12.5 Randomised Blocks and Incomplete Block Designs
12.6 Split-Plot and Nested Designs
12.7 Power of Experiments
13 ANOVA for Sensory and Consumer Data
13.1 Introduction
13.2 One-Way ANOVA
13.3 Single Replicate Two-Way ANOVA
13.4 Two-Way ANOVA with Randomised Replications
13.5 Multi-Way ANOVA
13.6 ANOVA for Fractional Factorial Designs
13.7 Fixed and Random Effects in ANOVA: Mixed Models
13.8 Nested and Split-Plot Models
13.9 Post Hoc Testing
14 Principal Component Analysis
14.1 Interpretation of Complex Data Sets by PCA
14.2 Data Structures for the PCA
14.3 PCA: Description of the Method
14.4 Projections and Linear Combinations
14.5 The Scores and Loadings Plots
14.6 Correlation Loadings Plot
14.7 Standardisation
14.8 Calculations and Missing Values
14.9 Validation
14.10 Outlier Diagnostics
14.11 Tucker-1
14.12 The Relation between PCA and Factor Analysis (FA)
15 Multiple Regression, Principal Components Regression and Partial Least Squares Regression
15.1 Introduction
15.2 Multivariate Linear Regression
15.3 The Relation between ANOVA and Regression Analysis
15.4 Linear Regression Used for Estimating Polynomial Models
15.5 Combining Continuous and Categorical Variables
15.6 Variable Selection for Multiple Linear Regression
15.7 Principal Components Regression (PCR)
15.8 Partial Least Squares (PLS) Regression
15.9 Model Validation: Prediction Performance
15.10 Model Diagnostics and Outlier Detection
15.11 Discriminant Analysis
15.12 Generalised Linear Models, Logistic Regression and Multinomial Regression
16 Cluster Analysis: Unsupervised Classification
16.1 Introduction
16.2 Hierarchical Clustering
16.3 Partitioning Methods
16.4 Cluster Analysis for Matrices
17 Miscellaneous Methodologies
17.1 Three-Way Analysis of Sensory Data
17.2 Relating Three-Way Data to Two-Way Data
17.3 Path Modelling
17.4 MDS-Multidimensional Scaling
17.5 Analysing Rank Data
17.6 The L-PLS Method
17.7 Missing Value Estimation
Nomenclature, Symbols and Abbreviations
Index
Nội dung
Statistics for Sensory and Consumer Science Statistics for Sensory and Consumer Science TORMOD NÆS Nofima Mat, Norway and PER B BROCKHOFF Danish Technical University, Denmark and OLIVER TOMIC Nofima Mat, Norway A John Wiley and Sons, Ltd., Publication This edition first published 2010 c 2010 John Wiley & Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose This work is sold with the understanding that the publisher is not engaged in rendering professional services The advice and strategies contained herein may not be suitable for every situation In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read No warranty may be created or extended by any promotional statements for this work Neither the publisher nor the author shall be liable for any damages arising herefrom Library of Congress Cataloging-in-Publication Data Næs, Tormod Statistics for sensory and consumer science / Tormod Næs and Per B Brockhoff and Oliver Tomic p cm Summary: “As we move further into the 21st Century, sensory and consumer studies continue to develop, playing an important role in food science and industry These studies are crucial for understanding the relation between food properties on one side and human liking and buying behaviour on the other This book by a group of established scientists gives a comprehensive, up-to-date overview of the most common statistical methods for handling data from both trained sensory panels and consumer studies of food It presents the topic in two distinct sections: problem-orientated (Part I) and method orientated (Part II), making it to appropriate for people at different levels with respect to their statistical skills This book succesfully makes a clear distinction between studies using a trained sensory panel and studies using consumers Concentrates on experimental studies with focus on how sensory assessors or consumers perceive and assess various product properties Focuses on relationships between methods and techniques and on considering all of them as special cases of more general statistical methodologies It is assumed that the reader has a basic knowledge of statistics and the most important data collection methods within sensory and consumer science This text is aimed at food scientists and food engineers working in research and industry, as well as food science students at master and PhD level In addition, applied statisticians with special interest in food science will also find relevant information within the book”– Provided by publisher Summary: “This book will describe the most basic and used statistical methods for analysis of data from trained sensory panels and consumer panels with a focus on applications of the methods It will start with a chapter discussing the differences and similarities between data from trained sensory and consumer tests”– Provided by publisher Includes bibliographical references and index ISBN 978-0-470-51821-2 (hardback) Food–Sensory evaluation New products I Brockhoff, Per B II Tomic, Oliver III Title TX546.N34 2010 664 07–dc22 2010016197 A catalogue record for this book is available from the British Library ISBN (Hbk) 9780470518212 Typeset in 10/12pt Times by Aptara Inc., New Delhi, India Printed and bound in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire Contents Preface Acknowledgements ix xi 1 Introduction 1.1 The Distinction between Trained Sensory Panels and Consumer Panels 1.2 The Need for Statistics in Experimental Planning and Analysis 1.3 Scales and Data Types 1.4 Organisation of the Book Important Data Collection Techniques for Sensory and Consumer Studies 2.1 Sensory Panel Methodologies 2.2 Consumer Tests PART I 3 5 PROBLEM DRIVEN Quality Control of Sensory Profile Data 3.1 General Introduction 3.2 Visual Inspection of Raw Data 3.3 Mixed Model ANOVA for Assessing the Importance of the Sensory Attributes 3.4 Overall Assessment of Assessor Differences Using All Variables Simultaneously 3.5 Methods for Detecting Differences in Use of the Scale 3.6 Comparing the Assessors’ Ability to Detect Differences between the Products 3.7 Relations between Individual Assessor Ratings and the Panel Average 3.8 Individual Line Plots for Detailed Inspection of Assessors 3.9 Miscellaneous Methods Correction Methods and Other Remedies for Improving Sensory Profile Data 4.1 Introduction 4.2 Correcting for Different Use of the Scale 11 11 15 18 19 24 27 29 33 34 39 39 40 vi Contents 4.3 4.4 Computing Improved Panel Averages Pre-processing of Data for Three-Way Analysis Detecting and Studying Sensory Differences and Similarities between Products 5.1 Introduction 5.2 Analysing Sensory Profile Data: Univariate Case 5.3 Analysing Sensory Profile Data: Multivariate Case 43 45 47 47 48 59 Relating Sensory Data to Other Measurements 6.1 Introduction 6.2 Estimating Relations between Consensus Profiles and External Data 6.3 Estimating Relations between Individual Sensory Profiles and External Data 67 67 68 Discrimination and Similarity Testing 7.1 Introduction 7.2 Analysis of Data from Basic Sensory Discrimination Tests 7.3 Examples of Basic Discrimination Testing 7.4 Power Calculations in Discrimination Tests 7.5 Thurstonian Modelling: What Is It Really? 7.6 Similarity versus Difference Testing 7.7 Replications: What to Do? 7.8 Designed Experiments, Extended Analysis and Other Test Protocols 79 79 80 81 85 86 87 89 93 Investigating Important Factors Influencing Food Acceptance and Choice 8.1 Introduction 8.2 Preliminary Analysis of Consumer Data Sets (Raw Data Overview) 8.3 Experimental Designs for Rating Based Consumer Studies 8.4 Analysis of Categorical Effect Variables 8.5 Incorporating Additional Information about Consumers 8.6 Modelling of Factors as Continuous Variables 8.7 Reliability/Validity Testing for Rating Based Methods 8.8 Rank Based Methodology 8.9 Choice Based Conjoint Analysis 8.10 Market Share Simulation 95 95 99 102 106 113 117 118 119 120 123 Preference Mapping for Understanding Relations between Sensory Product Attributes and Consumer Acceptance 9.1 Introduction 9.2 External and Internal Preference Mapping 9.3 Examples of Linear Preference Mapping 9.4 Ideal Point Preference Mapping 9.5 Selecting Samples for Preference Mapping 9.6 Incorporating Additional Consumer Attributes 127 128 129 136 141 146 147 74 Contents 9.7 10 Combining Preference Mapping with Additional Information about the Samples Segmentation of Consumer Data 10.1 Introduction 10.2 Segmentation of Rating Data 10.3 Relating Segments to Consumer Attributes vii 149 155 155 156 163 PART II METHOD ORIENTED 11 Basic Statistics 11.1 Basic Concepts and Principles 11.2 Histogram, Frequency and Probability 11.3 Some Basic Properties of a Distribution (Mean, Variance and Standard Deviation) 11.4 Hypothesis Testing and Confidence Intervals for the Mean µ 11.5 Statistical Process Control 11.6 Relationships between Two or More Variables 11.7 Simple Linear Regression 11.8 Binomial Distribution and Tests 11.9 Contingency Tables and Homogeneity Testing 165 165 166 12 Design of Experiments for Sensory and Consumer Data 12.1 Introduction 12.2 Important Concepts and Distinctions 12.3 Full Factorial Designs 12.4 Fractional Factorial Designs: Screening Designs 12.5 Randomised Blocks and Incomplete Block Designs 12.6 Split-Plot and Nested Designs 12.7 Power of Experiments 181 181 182 185 187 188 190 191 13 ANOVA for Sensory and Consumer Data 13.1 Introduction 13.2 One-Way ANOVA 13.3 Single Replicate Two-Way ANOVA 13.4 Two-Way ANOVA with Randomised Replications 13.5 Multi-Way ANOVA 13.6 ANOVA for Fractional Factorial Designs 13.7 Fixed and Random Effects in ANOVA: Mixed Models 13.8 Nested and Split-Plot Models 13.9 Post Hoc Testing 193 193 194 196 198 200 201 203 205 206 14 Principal Component Analysis 14.1 Interpretation of Complex Data Sets by PCA 14.2 Data Structures for the PCA 14.3 PCA: Description of the Method 209 209 210 211 168 169 172 173 175 177 178 viii Contents 14.4 Projections and Linear Combinations 14.5 The Scores and Loadings Plots 14.6 Correlation Loadings Plot 14.7 Standardisation 14.8 Calculations and Missing Values 14.9 Validation 14.10 Outlier Diagnostics 14.11 Tucker-1 14.12 The Relation between PCA and Factor Analysis (FA) 15 213 214 217 219 220 220 221 223 224 Multiple Regression, Principal Components Regression and Partial Least Squares Regression 15.1 Introduction 15.2 Multivariate Linear Regression 15.3 The Relation between ANOVA and Regression Analysis 15.4 Linear Regression Used for Estimating Polynomial Models 15.5 Combining Continuous and Categorical Variables 15.6 Variable Selection for Multiple Linear Regression 15.7 Principal Components Regression (PCR) 15.8 Partial Least Squares (PLS) Regression 15.9 Model Validation: Prediction Performance 15.10 Model Diagnostics and Outlier Detection 15.11 Discriminant Analysis 15.12 Generalised Linear Models, Logistic Regression and Multinomial Regression 245 16 Cluster Analysis: Unsupervised Classification 16.1 Introduction 16.2 Hierarchical Clustering 16.3 Partitioning Methods 16.4 Cluster Analysis for Matrices 249 249 251 254 259 17 Miscellaneous Methodologies 17.1 Three-Way Analysis of Sensory Data 17.2 Relating Three-Way Data to Two-Way Data 17.3 Path Modelling 17.4 MDS-Multidimensional Scaling 17.5 Analysing Rank Data 17.6 The L-PLS Method 17.7 Missing Value Estimation 263 263 269 269 271 271 273 273 Nomenclature, Symbols and Abbreviations Index 227 227 229 232 233 234 235 236 237 238 241 244 277 283 Miscellaneous Methodologies 273 for handling ranking data within for instance the framework of conjoint analysis We refer to Young (1981) for a thorough description of the method (see also Kruskal, 1965; Gifi, 1991; Green, 1973) 17.6 The L-PLS Method The L-PLS (Martens et al., 2005) method is a technique for combining three matrices which are linked according to the scheme in Figure 9.1 A typical situation where this is appropriate is in a preference mapping situation where the two horizontal matrices are the sensory data and the consumer ratings while the matrix on top is the matrix of additional consumer attributes The L-PLS method (Martens et al., 2005) uses the SVD on products of the three matrices involved and provide two-dimensional scatter plots of four different types The first type represents the samples, the next represents the sensory variables, the third the consumer hedonic scores and the last the additional consumer attributes All these pieces of information can be plotted separately or in the same plot and interpreted the same way as for regular bi-plots The method is an elegant extension of regular PLS regression For alternative, but similar approaches, we refer to Lengard and Kermit (2006) and Endrizzi et al (2009) 17.7 Missing Value Estimation In some situations there are missing values in large data sets If single elements are missing at random, there are techniques available for replacing them by estimated or imputed values Some of the techniques, for instance PLS and PCA solved by the NIPALS method, have this as an inbuilt feature, at least in many available implementations Walczak and Massart (2001) provides a discussion of various ways of solving the general problem of replacing missing values by estimates A possible procedure that seems to be used frequently is the iterative algorithm where one starts out by initial estimates of missing values (obtained by for instance averaging), then one calculates the SVD (or PCA) before reconstructing the Y by using a pre-defined number of components The missing values are then replaced by the reconstructed or predicted ones The procedure iterates until convergence Multivariate regression analysis of the missing values vs related variables in the data set is another possible strategy For an overview we refer to Donders et al (2006) (see also Grung and Manne, 1998) When there are systematic tendencies in the missing value pattern, one should always be careful when using these methods Many ANOVA models allow for imbalance and also incompleteness in the data set, and these methods should be used when possible Incomplete data with missing cells are, however, very difficult to handle in ANOVA if interactions are involved Note that if the number of missing values is high, there will always be problems with the validity of the imputation results References Arnold, G.M., Williams, A.A (1987) The use of generalised procrustes techniques in sensory analysis In J.R Piggott (ed.), Statistical Procedures in Food Research London: Elsevier Science Publishers, 244–53 274 Statistics for Sensory and Consumer Science Bro, R (1996) Multiway calibration Multilinear PLS Journal of Chemometrics 10, 47–61 Bro, R (1997) PARAFAC Tutorial and applications Chemometrics and Intelligent Laboratory Systems, 38, 149–71 Bro, R Smilde, A.K., de Jong, S (2001) On the difference between low-rank and subspace approximations: improved model for multilinear PLS regression Chemometrics and Intelligent Laboratory Systems 58, 3–13 Bro, R., Qannari, E.M, Kiers, H.A., Næs, T., Frøst, M.B (2008) Multi-way models for sensory profiling data J Chemometrics 22, 36–45 Brockhoff, P, Hirst, D., Næs, T (1996) In T Næs, E Risvik (eds), Multivariate Analysis of Data in Sensory Science Amsterdam: Elsevier Carroll, J.D 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consensus configuration from a genealised Procrustes analysis with the untransformed panel mean in sensory profile analysis Lebensmittel-Wissenschaft und Technologie 17, 296–8 Lavit, C., Escoufier, Y., Sabatier, R., Traissac, T (1994) The ACT (STATIS method) Computational Statistics and Data Analysis 18, 97–119 Lehmann, E.L (1975) Nonparametrics San Francisco: Holden-Day, Inc Lengard, V., Kermit, M (2006) 3-way and 3-block regression in consumer preference analysis Food Quality and Preference 17, 234–42 Miscellaneous Methodologies 275 L’Hermier del Plantes (1976) Structuration des tableaux a trios indices de la statistique Thesis, Montepellier II Mardia, K.V., Kent, J.T., Bibby, J.M (1979) Multivariate Analysis London: Academic Press Martens, H., Anderssen, E., Flatberg, A., et al (2005) Regression of a data matrix on descriptors of both rows and of its columns via latent variables: L-PLSR Comp Stat and Data Analysis 48, 103–23 Martens, M., Tenenhaus, M, Vinzi, V.E., Martens, H (2007) The use of partial least squares methods in new food product development In MacFie, H (ed.), Consumer-led Food Products Development Cambridge: Woodhead, 492–523 Næs, T., Kowalski B (1989) Predicting sensory profiles from external instrumental measurements Food Quality and Preference (4/5), 135–47 Pages, J and Tenenhaus, M (2001) Multiple factors analysis combined with path modelling Application to the analysis of relationships between physicochemical variables, sensory profiles and hedonic judgements Chemometrics and Intelligent Laboratory Systems 58, 261–73 Pages, J., Husson, F (2005) Multiple factors analysis with confidence ellipses: a methodology to study the relationships between sensory and instrumental data J Chemometrics 19, 138–44 Popper, R., Heymann, H (1996) Analyzing differences among products and panellists by multidimensional scaling In T Næs, E Risvik (eds), Multivariate Analysis of Data in Sensory Science Amsterdam: Elsevier, pp 159–84 Rayner, J.C.W., Best, D.J., Brockhoff, P.B., Rayner, G.D (2005) Nonparametrics for Sensory Science: A More Informative Approach Ames, USA: Blackwell Publishing Smilde A., Bro, R., Geladi, P (2004) Multi-Way Analysis Chichester: John Wiley & Sons, Ltd Tenenhaus, M., Vinzi, V.E., Chatelin Y-M., Lauro, C (2005) PLS path modelling Computational Statistics and Data Analysis, 48, 159–205 Tucker, L.R (1964) The extension of factor analysis to three-dimensional matrices In Frederiksen, N and Gulliksen, H (eds), Contributions to Mathematical Psychology New York: Holt, Rinehart & Winston, pp 110–82 Van der Kloot, Q.A., Kroonenberg, P.M (1985), External analysis with three-mode principal components models Psychometrika 50(4), 479–94 Vigneau E., Courcoux P., Semenou M (1999) Analysis of ranked preference data using latent class models, Food Quality and Preference 10(3), 201–7 Walczak, B., Massart, D.L (2001) Dealing with missing data, Part I Chemometrics and Intelligent Laboratory Systems 58, 15–27 Wedel, M., DeSarbo, W.S (1995) A mixture likelihood approach for generalized linear models Journal of Classification 12(1), 21–55 Wold, H (1982) Soft modelling: The basics and some extensions In K.G Jøreskog, H Wold (eds), Systems under Indirect Observation Amsterdam: North Holland Young, F (1981) Quantitative analysis of qualitative data Psychometrika 46(4), 357–88 Nomenclature, Symbols and Abbreviations Nomenclature and Distinctions Assessor Member of a sensory panel See also Judge May also be used for a consumer Acceptance Degree of acceptance for a product, either a food product or a combination of various extrinsic attributes Usually given as a score between a lower and an upper limit (for instance between and 9) The term is used for rating studies while the term preference is used when a consumer chooses one of several alternatives The term preference mapping is generally used for acceptance studies and represents as such a different use of the term For most cases, the distinction between the two terms has no influence on the understanding of the concepts presented A posteriori segmentation Segmentation done based on acceptance or preference data The term post hoc is also sometimes used A posteriori use of consumer attributes The consumer attributes are used after the primary analysis of the acceptance or preference data A priori segmentation Segmentation done before the acceptance data are analysed, based on for instance demographic variables A priori use of consumer attributes The consumer attributes are used in the primary analysis of the data Balanced design An experiment where the number of observations is the same for all possible levels of all factors involved Calibration Concept sometimes used for the process of estimating regression parameters Choice test When a consumer is asked to choose the product he prefers or likes best among several, this is a choice test Conjoint analysis Method for analysing the effect of a number of designed factors, for instance packaging and information factors, on consumer acceptance or choice Consumer Person selected from a larger population to participate in a hedonic or discrimination test Statistics for Sensory and Consumer Science C 2010 John Wiley & Sons, Ltd Tormod Næs, Per B Brockhoff and Oliver Tomic 278 Statistics for Sensory and Consumer Science Consumer attribute Aspect related to the consumer, demographic information or information related to attitudes, values and habits Descriptive sensory analysis Sensory analysis based on a trained panel of assessors Results in intensity score vales for a number of sensory attributes Experiment Either one single experiment or a series of experimental runs (in an experimental design) Experimental design A series of experimental runs constructed from statistical principles Experimental run One single element in a full experimental design Extrinsic/intrinsic attributes Intrinsic attributes are attributes related to the food product itself, while the extrinsic attributes are other properties related to the product, such as for instance health information and brand name Frobenius norm Norm for a matrix defined as the square root of the sum of squares of all elements Greek letters Used for describing parameters in models Incomplete data set A data set from for instance a sensory study where some of the cells (combinations of factors) are empty Interval scale data Numerical data Distances between numbers make sense No fixed zero point Typical for sensory and consumer liking data An interval scale with a fixed zero point is called a ratio scale Intrinsic/extrinsic attributes Intrinsic attributes are attributes related to the food product itself, while the extrinsic attributes are other properties related to the product, such as for instance health information and brand name Liking Generic term for describing consumers’ degree of pleasure Sometimes used instead of the concept acceptance Judge Member of a sensory panel See also Assessor Missing values Values that by some (random or other) reason have been left out of the data set Nominal scale data Categorical data without ordering For instance blue, red, white etc Object Physical product to be tested by some assessors, also used as a generic term for rows in a data table Ordinal scale data Categorical data with ordering Rank data are ordinal Preference Used in more or less the same way as choice A consumer chooses a product among several, i.e the one he/she prefers that product See also Acceptance Product Physical object to be tested by consumers See also Sample; Object Profile Generic name of a combination of attributes See also Product Ranking If the consumer is asked to rank the samples according to for instance liking, this gives ranking data, i.e ordinal data Repeatability Degree of similarity between sensory replicates of the same sample More generally used as a measure for similarity between replicated measurements taken of the same sample Nomenclature, Symbols and Abbreviations 279 Replicate If a product is tested several times by for instance a sensory panel, these measurements are sensory replicates If the samples are tested two times, we say that there are two replicates This is also called duplicates If there are three replicates, this is sometimes called triplicates There are different types of replicates Reproducibility Used here as similarity between assessors in a sensory panel More generally used when comparing full replicates, not only measurement replicates (repeated measurements) Rating This is used in situations where the consumers are asked to rate the products, i.e to give scores between a lower and an upper limit (for instance between and 9) See also Acceptance Sample Physical object to be tested A few places the word is also used for a set of observations, a statistical sample (see Chapter 11) When not described explicitly, the former use of the word is applied in this book Sensory analysis In this book used mainly as synonymous to descriptive sensory analysis See also Sensory science which is used in a broader context Sensory panel A group of trained humans used to judge properties or differences between products Sensory profiling See Descriptive sensory analysis Sensory science General term used for all studies where the human senses are used Can be used both for descriptive sensory analysis and consumer tests Scores Used for the values given by the sensory assessors (or by the consumers) Also used as a specific term related to scatter plots of the product information in PCA Self-explicated test This is a type of consumer test where the consumers are asked directly about which aspects of the product they like best This is different from conjoint analysis which is based on an experimental design Use of Lowercase, Italics, Boldface Boldface capital letter Matrix, for instance X Boldface lower-case letter Vector, when presented without transpose (T), it will always be a column vector, for instance x Lowercase, italics Scalar, for instance x Uppercase, italics Used as symbol for number of columns, number of observations etc Also used in expressions like Y -variable, Y -value etc Indices i=1, ,I Lower case when counting, upper case for the total number For instance xi , Symbols α and β Symbols used to describe regression coefficients Used both for scalars and vectors Also used as symbols for effects in ANOVA models 280 Statistics for Sensory and Consumer Science E and F Matrices of residuals for X and Y respectively (in PLS and PCR) ε Residual term in a regression model (used in a similar way as f ) γ Symbol used for effect in ANOVA model Hat, for instance αˆ Used for indicating estimated value See P, T and Q below for how the hat is used in connection with PCA, PCR and PLS H Transformation matrix i, I Number of assessors in sensory profiling j, J Number of samples/products tested in sensory profiling J also used as symbol for criterion to be optimised in fuzzy clustering k, K Number of variables in sensory analysis Also used as a generic term for columns of a matrix and as a generic term for the number of variables in a data matrix Number of factors in a design m, M Number of additional consumer attributes n, N Number of consumers in a consumer test Also used a generic term for rows in a matrix Also number of independent observations P, Pˆ Matrix of loadings for PCA and X-loadings for PLS Without hat when used in a model, with hat when estimated ˆ Symbol for Y-loadings in PCR and PLS Without hat when used in a model, with Q, Q hat when estimated ρ Population correlation coefficient r, R Replicates r Empirical correlation coefficient R Squared correlation coefficient – explained variance s, s Empirical standard deviation and variance θ Symbol used for effect in ANOVA model ˆ Matrix of scores for PCA and PLS Without hat when used in a model, with hat T, T when estimated T Transpose of a matrix or vector w Symbol for weight in weighted average X Uses to describe explanatory, input or independent variables Can be sensory, consumer, chemistry variable etc depending on application Only used in situations with input variables Can be used as matrix, vector or scalar Output or response variables Can be sensory, consumer, chemistry variables etc depending on application Also used as a generic term, when only one matrix of data is available Can be used as matrix, vector or scalar Y Abbreviations and Acronyms AIC ALS Akaike’s information criterion Alternating least squares Nomenclature, Symbols and Abbreviations ANOVA ANCOVA ASCA BIBD BIC CCA CCD CPCA CV CVA CVANOVA DF DoE EM FA FCM FCP GCA GLM GPA IIA INDSCAL LDA LS LS-PLS LSD L-PLSR MANOVA MDS MFA ML MLR MS MSC MSE MSEP NIPALS NIR N-PLS OS OVAT PARAFAC PCA PCR PLS PLSR Analysis of variance Analysis of covariance ANOVA-simultaneous component analysis Balanced incomplete block design Bayesian information criterion Canonical correlation analysis Central composite design Consensus PCA Cross-validation Canonical variate analysis ANOVA used for cross-validated results Degrees of freedom Design of experiments Algorithm – expectation minimisation algorithm Factor analysis Fuzzy clustering method Free choice profiling Generalised canonical correlation analysis Generalised linear model Generalised procrustes analysis Independent of irrelevant alternatives Method for individual MDS Linear discriminant analysis Least squares Method that combines LS and PLS regression Least significant differences PLS for L-type of data Multivariate ANOVA Multidimensional scaling Multiple factor analysis Maximum likelihood Multiple regression Mean square Multiplicative signal correction Mean square error The square of RMSEP Algorithm for PCA Near infrared N-way partial least squares regression Optimal scaling One variable at the time Parallel factor analysis Principal component analysis Principal component regression Partial last squares Partial least squares regression 281 282 Statistics for Sensory and Consumer Science Prefmap and MdPref QDA REML RMSEP RSM RV SD SE SS STATIS SVD VAF Names used for two different methods for preference mapping Quadratic discriminant analysis Restricted maximum likelihood Root means square error of prediction Used to describe prediction ability of an equation Response surface methodology Coefficient – criterion for comparison of matrices Standard deviation Standard error Sums of squares ` Trois Indices de la Statistique Structuration des Tableuax A Singular value decomposition Variance accounted for Index Page numbers in italics refer to figures a posteriori segmentation, 156 a priori segmentation, 155 A–not A method, 80, 93 abnormality detection, 99 acceptance data, 9–10, 112, 113, 115, 157–8 additional consumer attributes, 147–9, 163–4 additional sample attributes, 149–52 additive consumer effects, 135 AIC (Akaike’s information criterion), 123, 232 ANCOVA (analysis of covariance), 115 ANOVA (analysis of variance) (see also Brockhoff–Sommer ANOVA model, mixed model ANOVA, multi-way ANOVA, one-way ANOVA, three-way ANOVA, two-way ANOVA), 113–15, 193–4 model diagnostics, 241–4 relationship to regression analysis, 232–3 ASCA (ANOVA simultaneous component analysis), 64–5 assumptions (model diagnostics), 241 auctions, average preferences, 135 base alternatives, 121 Bayesian statistics, 258 beta-binomial models, 89, 90–2 bi-plots, 216–17 BIBDs (balanced incomplete block designs), 189 binomial distribution, 177–8 blind tasting, block experiment designs (blocking), 185, 188–90 box plots, 16, 17, 99, 169, 170 Bradley-Terry-Luce method, 123 Statistics for Sensory and Consumer Science C 2010 John Wiley & Sons, Ltd Brockhoff assessor model, 52 Brockhoff–Sommer ANOVA model 52 categorical models, 56 categorical variables, 115, 117, 178, 234–5 CCDs (central composite designs), 187 chaining effects (cluster analysis), 252 chemical methods, 68 chi-square test, 178, 179 choice based methods, 120–3 segmentation, 162–3 choice data, choice sets, 122 choice tests circular models, 143 cluster analysis/clustering, 147, 157–9, 160, 249–51 FCM (fuzzy clustering method), 161–2, 162, 254–6 finite mixture model clustering, 162, 258–9 hierarchical clustering, 251–3 K-means clustering, 254 noise clustering, 257–8 residual distances, 256–7 sequential clustering, 257, 258 simultaneous clustering, 160–3 cluster matrices, 259–60 collinearity problem (regression), 230–1, 237 complete block experiment designs, 189 complete linkage, 252 confidence intervals, 171 conjoint data, 106 consensus matrices, 265 consensuses, 70 consonance analysis, 34–5 consumer data, 219–20 consumer effects, 117 Tormod Næs, Per B Brockhoff and Oliver Tomic 284 Index consumer loadings, 132, 133–5, 137, 138, 142 consumer studies, 1, consumer variables, 115 contingency tables, 178 continuous models, 56–7 continuous variables, 115, 117–18, 166, 234–5 control charts, 172 control limits, 172 correction methods scaling differences, 40–3 unreliable assessors, 43–4 correlation, 173–5 correlation loadings, 22–3, 61, 134, 137, 138, 217–19 correlation plots, 30–2 covariance, 174 covariance criterion 238 covariance matrices, 175 cross-validation, 119, 232, 235, 239, 240–1 CVA (canonical variate analysis), 62 data (statistics), 165 data compression methods, 236 data matrices/tables, 210–11 data sets, 97–8, 128 degrees of freedom, 167–8, 194 dendrograms, 252–3 descriptive sensory analysis, 5–6, 67 design factors, 108 design of experiments (see experiment designs) difference tests (see discrimination tests) discriminant analysis, 244–5, 246 discrimination tests, 7, 79–6 comparison with similarity tests, 87–9 level analysis, 81–2 level analysis, 82 power analysis, 85–6 Thurstonian approach, 81, 82–5, 86–7 distributions (statistics), 166, 167–9 eggshell plots 33, 34 elliptic models, 143 empirical validation, 220–1 error variance, 176 estimates, 169 experiment designs, 102–6, 181–91 block designs, 188–90 fractional factorial designs, 103, 104–5, 187–8 full factorial designs, 104, 185–7 nested designs, 190–1 power, 191 product presentation, 105–6, 189–90 split-plot designs, 106, 190 explained variances, 213, 217, 221, 239 extended ANOVA tables, 53 external preference mapping (PREFMAP), 131, 132, 133, 137, 141, 159 external validation, 68, 72–4 F-distribution 167, 168 F-values, 28, 29, 43 factor analysis (FA), 224–5 factor effects, 98–9, 106–7, 108 FCM (fuzzy clustering method), 161–2, 162, 254–6, 260 filtered data, 44 finite mixture model clustering, 162, 258–9 formal equivalence testing, 88 fractional factorial experiment designs, 103, 104–5, 111, 150, 184, 187–8, 201–3 free choice profiling, 36 Friedman test, 120, 271 Frobenius norm, 265 full factorial experiment designs, 104, 184, 185–7, 188 fuzzy clustering method (FCM), 161–2, 162, 254–6, 260 GCA (generalised canonical analysis), 74, 75, 267–8 gender effects, 113–15 generators, 188 GLMs (generalised linear models), 245–6 GPA (generalised procrustes analysis), 74, 264–5 hierarchical clustering, 251–3 histograms, 16, 17, 99, 166 homogeneity tests, 179–80 hypothesis testing, 169–72, 176–8, 179–80 ideal point preference mapping, 130, 132, 141–3, 144, 162 IIA (independent of irrelevant alternatives) assumption, 122 incomplete block experiment designs, 189 individual assessor matrices, 36 individual effects, 116 individual line plots, 33–4, 35 INDSCAL (individual MDS) method, 271 industrial experiments, 186 interactions, 182, 185–6, 198, 199 internal preference mapping (MDPREF), 131–2, 133, 134 joint ANOVA approach, 108–11 joint modelling, 118 Index K-means clustering, 254, 256 Kruskal-Wallis test, 271 L-PLS regression, 149, 273 latent classes, 122 least squares, 175 level analysis, 81, 90, 93 level analysis, 80, 81–2, 90 level analysis, 80–1, 82, 90 level analysis (see Thurstonian approach) leverage, 222, 223, 242, 244 line plots, 16–18 linear combinations (vectors), 213 linear preference mapping, 129–41, 144 linear regression (see also multiple regression), 175–7 loading values, 212 loadings plots, 216 logit model, 122, 123 LS-PLS method, 71, 76 LSD (least significant differences) lines, 51 Mahalanobis distance, 251 Manhattan plots, 24, 25 Mann Whitney test, 271 MANOVA (multivariate ANOVA), 59–62 market share simulation, 123 maximum likelihood (ML) estimate, 246, 270 maximum utility model, 123 MDS (multidimensional scaling), 271 means (statistics) 168, 169, 170 medians, 169 MFA (multiple factor analysis), 268 missing cells/values, 18–19, 273 mixed model ANOVA, 18–19, 43, 203–5 model diagnostics, 241–4 model validation (see validation) MSC (multiplicative signal correction) transforms, 42 MSE (mean square error) values, 28, 29 multi-way ANOVA, 56, 57, 200–1 multinormal distributions, 258 multinomial regression, 121, 246 multiple regression, 68, 229–32 model diagnostics, 241–4 variable selection, 235 multivariate analysis, 15, 19 multivariate sensory data, 58–65 N-PLS (N-way PLS regression), 269 naive analysis, 91 nested experiment designs, 190–1, 205 NIPALS method, 220 NIR (near infrared) spectroscopy, 71, 74 285 noise clustering, 257–8 nonparametric methods, 52, 271 normal distribution 167 null hypothesis, 171 one-way ANOVA, 28–9, 56, 99, 102, 194–6, 232 optimal scaling (OS) method, 120, 272–3 optimisation (product properties), 72 order of product presentation, 105, 189–90 outliers, 16, 169, 221–2, 241, 242, 253 p-MSE plots, 28, 30 p-values, 28, 29, 50, 171 PARAFAC (parallel factor analysis), 74, 75, 76, 266, 271 parameters (statistics), 169 parametric models, 272 path modelling, 269–70 PC-ANOVA, 62–4, 65 PCA (principal component analysis), 58–9, 60–1, 115, 209–13 acceptance data, 112 relationship to factor analysis, 224–5 three-way PCA, 265–7 use of PCA in cluster analysis, 250 PCR (principal component regression), 69, 70, 71, 73, 133, 236–7 model diagnostics, 244 Placket-Burman experiment designs, 188 PLS (partial least square) regression, 69, 70, 71, 76, 133, 237–8, 269 model diagnostics, 244 PLS path modelling, 270 PLS/PCR modelling, 116 polynomial models, 117, 143, 233–4 populations (statistics), 165–6 post hoc testing, 206 power (experiment designs), 191 power analysis (discrimination tests), 85–6 pre-processing, 45 prediction testing, 221, 238–9 preference mapping, 68, 128–30 additional customer attributes, 147–9 additional sample attributes, 149–52 ideal point preference mapping, 141–3, 144 linear preference mapping, 129–41 sample selection, 146–7 preliminary analysis, 99–102 Proc mixed procedure, 205 Proclustrees method, 259 procrustes analysis, 35–6 Procrustes distance, 253, 259, 265 286 Index Procrustes rotations, 264 product combinations, 98, 150–2 product development, 135 product presentation, 105–6, 189–90 profile plots, 32–3 projection (vectors), 212, 213–14 q-q plots, 202, 242, 243 quadratic polynomial models, 130 quality control (sensory data), 11–36 quartiles, 169 random coefficient models, 52–3, 258 random consumer effects, 135 random errors, 108 randomisation, 185 randomness, 165 rank based studies, 119–20 ranking data, 3, 136, 271–3 ranking tests, 8–9 rating based studies, 8, 123 rating data, 156–63 raw data, 15–18 reduced experiment designs, 107, 109 regression analysis (see also linear regression, multiple regression), 57–8, 68–9, 70, 148, 227–9 relationship to ANOVA 232–3 regression coefficients, 229–30, 231–2, 236, 246, 272 relative frequencies, 166 relative utility model, 123 REML (restricted maximum likelihood) estimates, 65 repeated measurements, 185 replicates/replication, 53–6, 89–92, 184–5, 198 residual distances, 256–7 residuals, 19, 109, 221, 241–2, 244 resolution III/IV/V experiment designs, 187 RMSEP (root means square error of variation), 118–19, 239 RV coefficients, 45, 268 sample ranking, 29 sample selection, 146–7 sample spaces, 166, 167 Satterthwaite’s approximation, 205 scaling constants, 41, 42 scaling differences, 24–7, 40–3 scores plots, 214–16 segmentation, 112, 136, 155–64 self-explicated tests, 8, 96 sensory analysis, 219 sensory loadings, 132, 133, 135, 138, 141 sensory panels, 1, sensory profiling (see descriptive sensory analysis) sensory science, sequential clustering, 257, 258 sequential segmentation, 161 sign test, 271 similarity tests, 87–9 simple linear model 175 simultaneous clustering, 160–3 single linkage, 252 SPC (statistical process control), 172–3 Spearman rank order correlation, 272 split-plot experiment designs, 190, 205–6 standard deviations, 15–16, 40, 168, 170 standard errors, 169 standardisation (variables), 22, 219, 220 STATIS method, 44–5, 268 statistics, 165–80 contingency tables, 178 correlation, 173–5 distributions, 167–9 hypothesis testing, 169–72, 176–8, 179–80 linear regression, 175–6 SPC (statistical process control), 172–3 Stewart control chart, 172 stochastic variables, 166 structural equation modelling, 269 Student t-distribution 167, 171 subset regression, 235 supervised classification, 245 SVD (singular value decomposition), 220 Taguchi’s experiment designs, 188 tau-strategy, 86 Thurstonian approach, 81, 82–5, 86–7, 93 three-AFC test, 79, 80, 82, 83, 85, 87, 89 three-factor interactions, 185, 187 three-way analysis, 263–4, 269 three-way ANOVA, 45, 55, 204–5 three-way component methods, 36, 223 three-way PCA, 265–7 trained sensory panels (see sensory panels) triangle test, 79, 86, 91 true replicates, 185 Tucker-1 method, 21, 22, 24, 36, 71, 75, 224, 265, 266, 267, 268 Tucker-2 method, 74–5, 266–7, 269 Tukey’s correction, 51 Tukey’s test, 206 two-AFC test, 79, 80, 81, 82, 83, 85, 87 two-factor interactions, 185, 187 two-level factorial experiment designs, 186 two-way analysis, 269 two-way ANOVA, 48–52, 196–200, 203–4 Index type I errors, 172 type II errors, 172 type I/II/III tests, 201 unbalanced data, 18 unreliable assessors, 43–4 utilities, 113 287 VAF (variance accounted for) index, 35 validation, 118, 119, 123, 220–1, 232, 238–41 variable selection, 235 variances, 168, 169 weighted averages, 44 willingness to pay tests, ... Statistics for Sensory and Consumer Science Statistics for Sensory and Consumer Science TORMOD NÆS Nofima Mat, Norway and PER B BROCKHOFF Danish Technical University, Denmark and OLIVER... assessors Statistics for Sensory and Consumer Science C 2010 John Wiley & Sons, Ltd Tormod Næs, Per B Brockhoff and Oliver Tomic Statistics for Sensory and Consumer Science Figure 2.1 Three-way sensory. .. suitable for detecting outliers in the data A large standard deviation can for 16 Statistics for Sensory and Consumer Science 10 Score A B C D E F G H I Figure 3.3 Mean and standard deviation for