- Email: [email protected]
Using linear mixed models to handle variability of consumers’ liking
Abstracts / Food Quality and Preference 17 (2006) 658–668
Reference Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1992). Multivariate data analysis, with readings. New York: Maxwell Macmillan International.
Using linear mixed models to handle variability of consumers’ liking C. Chabanet a, N. Pineau b a
INRA UMR FLAVIC, 17 rue Sully, 21065 Dijon Cedex, France b Centre des sciences du gouˆt, 15 rue Hugues Picardet, 21000 Dijon, France E-mail addresses: [email protected] (C. Chabanet), [email protected] (N. Pineau) We address the question of variability of liking patterns among consumers. Firstly, consumers are clustered according to liking scores. A hierarchical clustering with Ward’s method of aggregation is performed on centred liking (individual median subtracted). This shows four main groups of
665
consumers. Then, mixed models (Diggle, Liang, & Zeger, 1994; Pinheiro & Bates, 2000) are used in order to highlight the relationship between liking and each of the physicochemical characteristics of the products, and its variability according to groups of consumers, or to consumers. The higher model M4 includes quadratic effects of the characteristic – say salt content– so as to handle the possible existence of an optimum, different intercept, linear and quadratic parameters according to group membership, and individual within group variability of intercept, linear and quadratic parameters (Table 1). This model includes both fixed effects, group, salt (intercept, linear and quadratic parameters), and interaction Table 1 Four mixed models to handle variability of liking pattern. (i: consumer, j: product, g(i) = 1 4: group membership of consumer i) M1: M2: M3: M4:
likingij ¼ li þ a saltj þ b salt2j þ eij li N ðl; r2l Þ; eij N ð0; r2e Þ likingij ¼ li þ ai saltj þ bi salt2j þ eij li N ðl; r2l Þ; ai N ða; r2a Þ; bi N ðb; r2b Þ; eij N ð0; r2e Þ likingij ¼ li þ agðiÞ saltj þ bgðiÞ salt2j þ eij li N ðlgðiÞ ; r2l Þ; eij N ð0; r2e Þ likingij ¼ li þ ai saltj þ bi salt2j þ eij li N ðlgðiÞ ; r2l Þ; ai N ðagðiÞ ; r2a Þ; bi N ðbgðiÞ ; r2b Þ; eij N ð0; r2e Þ
Fig. 1. Predicted liking for each group of consumers according to physicochemical properties of the products.
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Abstracts / Food Quality and Preference 17 (2006) 658–668
between them, and random effects, individual intercept, linear and quadratic terms that are supposed to be normally distributed, and allowed to be correlated. Three sub-models are considered. M3 is obtained by removing random linear and quadratic salt effects, M2 is obtained from M4 by removing group effect and group interaction with salt, M1 is obtained from M4 by removing random linear and quadratic term, group effect, and group interaction with salt, so that there remain only fixed linear and quadratic salt effects and random intercept. Parameters are estimated with REML estimator. The likelihood ratio test is used to compare two models with the same fixed part, and nested random terms. Fixed terms of a model are tested using conditional F-tests. Akaike Information Criterion (AIC) and Bayesian Schwarz Criterion (BIC) are used to compare non-nested models. The model with smaller AIC is M4 whereas the model with smaller BIC is M3. M4 and M3 are shown to be significantly different according to the likelihood ratio test (p = 0.01). Although random linear and quadratic terms are significant, they can be removed if one wants a model with a smaller number of parameters. Both group:salt interaction and group effects are shown to be highly significant (p < 0.0001). Therefore, the group membership captures the main variability although there remains a small individual variability around group pattern. Nonetheless, this small individual variability can be overlooked. This result is the same for each of the physicochemical variables, except for variable b*. For this variable, both AIC and BIC give M4 as the best model. There remains an individual variability that is not captured by the four groups. Predicted liking for each group of consumers are represented according to physicochemical properties of the products (Fig. 1). Consumers of group 4, who give rather low liking, search for optimal lipid and colour (L) levels whereas other groups are not or less sensitive to these characteristics. For groups 3 and 4, the lower the salt is, the lower is the liking. A low salt content is associated with high levels of TMA and TVBN. Therefore, the low liking may be due to low salt content or to high TMA and TVBN levels. One must pay attention to the correlations among characteristics during interpretation. This latter remark indicates that more work remains to be done in order to investigate how to handle several characteristics simultaneously or how to take into account all other characteristics of the products.
Analysis of salmon data using analysis of variance and preference-mapping-like analysis P.B. Brockhoff Informatics and Mathematical Modelling, The Technical University, Richard Petersens Plads, Build. 321, DK-2800 Kgs. Lyngby, Denmark E-mail address: [email protected] Analysis of variance techniques were used to decompose the 30 · 1063 = 31 890 consumer preference evaluations and to look for systematic patterns related to product characteristics and consumer characteristics. Overall, the data structure can be viewed as randomized block data with consumers as blocks and products as treatments. Internal preference mapping (McEwan, 1996) investigates the structure of the main effects of products together with the residual variation. External preference mapping (McEwan, 1996) using the 16 product characteristic variables can be seen as an attempt to decompose and interpret the product main effects together with the part of the residual variation that can be ascribed to interactions between consumers and the product characteristics. Interpreting the positions of the individual 1063 consumers in either internal or external preference mapping in light of consumer background/ usage/attitude information can be seen as an attempt to decompose these interactions between consumers and the product-effects. Hence it is of relevance to identify whether there are any indications of such interactions in the data, as it indicates consumer segmentation – and segmentation for which interpretations are readily available. Compared to the more common approach where segments of consumers are initially constructed based on the preference patterns, the present approach more directly tries to identify and summarize the potential consumer differences related to consumer demographics and usage and attitude questions. Using random effects/mixed model considerations the importance of the various effects is evaluated in light of the general consumer and/or product variations present in the data. Product characteristics investigated were principal components (and their squares) of the 16 quantitative variables together with the two categorical product variables: Origin and country. All consumer background,
doi:10.1016/j.foodqual.2006.03.008
References Diggle, P. J., Liang, K. Y., & Zeger, S. L. (1994). Analysis of longitudinal data. Oxford: Oxford University Press. Pinheiro, J. C., & Bates, D. M. (2000). Mixed-effects models in S and SPLUS. New York: Springer-Verlag.
Fig. 1. A diagram illustrating the structure for the analysis for each of the 79 consumer group variables (G). Prodvars stands for the inclusion of 10 product characteristic effects: PC1, PC2, PC3, PC4, PC12, PC22, PC32, PC42, origin and country. Cons = consumer, Prod = product. The underlined effects are considered as random.
Reference Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1992). Multivariate data analysis, with readings. New York: Maxwell Macmillan International.
Using linear mixed models to handle variability of consumers’ liking C. Chabanet a, N. Pineau b a
INRA UMR FLAVIC, 17 rue Sully, 21065 Dijon Cedex, France b Centre des sciences du gouˆt, 15 rue Hugues Picardet, 21000 Dijon, France E-mail addresses: [email protected] (C. Chabanet), [email protected] (N. Pineau) We address the question of variability of liking patterns among consumers. Firstly, consumers are clustered according to liking scores. A hierarchical clustering with Ward’s method of aggregation is performed on centred liking (individual median subtracted). This shows four main groups of
665
consumers. Then, mixed models (Diggle, Liang, & Zeger, 1994; Pinheiro & Bates, 2000) are used in order to highlight the relationship between liking and each of the physicochemical characteristics of the products, and its variability according to groups of consumers, or to consumers. The higher model M4 includes quadratic effects of the characteristic – say salt content– so as to handle the possible existence of an optimum, different intercept, linear and quadratic parameters according to group membership, and individual within group variability of intercept, linear and quadratic parameters (Table 1). This model includes both fixed effects, group, salt (intercept, linear and quadratic parameters), and interaction Table 1 Four mixed models to handle variability of liking pattern. (i: consumer, j: product, g(i) = 1 4: group membership of consumer i) M1: M2: M3: M4:
likingij ¼ li þ a saltj þ b salt2j þ eij li N ðl; r2l Þ; eij N ð0; r2e Þ likingij ¼ li þ ai saltj þ bi salt2j þ eij li N ðl; r2l Þ; ai N ða; r2a Þ; bi N ðb; r2b Þ; eij N ð0; r2e Þ likingij ¼ li þ agðiÞ saltj þ bgðiÞ salt2j þ eij li N ðlgðiÞ ; r2l Þ; eij N ð0; r2e Þ likingij ¼ li þ ai saltj þ bi salt2j þ eij li N ðlgðiÞ ; r2l Þ; ai N ðagðiÞ ; r2a Þ; bi N ðbgðiÞ ; r2b Þ; eij N ð0; r2e Þ
Fig. 1. Predicted liking for each group of consumers according to physicochemical properties of the products.
666
Abstracts / Food Quality and Preference 17 (2006) 658–668
between them, and random effects, individual intercept, linear and quadratic terms that are supposed to be normally distributed, and allowed to be correlated. Three sub-models are considered. M3 is obtained by removing random linear and quadratic salt effects, M2 is obtained from M4 by removing group effect and group interaction with salt, M1 is obtained from M4 by removing random linear and quadratic term, group effect, and group interaction with salt, so that there remain only fixed linear and quadratic salt effects and random intercept. Parameters are estimated with REML estimator. The likelihood ratio test is used to compare two models with the same fixed part, and nested random terms. Fixed terms of a model are tested using conditional F-tests. Akaike Information Criterion (AIC) and Bayesian Schwarz Criterion (BIC) are used to compare non-nested models. The model with smaller AIC is M4 whereas the model with smaller BIC is M3. M4 and M3 are shown to be significantly different according to the likelihood ratio test (p = 0.01). Although random linear and quadratic terms are significant, they can be removed if one wants a model with a smaller number of parameters. Both group:salt interaction and group effects are shown to be highly significant (p < 0.0001). Therefore, the group membership captures the main variability although there remains a small individual variability around group pattern. Nonetheless, this small individual variability can be overlooked. This result is the same for each of the physicochemical variables, except for variable b*. For this variable, both AIC and BIC give M4 as the best model. There remains an individual variability that is not captured by the four groups. Predicted liking for each group of consumers are represented according to physicochemical properties of the products (Fig. 1). Consumers of group 4, who give rather low liking, search for optimal lipid and colour (L) levels whereas other groups are not or less sensitive to these characteristics. For groups 3 and 4, the lower the salt is, the lower is the liking. A low salt content is associated with high levels of TMA and TVBN. Therefore, the low liking may be due to low salt content or to high TMA and TVBN levels. One must pay attention to the correlations among characteristics during interpretation. This latter remark indicates that more work remains to be done in order to investigate how to handle several characteristics simultaneously or how to take into account all other characteristics of the products.
Analysis of salmon data using analysis of variance and preference-mapping-like analysis P.B. Brockhoff Informatics and Mathematical Modelling, The Technical University, Richard Petersens Plads, Build. 321, DK-2800 Kgs. Lyngby, Denmark E-mail address: [email protected] Analysis of variance techniques were used to decompose the 30 · 1063 = 31 890 consumer preference evaluations and to look for systematic patterns related to product characteristics and consumer characteristics. Overall, the data structure can be viewed as randomized block data with consumers as blocks and products as treatments. Internal preference mapping (McEwan, 1996) investigates the structure of the main effects of products together with the residual variation. External preference mapping (McEwan, 1996) using the 16 product characteristic variables can be seen as an attempt to decompose and interpret the product main effects together with the part of the residual variation that can be ascribed to interactions between consumers and the product characteristics. Interpreting the positions of the individual 1063 consumers in either internal or external preference mapping in light of consumer background/ usage/attitude information can be seen as an attempt to decompose these interactions between consumers and the product-effects. Hence it is of relevance to identify whether there are any indications of such interactions in the data, as it indicates consumer segmentation – and segmentation for which interpretations are readily available. Compared to the more common approach where segments of consumers are initially constructed based on the preference patterns, the present approach more directly tries to identify and summarize the potential consumer differences related to consumer demographics and usage and attitude questions. Using random effects/mixed model considerations the importance of the various effects is evaluated in light of the general consumer and/or product variations present in the data. Product characteristics investigated were principal components (and their squares) of the 16 quantitative variables together with the two categorical product variables: Origin and country. All consumer background,
doi:10.1016/j.foodqual.2006.03.008
References Diggle, P. J., Liang, K. Y., & Zeger, S. L. (1994). Analysis of longitudinal data. Oxford: Oxford University Press. Pinheiro, J. C., & Bates, D. M. (2000). Mixed-effects models in S and SPLUS. New York: Springer-Verlag.
Fig. 1. A diagram illustrating the structure for the analysis for each of the 79 consumer group variables (G). Prodvars stands for the inclusion of 10 product characteristic effects: PC1, PC2, PC3, PC4, PC12, PC22, PC32, PC42, origin and country. Cons = consumer, Prod = product. The underlined effects are considered as random.