Analysis of Variance for Sensory Data

Analysis of Variance (ANOVA) is a statistical technique used in a number of chemical areas including the food industry. This handbook is a highly readable guide to the uses of the very important technique of ANOVA applied to sensory analysis. It is unique in stressing the practical implications of the topic rather than the theoretical background. This is important since ANOVA is a complex statistical technique which can be misunderstood and misused by sensory analysts. Typical problems from sensory analysis are used as examples.

About the Authors Per Lea and Tormod Næs are both Research Scientists in Statistics at Matforsk, the Norwegian Food Research Institute; Marit Rødbotten is Senior Sensory Analyst at Matforsk.

Preface

ix

(4)

Acknowledgments

xiii

1 Introduction

1

(5)

1.1 Sensory Analysis--Problems and Distinctions

3

(1)

1.2 Use of Different Measurement Scales

4

(2)

2 Factorial designs

6

(11)

2.1 Introduction

6

(1)

2.2 A Simple Example

6

(1)

2.3 A Good and a Poor Design Strategy

7

(1)

2.4 Important Quantities to Compute

8

(3)

2.4.1 Main effects

8

(2)

2.4.2 Interaction

10

(1)

2.5 Replicates of the Experiment

11

(2)

2.6 Main Effects and Interactions in More General Factorial Experiments

13

(2)

2.7 Statistical Models for Factorial Experiments

15

(1)

2.8 Some Final Remarks

16

(1)

3 Further Aspects of Design and Modelling

17

(6)

3.1 Fractional Factorial Designs

17

(1)

3.2 Randomisation and Blocking

18

(1)

3.3 Unfolding Factors

19

(1)

3.4 The Effect of Averaging Over Assessors

19

(1)

3.5 Reasons for Individual Differences Among Assessors

20

(1)

3.6 Nested Effects

21

(2)

4. Analysis of Variance

23

(14)

4.1 Introduction

23

(1)

4.2 Notation and Terminology

24

(1)

4.3 Example 1: One-Way ANOVA

24

(2)

4.4 Example 2: Two-Way ANOVA

26

(1)

4.5 Total Variability Split Up According to Factors of Variation

27

(1)

4.6 Hypothesis Testing

28

(5)

4.6.1 Analysis of variance table

29

(2)

4.6.2 Selecting a significance level for the tests

31

(2)

4.7 A Measure of Model Fit (R2)

33

(1)

4.8 Tools for Detecting Differences Among Assessors and Replicates

33

(3)

4.9 An Example of the Analysis of Assessor Performance

36

(1)

5 Random or Fixed Assessors in Analysis of Variance

37

(9)

5.1 Introduction

37

(2)

5.2 Examples and ANOVA Table

39

(1)

5.3 Fixed or Random Assessor Effects

40

(1)

5.4 Testing in Mixed Models

41

(2)

5.5 Violation of Standard Mixed Model Assumptions

43

(1)

5.6 A Possible Hypothesis That Can Solve the Problem

44

(2)

6 More Complex ANOVA Situations

46

(10)

6.1 Introduction

46

(1)

6.2 General Models

46

(2)

6.3 The Analysis of Variance Table

48

(1)

6.4 Example of a Four-Way Analysis of Variance

49

(1)

6.5 F-tests Developed from Expected Mean Squares

50

(4)

6.6 The Underlying Assumptions for ANOVA Tests

54

(2)

7 Replicates in Sensory Analysis

56

(8)

7.1 Introduction

56

(1)

7.2 Replicates

56

(5)

7.2.1 Replicate structure 1

57

(2)

7.2.2 Replicate structue 2

59

(1)

7.2.3 Other replicate structure (structures 3 and 4)

60

(1)

7.3 Different Ways of Selecting Samples from Units

61

(1)

7.4 How Many Replicates Should We Use?

62

(2)

8 Multiple Comparisons

64

(7)

8.1 Introduction

64

(1)

8.2 Strategies for Multiple Comparisons

65

(2)

8.3 Methods for Multiple Comparisons

67

(3)

8.3.1 Tukey's test

67

(1)

8.3.2 Newman-Keul's test

67

(2)

8.3.3 Duncan's test

69

(1)

8.3.4 Comparing the three test

69

(1)

8.4 Multiple Comparisons in Multiway and Mixed Model ANOVAs

70

(1)

9 Two Detailed Examples

71

(7)

9.1 Choice of Examples

71

(1)

9.2 Example 1

71

(4)

9.3 Example 2

75

(2)

References and Relevant Literature

78

(2)

Appendix 1 The Sums of Squares for One-Way and Two-Way ANOVA

80

(2)

A.1.1 Two-Way ANOVA

80

(1)

A.1.2 One-Way ANOVA

81

(1)

Appendix 2 Interpretations of Degrees of Freedom

82

(1)

Appendix 3 Computer Programs for Example 1 in Chapter 9

83

(4)

A.3.1 SAS (Version 6)

83

(1)

A.3.2 SYSTAT (Version 6 for DOS)

84

(1)

A.3.3 STATISTIX (Version 4.1)

85

(1)

A.3.4 MINITAB (release 10.51 Xtra)

86

(1)

Appendix 4 Computer Programs for Example 2 in Chapter 9

87

(3)

A.4.1 SAS (Version 6)

87

(1)

A.4.2 SYSTAT (Version 6 for DOS)

87

(1)

A.4.3 STATISTIX (Version 4.1)

88

(1)

A.4.4 MINITAB (Release 10.51 Xtra)

88

(2)

Appendix 5 Selected Words and Expressions Explained

90

(7)

Appendix 6 Symbols

97

(2)

Index

99

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Analysis of Variance for Sensory Data

Summary

Author Biography

Table of Contents

Supplemental Materials