Sensory Discrimination Tests and Measurements Statistical Principles, Procedures and Tables Sensory Discrimination Tests and Measurements Statistical Principles, Procedures and Tables Jian Bi Sensometrics Research and Service Richmond, Virginia, USA Jian Bi is a Senior Statistician and the President of Sensometrics Research and Service, Richmond, Virginia C 2006 Jian Bi All rights reserved Blackwell Publishing Professional 2121 State Avenue, Ames, Iowa 50014, USA Orders: 1-800-862-6657 Office: 1-515-292-0140 Fax: 1-515-292-3348 Web site: www.blackwellprofessional.com Blackwell Publishing Ltd 9600 Garsington Road, Oxford OX4 2DQ, UK Tel.: +44 (0)1865 776868 Blackwell Publishing Asia 550 Swanston Street, Carlton, Victoria 3053, Australia Tel.: +61 (0)3 8359 1011 Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by Jian Bi, provided that the base fee of $.10 per copy is paid directly to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 For those organizations that have been granted a photocopy license by CCC, a separate system of payments has been arranged The fee codes for users of the Transactional Reporting Service are ISBN-13: 978-0-8138-1111-6; ISBN-10: 0-8138-1111-2/2005 $.10 First edition, 2006 Library of Congress Cataloging-in-Publication Data Bi, Jian, 1949– Sensory discrimination tests and measurements : statistical principles, procedures, and tables / Jian Bi.– 1st ed p cm Includes bibliographical references ISBN-13: 978-0-8138-1111-6 (alk paper) ISBN-10: 0-8138-1111-2 Agriculture—Statistical methods Sensory discrimination—Statistical methods I Title S566.55B55 2006 630 72 7—dc22 2005017101 The last digit is the print number: To Yulin Contents Preface Introduction 1.1 A Brief Review of Sensory Analysis Methodologies 1.2 Method, Test, and Measurement 1.3 Standard Discrimination Methods 1.4 Classification of Sensory Discrimination Methods References ix 1 2 Standard Discrimination Tests 2.1 Binomial Model for Discrimination Testing 2.2 Discrimination Tests Using Forced-Choice Methods 2.3 Discrimination Tests Using the Methods with Response Bias References 6 12 20 Statistical Power Analysis for Standard Discrimination Tests 3.1 Introduction 3.2 Power and Sample Size for Forced-Choice Methods 3.3 Power and Sample Size for the Methods with Response Bias 3.4 Efficiency Comparisons of Discrimination Tests References 21 21 22 27 38 44 Modified Discrimination Tests 4.1 The Modified Triangle Test 4.2 The Degree of Difference Test 4.3 The Double Discrimination Test 4.4 The Preference Test with “No Preference” Option References 45 45 56 61 72 76 Multiple-Sample Discrimination Tests 5.1 Multiple-Sample Comparison Based on Proportions 5.2 Multiple-Sample Comparison Based on Ranks 5.3 Multiple-Sample Comparison Based on Categorical Scales References 78 78 82 98 104 Replicated Discrimination Tests: Beta-Binomial Model 6.1 Introduction 6.2 The Beta-Binomial Distribution 6.3 Estimation of Parameters of Beta-Binomial Model 106 106 108 109 vii viii contents 6.4 Applications of Beta-Binomial Model in Replicated Tests 6.5 Testing Power and Sample Size for Beta-Binomial Tests References Appendix 6A 113 122 127 129 Replicated Discrimination Tests: Corrected Beta-Binomial Model 7.1 Introduction 7.2 The Corrected Beta-Binomial Distribution 7.3 Estimation of Parameters of Corrected Beta-Binomial Model 7.4 Statistical Testing for Parameters in Corrected Beta-Binomial Model 7.5 Testing Power and Sample Size References Appendix 7A 138 138 138 142 146 148 150 151 Replicated Discrimination Tests: Dirichlet-Multinomial Model 8.1 The Dirichlet-Multinomial Distribution 8.2 Estimation of Parameters of Dirichlet-Multinomial Model 8.3 Applications of DM model in Replicated Tests 8.4 Testing Power for Dirichlet-Multinomial Model References 163 163 165 167 179 182 Measurements of Sensory Difference: Thurstonian Model 9.1 Introduction 9.2 Thurstonian ␦ 9.3 Variance of d 9.4 Tables for d and Variance of d References 184 184 185 190 237 240 10 Statistical Analysis for d Data 10.1 Estimates of Population or Group d 10.2 Statistical Inference for d Data References 243 243 248 254 11 Similarity Testing 11.1 Introduction 11.2 Similarity Testing for Preference 11.3 Similarity Testing Using Forced-Choice Methods 11.4 Similarity Testing Using the A–Not A and the Same–Different Methods References Appendix 11A 255 255 256 259 Appendix A List of S-Plus Codes Author Index Subject Index 261 268 269 287 289 293 Preface Discriminative analysis, including discrimination tests and measurements, is the most fundamental type of methodology in sensory science The validation of the methodology depends to some extent on sound statistical models The objective of this book is to deal with statistical aspects of the methodology and to provide the reader with statistical principles, procedures and tables for some methods The book attempts to give a unified picture of the state of the subject and to reflect some features of advanced sensory discriminative analysis This book consists of eleven chapters It is organized as follows: Chapter briefly reviews sensory methodologies with emphasis on six standard, widely used discrimination methods: the 2-AFC, 3-AFC, Duo–Trio, Triangle, A–Not A, and the Same–Different methods Chapters to discuss discrimination testing including standard discrimination tests (Chapters 2–3), modified discrimination tests (Chapter 4), and multiple-sample discrimination tests (Chapter 5) under the conventional assumption that the consumer population is composed of “discriminator” and “non-discriminator” and panelists of a laboratory panel have the same discrimination ability Chapters to present a unified approach to replicated discrimination tests using a beta-binomial framework under the assumption that discrimination ability or preference for each individual consumer and panelist is not a constant but a random variable The assumptions under discrimination testing discussed in Chapters to and Chapters to are philosophically different Chapters to 10 are devoted to a discussion on sensory measurement using Thurstonian discriminal distance ␦ (or d ) Chapter 11, the last chapter, discusses similarity testing, which is practically and theoretically important but often confusing The book is intended for researchers and practitioners in the sensory and consumer field and has been written keeping both the statistical and non-statistical readers in mind It is not difficult to apply most of the methods by following the numerical examples using the corresponding formulas and tables provided in the book For some of the methods involving complicated calculations, computer programs are needed Thanks to modern computer technology, calculations are much easier than before The extent of computational complication involved in a method should not be regarded as a major concern in the selection of methods For some statistical considerations behind the methodology and some mathematical derivations in the book, readers with a more statistical background will understand them without major difficulty Some S-PLUS codes, which appear in the book and are listed in Appendix A, are available from the author on request The author may be contacted via e-mail at BBDJCY@aol.com ix x preface Acknowledgments I am greatly indebted to the Series Editor, Dr Max Gacula, who encouraged me to write this book, reviewed the manuscript, and provided insightful comments I wish to express my gratitude to Professor Edgar Chambers, Dr Morten Meilgaard, Professor Michael O’Mahony, and Dr Daniel Ennis for their valuable support and help for the past years I would like to thank the publisher and my editors Mark Barrett, Dede Pedersen, Susan Borts, and Judi Brown at Blackwell Publishing and Suditi Srivastava at TechBooks for publishing my book and bringing the project to completion Finally, I wish to thank deeply my wife, Yulin, for her patience, understanding, and encouragement during the preparation of this book Jian Bi Sensory Discrimination Tests and Measurements: Statistical Principles, Procedures and Tables Jian Bi Copyright © 2006 by Jian Bi Introduction 1.1 A brief review of sensory analysis methodologies To conduct valid tests and to provide reliable sensory measurements are the main functions of sensory analysis Statistical inference is the theoretical basis of sensory tests Psychometrics, which provides invariable indexes, which is independent of methods, is the theoretical basis of sensory measurements Sensory analysis can be divided into two parts: laboratory sensory analysis and consumer sensory analysis In the laboratory sensory analysis, a trained panel is used as an analytical instrument to measure sensory properties of products In the consumer sensory analysis, a sample of specified consumer population is used to test and predict consumer responses for products The two types of sensory analysis have different goals and functions, but they share some of the same methodologies Discriminative analysis and descriptive analysis are the main classes of methodology for both the laboratory and consumer sensory analyses Discriminative analysis includes discrimination tests and measurements Discrimination tests are used to determine, usually using a 2-point scale or a ranking scale, whether a difference exists between treatments for confusable sensory properties of products Discrimination measurements are used to measure, using an index, the extent of the difference There are two sources of sensory differences: intensity and preference Discriminative analysis is referred to difference test when testing difference of intensity Discriminative analysis is referred to preference test when testing difference of preference Descriptive analysis is to determine, using a rating scale, how much a specific characteristic difference exists among products, which is quantitative descriptive analysis, and to characterize a product’s sensory attributes, which is qualitative descriptive analysis Quantitative descriptive analysis for preference is also called acceptance testing Acceptance or preference testing for a laboratory panel is of very limited value (Amerine et al 1965) However, the consumer discriminative and descriptive analyses for both intensity and references are valuable The laboratory difference testing, using a trained panel under controlled conditions, has been called the Sensory Evaluation I, whereas the consumer difference testing, using a sample of untrained consumers under ordinary using (eating) conditions, has been called the Sensory Evaluation II (O’Mahony 1995) They are different types of difference testing Misusing the two types of difference testing will lead to misleading conclusions The controversy over whether the consumer can be used for difference testing may ignore the fact that the laboratory and consumer difference tests have different goals and functions The distinction between the discriminative analysis and the quantitative descriptive analysis is not absolute from the viewpoint of modern sensory analysis The Thurstonian model that will be discussed in Chapters 9–10 of this book can be used for both discriminative sensory discrimination tests and measurements analysis and quantitative descriptive analysis The Thurstonian ␦ (or d ), which is a measure of sensory difference, can be obtained from any kind of scales used in discriminative and descriptive analyses In addition, rating scale, which is typically used in descriptive analysis, is also used in some modified discrimination tests Besides discriminative analysis and descriptive analysis, there are other classes of sensory methodologies, i.e., sensitivity analysis, time-intensity (TI) analysis, and similarity testing Sensitivity analysis is to determine sensory thresholds, including individual and population thresholds Threshold is a statistical concept It is an intensity that produces a response with a 0.5 probability There are many specific statistical methods for estimating and testing thresholds (for review, see, e.g., Bi and Ennis 1997) Time-intensity analysis or shelf-life analysis is used to determine the relationship between sensory intensity and time Survival analysis, which is a well-developed field, provides sound statistical methodology for TI analysis Time-intensity analysis is conventionally included in the descriptive analysis Considering the specifications of the methodology, it seems that TI analysis should be separated from the conventional descriptive analysis Similarity testing is relatively new and is not well developed in the sensory field Unlike discrimination testing, the objective of similarity testing is to demonstrate similarity rather than difference Similarity testing uses the same sensory analysis methods for discrimination tests, but different statistical models This book is primarily concerned with methodology, mainly in statistical aspects, of sensory discriminative analysis including laboratory and consumer discriminative analyses Similarity testing is briefly discussed in Chapter 11 1.2 Method, test, and measurement In this book, a distinction is made among the three terms: “sensory discrimination method,” “sensory discrimination test,” and “sensory discrimination measurement.” In sensory discriminative analysis, some procedures are used for experiments The procedures are called discrimination methods, e.g., the Duo–Trio method, the Triangular method When the discrimination procedures are used for statistical hypothesis testing, or in other words, when statistical testing is conducted for the data from a discrimination procedure, the procedure is called discrimination testing, e.g., the Duo–Trio test, the Triangular test When the discrimination procedures are used for measurement, or in other words, when an index, e.g., Thurstonian ␦ (or d ), is produced using the data from a discrimination procedure, the procedure is called discrimination measurement, e.g., the Duo–Trio measurement, the Triangular measurement 1.3 Standard discrimination methods Six standard and basic discrimination methods are the focus of this book They are: (a) The 2-Alternative Forced-Choice method (2-AFC) (Green and Swets 1966): This method is also called the paired comparison method (Dawson and Harris 1951, Peryam 1958) In this method, the panelist receives a pair of coded samples, A and B, for comparison on the basis of some specified sensory characteristic The possible pairs are AB and BA The panelist is asked to select the sample with the similarity testing Table 11A.15 n 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 281 Contd 45 52 58 64 70 77 83 89 96 102 109 115 121 128 134 140 147 153 160 166 173 179 185 192 198 205 211 218 224 231 237 244 46 52 58 65 71 77 84 90 96 103 109 116 122 128 135 141 148 154 160 167 173 180 186 193 199 205 212 218 225 231 238 244 47 53 59 65 72 78 84 91 97 103 110 116 123 129 135 142 148 155 161 167 174 180 187 193 200 206 213 219 225 232 238 245 47 53 60 66 72 79 85 91 98 104 110 117 123 130 136 142 149 155 162 168 175 181 187 194 200 207 213 220 226 233 239 246 48 54 60 67 73 79 86 92 98 105 111 117 124 130 137 143 149 156 162 169 175 182 188 195 201 207 214 220 227 233 240 246 48 55 61 67 74 80 86 93 99 105 112 118 124 131 137 144 150 157 163 169 176 182 189 195 202 208 215 221 227 234 240 247 49 55 62 68 74 81 87 93 100 106 112 119 125 132 138 144 151 157 164 170 176 183 189 196 202 209 215 222 228 235 241 247 50 56 62 69 75 81 88 94 100 107 113 119 126 132 139 145 151 158 164 171 177 184 190 196 203 209 216 222 229 235 242 248 50 57 63 69 75 82 88 95 101 107 114 120 126 133 139 146 152 158 165 171 178 184 191 197 204 210 216 223 229 236 242 249 51 57 63 70 76 82 89 95 102 108 114 121 127 133 140 146 153 159 166 172 178 185 191 198 204 211 217 224 230 236 243 249 Table 11A.16 Maximum number of correct responses for similarity testing using the 3-AFC and Triangular methods (␣ = 0.1, pd = 0.1) n 10 20 30 40 50 60 70 80 90 100 110 120 130 140 11 15 18 22 25 29 33 36 40 44 48 11 15 19 22 26 29 33 37 41 44 48 12 15 19 22 26 30 33 37 41 45 48 12 16 19 23 27 30 34 38 41 45 49 12 16 20 23 27 31 34 38 42 45 49 13 16 20 24 27 31 35 38 42 46 49 10 13 17 20 24 28 31 35 39 42 46 50 10 14 17 21 24 28 32 35 39 43 46 50 10 14 17 21 25 28 32 36 39 43 47 51 11 14 18 21 25 29 32 36 40 43 47 51 Contd 282 sensory discrimination tests and measurements Table 11A.16 n 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 Contd 51 55 59 63 66 70 74 78 82 85 89 93 97 101 104 108 112 116 120 123 127 131 135 139 143 52 55 59 63 67 71 74 78 82 86 89 93 97 101 105 109 112 116 120 124 128 131 135 139 143 52 56 60 63 67 71 75 78 82 86 90 94 97 101 105 109 113 117 120 124 128 132 136 140 143 52 56 60 64 68 71 75 79 83 86 90 94 98 102 105 109 113 117 121 125 128 132 136 140 144 53 57 60 64 68 72 75 79 83 87 91 94 98 102 106 110 113 117 121 125 129 133 136 140 144 53 57 61 64 68 72 76 80 83 87 91 95 99 102 106 110 114 118 122 125 129 133 137 141 145 54 57 61 65 69 72 76 80 84 88 91 95 99 103 107 110 114 118 122 126 130 133 137 141 145 54 58 61 65 69 73 77 80 84 88 92 96 99 103 107 111 115 118 122 126 130 134 138 141 145 54 58 62 66 69 73 77 81 85 88 92 96 100 104 107 111 115 119 123 127 130 134 138 142 146 55 58 62 66 70 74 77 81 85 89 93 96 100 104 108 112 115 119 123 127 131 135 138 142 146 Table 11A.17 Maximum number of correct responses for similarity testing using the 3-AFC and Triangular methods (␣ = 0.1, pd = 0.2) n 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 10 14 18 22 26 31 35 39 44 48 52 57 61 66 70 74 79 83 88 92 10 14 18 22 27 31 35 40 44 48 53 57 62 66 70 75 79 84 88 93 10 14 19 23 27 31 36 40 45 49 53 58 62 66 71 75 80 84 89 93 11 15 19 23 28 32 36 41 45 49 54 58 63 67 71 76 80 85 89 94 11 15 20 24 28 32 37 41 45 50 54 59 63 67 72 76 81 85 90 94 12 16 20 24 28 33 37 41 46 50 55 59 63 68 72 77 81 86 90 94 12 16 20 25 29 33 38 42 46 51 55 59 64 68 73 77 82 86 90 95 12 17 21 25 29 34 38 42 47 51 55 60 64 69 73 78 82 86 91 95 13 17 21 25 30 34 38 43 47 52 56 60 65 69 74 78 82 87 91 96 13 17 22 26 30 35 39 43 48 52 56 61 65 70 74 78 83 87 92 96 similarity testing Table 11A.17 n 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 283 Contd 97 101 106 110 115 119 123 128 132 137 141 146 150 155 159 164 168 97 102 106 110 115 119 124 128 133 137 142 146 151 155 160 164 169 98 102 106 111 115 120 124 129 133 138 142 147 151 156 160 165 169 98 102 107 111 116 120 125 129 134 138 143 147 152 156 161 165 170 98 103 107 112 116 121 125 130 134 139 143 148 152 157 161 166 170 99 103 108 112 117 121 126 130 135 139 144 148 153 157 162 166 171 99 104 108 113 117 122 126 131 135 140 144 149 153 158 162 167 171 100 104 109 113 118 122 127 131 136 140 145 149 154 158 163 167 172 100 105 109 114 118 123 127 132 136 140 145 149 154 158 163 167 172 101 105 110 114 119 123 127 132 136 141 145 150 154 159 163 168 172 Table 11A.18 Maximum number of correct responses for similarity testing using the 3-AFC and Triangular methods (␣ = 0.1, pd = 0.3) n 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 92 97 102 107 112 117 122 127 132 138 143 12 17 22 27 31 36 41 46 51 56 62 67 72 77 82 87 92 97 102 107 112 118 123 128 133 138 143 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 93 98 103 108 113 118 123 128 134 139 144 13 18 23 28 32 37 42 47 52 58 63 68 73 78 83 88 93 98 103 108 114 119 124 129 134 139 144 13 18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 94 99 104 109 114 119 124 129 135 140 145 14 19 24 29 33 38 43 48 53 59 64 69 74 79 84 89 94 99 104 109 115 120 125 130 135 140 145 10 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 95 100 105 110 115 120 125 130 136 141 146 10 15 20 25 29 34 39 44 49 54 60 65 70 75 80 85 90 95 100 105 110 116 121 126 131 136 141 146 11 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 96 101 106 111 116 121 126 131 137 142 147 11 16 21 26 30 35 40 45 50 55 61 66 71 76 81 86 91 96 101 106 111 117 122 127 132 137 142 147 Contd 284 sensory discrimination tests and measurements Table 11A.18 n 300 310 320 330 340 350 360 370 380 390 Contd 148 153 158 163 169 174 179 184 189 194 148 154 159 164 169 174 179 185 190 195 149 154 159 164 170 175 180 185 190 195 149 155 160 165 170 175 180 186 191 196 150 155 160 165 171 176 181 186 191 196 150 156 161 166 171 176 181 187 192 197 151 156 161 166 172 177 182 187 192 197 152 157 162 167 172 177 182 188 193 198 152 157 162 168 173 178 183 188 193 199 153 158 163 168 173 178 184 189 194 199 Table 11A.19 Maximum number of correct responses for similarity testing using the 3-AFC and Triangular methods (␣ = 0.1, pd = 0.4) n 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 14 19 25 30 36 41 47 53 58 64 70 76 81 87 93 99 104 110 116 122 127 133 139 145 151 156 162 168 174 180 186 191 197 203 209 215 221 14 20 25 31 36 42 48 53 59 65 70 76 82 88 93 99 105 111 116 122 128 134 140 145 151 157 163 169 175 180 186 192 198 204 209 215 221 15 20 26 31 37 42 48 54 60 65 71 77 82 88 94 100 105 111 117 123 129 134 140 146 152 158 163 169 175 181 187 193 198 204 210 216 222 10 15 21 26 32 37 43 49 54 60 66 72 77 83 89 95 100 106 112 118 123 129 135 141 147 152 158 164 170 176 181 187 193 199 205 211 216 222 10 16 21 27 32 38 44 49 55 61 66 72 78 84 89 95 101 107 112 118 124 130 136 141 147 153 159 165 170 176 182 188 194 200 205 211 217 223 11 16 22 27 33 39 44 50 56 61 67 73 78 84 90 96 101 107 113 119 125 130 136 142 148 154 159 165 171 177 183 188 194 200 206 212 218 224 11 17 22 28 33 39 45 50 56 62 68 73 79 85 90 96 102 108 114 119 125 131 137 143 148 154 160 166 172 177 183 189 195 201 207 212 218 224 12 17 23 28 34 40 45 51 57 62 68 74 80 85 91 97 103 108 114 120 126 132 137 143 149 155 161 166 172 178 184 190 195 201 207 213 219 225 12 18 23 29 35 40 46 52 57 63 69 74 80 86 92 97 103 109 115 121 126 132 138 144 150 155 161 167 173 179 184 190 196 202 208 214 219 225 13 18 24 30 35 41 46 52 58 64 69 75 81 86 92 98 104 110 115 121 127 133 138 144 150 156 162 168 173 179 185 191 197 202 208 214 220 226 similarity testing 285 Table 11A.20 Maximum number of correct responses for similarity testing using the 3-AFC and Triangular methods (␣ = 0.1, pd = 0.5) n 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 10 16 22 28 34 41 47 53 60 66 72 79 85 92 98 104 111 117 124 130 137 143 150 156 163 169 176 182 189 195 201 208 214 221 228 234 241 247 10 16 22 29 35 41 48 54 60 67 73 79 86 92 99 105 112 118 124 131 137 144 150 157 163 170 176 183 189 196 202 209 215 222 228 235 241 248 11 17 23 29 36 42 48 55 61 67 74 80 86 93 99 106 112 119 125 132 138 144 151 157 164 170 177 183 190 196 203 209 216 222 229 235 242 248 11 18 24 30 36 42 49 55 62 68 74 81 87 93 100 106 113 119 126 132 139 145 152 158 165 171 177 184 190 197 203 210 216 223 229 236 242 249 12 18 24 31 37 43 49 56 62 69 75 81 88 94 101 107 113 120 126 133 139 146 152 159 165 172 178 185 191 198 204 211 217 224 230 237 243 250 13 19 25 31 37 44 50 56 63 69 76 82 88 95 101 108 114 121 127 133 140 146 153 159 166 172 179 185 192 198 205 211 218 224 231 237 244 250 13 19 26 32 38 44 51 57 63 70 76 83 89 95 102 108 115 121 128 134 141 147 153 160 166 173 179 186 192 199 205 212 218 225 231 238 244 251 14 20 26 32 39 45 51 58 64 70 77 83 90 96 102 109 115 122 128 135 141 148 154 161 167 174 180 187 193 200 206 213 219 226 232 239 245 252 14 21 27 33 39 46 52 58 65 71 77 84 90 97 103 110 116 122 129 135 142 148 155 161 168 174 181 187 194 200 207 213 220 226 233 239 246 252 15 21 27 34 40 46 53 59 65 72 78 85 91 97 104 110 117 123 130 136 142 149 155 162 168 175 181 188 194 201 207 214 220 227 233 240 246 253 Sensory Discrimination Tests and Measurements: Statistical Principles, Procedures and Tables Jian Bi Copyright © 2006 by Jian Bi Appendix A List of a part of S-PLUS codes used in the book No Name Objective Page 10 11 12 13 14 fisherpower moditr forcpref kvisit extrank taplinst bradley bradleyv partchi smtest gsmtest bbest2 bbmaxg bbmax2 34 47 71 74 89 92 96 96 101 61 104 111 113 112 15 16 17 18 19 20 21 22 23 24 25 26 27 28 bbtest2 cbbpdf cbbmaxg dmf dmff cbval recbtest chipow3 ratdelg anadvn sddvn popudv dstest simanatest The power of the Fisher’s exact test Modified Triangle test based on Bradley–Harmon model Estimate of parameters in the double preference test Preference testing based on Ferris k-visit mode Preference testing based on Anderson statistic Preference testing based on Taplin statistic Bradley–Terry model for multiple paired comparisons Covariance matrix of estimators based on Bradley–Terry model Partition of chi-square test Stuart–Maxwell test for two correlated frequency vectors Generalized Stuart–Maxwell test for multiple correlated frequency vectors Moment estimate of parameters in BB mode with unequal replications Maximum likelihood estimate of parameters in BB mode using original data Maximum likelihood estimate of parameters in BB mode with equal replications using cumulative frequencies Test two sets of parameters in two BB populations The probability function of the corrected beta-binomial distribution Maximum likelihood estimate of parameters in CBB mode The probability function of the Dirichlet-multinomial distribution Log likelihood function for Dirichlet-multinomial model Moment estimate of C value in Dirichlet-multinomial model Replicated testing for multiple correlated proportions Power of replicated mixed designed A–Not A test Estimating d from rating data d and variance of d for the A–Not A method d and variance of d for the Same–Different method Estimating population or group d Test for multiple d s Similarity test using A–Not A method based on Dunnett and Gent’s model 117 140 145 164 167 166 179 182 189 240 240 246 250 265 287 Sensory Discrimination Tests and Measurements: Statistical Principles, Procedures and Tables Jian Bi Copyright © 2006 by Jian Bi Author Index A Abrams, D., 184, 241 Alf, E., Jr., 188, 241 Altham, P.M.E., 108, 127 Amerine, M.A., 1, 3, Anderson, D.A., 106, 127 Anderson, R.L., 86, 88, 104 Aust, L.B., 56, 57, 76 B Beard, S.A., 76 Beaver, R.J., 93, 104 Bement, T.R., 244, 254 Bennett, B.M., 18, 20, 33, 36, 37, 38, 44, 174, 182 Best, D.J., 87, 99, 100, 101, 104–105 Bi, J., 2, 5, 21, 27, 29, 39, 43, 44, 56, 58, 61, 76, 107, 117, 127, 138, 150, 163, 169, 170, 174, 182, 190, 215, 240 Birdsall, T.G., 185, 240, 241 Blackwelder, W.C., 255, 268 Bradley, J.L., 48, 77 Bradley, R.D., 21, 41, 44, 45, 46, 76, 93, 94, 96, 104, 185, 240, 241 Brier, S.S., 168, 172, 175, 182 Brittain, E., 29, 44 Brockhoff, P.B., 107, 127, 138, 150 Byer, A.J., 184, 241 Civille, G.V., Cochran, W.G., 80, 104, 174, 182, 244, 245, 254 Com-Nougue, C., 268 Conover, W.J., 90, 104 Cooper, M.M., 105 Cox, D.R., 108, 127 Craven, B.J., 186, 241 Creelman, C.D., 4, 5, 185, 188, 241 D Dai, H., 242 David, H.A., 39, 44, 185, 241 Davidson, R.R., 94, 104 Dawson, E.H., 2, 3, Delwiche, J., 184, 241 Dorfman, D.D., 188, 241 Dunnett, C.W., 255, 261, 268 Durbin, J., 85, 104 Dykstra, O., Jr., 93–94, 104 E Ehrenberg, A.S.C., 120, 127 Elliott, P.B., 186, 187, 241 Ennis, D.M., 2, 5, 21, 27, 29, 38, 39, 44, 107, 117, 127, 138, 150, 163, 182, 186, 240, 241, 242 Everitt, B.S., 60, 76 C F Carr, B.T., Carroll, S.P., 244, 254 Chapman, D.G., 34, 44, 181, 182 Chatfield, C., 120, 127 FDA, 244, 254, 255, 268 Ferris, G.E., 72, 73, 76, 106, 127 Fisher, R.A., 15, 20 Fleiss, J.L., 19, 20, 60, 76 289 290 author index Friedman, M., 82, 104 Frijters, J.E.R., 21, 39, 44, 184, 185, 186, 241 G Gacula, M.C., Jr., 45, 76, 87, 93, 104 Galanter, E., 190, 241 Geelhoed, E.N., 184, 241 Gent, M., 255, 261, 268 Goodhardt, G.J., 120, 127 Gourevitch, V., 190, 241 Green, D M., 2, 3, 4, 5, 185, 187, 241, 242 Gregson, R.A.M., 48, 77 Grey, D.R., 188, 241 Gridgeman, N.T., 21, 39, 44, 45, 48, 49, 72, 76, 77, 93, 104, 184, 241 Griffiths, D.A., 112, 127 H Hacker, M.J., 186, 241 Harmon, T.J., 45, 46, 76 Harries, J.K., 107, 127, 138, 150 Harris, B.L., 2, 3, Harter, H.L., 79, 84, 104 Haseman, J.W., 33, 44 Hatjopoulos, D., 241 Hayakawa, R., 108, 127 Hill, P., 105 Hochberg, Y., 79, 80, 84, 104 Hopkins, J.W., 21, 39, 44, 93, 104, 184, 241 Horsnell, G., 72, 76, 107, 127 Hsu, P., 33, 44 Hunter, E.A., 107, 127 I Irwin, J.O., 15, 20 Ishii, G., 127 Ishii, R., Islam, A.S., 109, 128 ISO, 82, 105 J Johnson, N.L., 108, 127 K Kaplan, H.L., 3, 5, 187, 241 Keene, O.N., 244, 254 Kennedy, T., 105 Kleinman, J.C., 108, 127 Koch, G.C., 105 Koehler, K.J., 168, 169, 175, 182 Kooistra, A., 241 Kotz, S., 108, 127 Kronman, H.B., 115, 127 Kunert, J., 107, 127 L Lachin, J.M., 34, 44, 181, 182 Lancaster, H.O., 99, 100, 105 Landis, R.J., 101, 105 Larmond, E., 93, 105 Lee, K.Y.M., 127 Liang, K.Y., 128 M Macmillan, N.A., 3, 4, 5, 185, 187, 241 MacRae, A.W., 184, 241 Mandel, J., 245, 254 Marascuilo, L.A., 250, 252, 254 Masuoka, S., 184, 241 Maxwell, A.E., 60, 77, 101, 105, 170, 182 McNemar, Q., 18, 20, 174, 182 McSweeney, M., 252, 254 Meier, P., 244, 254 Meilgaard, M., 3, Meng, R.C., 34, 44, 181, 182 Metz, C.E., 115, 127 Meyners, M., 107, 127 Miettinen, O.S., 36, 38, 44 Moore, D.F., 108, 127 Moore, D.S., 167, 182 Moran, P.A.P., 108, 127 Morgan, B.J.T., 188, 241 Morrison, D.G., 107, 127, 138, 150 Morrison, G.R., 39, 44 Mosemann, J.E., 163, 183 Mosteller, F., 185, 241 author index N Nair, V.N., 99, 105 Nemenyi, P., 84, 105 Neyman, J., 99, 105 Nonaka, R., 242 O O’Mahony, M., 1, 4, 5, 38, 44, 184, 241, 242 Odbert, N 38, 44 Odeh, R.E., 83, 105 Ogilvie, J.C., 188, 241 Ord, J.K., 180, 183 Owen, D.B., 84, 104 P Pack, S.E., 108, 127 Pangborn, R.M., Paul, S.R., 109, 114, 128, 164, 183 Paule, R.C., 245, 254 Peryam, D.R., 2, 3, Peterson, W.W., 185, 240 Petrasovits, A., 105 Pfaffmann, C., 3, Piggot, J.R., 44, 127 Pilgrim, F.J., 184, 241 Prentice, R.L., 109, 128 Schlesselman, J.J., 29, 44 Schlich, P., 107, 127, 138, 150 Schuirmann, D.J., 255, 256, 268 Schutz, H.G., 48, 77 Self, S.G., 128 Singh, J., 45, 76, 87, 93, 104 Skellam J.G., 108, 128 Smith, D.M., 112, 128 Smith, G.L., 107, 127, 138, 150 Stillman, J.A., 184, 242 Stuart, A., 60, 77, 101, 105, 170, 180, 183 Susumu, M., Swets, J.A., 2, 3, 4, 5, 185, 187, 241 T Tamhane, A.C., 79, 80, 84, 104 Taplin, R.H., 89, 90, 105 Tarone, R.E., 114, 128, 165, 183 Tedja, S., 184, 242 Terry, M.E., 93, 96, 104 Thurstone, L.L., 185, 242 Torgerson, W.S., 4, Tournade, M-F., 268 Trivedi, M.C., 39, 44, 185, 241 U Underwood, R.E., 18, 20, 36, 37, 38, 44 Ura, S., 39, 44, 185 R Rao, C.R., 139, 150 Raffensberger, E.L., 184, 241 Ratcliff, R., 186, 241 Rayner, J.C.W., 87, 99, 101, 104, 105 Rodary, C., 264, 268 Roessler, E.B., Rousseau, B., 184, 242 Rukhin, A.L., 246, 254 S Sachs, L., 7, 18, 20 Scheffe, H., 252, 254 291 V Vangel, M.G., 246, 254 Vereijken, P.F.G., 241 Versfeld, N.J., 186, 242 W Washam, R.W., II, 76 Wellek, S., 255, 268 Westlake, W.J., 255, 268 Wierenga, B., 107, 128 Williams, J.S., 244, 254 Wilson, J.R., 168, 169, 175, 182, 183 Sensory Discrimination Tests and Measurements: Statistical Principles, Procedures and Tables Jian Bi Copyright © 2006 by Jian Bi Subject Index A Acceptance testing, Adjusted Bennett’s statistic, 174–179 Adjusted Pearson’s chi-square statistic, 118, 168 Adjusted Stuart–Maxwell statistic, 170 Allocation, 28–29, 215 Allowed difference (defining similarity), 255, 256, 259, 261 Anderson statistic, 86–89 AUC, 244 B B (or BA , BN , Bs , Bd ) value (in estimate of variance of d ), 43, 47, 190–240 Balanced incomplete block (BIB) design, 83, 85, 93 Bennett’s statistic, 174 Beta distribution density function, 108, 139 Beta function, 108, 139 Beta-binomial (BB) model probability function, 108 mean and variance, 109 moment estimate, 109–110 maximum likelihood estimate, 111–113 application of, replicated difference and preference tests using two sided paired comparison method, 113–117 replicated monadic A–Not A and Same–Different tests, 117–119 consumer repeat buying behavior, 119–122 power and sample size, 122–127 Binomial coefficients, 15 Binomial distribution probability function, cumulative distribution function, mean, variance, Binomial expansion, 139, 141 Binomial experiment, Binomial mixture model, 107 Binomial variable, Bioequivalence, 255, 256 Bradley–Harmon model (modified triangle test), 45–48 Bradley–Terry model (multiple paired comparisons) maximum likelihood estimation, 93 likelihood ratio test, 94 covariance matrix, 94 simultaneous confidence interval, 95 combination of experiments, 96–98 Brand effect, 18, 38 Brand choice, 120 Brand loyalty, 120 B-value (in Bradley–Terry model), 94–97 C C value: see Dirichlet-multinomial (DM) mode Chi-square statistic (test, distribution), 13, 14, 17, 18, 34–37, 46, 47, 58–61, 76, 79, 80, 82, 83, 85–92, 94, 95, 97–102, 115–119, 167–170, 172–176, 178–182, 250, 252–254, 264, 265 Cochran’s Q test, 80–81, 174, 176 293 294 subject index Comparison for multiple matched proportions, 80–82; see also Cochran’s Q test Comparison for multiple independent proportions, 78–80 “Comparison of distance” strategy, 4, 44 Complete block design, 82 Composite model, 72, 107 Compound distribution, see Beta-binomial (BB) model; Dirichlet-multinomial (DM) mode Consumer panel data, 120 Consumer repeat buying behavior, 119–122 Continuity correction, 14, 18, 23–26, 28–30, 33, 66, 67, 69, 118–119, 172, 264 Corrected beta-binomial (CBB) model probability function, 139–141 mean and variance of X, Pˆc , ␲ˆ c , 139–141 moment estimation, 142–145 maximum likelihood, 145, 146 testing for parameters, 146–147 testing power and sample size, 148–150 Covariance matrix, 47, 60, 61, 71, 74, 76, 90, 91, 94, 95, 96, 112, 113, 115–117, 145, 146, 167, 170, 188, 189, 258 D Decision criteria, see Response bias Degree of difference test ratings are regarded as continuous data, 57–58 ratings are regarded as categorical data, monadic design, 58–59 mixed design, 59 paired design, 60–61 Descriptive analysis, Difference test, Dirichlet distribution, 163 Dirichlet-multinomial (DM) model probability function, 163 goodness of fit testing, 164–165 moment estimation of parameters, 165–166 maximum likelihood estimation, 167 C-value, 163–173, 175–176, 179–182 application of for comparison with a specified proportion vector, 167–168 for comparison among several independent proportion vectors, 168–169 for comparison between two dependent proportion vectors, 170 for testing independence of two classifications in contingency table, 172 for comparison of multiple correlated proportions, 174–179 testing power of for comparison with a specified proportion vector, 179–180 for comparison among several independent proportion vectors, 180–181 for test of independence in contingency table, 181–182 Discrimination method, test, measurement, Discriminative analysis, Discriminators, 7, 12, 62, 69–70, 107, 259–261 Distance of difference, 4–5 Double difference test guessing model, 62 critical values for statistical tests, 62–66 power and sample size, 66–69 estimate of discriminators, 69–70 Double preference test guessing model, 70 moment estimates of pa , pb , pn , 70 maximum likelihood estimate, 71 testing, 71–72 Dunnett and Gent’s chi-square test (for similarity), 261, 264–268 Durbin statistic, 85–86 subject index E 295 for double preference test, 70 in Ferris model, 72 ED50 , 244; see also Threshold F H Ferris model (for preference test with “no preference” option) Ferris 2-visit method, 72–73 maximum likelihood estimation, 73–75 statistical inference for parameters, 75–76 Ferris k-visit method, 72, 106, 257; see also Ferris model Fisher’s exact test, 15–16, 33–34 Fisher–Irwin test: see Fisher’s exact test Fixed effect model, 243 Forced-choice procedures, Friedman rank sum test Friedman statistic, 82–83 corrected Friedman statistic for tied ranks, 83 pairwise comparisons based on rank sums, 84 Hypergeometric distribution, 15 Hypothesis test null hypothesis, alternative hypothesis, one-sided, two-sided, Type I error, ␣, 8; see also Significance level Type II error, ␤, 8; see also Power of discrimination testing; Power of similarity testing G Gamma function, 163 Generalized inverse of a matrix, 167–168 Generalized linear model, 107 Generalized power efficiency, 42–44 Generalized Stuart–Maxwell test for multiple dependent samples, 101–104 Generalized Tarone’s Z statistic, 165 Goodness of fit testing: see Tarone’s Z statistic; Generalized Tarone’s Z statistic Gridgeman model (modified triangle test), data for, 48 hypotheses under, 48 extended table for test, 52–53 mean and variance of weights, 49 power and sample size, 50–51, 54–56 Group sensitivity, 243, 247 Guessing models for difference tests, for preference test, for double difference test, 62 I Incomplete standard normal distribution, 46 Inter-trial (panelist) variation, 106, 148 Interval hypothesis testing, 256, 258, 259, 261 Intra-trial (panelist) variation, 106, 148 Iterative procedure, 93, 110, 246 J Just About Right (JAR) scale, 98, 102, 167–168, 180 L Likelihood ratio test, 46, 94, 97 Location effect, 87, 90, 100; see also Partition of chi-square Log odds ratio, 20 Lognormal distribution, 244 M Mandel–Paule algorithm, 245–246 McNemar test statistic, 18 power and sample size Bennett and Underwood’s approach, 37 Miettinen’s approach, 36 296 subject index Meaningful difference, 146–147 Measurements of sensory difference: see Thurstonial model Mid-rank, 100 Mixed design, 16, 34, 59, 172–173, 181–182 Modified discrimination tests modified triangle test, 45–56 degree of difference test, 56–61 double discrimination test, 61–72 preference test with “no preference” option, 72–76 Modified triangle method: see Bradley–Harmon model; Gridgeman model Monadic design, 13, 27, 58, 117, 125 Multinomial distribution, 49, 50, 100, 163–165, 167, 170, 172, 175 Multiple comparisons: see Simultaneous confidence interval Multiple-sample discrimination tests based on proportions, 78–82 based on ranks, 82–98 based on categorical scales, 98–104 N Nature of difference, 4–5 Newton’s method, 246 Non-central chi-square distribution, 34, 37, 180, 181 Non-discriminator, 7, 107 Non-linear effects, 89–90, 92; see also Partition of chi-square Normal distribution, see Z statistic O Odds ratio, 19–20 Orthogonal polynomials, 99, 100 Overdispersion, 106–107, 109, 118, 146, 164, 168, 175, 180 Overdispersed binomial (multinomial) data: see Overdispersion P Paired design, 17, 36, 60 Paradox of discriminatory non-discriminators, 184 Partition of chi-square, 87, 99, 100 “Penetration”, 121 Population sensitivity, 243, 245–246 “Power approach” (for similarity), 255 Power of discrimination testing, 21; see also Standard discrimination tests Power of similarity testing, 258–259, 260–261, 265–268, see also Similarity testing Power comparison for forced-choice methods, 39–40 for the methods with response bias, 40 for different types of discrimination methods, 40–41 for conventional and double discrimination methods, 68–69 power efficiency, 41–42 generalized power efficiency, 42–44 Preference proportion (in Taplin statistic), 89–90 Preference test with “no preference” option, see Ferris model Preference test, Probability (proportion) of correct guessing, (Po ), 7–9, 12, 62, 259–261 Probability (proportion) of correct responses, (Pc ), 7–8, 12, 26–27, 62, 185–186, 189–192, 215, 237–239, 260 Probability (proportion) of discrimination or proportion of discriminators, (Pd ): see Discriminators Proportions of preference, 12, 70–71, 72–74, 165–166 Psychometric functions for 2-AFC, 186 for 3-AFC, 186 for Triangular, 186 for Duo–Trio, 186 for A–Not A, 187 subject index for Same–Different, 187 for double discrimination methods, 189–190 for rating method, 188 Psychometrics, Purchase intent, 18 “Purchasing week”, 120 Q Quantitative descriptive analysis, 1; see also Acceptance testing R Random effect model, 243 Response bias, 4–5, 12, 27, 40, 44 Response patterns, 174–176, 178–179 S Scheffe’s Theorem, 252 Schuirmann’s two one-sided tests, 256 Semi-weighted mean of d s, 245 Sensitivity analysis, 2, see also Threshold; ED50 Sensory analysis methodologies laboratory sensory analysis, consumer sensory analysis, discriminative analysis, descriptive analysis, Sensitivity analysis, time-intensity analysis, preference testing, difference testing, Similarty testing, tests, measurements, Sensory discrimination methods requiring and not requiring the nature of difference, 3–5 with and without response bias, 4–5 Sensory Evaluation I, 1, Sensory Evaluation II, 1, Shelf life analysis, Signal Detection Theory (SDT): see Thurstonian model 297 Similarity testing for preference, 256–258 using forced-choice methods, 259–260 using the A Not–A and Same–Different methods, 261, 264, 265 power and sample size, 258–259, 260–261, 265–268 Significance level, 8; see also Hypothesis test Simultaneous confidence interval, 79, 80, 82, 84, 85, 95–96 “Skimming” strategy, 4, 44 S-PLUS codes, 287 Spread effect, 87–88, 100; see also Partition of chi-square Standard discrimination methods 2-Alternative Forced-Choice method (2-AFC) 3-Alternative Forced-Choice method (3-AFC), Duo–Trio, Triangular, A–Not A, Same–Different, Standard discrimination tests discrimination tests using forced-choice methods test statistic and critical value, 9–11 power and sample size, 22–27 discrimination tests using the methods with response bias chi-square test for homogeneity, 13–14 Z-test for difference of two proportions, 14–15 power and sample size based on a normal approximation, 28–33 Fisher’s exact test, 15–16 power and sample size based on Fisher’s exact test, 33–34 chi-square test for independence, 16–17 power and sample size for chi-square test for independence, 34–36 McNemar test, 18; see also McNemar test 298 subject index power and sample size for McNemar’s test Miettinen’s approach, 36–37 Bennett and Underwood’s approach, 37–38 Statistical analysis for d data estimate of population or group sensitivity, 243–248 confidence interval, 248 compare with a specified value, 248 comparison of two d s, 249 comparison of multiple d s, 250 comparison of multiple sets of d s, 250 multiple comparisons, 252–254 Statistical inference, Stochastic model, 107, 119 Stuart–Maxwell statistic, 60, 101 Survival analysis, T Tables of d and B value for 2-AFC, 191 for 3-AFC, 191 for Triangular, 192 for Duo–Trio, 192 for A–Not A, 194–215 for Same–Different, 216–237 for double discrimination methods, 237–239 Taplin statistic, 89–92 Tarone’s Z statistic, 114 Taylor series, 12, 70, 190, 244 Test for non-inferiority, 256 Test for non-superiority, 256 The × table for data from a monadic A–Not A test, 13 for data from a mixed A–Not A test, 16 for data from a paired A–Not A test, 17 Theorem on total probabilities, Threshold, 2; see also ED50 Thurstonial model Thurstonian ␦, 185–190 variance of d , 190–193, 215, 237 tables 191–192, 194–215, 216–237, 237–239 Thurstonian ␦ (or d ), 2, 25–27, 29–32, 46–48, 185–190; see also Thurstonial model Thurstonian scale: see Thurstonian model Time-intensity (IT) analysis, Trained panel, Types of buyers repeat buyers, 121 new buyers, 121 lost buyers, 121 non-buyers, 121 Two-stage triangle test: see Modified triangle method U Under-dispersion, 109 Untrained consumers, V Variance (B value) of d for 2-AFC, 190 for 3-AFC, 191 for Triangular, 192 for Duo–Trio, 191 for A–Not A, 193 for Same–Different, 193, 215 for double discrimination methods, 215, 237 for a group d , 247 for a population d , 246 W Weights, 49, 51, 56, 110, 245 Weighted mean of d s, 244–247 Z Z statistic, 9, 14–15, 18, 49, 53, 72, 75, 114–115, 118, 147, 165, 248–249, 257–258, 264–265 ... understanding, and encouragement during the preparation of this book Jian Bi Sensory Discrimination Tests and Measurements: Statistical Principles, Procedures and Tables Jian Bi Copyright © 2006. .. Handbook of Techniques Translated by Z Reynarowych Springer-Verlag, New York Sensory Discrimination Tests and Measurements: Statistical Principles, Procedures and Tables Jian Bi Copyright © 2006. . .Sensory Discrimination Tests and Measurements Statistical Principles, Procedures and Tables Jian Bi Sensometrics Research and Service Richmond, Virginia, USA