ANOVA for the Behavioral Sciences Researcher

by Cardinal; Rudolf N.

ISBN13: 9780805855869

ISBN10: 0805855866

Format: Nonspecific Binding

Pub. Date: 2005-11-17

Publisher(s): Psychology Pres

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Summary

This new book provides a theoretical and practical guide to analysis of variance (ANOVA) for those who have not had a formal course in this technique, but need to use this 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.

Author Biography

Rudolf Cardinal was born in Norwich in 1975. He received a B.A. with first-class honors in medical sciences and neuroscience (1993–1996), M.B. and BChir degrees in clinical medicine and surgery (1996–2001), and a Ph.D. in behavioural neuroscience (1997–2000), all from the University of Cambridge. He has worked as a house physician and surgeon in Cambridge and Norwich (2001–2002) and then as a lecturer in neuroscience in the Department of Experimental Psychology, University of Cambridge (2002–2005), conducting neuroscience research and teaching in neuroscience, psychology, and statistics. His interests within neuroscience include the neural mechanisms of emotion, perception, cognition, and the control of behaviour; reinforcement and reinforcement learning; the nature and neural basis of consciousness; normative decision-making and probability; and artificial intelligence. He is currently a clinical fellow at Addenbrooke’s Hospital, Cambridge.
Mike Aitken was born in Cheltenham in 1970. He is currently a Department Lecturer based at the MRC/Wellcome Behavioural and Clinical Neurosciences Institute within the Department of Experimental Psychology at the University of Cambridge (where he completed both his undergraduate degree and Ph.D.). His research interests include associative theory of categorization and causal induction, functional neuroimaging of associative learning processes, and mechanisms of self-control and response selection.

PREFACE

CHAPTER 1: QUICK SUMMARY

(6)

1.1 Overview of this book

(1)

1.2 Background knowledge

(2)

1.3 Supporting Web site

(1)

1.4 Quick summary: choosing and performing an ANOVA

(3)

CHAPTER 2: UNDERSTANDING THE BASICS

(54)

2.1 The basic logic and assumptions of ANOVA

(6)

2.1.1 A 'model' that describes and predicts some data

(1)

2.1.2 An example: data and a structural model

(2)

2.1.3 The null hypothesis

(1)

2.1.4 The assumptions of ANOVA

(1)

2.1.5 The logic of ANOVA

(2)

2.1.6 Expected mean squares (EMS)

(1)

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

(7)

2.2.1 Calculations using means (preferred) or totals

(1)

2.2.2 Sums of squares: calculating SStotal, SStreatment, and SSerror

(3)

2.2.3 Degrees of freedom

(1)

2.2.4 Mean squares

(1)

2.2.5 The F test

(1)

2.2.6 ANOVA summary table

(1)

2.2.7 SStreatment for unequal sample sizes

(1)

2.2.8 Pictorial representation

(1)

2.3 Regression ANOVA: the other way to understand the logic

(4)

2.3.1 Linear regression in terms of sums of squares

(3)

2.3.2 Pictorial representation

(1)

2.3.3 Linear regression as an ANOVA

(1)

2.4 Factors versus covariates

(1)

2.5 Assumptions of ANOVA involving covariates

(1)

2.6 ANOVA with two between-subjects factors

(9)

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

(2)

2.6.2 Expected mean squares

(1)

2.6.3 Degrees of freedom

(1)

2.6.4 Sums of squares

(2)

2.6.5 Relating SS calculations to the structural model

(1)

2.6.6 ANOVA table

(2)

2.6.7 Pictorial representation

(1)

2.7 Within-subjects (repeated measures) ANOVA

(4)

2.7.1 Structural model

(1)

2.7.2 Degrees of freedom

(1)

2.7.3 Sums of squares

(1)

2.7.4 EMS and ANOVA summary table

(2)

2.8 Assumptions of within-subjects ANOVA: 'sphericity'

(3)

2.8.1 Short version

(1)

2.8.2 Long version

(2)

2.9 Missing data in designs involving within-subjects factors

(1)

2.10 Mixed ANOVA (with both BS and WS factors)

(5)

2.10.1 Structural model

(1)

2.10.2 Degrees of freedom

(1)

2.10.3 Sums of squares

(2)

2.10.4 ANOVA table

(1)

2.11 Fixed and random factors

(1)

2.12 Additional material (ADVANCED)

(12)

2.12.1 Notation for variances and mean squares in EMS expressions

(1)

2.12.2 Expected value of F

(1)

2.12.3 A χ² distribution is the sum of squared z scores

(2)

2.12.4 Relationship between the sample variance and the χ² distribution

(1)

2.12.5 The F distribution

(1)

2.12.6 Comparing two variances with an F test

(1)

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

(2)

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

(3)

CHAPTER 3: PRACTICAL ANALYSIS

(48)

3.1 Reminder: assumptions of ANOVA

(2)

3.2 Reminder: assumption of ANOVA with WS factors

(1)

3.3 Consequences of violating the assumptions of ANOVA

(1)

3.4 Exploratory data analysis and transformations

(6)

3.4.1 Plot your data

(1)

3.4.2 Outliers

(1)

3.4.3 Transformations

(4)

3.5 Performing the ANOVA

(1)

3.6 Residuals

(5)

3.7 Further analysis: after the ANOVA has been run

(17)

3.7.1 Main effects, interactions, and simple effects revisited

(1)

3.7.2 Conducting simple-effects analysis

(1)

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

(1)

3.7.4 A fallacy to avoid: simple effects without interactions

(2)

3.7.5 Determining the effects of a factor with >2 levels

(1)

3.7.6 Multiple comparisons: a problem

(1)

3.7.7 Post hoc tests: a problem

(1)

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

(2)

3.7.9 Otherwise: a variety of post hoc tests

(4)

3.7.10 Post hoc tests for within-subject factors

(1)

3.7.11 A priori tests: planned contrasts

(1)

3.7.12 Apparent inconsistency between the F test and post hoc tests

(1)

3.7.13 SPSS's default pairwise comparison post hoc tests

(3)

3.8 Drawing pictures: error bars for different comparisons

(5)

3.8.1 Error bars for t tests: between-subjects comparisons

(1)

3.8.2 Error bars for t tests: within-subjects comparisons

(3)

3.8.3 Error bars for an ANOVA: between-subjects designs

(1)

3.8.4 Error bars for an ANOVA: effects in mixed designs

(1)

3.9 Summarizing your methods: a writing guide

(2)

3.10 Additional material (ADVANCED)

101

(8)

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

101

(2)

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

103

(2)

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

105

(1)

3.10.4 Obtaining SEDs from an ANOVA table

105

(4)

CHAPTER 4: PITFALLS AND COMMON ISSUES

109

(10)

4.1 Time in within-subjects (repeated measures) designs

109

(1)

4.2 Analysis of pre-test versus post-test data

109

(1)

4.3 Observing subjects repeatedly to increase power

110

(2)

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

112

(2)

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

114

(1)

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

115

(4)

4.6.1 Adding and removing levels by adding new observations

116

(1)

4.6.2 Collapsing over or subdividing levels

117

(2)

CHAPTER 5: USING SPSS FOR ANOVA

119

(28)

5.1 Running ANOVAs using SPSS

119

(5)

5.1.1 Analysis of variance

119

(1)

5.1.2 Organizing and reorganizing your data

120

(1)

5.1.3 Syntax

120

(1)

5.1.4 Plots

121

(1)

5.1.5 Options, including homogeneity-of-variance tests

121

(2)

5.1.6 Post hoc tests

123

(1)

5.2 Interpreting the output

124

(15)

Tip: pairwise comparisons for interactions

136

(3)

5.3. Further analysis: selecting cases

139

(3)

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

142

(5)

CHAPTER 6: CONTRASTS AND TRENDS

147

(14)

6.1 Contrasts

147

(8)

6.1.1 About linear contrasts

147

(1)

6.1.2 Type I error rates with planned contrasts

148

(2)

6.1.3 Orthogonal contrasts

150

(1)

6.1.4 Linear contrasts in SPSS

151

(3)

6.1.5 Contrasts in multifactor designs—an overview

154

(1)

6.2 Trend analysis: the effects of quantitative factors

155

(6)

6.2.1 Trends

155

(2)

6.2.2 Trend analysis in SPSS

157

(1)

6.2.3 Trend analysis, multiple regression, and polynomial ANCOVA

158

(3)

CHAPTER 7: ADVANCED TOPICS

161

(54)

7.1 Rules for calculating sums of squares

161

(2)

7.1.1 Partitioning sums of squares

161

(1)

7.1.2 General rule for calculating sums of squares

161

(2)

7.2 Rules for calculating degrees of freedom

163

(1)

7.3 Expected mean squares (EMS) and error terms

164

(10)

7.3.1 Rules for obtaining expected mean squares (EMS)

165

(3)

7.3.2 Choosing an error term

168

(2)

7.3.3 Error terms in models including random factors (complicated)

170

(3)

7.3.4 Pooling error terms

173

(1)

7.4 Unequal group sizes and non-orthogonal sums of squares

174

(5)

7.4.1 Proportional cell frequencies

174

(1)

7.4.2 Disproportionate cell frequencies—a problem

175

(2)

7.4.3 Correlated predictors in general—a problem

177

(2)

7.5 How computers perform ANOVA: general linear models

179

(23)

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

180

(1)

7.5.2 Using a GLM for simple ANOVA: the design matrix

181

(2)

7.5.3 Example of a GLM for a one-way ANOVA

183

(2)

7.5.4 GLM for two-way ANOVA and beyond

185

(2)

7.5.5 F statistics for GLMs: comparing full and reduced models

187

(1)

7.5.6 An overview of GLM designs

188

(6)

7.5.7 GLM designs involving random effects

194

(1)

7.5.8 A hint at multivariate analysis: MANOVA

195

(1)

7.5.9 Linear contrasts with a GLM

196

(1)

7.5.10 GLMs and custom contrasts in SPSS

197

(5)

7.6 Effect size

202

(13)

7.6.1 Effect size in the language of multiple regression

203

(6)

7.6.2 Effect size in the language of ANOVA

209

(6)

CHAPTER 8: SPECIFIC DESIGNS

215

(152)

8.1 One between-subjects (BS) factor

217

(5)

8.2 Two BS factors

222

(3)

8.3 Three BS factors

225

(3)

8.4 One within-subjects (WS) factor

228

(5)

8.5 Two WS factors

233

(5)

8.6 Three WS factors

238

(6)

8.7 One BS and one WS factor

244

(7)

8.8 Two BS factors and one WS factor

251

(5)

8.9 One BS factor and two WS factors

256

(5)

8.10 Other ANOVA designs with BS and/or WS factors

261

(3)

8.11 One BS covariate (linear regression)

264

(7)

8.12 One BS covariate and one BS factor

271

(15)

8.12.1 The covariate and factor do not interact

271

(10)

8.12.2 The covariate and factor interact

281

(5)

8.13 One BS covariate and two BS factors

286

(3)

8.14 Two or more BS covariates (multiple regression)

289

(3)

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

292

(3)

8.16 One WS covariate

295

(4)

8.17 One WS covariate and one BS factor

299

(8)

8.17.1 The covariate and factor do not interact

299

(4)

8.17.2 The covariate and factor interact

303

(4)

8.18 Hierarchical designs

307

(21)

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

307

(4)

8.18.2 Groups versus individuals

311

(1)

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

312

(2)

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

314

(2)

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

316

(3)

8.18.6 The split-split plot design

319

(5)

8.18.7 Three levels of relatedness

324

(4)

8.19 Latin square designs

328

(23)

8.19.1 Latin squares in experimental design

328

(1)

8.19.2 The analysis of a basic Latin square

329

(3)

8.19.3 A x B interactions in a single Latin square

332

(2)

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

334

(3)

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

337

(4)

8.19.6 BS designs using Latin squares (fractional factorial designs)

341

(3)

8.19.7 Several-squares design with a BS factor

344

(3)

8.19.8 Replicated-squares design with a BS factor

347

(4)

8.20 Agricultural terminology and designs

351

(16)

CHAPTER 9: MATHEMATICS

367

(36)

9.1 Matrices

367

(7)

9.1.1 Matrix notation

367

(2)

9.1.2 Matrix algebra

369

(3)

9.1.3 The inverse of a matrix

372

(1)

9.1.4 Matrix transposition

373

(1)

9.2 Calculus

374

(5)

9.2.1 Derivatives

374

(1)

9.2.2 Simple, non-trigonometric derivatives

375

(1)

9.2.3 Rules for differentiation

375

(1)

9.2.4 Derivatives of a vector function

376

(1)

9.2.5 Partial derivatives

376

(1)

9.2.6 The chain rule for partial derivatives

377

(1)

9.2.7 Illustrations of partial derivatives

377

(2)

9.3 Solving a GLM (an overdetermined system of equations)

379

(3)

9.4 Singular value decomposition to solve GLMs

382

(5)

9.4.1 Eigenvectors and eigenvalues

384

(1)

9.4.2 Singular value decomposition

385

(1)

9.4.3 An underdetermined set of equations: the role of expectations

386

(1)

9.5 Random variables, means, and variances

387

(10)

9.5.1 Summation

387

(1)

9.5.2 Random variables; definition of mean and variance

388

(1)

9.5.3 Continuous random variables

389

(1)

9.5.4 Expected values

390

(1)

9.5.5 Variance laws

391

(1)

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

392

(3)

9.5.7 The sample mean and SD are unbiased estimators of μ and σ²

395

(2)

9.6 The harmonic mean

397

(1)

9.7 Rules for powers and logarithms

397

(1)

9.8 Probability

398

(5)

Basic notation in probability

398

(1)

Basic laws of probability

398

(1)

Odds

399

(1)

Bayes' theorem and Bayesian inference

400

(3)

CHAPTER 10: STATISTICAL TABLES

403

(6)

10.1 Critical values of t

403

(1)

10.2 Critical values of F

404

(4)

10.3 Polynomial trend coefficients

408

(1)

GLOSSARY

409

(24)

Symbols

409

(2)

Abbreviations

411

(1)

Terms

412

(21)

ANOVA for the Behavioral Sciences Researcher

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