Level
Means 704
16.8 Regression Approach to Single-Factor
Analysis of Variance 704
Factor Effects Model with Unweighted
Mean 705
Factor Effects Model with Weighted
Mean 709
Cell Means Model 710
16.9 Randomization Tests 712
16.10 Planning of Sample Sizes with Power
Approach 716
Power of F Test 716
Use of Table B.12 for Single-Factor
Studies 718
Some Further Observations on Use
of Table B.12 720
16.11 Planning of Sample Sizes to Find “Best”
Treatment 721
Cited Reference 722
Contents xxi
Problems 722
Exercises 730
Projects 730
Case Studies 732
Chapter 17
Analysis of Factor Level Means 733
17.1 Introduction 733
17.2 Plots of Estimated Factor Level
Means 735
Line Plot 735
Bar Graph and Main Effects Plot 736
17.3 Estimation and Testing of Factor Level
Means 737
Inferences for Single Factor Level
Mean 737
Inferences for Difference between Two
Factor Level Means 739
Inferences for Contrast of Factor Level
Means 741
Inferences for Linear Combination of
Factor Level Means 743
17.4 Need for Simultaneous Inference
Procedures 744
17.5 Tukey Multiple Comparison
Procedure 746
Studentized Range Distribution 746
Simultaneous Estimation 747
Simultaneous Testing 747
Example 1—Equal Sample Sizes 748
Example 2—Unequal Sample Sizes 750
17.6 Scheff´e Multiple Comparison
Procedure 753
Simultaneous Estimation 753
Simultaneous Testing 754
Comparison of Scheff ´ e and Tukey
Procedures 755
17.7 Bonferroni Multiple Comparison
Procedure 756
Simultaneous Estimation 756
Simultaneous Testing 756
Comparison of Bonferroni Procedure with
Scheff´e
and Tukey Procedures 757
Analysis of Means 758
17.8 Planning of Sample Sizes with Estimation
Approach 759
Example 1—Equal Sample Sizes 759
Example 2—Unequal Sample Sizes 761
17.9 Analysis of Factor Effects when Factor
Is Quantitative 762
Cited References 766
Problems 767
Exercises 773
Projects 774
Case Studies 774
Chapter 18
ANOVA Diagnostics and Remedial
Measures 775
18.1 Residual Analysis 775
Residuals 776
Residual Plots 776
Diagnosis of Departures from ANOVA
Model 778
18.2 Tests for Constancy of Error
Variance 781
Hartley Test 782
Brown-Forsythe Test 784
18.3 Overview of Remedial Measures 786
18.4 Weighted Least Squares 786
18.5 Transformations of Response
Variable 789
Simple Guides to Finding a
Transformation 789
Box-Cox Procedure 791
18.6 Effects of Departures from Model 793
Nonnormality 793
Unequal Error Variances 794
Nonindependence of Error Terms 794
18.7 Nonparametric Rank F Test 795
Test Procedure 795
Multiple Pairwise Testing
Procedure 797
18.8 Case Example—Heart Transplant 798
Cited References 801
Problems 801
Exercises 807
Projects 807
Case Studies 809
xxii Contents
PART FIVE
MULTI-FACTOR STUDIES 811
Chapter 19
Two-Factor Studies with Equal
Sample Sizes 812
19.1 Two-Factor Observational and
Experimental Studies 812
Examples of Two-Factor Experiments and
Observational Studies 812
The One-Factor-at-a-Time (OFAAT)
Approach to Experimentation 815
Advantages of Crossed, Multi-Factor
Designs 816
19.2 Meaning of ANOVA Model
Elements 817
Illustration 817
Treatment Means 817
Factor Level Means 818
Main Effects 818
Additive Factor Effects 819
Interacting Factor Effects 822
Important and Unimportant
Interactions 824
Transformable and Nontransformable
Interactions 826
Interpretation of Interactions 827
19.3 Model I (Fixed Factor Levels) for
Two-Factor Studies 829
Cell Means Model 830
Factor Effects Model 831
19.4 Analysis of Variance 833
Illustration 833
Notation 834
Fitting of ANOVA Model 834
Partitioning of Total Sum
of Squares 836
Partitioning of Degrees of Freedom 839
Mean Squares 839
Expected Mean Squares 840
Analysis of Variance Table 840
19.5 Evaluation of Appropriateness of
ANOVA Model 842
19.6 F Tests 843
Test for Interactions 844
Test for Factor A Main Effects 844
Test for Factor B Main Effects 845
Kimball Inequality 846
19.7 Strategy for Analysis 847
19.8 Analysis of Factor Effects when Factors
Do Not Interact 848
Estimation of Factor Level Mean 848
Estimation of Contrast of Factor Level
Means 849
Estimation of Linear Combination of
Factor Level Means 850
Multiple Pairwise Comparisons of Factor
Level Means 850
Multiple Contrasts of Factor Level
Means 852
Estimates Based on Treatment
Means 853
Example 1—Pairwise Comparisons
of Factor Level Means 853
Example 2—Estimation of Treatment
Means 855
19.9 Analysis of Factor Effects when
Interactions Are Important 856
Multiple Pairwise Comparisons
of Treatment Means 856
Multiple Contrasts of Treatment
Means 857
Example 1—Pairwise Comparisons
of Treatment Means 857
Example 2—Contrasts of Treatment
Means 860
19.10 Pooling Sums of Squares in Two-Factor
Analysis of Variance 861
19.11 Planning of Sample Sizes for Two-Factor
Studies 862
Power Approach 862
Estimation Approach 863
Finding the “Best” Treatment
864
Problems 864
Exercises 876
Projects 876
Case Studies 879
Contents xxiii
Chapter 20
Two-Factor Studies—One Case
per Treatment 880
20.1 No-Interaction Model 880
Model 881
Analysis of Variance 881
Inference Procedures 881
Estimation of Treatment Mean 884
20.2 Tukey Test for Additivity 886
Development of Test Statistic 886
Remedial Actions if Interaction Effects
Are Present 888
Cited Reference 889
Problems 889
Exercises 891
Case Study 891
Chapter 21
Randomized Complete Block
Designs 892
21.1 Elements of Randomized Complete Block
Designs 892
Description of Designs 892
Criteria for Blocking 893
Advantages and Disadvantages 894
How to Randomize 895
Illustration 895
21.2 Model for Randomized Complete Block
Designs 897
21.3 Analysis of Variance and Tests 898
Fitting of Randomized Complete
Block Model 898
Analysis of Variance 898
21.4 Evaluation of Appropriateness
of Randomized Complete Block
Model 901
Diagnostic Plots 901
Tukey Test for Additivity 903
21.5 Analysis of Treatment Effects 904
21.6 Use of More than One Blocking
Variable 905
21.7 Use of More than One Replicate in Each
Block 906
21.8 Factorial Treatments 908
21.9 Planning Randomized Complete Block
Experiments 909
Power Approach 909
Estimation Approach 910
Efficiency of Blocking Variable 911
Problems 912
Exercises 916
Chapter 22
Analysis of Covariance 917
22.1 Basic Ideas 917
How Covariance Analysis Reduces Error
Variability 917
Concomitant Variables 919
22.2 Single-Factor Covariance Model 920
Notation 921
Development of Covariance Model 921
Properties of Covariance Model 922
Generalizations of Covariance
Model 923
Regression Formula of Covariance
Model 924
Appropriateness of Covariance
Model 925
Inferences of Interest 925
22.3 Example of Single-Factor Covariance
Analysis 926
Development of Model 926
Test for Treatment Effects 928
Estimation of Treatment Effects 930
Test for Parallel Slopes 932
22.4 Two-Factor Covariance Analysis 933
Covariance Model for Two-Factor
Studies 933
Regression Approach 934
Covariance Analysis for Randomized
Complete Block Designs 937
22.5 Additional Considerations for the Use
of Covariance Analysis 939
Covariance Analysis as Alternative
to Blocking 939
Use of Differences 939
Correction for Bias 940
xxiv Contents
Interest in Nature of Treatment
Effects 940
Problems 941
Exercise 947
Projects 947
Case Studies 950
Chapter 23
Two-Factor Studies with Unequal
Sample Sizes 951
23.1 Unequal Sample Sizes 951
Notation 952
23.2 Use of Regression Approach for Testing
Factor Effects when Sample Sizes Are
Unequal 953
Regression Approach to Two-Factor
Analysis of Variance 953
23.3 Inferences about Factor Effects when
Sample Sizes Are Unequal 959
Example 1—Pairwise Comparisons
of Factor Level Means 962
Example 2—Single-Degree-of-Freedom
Test 964
23.4 Empty Cells in Two-Factor Studies 964
Partial Analysis of Factor Effects 965
Analysis if Model with No Interactions Can
Be Employed 966
Missing Observations in Randomized
Complete Block Designs 967
23.5 ANOVA Inferences when Treatment
Means Are of Unequal Importance 970
Estimation of Treatment Means and Factor
Effects 971
Test for Interactions 972
Tests for Factor Main Effects by Use
of Equivalent Regression Models 972
Tests for Factor Main Effects by Use
of Matrix Formulation 975
Tests for Factor Effects when Weights Are
Proportional to Sample Sizes 977
23.6 Statistical Computing Packages 980
Problems 981
Exercises 988
Projects 988
Case Studies 990
Chapter 24
Multi-Factor Studies 992
24.1 ANOVA Model for Three-Factor
Studies 992
Notation 992
Illustration 993
Main Effects 993
Two-Factor Interactions 995
Three-Factor Interactions 996
Cell Means Model 996
Factor Effects Model 997
24.2 Interpretation of Interactions
in Three-Factor Studies 998
Learning Time Example 1: Interpretation
of Three-Factor Interactions 998
Learning Time Example 2: Interpretation
of Multiple Two-Factor Interactions 999
Learning Time Example 3: Interpretation
of a Single Two-Factor Interaction 1000
24.3 Fitting of ANOVA Model 1003
Notation 1003
Fitting of ANOVA Model 1003
Evaluation of Appropriateness of ANOVA
Model 1005
24.4 Analysis of Variance 1008
Partitioning of Total Sum of Squares 1008
Degrees of Freedom and Mean
Squares 1009
Tests for Factor Effects 1009
24.5 Analysis of Factor Effects 1013
Strategy for Analysis 1013
Analysis of Factor Effects when Factors Do
Not Interact 1014
Analysis of Factor Effects with Multiple
Two-Factor Interactions or Three-Factor
Interaction 1016
Analysis of Factor Effects with Single
Two-Factor Interaction 1016
Example—Estimation of Contrasts
of Treatment Means 1018
24.6 Unequal Sample Sizes in Multi-Factor
Studies 1019
Tests for Factor Effects 1019
Inferences for Contrasts of Factor Level
Means 1020
Contents xxv
24.7 Planning of Sample Sizes 1021
Power of F Test for Multi-Factor
Studies 1021
Use of Table B.12 for Multi-Factor
Studies 1021
Cited Reference 1022
Problems 1022
Exercises 1027
Projects 1027
Case Studies 1028
Chapter 25
Random and Mixed Effects Models 1030
25.1 Single-Factor Studies—ANOVA
Model II 1031
Random Cell Means Model 1031
Questions of Interest 1034
Test whether •2
µ = 0 1035
Estimation of µ• 1038
Estimation of •2
µ/_•2
µ •2_ 1040
Estimation of •2 1041
Point Estimation of •2
µ 1042
Interval Estimation of •2
µ 1042
Random Factor Effects Model 1047
25.2 Two-Factor Studies—ANOVA Models II
and III 1047
ANOVA Model II—Random Factor
Effects 1047
ANOVA Model III—Mixed Factor
Effects 1049
25.3 Two-Factor Studies—ANOVA Tests for
Models II and III 1052
Expected Mean Squares 1052
Construction of Test Statistics 1053
25.4 Two-Factor Studies—Estimation
of Factor Effects for Models II
and III 1055
Estimation of Variance Components 1055
Estimation of Fixed Effects in Mixed
Model 1056
25.5 Randomized Complete Block Design:
Random Block Effects 1060
Additive Model 1061
Interaction Model 1064
25.6 Three-Factor Studies—ANOVA
Models II and III 1066
ANOVA Model II—Random Factor
Effects 1066
ANOVA Model III—Mixed Factor
Effects 1066
Appropriate Test Statistics 1067
Estimation of Effects 1069
25.7 ANOVA Models II and III with Unequal
Sample Sizes 1070
Maximum Likelihood Approach 1072
Cited References 1077
Problems 1077
Exercises 1085
Projects 1085
PART SIX
SPECIALIZED STUDY
DESIGNS 1087
Chapter 26
Nested Designs, Subsampling, and
Partially Nested Designs 1088
26.1 Distinction between Nested and Crossed
Factors 1088
26.2 Two-Factor Nested Designs 1091
Development of Model Elements 1091
Nested Design Model 1092
Random Factor Effects 1093
26.3 Analysis of Variance for Two-Factor
Nested Designs 1093
Fitting of Model 1093
Sums of Squares 1094
Degrees of Freedom 1095
Tests for Factor Effects 1097
Random Factor Effects 1099
26.4 Evaluation of Appropriateness of Nested
Design Model 1099
26.5 Analysis of Factor Effects in Two-Factor
Nested Designs 1100
Estimation of Factor Level Means
µi . 1100
Estimation of Treatment Means µi j 1102
Estimation of Overall Mean µ.. 1103
Estimation of Variance Components 1103
xxvi Contents
26.6 Unbalanced Nested Two-Factor
Designs 1104
26.7 Subsampling in Single-Factor Study with
Completely Randomized Design 1106
Model 1107
Analysis of Variance and Tests of
Effects 1108
Estimation of Treatment Effects 1110
Estimation of Variances 1111
26.8 Pure Subsampling in Three Stages 1113
Model 1113
Analysis of Variance 1113
Estimation of µ.. 1113
26.9 Three-Factor Partially Nested
Designs 1114
Development of Model 1114
Analysis of Variance 1115
Cited Reference 1119
Problems 1119
Exercises 1125
Projects 1125
Chapter 27
Repeated Measures and Related
Designs 1127
27.1 Elements of Repeated Measures
Designs 1127
Description of Designs 1127
Advantages and Disadvantages 1128
How to Randomize 1128
27.2 Single-Factor Experiments with Repeated
Measures on All Treatments 1129
Model 1129
Analysis of Variance and Tests 1130
Evaluation of Appropriateness of Repeated
Measures Model 1134
Analysis of Treatment Effects 1137
Ranked Data 1138
Multiple Pairwise Testing
Procedure 1138
27.3 Two-Factor Experiments with Repeated
Measures on One Factor 1140
Description of Design 1140
Model 1141
Analysis of Variance and Tests 1142
Evaluation of Appropriateness of Repeated
Measures Model 1144
Analysis of Factor Effects: Without
Interaction 1145
Analysis of Factor Effects: With
Interaction 1148
Blocking of Subjects in Repeated Measures
Designs 1153
27.4 Two-Factor Experiments with Repeated
Measures on Both Factors 1153
Model 1154
Analysis of Variance and Tests 1155
Evaluation of Appropriateness of Repeated
Measures Model 1157
Analysis of Factor Effects 1157
27.5 Regression Approach to Repeated
Measures Designs 1161
27.6 Split-Plot Designs 1162
Cited References 1164
Problems 1164
Exercise 1171
Projects 1171
Chapter 28
Balanced Incomplete Block, Latin Square,
and Related Designs 1173
28.1 Balanced Incomplete Block
Designs 1173
Advantages and Disadvantages
of BIBDs 1175
28.2 Analysis of Balanced Incomplete Block
Designs 1177
BIBD Model 1177
Regression Approach to Analysis of
Balanced Incomplete Block Designs 1177
Analysis of Treatment Effects 1180
Planning of Sample Sizes with Estimation
Approach 1182
28.3 Latin Square Designs 1183
Basic Ideas 1183
Description of Latin Square
Designs 1184
Advantages and Disadvantages of Latin
Square Designs 1185
Contents xxvii
Randomization of Latin Square
Design 1185
28.4 Latin Square Model 1187
28.5 Analysis of Latin Square
Experiments 1188
Notation 1188
Fitting of Model 1188
Analysis of Variance 1188
Test for Treatment Effects 1190
Analysis of Treatment Effects 1190
Residual Analysis 1191
Factorial Treatments 1192
Random Blocking Variable Effects 1193
Missing Observations 1193
28.6 Planning Latin Square
Experiments 1193
Power of F Test 1193
Necessary Number of Replications 1193
Efficiency of Blocking Variables 1193
28.7 Additional Replications with Latin
Square Designs 1195
Replications within Cells 1195
Additional Latin Squares 1196
28.8 Replications in Repeated Measures
Studies 1198
Latin Square Crossover Designs 1198
Use of Independent Latin Squares 1200
Carryover Effects 1201
Cited References 1202
Problems 1202
Chapter 29
Exploratory Experiments: Two-Level
Factorial and Fractional Factorial
Designs 1209
29.1 Two-Level Full Factorial
Experiments 1210
Design of Two-Level Studies 1210
Notation 1210
Estimation of Factor Effects 1212
Inferences about Factor Effects 1214
29.2 Analysis of Unreplicated Two-Level
Studies 1216
Pooling of Interactions 1218
Pareto Plot 1219
Dot Plot 1220
Normal Probability Plot 1221
Center Point Replications 1222
29.3 Two-Level Fractional Factorial
Designs 1223
Confounding 1224
Defining Relation 1227
Half-Fraction Designs 1228
Quarter-Fraction and Smaller-Fraction
Designs 1229
Resolution 1231
Selecting a Fraction of Highest
Resolution 1232
29.4 Screening Experiments 1239
2k- f
III Fractional Factorial Designs 1239
Plackett-Burman Designs 1240
29.5 Incomplete Block Designs for Two-Level
Factorial Experiments 1240
Assignment of Treatments to Blocks 1241
Use of Center Point Replications 1243
29.6 Robust Product and Process
Design 1244
Location and Dispersion Modeling 1246
Incorporating Noise Factors 1250
Case Study—Clutch Slave Cylinder
Experiment 1252
Cited References 1256
Problems 1256
Exercises 1266
Chapter 30
Response Surface Methodology 1267
30.1 Response Surface Experiments 1267
30.2 Central Composite Response Surface
Designs 1268
Structure of Central Composite
Designs 1268
Commonly Used Central Composite
Designs 1270
Rotatable Central Composite
Designs 1271
Other Criteria for Choosing a Central
Composite Design 1273
Blocking Central Composite
Designs 1275
xxviii Contents
Additional General-Purpose Response
Surface Designs 1276
30.3 Optimal Response Surface
Designs 1276
Purpose of Optimal Designs 1276
Optimal Design Approach 1278
Design Criteria for Optimal Design
Selection 1279
Construction of Optimal Response Surface
Designs 1282
Some Final Cautions 1283
30.4 Analysis of Response Surface
Experiments 1284
Model Interpretation and
Visualization 1284
Response Surface Optimum
Conditions 1286
30.5 Sequential Search for Optimum
Conditions—Method of Steepest
Ascent 1290
Cited References 1292
Problems 1292
Projects 1295
Appendix A
Some Basic Results in Probability
and Statistics 1297
Appendix B
Tables 1315
Appendix C
Data Sets 1348
Appendix D
Rules for Developing ANOVA Models and
Tables for Balanced Designs 1358
Appendix E
Selected Bibliography 1374
Index 1385