ANOVA for the Behavioral Sciences Researcher

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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.

Table of Contents

PREFACE xv
CHAPTER 1: QUICK SUMMARY 1(6)
1.1 Overview of this book
1(1)
1.2 Background knowledge
2(2)
1.3 Supporting Web site
4(1)
1.4 Quick summary: choosing and performing an ANOVA
4(3)
CHAPTER 2: UNDERSTANDING THE BASICS 7(54)
2.1 The basic logic and assumptions of ANOVA
7(6)
2.1.1 A 'model' that describes and predicts some data
7(1)
2.1.2 An example: data and a structural model
7(2)
2.1.3 The null hypothesis
9(1)
2.1.4 The assumptions of ANOVA
9(1)
2.1.5 The logic of ANOVA
10(2)
2.1.6 Expected mean squares (EMS)
12(1)
2.2 The calculations behind a one-way ANOVA (one BS factor)
13(7)
2.2.1 Calculations using means (preferred) or totals
13(1)
2.2.2 Sums of squares: calculating SStotal, SStreatment, and SSerror
13(3)
2.2.3 Degrees of freedom
16(1)
2.2.4 Mean squares
17(1)
2.2.5 The F test
17(1)
2.2.6 ANOVA summary table
18(1)
2.2.7 SStreatment for unequal sample sizes
19(1)
2.2.8 Pictorial representation
19(1)
2.3 Regression ANOVA: the other way to understand the logic
20(4)
2.3.1 Linear regression in terms of sums of squares
20(3)
2.3.2 Pictorial representation
23(1)
2.3.3 Linear regression as an ANOVA
23(1)
2.4 Factors versus covariates
24(1)
2.5 Assumptions of ANOVA involving covariates
25(1)
2.6 ANOVA with two between-subjects factors
26(9)
2.6.1 Main effects, interactions, simple effects, and a structural model
27(2)
2.6.2 Expected mean squares
29(1)
2.6.3 Degrees of freedom
30(1)
2.6.4 Sums of squares
30(2)
2.6.5 Relating SS calculations to the structural model
32(1)
2.6.6 ANOVA table
32(2)
2.6.7 Pictorial representation
34(1)
2.7 Within-subjects (repeated measures) ANOVA
35(4)
2.7.1 Structural model
35(1)
2.7.2 Degrees of freedom
36(1)
2.7.3 Sums of squares
37(1)
2.7.4 EMS and ANOVA summary table
37(2)
2.8 Assumptions of within-subjects ANOVA: 'sphericity'
39(3)
2.8.1 Short version
39(1)
2.8.2 Long version
40(2)
2.9 Missing data in designs involving within-subjects factors
42(1)
2.10 Mixed ANOVA (with both BS and WS factors)
43(5)
2.10.1 Structural model
44(1)
2.10.2 Degrees of freedom
45(1)
2.10.3 Sums of squares
45(2)
2.10.4 ANOVA table
47(1)
2.11 Fixed and random factors
48(1)
2.12 Additional material (ADVANCED)
49(12)
2.12.1 Notation for variances and mean squares in EMS expressions
49(1)
2.12.2 Expected value of F
50(1)
2.12.3 A χ² distribution is the sum of squared z scores
51(2)
2.12.4 Relationship between the sample variance and the χ² distribution
53(1)
2.12.5 The F distribution
54(1)
2.12.6 Comparing two variances with an F test
55(1)
2.12.7 ANOVA: comparing two mean-square values with an F test
56(2)
2.12.8 Relating SS calculations to the model for one-way ANOVA
58(3)
CHAPTER 3: PRACTICAL ANALYSIS 61(48)
3.1 Reminder: assumptions of ANOVA
61(2)
3.2 Reminder: assumption of ANOVA with WS factors
63(1)
3.3 Consequences of violating the assumptions of ANOVA
64(1)
3.4 Exploratory data analysis and transformations
65(6)
3.4.1 Plot your data
65(1)
3.4.2 Outliers
66(1)
3.4.3 Transformations
67(4)
3.5 Performing the ANOVA
71(1)
3.6 Residuals
72(5)
3.7 Further analysis: after the ANOVA has been run
77(17)
3.7.1 Main effects, interactions, and simple effects revisited
77(1)
3.7.2 Conducting simple-effects analysis
78(1)
3.7.3 A fallacy to avoid: when A differs from C but B doesn't
79(1)
3.7.4 A fallacy to avoid: simple effects without interactions
80(2)
3.7.5 Determining the effects of a factor with >2 levels
82(1)
3.7.6 Multiple comparisons: a problem
82(1)
3.7.7 Post hoc tests: a problem
83(1)
3.7.8 The special case of three groups: multiple t tests are OK
84(2)
3.7.9 Otherwise: a variety of post hoc tests
86(4)
3.7.10 Post hoc tests for within-subject factors
90(1)
3.7.11 A priori tests: planned contrasts
90(1)
3.7.12 Apparent inconsistency between the F test and post hoc tests
91(1)
3.7.13 SPSS's default pairwise comparison post hoc tests
91(3)
3.8 Drawing pictures: error bars for different comparisons
94(5)
3.8.1 Error bars for t tests: between-subjects comparisons
94(1)
3.8.2 Error bars for t tests: within-subjects comparisons
95(3)
3.8.3 Error bars for an ANOVA: between-subjects designs
98(1)
3.8.4 Error bars for an ANOVA: effects in mixed designs
98(1)
3.9 Summarizing your methods: a writing guide
99(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)
FURTHER READING 433(2)
REFERENCES 435(4)
INDEX 439

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