PREFACE |
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xv | |
CHAPTER 1: QUICK SUMMARY |
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1 | (6) |
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1.1 Overview of this book |
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1 | (1) |
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2 | (2) |
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4 | (1) |
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1.4 Quick summary: choosing and performing an ANOVA |
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4 | (3) |
CHAPTER 2: UNDERSTANDING THE BASICS |
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7 | (54) |
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2.1 The basic logic and assumptions of ANOVA |
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7 | (6) |
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2.1.1 A 'model' that describes and predicts some data |
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7 | (1) |
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2.1.2 An example: data and a structural model |
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7 | (2) |
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2.1.3 The null hypothesis |
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9 | (1) |
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2.1.4 The assumptions of ANOVA |
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9 | (1) |
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10 | (2) |
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2.1.6 Expected mean squares (EMS) |
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12 | (1) |
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2.2 The calculations behind a one-way ANOVA (one BS factor) |
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13 | (7) |
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2.2.1 Calculations using means (preferred) or totals |
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13 | (1) |
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2.2.2 Sums of squares: calculating SStotal, SStreatment, and SSerror |
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13 | (3) |
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16 | (1) |
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17 | (1) |
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17 | (1) |
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2.2.6 ANOVA summary table |
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18 | (1) |
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2.2.7 SStreatment for unequal sample sizes |
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19 | (1) |
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2.2.8 Pictorial representation |
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19 | (1) |
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2.3 Regression ANOVA: the other way to understand the logic |
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20 | (4) |
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2.3.1 Linear regression in terms of sums of squares |
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20 | (3) |
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2.3.2 Pictorial representation |
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23 | (1) |
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2.3.3 Linear regression as an ANOVA |
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23 | (1) |
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2.4 Factors versus covariates |
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24 | (1) |
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2.5 Assumptions of ANOVA involving covariates |
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25 | (1) |
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2.6 ANOVA with two between-subjects factors |
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26 | (9) |
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2.6.1 Main effects, interactions, simple effects, and a structural model |
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27 | (2) |
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2.6.2 Expected mean squares |
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29 | (1) |
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30 | (1) |
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30 | (2) |
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2.6.5 Relating SS calculations to the structural model |
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32 | (1) |
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32 | (2) |
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2.6.7 Pictorial representation |
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34 | (1) |
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2.7 Within-subjects (repeated measures) ANOVA |
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35 | (4) |
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35 | (1) |
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36 | (1) |
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37 | (1) |
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2.7.4 EMS and ANOVA summary table |
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37 | (2) |
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2.8 Assumptions of within-subjects ANOVA: 'sphericity' |
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39 | (3) |
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39 | (1) |
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40 | (2) |
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2.9 Missing data in designs involving within-subjects factors |
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42 | (1) |
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2.10 Mixed ANOVA (with both BS and WS factors) |
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43 | (5) |
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44 | (1) |
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2.10.2 Degrees of freedom |
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45 | (1) |
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45 | (2) |
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47 | (1) |
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2.11 Fixed and random factors |
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48 | (1) |
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2.12 Additional material (ADVANCED) |
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49 | (12) |
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2.12.1 Notation for variances and mean squares in EMS expressions |
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49 | (1) |
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2.12.2 Expected value of F |
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50 | (1) |
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2.12.3 A χ² distribution is the sum of squared z scores |
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51 | (2) |
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2.12.4 Relationship between the sample variance and the χ² distribution |
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53 | (1) |
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2.12.5 The F distribution |
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54 | (1) |
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2.12.6 Comparing two variances with an F test |
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55 | (1) |
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2.12.7 ANOVA: comparing two mean-square values with an F test |
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56 | (2) |
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2.12.8 Relating SS calculations to the model for one-way ANOVA |
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58 | (3) |
CHAPTER 3: PRACTICAL ANALYSIS |
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61 | (48) |
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3.1 Reminder: assumptions of ANOVA |
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61 | (2) |
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3.2 Reminder: assumption of ANOVA with WS factors |
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63 | (1) |
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3.3 Consequences of violating the assumptions of ANOVA |
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64 | (1) |
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3.4 Exploratory data analysis and transformations |
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65 | (6) |
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65 | (1) |
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66 | (1) |
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67 | (4) |
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71 | (1) |
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72 | (5) |
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3.7 Further analysis: after the ANOVA has been run |
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77 | (17) |
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3.7.1 Main effects, interactions, and simple effects revisited |
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77 | (1) |
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3.7.2 Conducting simple-effects analysis |
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78 | (1) |
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3.7.3 A fallacy to avoid: when A differs from C but B doesn't |
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79 | (1) |
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3.7.4 A fallacy to avoid: simple effects without interactions |
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80 | (2) |
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3.7.5 Determining the effects of a factor with >2 levels |
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82 | (1) |
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3.7.6 Multiple comparisons: a problem |
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82 | (1) |
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3.7.7 Post hoc tests: a problem |
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83 | (1) |
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3.7.8 The special case of three groups: multiple t tests are OK |
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84 | (2) |
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3.7.9 Otherwise: a variety of post hoc tests |
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86 | (4) |
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3.7.10 Post hoc tests for within-subject factors |
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90 | (1) |
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3.7.11 A priori tests: planned contrasts |
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90 | (1) |
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3.7.12 Apparent inconsistency between the F test and post hoc tests |
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91 | (1) |
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3.7.13 SPSS's default pairwise comparison post hoc tests |
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91 | (3) |
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3.8 Drawing pictures: error bars for different comparisons |
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94 | (5) |
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3.8.1 Error bars for t tests: between-subjects comparisons |
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94 | (1) |
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3.8.2 Error bars for t tests: within-subjects comparisons |
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95 | (3) |
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3.8.3 Error bars for an ANOVA: between-subjects designs |
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98 | (1) |
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3.8.4 Error bars for an ANOVA: effects in mixed designs |
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98 | (1) |
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3.9 Summarizing your methods: a writing guide |
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99 | (2) |
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3.10 Additional material (ADVANCED) |
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101 | (8) |
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3.10.1 Error bars for t tests: between-subjects comparisons: SEMs |
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101 | (2) |
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3.10.2 Error bars for t tests: between-subjects comparisons: CIs |
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103 | (2) |
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3.10.3 Error bars for t tests: between-subjects comparisons: SDs |
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105 | (1) |
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3.10.4 Obtaining SEDs from an ANOVA table |
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105 | (4) |
CHAPTER 4: PITFALLS AND COMMON ISSUES |
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109 | (10) |
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4.1 Time in within-subjects (repeated measures) designs |
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109 | (1) |
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4.2 Analysis of pre-test versus post-test data |
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109 | (1) |
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4.3 Observing subjects repeatedly to increase power |
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110 | (2) |
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4.4 'It's significant in this subject...' |
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112 | (2) |
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4.5 Should I add/remove a factor? Full and reduced models |
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114 | (1) |
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4.6 Should I add/remove/collapse over levels of a factor? |
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115 | (4) |
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4.6.1 Adding and removing levels by adding new observations |
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116 | (1) |
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4.6.2 Collapsing over or subdividing levels |
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117 | (2) |
CHAPTER 5: USING SPSS FOR ANOVA |
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119 | (28) |
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5.1 Running ANOVAs using SPSS |
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119 | (5) |
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5.1.1 Analysis of variance |
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119 | (1) |
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5.1.2 Organizing and reorganizing your data |
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120 | (1) |
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120 | (1) |
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121 | (1) |
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5.1.5 Options, including homogeneity-of-variance tests |
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121 | (2) |
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123 | (1) |
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5.2 Interpreting the output |
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124 | (15) |
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Tip: pairwise comparisons for interactions |
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136 | (3) |
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5.3. Further analysis: selecting cases |
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139 | (3) |
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5.4 The 'intercept', 'total', and 'corrected total' terms |
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142 | (5) |
CHAPTER 6: CONTRASTS AND TRENDS |
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147 | (14) |
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147 | (8) |
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6.1.1 About linear contrasts |
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147 | (1) |
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6.1.2 Type I error rates with planned contrasts |
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148 | (2) |
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6.1.3 Orthogonal contrasts |
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150 | (1) |
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6.1.4 Linear contrasts in SPSS |
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151 | (3) |
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6.1.5 Contrasts in multifactor designs—an overview |
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154 | (1) |
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6.2 Trend analysis: the effects of quantitative factors |
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155 | (6) |
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155 | (2) |
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6.2.2 Trend analysis in SPSS |
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157 | (1) |
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6.2.3 Trend analysis, multiple regression, and polynomial ANCOVA |
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158 | (3) |
CHAPTER 7: ADVANCED TOPICS |
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161 | (54) |
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7.1 Rules for calculating sums of squares |
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161 | (2) |
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7.1.1 Partitioning sums of squares |
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161 | (1) |
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7.1.2 General rule for calculating sums of squares |
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161 | (2) |
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7.2 Rules for calculating degrees of freedom |
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163 | (1) |
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7.3 Expected mean squares (EMS) and error terms |
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164 | (10) |
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7.3.1 Rules for obtaining expected mean squares (EMS) |
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165 | (3) |
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7.3.2 Choosing an error term |
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168 | (2) |
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7.3.3 Error terms in models including random factors (complicated) |
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170 | (3) |
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7.3.4 Pooling error terms |
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173 | (1) |
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7.4 Unequal group sizes and non-orthogonal sums of squares |
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174 | (5) |
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7.4.1 Proportional cell frequencies |
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174 | (1) |
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7.4.2 Disproportionate cell frequencies—a problem |
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175 | (2) |
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7.4.3 Correlated predictors in general—a problem |
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177 | (2) |
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7.5 How computers perform ANOVA: general linear models |
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179 | (23) |
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7.5.1 The basic idea of a GLM, illustrated with multiple regression |
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180 | (1) |
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7.5.2 Using a GLM for simple ANOVA: the design matrix |
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181 | (2) |
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7.5.3 Example of a GLM for a one-way ANOVA |
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183 | (2) |
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7.5.4 GLM for two-way ANOVA and beyond |
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185 | (2) |
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7.5.5 F statistics for GLMs: comparing full and reduced models |
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187 | (1) |
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7.5.6 An overview of GLM designs |
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188 | (6) |
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7.5.7 GLM designs involving random effects |
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194 | (1) |
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7.5.8 A hint at multivariate analysis: MANOVA |
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195 | (1) |
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7.5.9 Linear contrasts with a GLM |
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196 | (1) |
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7.5.10 GLMs and custom contrasts in SPSS |
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197 | (5) |
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202 | (13) |
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7.6.1 Effect size in the language of multiple regression |
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203 | (6) |
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7.6.2 Effect size in the language of ANOVA |
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209 | (6) |
CHAPTER 8: SPECIFIC DESIGNS |
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215 | (152) |
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8.1 One between-subjects (BS) factor |
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217 | (5) |
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222 | (3) |
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225 | (3) |
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8.4 One within-subjects (WS) factor |
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228 | (5) |
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233 | (5) |
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238 | (6) |
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8.7 One BS and one WS factor |
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244 | (7) |
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8.8 Two BS factors and one WS factor |
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251 | (5) |
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8.9 One BS factor and two WS factors |
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256 | (5) |
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8.10 Other ANOVA designs with BS and/or WS factors |
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261 | (3) |
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8.11 One BS covariate (linear regression) |
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264 | (7) |
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8.12 One BS covariate and one BS factor |
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271 | (15) |
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8.12.1 The covariate and factor do not interact |
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271 | (10) |
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8.12.2 The covariate and factor interact |
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281 | (5) |
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8.13 One BS covariate and two BS factors |
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286 | (3) |
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8.14 Two or more BS covariates (multiple regression) |
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289 | (3) |
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8.15 Two or more BS covariates and one or more BS factors |
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292 | (3) |
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295 | (4) |
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8.17 One WS covariate and one BS factor |
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299 | (8) |
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8.17.1 The covariate and factor do not interact |
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299 | (4) |
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8.17.2 The covariate and factor interact |
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303 | (4) |
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8.18 Hierarchical designs |
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307 | (21) |
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8.18.1 Subjects within groups within treatments (S/G/A) |
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307 | (4) |
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8.18.2 Groups versus individuals |
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311 | (1) |
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8.18.3 Adding a further within-group, BS variable (S/GB/A) |
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312 | (2) |
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8.18.4 Adding a within-subjects variable (US/GB/A) |
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314 | (2) |
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8.18.5 Nesting within-subjects variables, such as V/US/A |
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316 | (3) |
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8.18.6 The split-split plot design |
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319 | (5) |
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8.18.7 Three levels of relatedness |
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324 | (4) |
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8.19 Latin square designs |
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328 | (23) |
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8.19.1 Latin squares in experimental design |
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328 | (1) |
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8.19.2 The analysis of a basic Latin square |
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329 | (3) |
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8.19.3 A x B interactions in a single Latin square |
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332 | (2) |
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8.19.4 More subjects than rows: (a) using several squares |
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334 | (3) |
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8.19.5 More subjects than rows: (b) using one square several times |
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337 | (4) |
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8.19.6 BS designs using Latin squares (fractional factorial designs) |
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341 | (3) |
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8.19.7 Several-squares design with a BS factor |
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344 | (3) |
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8.19.8 Replicated-squares design with a BS factor |
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347 | (4) |
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8.20 Agricultural terminology and designs |
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351 | (16) |
CHAPTER 9: MATHEMATICS |
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367 | (36) |
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367 | (7) |
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367 | (2) |
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369 | (3) |
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9.1.3 The inverse of a matrix |
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372 | (1) |
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9.1.4 Matrix transposition |
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373 | (1) |
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374 | (5) |
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374 | (1) |
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9.2.2 Simple, non-trigonometric derivatives |
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375 | (1) |
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9.2.3 Rules for differentiation |
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375 | (1) |
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9.2.4 Derivatives of a vector function |
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376 | (1) |
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9.2.5 Partial derivatives |
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376 | (1) |
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9.2.6 The chain rule for partial derivatives |
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377 | (1) |
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9.2.7 Illustrations of partial derivatives |
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377 | (2) |
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9.3 Solving a GLM (an overdetermined system of equations) |
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379 | (3) |
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9.4 Singular value decomposition to solve GLMs |
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382 | (5) |
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9.4.1 Eigenvectors and eigenvalues |
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384 | (1) |
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9.4.2 Singular value decomposition |
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385 | (1) |
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9.4.3 An underdetermined set of equations: the role of expectations |
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386 | (1) |
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9.5 Random variables, means, and variances |
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387 | (10) |
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387 | (1) |
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9.5.2 Random variables; definition of mean and variance |
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388 | (1) |
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9.5.3 Continuous random variables |
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389 | (1) |
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390 | (1) |
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391 | (1) |
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9.5.6 Distribution of a set of means: the standard error of the mean |
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392 | (3) |
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9.5.7 The sample mean and SD are unbiased estimators of μ and σ² |
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395 | (2) |
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397 | (1) |
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9.7 Rules for powers and logarithms |
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397 | (1) |
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398 | (5) |
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Basic notation in probability |
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398 | (1) |
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Basic laws of probability |
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398 | (1) |
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399 | (1) |
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Bayes' theorem and Bayesian inference |
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400 | (3) |
CHAPTER 10: STATISTICAL TABLES |
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403 | (6) |
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10.1 Critical values of t |
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403 | (1) |
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10.2 Critical values of F |
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404 | (4) |
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10.3 Polynomial trend coefficients |
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408 | (1) |
GLOSSARY |
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409 | (24) |
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409 | (2) |
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411 | (1) |
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412 | (21) |
FURTHER READING |
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433 | (2) |
REFERENCES |
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435 | (4) |
INDEX |
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439 | |