xvContents
10.1.2 Notation .........................................................................................................422
10.2 One-Way ANOVA .....................................................................................................423
10.2.1 lm Revisited .................................................................................................. 426
10.3 Inferences for Factor Levels ..................................................................................... 429
10.3.1 Introduction to Contrasts............................................................................429
10.3.2 Orthogonality ............................................................................................... 431
10.3.3 lm contrast...............................................................................................432
10.3.3.1 Treatment Contrasts .................................................................... 432
10.3.3.2 Helmert Contrasts .......................................................................433
10.3.3.3 Sum Contrasts .............................................................................. 435
10.3.4 Issues with Multiple Comparisons ............................................................436
10.3.5 Simultaneous Inference Procedures .........................................................436
10.3.5.1 Bonferroni Correction .................................................................437
10.3.5.2 Least Signicant Difference ....................................................... 437
10.3.5.3 Scheffé’s Procedure .....................................................................437
10.3.5.4 Tukey–Kramer Method...............................................................438
10.3.5.5 Dunnett’s Method .......................................................................439
10.3.5.6 Comparing Simultaneous Inference Procedures ....................439
10.3.6 General Methods for P-Value Adjustment ...............................................441
10.3.7 Depicting Factor-Level Comparisons .......................................................442
10.4 ANOVA as a General Linear Model .......................................................................442
10.5 Random Effects .........................................................................................................445
10.5.1 Variance Components .................................................................................447
10.5.2 Hypothesis Testing ......................................................................................449
10.5.2.1 Likelihood Ratio Test ..................................................................449
10.5.3 lme and lmer ............................................................................................... 450
10.6 Power, Sample Size, and Effect Size .......................................................................453
10.7 ANOVA Diagnostics and Assumptions ................................................................454
10.7.1 Independence of Error Terms ....................................................................454
10.7.2 Normality of Error Terms ...........................................................................456
10.7.3 Constancy of Error Variance ......................................................................456
10.7.4 Outliers ..........................................................................................................457
10.7.5 Random and Mixed Effect Models ............................................................457
10.8 Two-Way Factorial Design ....................................................................................... 459
10.8.1 Random Effects ............................................................................................463
10.9 Randomized Block Design ......................................................................................466
10.9.1 Random Block Effects .................................................................................469
10.10 Nested Design ...........................................................................................................471
10.11 Split-Plot Design ........................................................................................................ 474
10.11.1 aov .................................................................................................................475
10.12 Repeated Measures Design ..................................................................................... 478
10.12.1 Analysis of Contrasts .................................................................................. 479
10.12.2 Modern Mixed-Model Approaches ...........................................................482
10.13 ANCOVA .................................................................................................................... 483
10.14 Unbalanced Designs ................................................................................................. 488
10.14.1 Type II and III Sums of Squares .................................................................488
10.15 Robust ANOVA .........................................................................................................492
10.15.1 Permutation Tests .........................................................................................493
10.15.2 Rank-Based Permutation Tests ...................................................................493