Christine Borgen Linander 1,, Rune Haubo Bojesen Christensen 1, Rebecca Evans 2, Graham Cleaver 2, Per Bruun Brockhoff 1. University of Denmark
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1 Individual differences in replicated multi-product 2-AFC data with and without supplementary difference scoring: Comparing Thurstonian mixed regression models for binary and ordinal data with linear mixed models Christine Borgen Linander 1,, Rune Haubo Bojesen Christensen 1, Rebecca Evans 2, Graham Cleaver 2, Per Bruun Brockhoff 1 1 Department of Informatics and Mathematical Modeling, Section for Statistics, Technical University of Denmark 2 Unilever Research and Development Port Sunlight, UK *Contact author: chjo@imm.dtu.dk Christine Borgen Linander Sensometrics 2012 July 13th /30
2 Outline 1 Introduction 2 Results 3 Final remarks Christine Borgen Linander Sensometrics 2012 July 13th /30
3 Introduction Experiment Data: 12 Products - denoted A, B, C,..., L 24 Assessors 8 Attributes 5 different attributes - denoted A, B, C, D, E 3 of those evaluated again after 5 minutes - denoted C.5m, D.5m, E.5m 2 Sessions leading to replications Each Product was compared to a control. Not all combinations of Assessor and Product are present. Hence incomplete data. (Have included Assessors with at least 50% of the observations) For each Attribute a maximum of 24 observations for each Assessor. Christine Borgen Linander Sensometrics 2012 July 13th /30
4 Introduction Experiment Test protocol: 2-AFC protocol With an additional task to score the difference from not different to extremely different with 5 categories The data from the additional task is transformed into a scale from -4 to 4 where: 0 corresponds to no difference negative values are in favor of the control positive values are in favor of the Product Thus possibility to consider the response in 3 ways: Binary with values 0, 1 Quantitative with values -4, 3,..., 3, 4 Ordinal with values -4, 3,..., 3, 4 Christine Borgen Linander Sensometrics 2012 July 13th /30
5 Introduction Aims for the analysis Analytical aims: We want to illustrate that it is possible to fit a Thurstonian mixed model with the binary response We want to have the capability to take account of other sources of variation when analysing data from the 2-AFC protocol To be more specific - we want to model the individual assessors We want to analyse the results from a set of products in one analysis rather than one product comparison at a time Business oriented aim: We want to investigate if we gain valuable extra information adding the additional task to the usual 2-AFC protocol. Christine Borgen Linander Sensometrics 2012 July 13th /30
6 Introduction Models We consider 2 types of models: Naive aggregation model - ignoring the replicated structure Mixed model - including Assessor as a random effect, modelling the replicated structure Where the naive model will be fitted with the binary response. And the mixed model will be fitted with the 3 different responses. Christine Borgen Linander Sensometrics 2012 July 13th /30
7 F test - mixed model - Quantitative response The PanelCheck way by the R-package lmertest Assessor by Product interaction effects F statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
8 F test - mixed model - Quantitative response Assessor main effects F statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
9 F test - mixed model - Quantitative response Product main effects F statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
10 Likelihood Ratio test - mixed model - Binary response Assessor by Product interaction effects Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
11 Likelihood Ratio test - mixed model - Binary response Assessor main effects Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
12 Likelihood Ratio test - mixed model - Binary response Product main effects mixed model Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
13 Likelihood Ratio test - naive model - Binary response Product main effects naive model Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
14 Likelihood Ratio test - comparison of Product - Binary response Comparing test statistics for the mixed and naive model Likelihood Ratio statistic Mixed model Naive model A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
15 Comparing the two models for Product - Binary response Not a big difference in values between the methods. This fits nicely with the Assessor-by-Product interaction being non-significant In general you would expect the values from the mixed model to be less than the values from the naive model For some attributes this is not the case. These are the ones with the highest values of the test of Assessor main effect Looking at the plot of the Assessor main effect you would expect that this was also the case for for the D.5m attribute. But looking at the plot of the Assessor-by-Product interaction this is the attribute with the highest value Christine Borgen Linander Sensometrics 2012 July 13th /30
16 Post hoc - Product differences in terms of d-prime Product estimates for the attribute E.5m (95% confidence interval) A B C D E F G H I J K L Products Christine Borgen Linander Sensometrics 2012 July 13th /30
17 Post hoc - Product differences in terms of d-prime A B C D A D G J A D G J A D G J A D G J Products Products Products Products E C.5m D.5m E.5m A D G J A D G J A D G J A D G J Products Products Products Products Christine Borgen Linander Sensometrics 2012 July 13th /30
18 Post hoc - PCA of product d-prime values Bi plot Principal component 2 (11.4%) L E H A J C K B B E A D E.5m D.5m G C C.5m D I F Principal component 1 (74.4%) Christine Borgen Linander Sensometrics 2012 July 13th /30
19 Post hoc - Assessor performance in terms of d-prime Assessor estimates for the attribute E.5m (95% confidence interval) Christine Borgen Linander Sensometrics 2012 July 13th /30
20 Post hoc - Assessor performance in terms of d-prime A B C D Assessors Assessors Assessors Assessors E C.5m D.5m E.5m Assessors Assessors Assessors Assessors Christine Borgen Linander Sensometrics 2012 July 13th /30
21 Likelihood Ratio test - mixed model - Ordinal response Using the R-package ordinal Assessor by Product interaction effects Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
22 Likelihood Ratio test - mixed model - Ordinal response Assessor main effects Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
23 Likelihood Ratio test - mixed model - Ordinal response Product main effects Likelihood Ratio statistic < p value 0.01 < p value < < p value < 0.01 p value < A B C D E C.5m D.5m E.5m Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
24 Likelihood Ratio test - comparison Assessor Likelihood Ratio test Assessor main effects Likelihood ratio statistic Binary Ordinal Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
25 Likelihood Ratio test - comparison Product Likelihood Ratio test Product main effects Likelihood ratio statistic Binary Ordinal Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
26 Likelihood Ratio test - comparison Assessor-by-Product Likelihood Ratio test Assessor by Product interaction effects Likelihood ratio statistic Binary Ordinal Attributes Christine Borgen Linander Sensometrics 2012 July 13th /30
27 Final remarks Final remarks We can handle incomplete observations We can fit the Thurstonian mixed model with binary data The Thurstonian mixed model provides additional information (compared to the naive aggregation model) d-prime interpretations of Product estimates d-prime interpretations of Assessor estimates These preliminary results indicate that the ordinal information is more sensitive than the binary information Christine Borgen Linander Sensometrics 2012 July 13th /30
28 Future work Final remarks At some point in the future: Investigation of how much binary data is needed to find a significant Assessor-by-Product interaction Come up with a proper Thurstonian interpretation of the ordinal approach General modelling - for instance how to handle order effects Christine Borgen Linander Sensometrics 2012 July 13th /30
29 Final remarks Thank you for your attention! Christine Borgen Linander Sensometrics 2012 July 13th /30
30 References Final remarks PanelCheck: Open source software for sensory profile data, Christensen, R. H. B. (2012). Ordinal - Regression Models for Ordinal Data. R package version Christensen, R. H. B. and P. B. Brockhoff (2012). Analysis of replicated categorical ratings data from sensory experiments. Working paper. Alexandra Kuznetsova, Per Bruun Brockhoff and Rune Haubo Bojesen Christensen (2012). lmertest: Tests for random and fixed effects for linear mixed effect models (lmer objects of lme4 package).. R package version 1.0. Brockhoff, P.B. and Christensen, R.H.B. (2010). Thurstonian models for sensory discrimination tests as generalized linear models. Food Quality and Preference 21(3), Christine Borgen Linander Sensometrics 2012 July 13th /30
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