ANOVA and ANCOVA: A GLM Approach

ANOVA and ANCOVA A GLM Approach......Page 5
Contents......Page 7
Acknowledgments......Page 15
1.1 Regression, Analysis of Variance, and Analysis of Covariance......Page 17
1.2 A Pocket History of Regression, ANOVA, and ANCOVA......Page 18
1.3 An Outline of General Linear Models (GLMs)......Page 19
1.3.1 Regression......Page 20
1.3.3 Analysis of Covariance......Page 21
1.4 The \"General\" in GLM......Page 22
1.5 The \"Linear\" in GLM......Page 24
1.6 Least Squares Estimates......Page 27
1.7 Fixed, Random, and Mixed Effects Analyses......Page 28
1.8 The Benefits of a GLM Approach to ANOVA and ANCOVA......Page 29
1.9 The GLM Presentation......Page 30
1.10 Statistical Packages for Computers......Page 31
2.1 Independent Measures Designs......Page 33
2.2 Balanced Data Designs......Page 35
2.3 Factors and Independent Variables......Page 36
2.4 An Outline of Traditional ANOVA for Single Factor Designs......Page 37
2.5 Variance......Page 39
2.6 Traditional ANOVA Calculations for Single Factor Designs......Page 41
2.7 Confidence Intervals......Page 46
2.8.1 Experimental Design GLMs......Page 47
2.8.2 Estimating Effects by Comparing Full and Reduced Experimental Design GLMs......Page 53
2.8.4.1 Dummy Coding......Page 57
2.8.4.2 Why Only (p … 1) Variables Are Used to Represent All Experimental Conditions?......Page 60
2.8.4.3 Effect Coding......Page 63
2.8.6 Cell Mean GLMs......Page 66
2.8.7 Experimental Design Regression and Cell Mean GLMs......Page 67
3.1 Introduction......Page 69
3.2 Comparisons Between Experimental Condition Means......Page 71
3.3 Linear Contrasts......Page 72
3.4 Comparison Sum of Squares......Page 73
3.5 Orthogonal Contrasts......Page 74
3.6 Testing Multiple Hypotheses......Page 78
3.6.1 Type 1 and Type 2 Errors......Page 79
3.6.2 Type 1 Error Rate Inflation with Multiple Hypothesis Testing......Page 81
3.6.3 Type 1 Error Rate Control and Analysis Power......Page 82
3.6.4.1 Testwise Type 1 Error Rate......Page 84
3.6.4.2 Family wise Type 1 Error Rate......Page 85
3.6.4.4 False Discovery Rate......Page 86
3.6.5 Identifying the \"Family\" in Familywise Type 1 Error Rate Control......Page 87
3.6.6.1 Logical Relations......Page 88
3.6.6.2 Empirical Relations......Page 90
3.7 Planned and Unplanned Comparisons......Page 92
3.7.1 Direct Assessment of Planned Comparisons......Page 93
3.7.2 Contradictory Results with ANOVA Omnibus F-tests and Direct Planned Comparisons......Page 94
3.8.1 ANOVA First?......Page 95
3.8.2 Strong and Weak Type 1 Error Control......Page 96
3.8.3 Stepwise Tests......Page 97
3.8.4 Test Power......Page 98
3.9.2.1 Rom\'s Test......Page 99
3.9.2.2 Shaffer\'s R Test......Page 100
3.9.2.3 Applying Shaffer\'s R Test After a Significant F-test......Page 102
3.9.3 Stage 3......Page 105
3.10 The Role of the Omnibus F-Test......Page 107
4.1 Introduction......Page 109
4.2 Effect Size as a Standardized Mean Difference......Page 110
4.3 Effect Size as Strength of Association (SOA)......Page 112
4.3.1 SOA for Specific Comparisons......Page 114
4.5 Effect Size in Related Measures Designs......Page 115
4.6 Overview of Standardized Mean Difference and SOA Measures of Effect Size......Page 116
4.7.1 Influences on Power......Page 117
4.7.2 Uses of Power Analysis......Page 119
4.7.3 Determining the Sample Size Needed to Detect the Omnibus Effect......Page 120
4.7.4 Determining the Sample Size Needed to Detect Specific Effects......Page 123
4.7.5 Determining the Power Level of a Planned or Completed Study......Page 125
4.7.6 The Fallacy of Observed Power......Page 126
5.1 Factorial Designs......Page 127
5.2 Factor Main Effects and Factor Interactions......Page 128
5.2.1 Estimating Effects by Comparing Full and Reduced Experimental Design GLMs......Page 133
5.3 Regression GLMs for Factorial ANOVA......Page 137
5.4 Estimating Effects with Incremental Analysis......Page 139
5.4.1.2 Step 2......Page 140
5.4.1.3 Step 3......Page 141
5.5.1.1 Complete ?2 for Main and Interaction Effects......Page 142
5.5.2 Partial ?2 for Specific Comparisons......Page 143
5.6.1 Main Effects: Encoding Instructions and Study Time......Page 144
5.6.2 Interaction Effect: Encoding Instructions × Study Time......Page 147
5.6.2.1 Simple Effects: Comparing the Three Levels of Factor B at a1, and at a2......Page 148
5.6.2.2 Simple Effects: Comparing the Two Levels of Factor A at b1, at b2, and at b3......Page 151
5.7.1 Determining the Sample Size Needed to Detect Omnibus Main Effects and Interactions......Page 152
5.7.2 Determining the Sample Size Needed to Detect Specific Effects......Page 154
6.1 Introduction......Page 155
6.1.1 Randomized Block Designs......Page 156
6.1.3 Repeated Measures Designs......Page 157
6.2.2.1 Crossover Designs......Page 160
6.2.2.2 Latin Square Designs......Page 161
6.3 The GLM Approach to Single Factor Repeated Measures Designs......Page 162
6.4 Estimating Effects by Comparing Full and Reduced Repeated Measures Design GLMs......Page 169
6.5 Regression GLMs for Single Factor Repeated Measures Designs......Page 172
6.6.1 A Complete ?2 SOA for the Omnibus Effect Comparable Across Repeated and Independent Measures Designs......Page 176
6.6.2 A Partial ?2 SOA for the Omnibus Effect Appropriate for Repeated Measures Designs......Page 177
6.7 Further Analyses......Page 178
6.8.1 Determining the Sample Size Needed to Detect the Omnibus Effect......Page 184
6.8.2 Determining the Sample Size Needed to Detect Specific Effects......Page 185
7.1 Factorial Related and Repeated Measures Designs......Page 187
7.2 Fully Repeated Measures Factorial Designs......Page 188
7.3 Estimating Effects by Comparing Full and Reduced Experimental Design GLMs......Page 195
7.4 Regression GLMs for the Fully Repeated Measures Factorial ANOVA......Page 196
7.5.1 A Complete ?2 SOA for Main and Interaction Omnibus Effects Comparable Across Repeated Measures and Independent Designs......Page 202
7.5.2 A Partial ?2 SOA for the Main and Interaction Omnibus Effects Appropriate for Repeated Measures Designs......Page 203
7.6.1 Main Effects: Encoding Instructions and Study Time......Page 204
7.6.2.1 Simple Effects: Comparison of Differences Between the Three Levels of Factor B (Study Time) at Each Level of Factor A (Encoding Instructions)......Page 207
7.6.2.2 Simple Effects: Comparison of Differences Between the Two Levels of Factor A (Encoding Instructions) at Each Level of Factor B (Study Time)......Page 209
7.7 Power......Page 213
8.1 Mixed Measures and Split-Plot Designs......Page 215
8.2 Factorial Mixed Measures Designs......Page 216
8.3 Estimating Effects by Comparing Full and Reduced Experimental Design GLMs......Page 221
8.4 Regression GLM for the Two-Factor Mixed Measures ANOVA......Page 222
8.6.1 Main Effects: Independent Factor„Encoding Instructions......Page 227
8.6.3.2 Simple Effects: Comparing Differences Between the Two Levels of Factor A (Encoding Instructions) at Each Level of Factor B (Study Time)......Page 228
8.7 Power......Page 230
9.1 The Nature of ANCOVA......Page 231
9.2 Single Factor Independent Measures ANCOVA Designs......Page 232
9.3 Estimating Effects by Comparing Full and Reduced ANCOVA GLMs......Page 237
9.4 Regression GLMs for the Single Factor, Single-Covariate ANCOVA......Page 242
9.5 Further Analyses......Page 245
9.6.1 A Partial ?2 SOA for the Omnibus Effect......Page 247
9.7 Power......Page 248
9.8.2 Mixed Measures Factorial ANCOVA......Page 249
10.2 ANOVA and GLM Assumptions......Page 251
10.2.1 Independent Measures Designs......Page 252
10.2.2.1 Assessing and Dealing with Sphericity Violations......Page 254
10.2.3 Traditional ANCOVA......Page 256
10.3 A Strategy for Checking GLM and Traditional ANCOVA Assumptions......Page 257
10.4 Assumption Checks and Some Assumption Violation Consequences......Page 258
10.4.1.1 Random Sampling......Page 259
10.4.1.2 Independence......Page 260
10.4.1.3 Normality......Page 261
10.4.1.4 Homoscedasticity: Homogeneity of Variance......Page 264
10.4.2.1 Covariate Independent of Experimental Conditions......Page 266
10.4.2.2 Linear Regression......Page 268
10.4.2.3 Homogeneous Regression......Page 272
10.5 Should Assumptions be Checked?......Page 275
11.1 Alternatives to Traditional ANCOVA......Page 279
11.2 The Heterogeneous Regression Problem......Page 280
11.3 The Heterogeneous Regression ANCOVA GLM......Page 281
11.4 Single Factor Independent Measures Heterogeneous Regression ANCOVA......Page 282
11.5 Estimating Heterogeneous Regression ANCOVA Effects......Page 284
11.6 Regression GLMs for Heterogeneous Regression ANCOVA......Page 289
11.7.1 Adjustments Based on the General Covariate Mean......Page 292
11.7.2 Multicolinearity......Page 293
11.8.1 Stratification (Blocking)......Page 294
11.8.2 Replacing the Experimental Conditions with the Covariate......Page 295
11.9 The Role of Heterogeneous Regression ANCOVA......Page 296
12.1 Introduction......Page 297
12.2 Review of the Single Factor Repeated Measures Experimental Design GLM and ANOVA......Page 298
12.3 The Multilevel Approach to the Single Factor Repeated Measures Experimental Design......Page 299
12.4 Parameter Estimation in Multilevel Analysis......Page 304
12.5.1 Using SYSTAT to Apply the Multilevel GLM of the Repeated Measures Experimental Design GLM......Page 305
12.5.1.1 The Linear Mixed Model......Page 307
12.5.1.2 The Hierarchical Linear Mixed Model......Page 311
12.5.2 Applying Alternative Multilevel GLMs to the Repeated Measures Data......Page 314
12.6 Empirically Assessing Different Multilevel Models......Page 319
Appendix A......Page 321
Appendix B......Page 323
Appendix C......Page 331
References......Page 341
Index......Page 355