ANOVA for the behavioural sciences researcher / Rudolf N. Cardinal, Michael R. F. Aitken.

Format:

Book

Author/Creator:

Cardinal, Rudolf N.

Contributor:

Aitken, Michael R. F.

Language:

English

Subjects (All):

Analysis of variance.

Social sciences--Statistical methods.

Social sciences.

Analysis of Variance.

Social Sciences--methods.

Social Sciences--statistical & numerical data.

Medical Subjects:

Analysis of Variance.

Social Sciences--methods.

Social Sciences--statistical & numerical data.

Physical Description:

xvi, 448 pages : illustrations ; 23 cm

Other Title:

Analysis of variance for the behavioural sciences researcher

Place of Publication:

Mahwah, N.J. : Lawrence Erlbaum, 2006.

Summary:

The authors have endeavoured to put together their own theoretical and practical guide to analysis of variance (ANOVA) for the use of students and researchers who have not had a formal course in this technique, but nevertheless require to employ this form of analysis as part of their research.

From their experience in teaching this material and applying it to research problems, the authors have created a summary of the statistical theory underlying ANOVA, together with important issues, guidance, practical methods, references, and hints about using statistical software. These have been organized so that the student can learn the logic of the analytical techniques but also use the book as a reference guide to experimental designs, realizing along the way what pitfalls are likely to be encountered.

Contents:

1.2 Background knowledge 2

1.3 Supporting Web site 4

1.4 Quick summary: choosing and performing an ANOVA 4

Chapter 2 Understanding the Basics 7

2.1 The basic logic and assumptions of ANOVA 7

2.1.1 A 'model' that describes and predicts some data 7

2.1.2 An example: data and a structural model 7

2.1.3 The null hypothesis 9

2.1.4 The assumptions of ANOVA 9

2.1.5 The logic of ANOVA 10

2.1.6 Expected mean squares (EMS) 12

2.2 The calculations behind a one-way ANOVA (one BS factor) 13

2.2.1 Calculations using means (preferred) or totals 13

2.2.2 Sums of squares: calculating SS[subscript total], SS[subscript treatment], and SS[subscript error] 13

2.2.3 Degrees of freedom 16

2.2.4 Mean squares 17

2.2.5 The F test 17

2.2.6 ANOVA summary table 18

2.2.7 SS[subscript treatment] for unequal sample sizes 19

2.2.8 Pictorial representation 19

2.3 Regression ANOVA: the other way to understand the logic 20

2.3.1 Linear regression in terms of sums of squares 20

2.3.2 Pictorial representation 23

2.3.3 Linear regression as an ANOVA 23

2.4 Factors versus covariates 24

2.5 Assumptions of ANOVA involving covariates 25

2.6 ANOVA with two between-subjects factors 26

2.6.1 Main effects, interactions, simple effects, and a structural model 27

2.6.2 Expected mean squares 29

2.6.3 Degrees of freedom 30

2.6.4 Sums of squares 30

2.6.5 Relating SS calculations to the structural model 32

2.6.6 ANOVA table 32

2.6.7 Pictorial representation 34

2.7 Within-subjects (repeated measures) ANOVA 35

2.7.1 Structural model 35

2.7.2 Degrees of freedom 36

2.7.3 Sums of squares 37

2.7.4 EMS and ANOVA summary table 37

2.8 Assumptions of within-subjects ANOVA: 'sphericity' 39

2.8.1 Short version 39

2.8.2 Long version 40

2.9 Missing data in designs involving within-subjects factors 42

2.10 Mixed ANOVA (with both BS and WS factors) 43

2.10.1 Structural model 44

2.10.2 Degrees of freedom 45

2.10.3 Sums of squares 45

2.10.4 ANOVA table 47

2.11 Fixed and random factors 48

2.12 Additional material (ADVANCED) 49

2.12.1 Notation for variances and mean squares in EMS expressions 49

2.12.2 Expected value of F 50

2.12.3 A X[superscript 2] distribution is the sum of squared z scores 51

2.12.4 Relationship between the sample variance and the x[superscript 2] distribution 53

2.12.5 The F distribution 54

2.12.6 Comparing two variances with an F test 55

2.12.7 ANOVA: comparing two mean-square values with an F test 56

2.12.8 Relating SS calculations to the model for one-way ANOVA 58

Chapter 3 Practical Analysis 61

3.1 Reminder: assumptions of ANOVA 61

3.2 Reminder: assumption of ANOVA with WS factors 63

3.3 Consequences of violating the assumptions of ANOVA 64

3.4 Exploratory data analysis and transformations 65

3.4.1 Plot your data 65

3.4.2 Outliers 66

3.4.3 Transformations 67

3.5 Performing the ANOVA 71

3.6 Residuals 72

3.7 Further analysis: after the ANOVA has been run 77

3.7.1 Main effects, interactions, and simple effects revisited 77

3.7.2 Conducting simple-effects analysis 78

3.7.3 A fallacy to avoid: when A differs from C but B doesn't 79

3.7.4 A fallacy to avoid: simple effects without interactions 80

3.7.5 Determining the effects of a factor with >2 levels 82

3.7.6 Multiple comparisons: a problem 82

3.7.7 Post hoc tests: a problem 83

3.7.8 The special case of three groups: multiple t tests are OK 84

3.7.9 Otherwise: a variety of post hoc tests 86

3.7.10 Post hoc tests for within-subject factors 90

3.7.11 A priori tests: planned contrasts 90

3.7.12 Apparent inconsistency between the F test and post hoc tests 91

3.7.13 SPSS's default pairwise comparison post hoc tests 91

3.8 Drawing pictures: error bars for different comparisons 94

3.8.1 Error bars for t tests: between-subjects comparisons 94

3.8.2 Error bars for t tests: within-subjects comparisons 95

3.8.3 Error bars for an ANOVA: between-subjects designs 98

3.8.4 Error bars for an ANOVA: effects in mixed designs 98

3.9 Summarizing your methods: a writing guide 99

3.10 Additional material (ADVANCED) 101

3.10.1 Error bars for t tests: between-subjects comparisons: SEMs 101

3.10.2 Error bars for t tests: between-subjects comparisons: CIs 103

3.10.3 Error bars for t tests: between-subjects comparisons: SDs 105

3.10.4 Obtaining SEDs from an ANOVA table 105

Chapter 4 Pitfalls and Common Issues 109

4.1 Time in within-subjects (repeated measures) designs 109

4.2 Analysis of pre-test versus post-test data 109

4.3 Observing subjects repeatedly to increase power 110

4.4 'It's significant in this subject...' 112

4.5 Should I add/remove a factor? Full and reduced models 114

4.6 Should I add/remove/collapse over levels of a factor? 115

4.6.1 Adding and removing levels by adding new observations 116

4.6.2 Collapsing over or subdividing levels 117

Chapter 5 Using SPSS for ANOVA 119

5.1 Running ANOVAs using SPSS 119

5.1.1 Analysis of variance 119

5.1.2 Organizing and reorganizing your data 120

5.1.3 Syntax 120

5.1.4 Plots 121

5.1.5 Options, including homogeneity-of-variance tests 121

5.1.6 Post hoc tests 123

5.2 Interpreting the output 124

Tip: pairwise comparisons for interactions 136

5.3 Further analysis: selecting cases 139

5.4 The 'intercept', 'total', and 'corrected total' terms 142

Chapter 6 Contrasts and Trends 147

6.1 Contrasts 147

6.1.1 About linear contrasts 147

6.1.2 Type I error rates with planned contrasts 148

6.1.3 Orthogonal contrasts 150

6.1.4 Linear contrasts in SPSS 151

6.1.5 Contrasts in multifactor designs-an overview 154

6.2 Trend analysis: the effects of quantitative factors 155

6.2.1 Trends 155

6.2.2 Trend analysis in SPSS 157

6.2.3 Trend analysis, multiple regression, and polynomial ANCOVA 158

Chapter 7 Advanced Topics 161

7.1 Rules for calculating sums of squares 161

7.1.1 Partitioning sums of squares 161

7.1.2 General rule for calculating sums of squares 161

7.2 Rules for calculating degrees of freedom 163

7.3 Expected mean squares (EMS) and error terms 164

7.3.1 Rules for obtaining expected mean squares (EMS) 165

7.3.2 Choosing an error term 168

7.3.3 Error terms in models including random factors (complicated) 170

7.3.4 Pooling error terms 173

7.4 Unequal group sizes and non-orthogonal sums of squares 174

7.4.1 Proportional cell frequencies 174

7.4.2 Disproportionate cell frequencies-a problem 175

7.4.3 Correlated predictors in general-a problem 177

7.5 How computers perform ANOVA: general linear models 179

7.5.1 The basic idea of a GLM, illustrated with multiple regression 180

7.5.2 Using a GLM for simple ANOVA: the design matrix 181

7.5.3 Example of a GLM for a one-way ANOVA 183

7.5.4 GLM for two-way ANOVA and beyond 185

7.5.5 F statistics for GLMs: comparing full and reduced models 187

7.5.6 An overview of GLM designs 188

7.5.7 GLM designs involving random effects 194

7.5.8 A hint at multivariate analysis: MANOVA 195

7.5.9 Linear contrasts with a GLM 196

7.5.10 GLMs and custom contrasts in SPSS 197

7.6 Effect size 202

7.6.1 Effect size in the language of multiple regression 203

7.6.2 Effect size in the language of ANOVA 209

Chapter 8 Specific Designs 215

8.1 One between-subjects (BS) factor 217

8.2 Two BS factors 222

8.3 Three BS factors 225

8.4 One within-subjects (WS) factor 228

8.5 Two WS factors 233

8.6 Three WS factors 238

8.7 One BS and one WS factor 244

8.8 Two BS factors and one WS factor 251

8.9 One BS factor and two WS factors 256

8.10 Other ANOVA designs with BS and/or WS factors 261

8.11 One BS covariate (linear regression) 264

8.12 One BS covariate and one BS factor 271

8.12.1 The covariate and factor do not interact 271

8.12.2 The covariate and factor interact 281

8.13 One BS covariate and two BS factors 286

8.14 Two or more BS covariates (multiple regression) 289

8.15 Two or more BS covariates and one or more BS factors 292

8.16 One WS covariate 295

8.17 One WS covariate and one BS factor 299

8.17.1 The covariate and factor do not interact 299

8.17.2 The covariate and factor interact 303

8.18 Hierarchical designs 307

8.18.1 Subjects within groups within treatments (S/G/A) 307

8.18.2 Groups versus individuals 311

8.18.3 Adding a further within-group, BS variable (S/GB/A) 312

8.18.4 Adding a within-subjects variable (US/GB/A) 314

8.18.5 Nesting within-subjects variables, such as V/US/A 316

8.18.6 The split-split plot design 319

8.18.7 Three levels of relatedness 324

8.19 Latin square designs 328

8.19.1 Latin squares in experimental design 328

8.19.2 The analysis of a basic Latin square 329

8.19.3 A x B interactions in a single Latin square 332

8.19.4 More subjects than rows: (a) using several squares 334

8.19.5 More subjects than rows: (b) using one square several times 337

8.19.6 BS designs using Latin squares (fractional factorial designs) 341

8.19.7 Several-squares design with a BS factor 344

8.19.8 Replicated-squares design with a BS factor 347

8.20 Agricultural terminology and designs 351

Chapter 9 Mathematics 367

9.1 Matrices 367

9.1.1 Matrix notation 367

9.1.2 Matrix algebra 369

9.1.3 The inverse of a matrix 372

9.1.4 Matrix transposition 373

9.2 Calculus 374

9.2.1 Derivatives 374

9.2.2 Simple, non-trigonometric derivatives 375

9.2.3 Rules for differentiation 375

9.2.4 Derivatives of a vector function 376

9.2.5 Partial derivatives 376

9.2.6 The chain rule for partial derivatives 377

9.2.7 Illustrations of partial derivatives 377

9.3 Solving a GLM (an overdetermined system of equations) 379

9.4 Singular value decomposition to solve GLMs 382

9.4.1 Eigenvectors and eigenvalues 384

9.4.2 Singular value decomposition 385

9.4.3 An underdetermined set of equations: the role of expectations 386

9.5 Random variables, means, and variances 387

9.5.1 Summation 387

9.5.2 Random variables; definition of mean and variance 388

9.5.3 Continuous random variables 389

9.5.4 Expected values 390

9.5.5 Variance laws 391

9.5.6 Distribution of a set of means: the standard error of the mean 392

9.5.7 The sample mean and SD are unbiased estimators of [mu] and [sigma superscript 2] 395

9.6 The harmonic mean 397

9.7 Rules for powers and logarithms 397

9.8 Probability 398

Basic notation in probability 398

Basic laws of probability 398

Odds 399

Bayes' theorem and Bayesian inference 400

Chapter 10 Statistical Tables 403

10.1 Critical values of t 403

10.2 Critical values of F 404

10.3 Polynomial trend coefficients 408.

Notes:

Includes bibliographical references (pages 435-438) and index.

ISBN:

0805855858

0805855866

OCLC:

63145336

Online:

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