A bayesian mixed Beta model applied to multivariate sensory data
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Resumo
Sensory evaluation studies often rely on hedonic scales that yield ordinal and bounded data. Traditional statistical techniques, although widely applied, often fail to account to the boundedness and interdependencies characteristic of sensory responses. This study proposes a Bayesian beta regression framework specifically designed for sensory data, aiming to extend existing methodologies by addressing the joint behavior of multiple attributes. The approach models scores rescaled to the (0,1) interval via a beta distribution, effectively capturing formulation effects and correlations among sensory attributes. To this end, we assumed a hierarchical structure for the regression coefficients. We chose weak prior distributions to avoid strong or subjective assumptions that might distort the results thereby, making the analysis more robust. Without strong restrictions on the prior model, the posterior inference was conducted via Markov Chain Monte Carlo (MCMC) methods. By jointly analyzing all attributes within a hierarchical structure, the method enables direct estimation of inter-attribute associations and offers a more integrated interpretation of product performance. Applied to a grape juice acceptance study, the model not only identified the most preferred formulations but also unveiled meaningful patterns across sensory dimensions. With this Bayesian construction, we present a singular contribution that provides a reliable alternative for researchers seeking to analyze bounded sensory responses while simultaneously exploring the multivariate nature of consumer perception.
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Referências
1. ABNT. ABNT NBR ISO 4121:2018 – Análise sensorial – Guia geral para o uso de escalas de
respostas quantitativas Rio de Janeiro, 2018.
2. Agresti, A. Categorical data analysis (JohnWiley & Sons, 2012).
3. Alves, J. C. R., Palma, G. R. & de Lara, I. A. R. Unified Beta Regression Model with Random
Effects for the Analysis of Sensory Attributes. arXiv preprint arXiv:2504.05996 (2025).
4. Bardsley, E. Tendency toward negative correlations for positively-skewed independent random
variables. Journal of Hydrology (New Zealand) 53, 175–177 (2014).
5. Barrientos, A. F., Jara, A. & Quintana, F. A. Fully nonparametric regression for bounded data
using dependent Bernstein polynomials. Journal of the American Statistical Association 112, 806–
825 (2017).
6. Bayes, C. L., Bazán, J. L. & García, C. A new robust regression model for proportions. Bayesian
Analysis 7, 841–866 (2012).
7. Bernardo, J. M. & Smith, A. F. Bayesian theory (JohnWiley & Sons, 2009).
8. Branscum, A. J., Johnson, W. O. & Thurmond, M. C. Bayesian beta regression: applications
to household expenditure data and genetic distance between foot-and-mouth disease viruses.
Australian & New Zealand Journal of Statistics 49, 287–301 (2007).
9. Brooks, S. P. & Gelman, A. General methods for monitoring convergence of iterative simulations.
Journal of computational and graphical statistics 7, 434–455 (1998).
10. Christensen, R. H. B. & Brockhoff, P. B. Analysis of sensory ratings data with cumulative link
models. Journal de la Société Française de Statistique 154, 58–79 (2013).
11. Chung, Y., Gelman, A., Rabe-Hesketh, S., Liu, J. & Dorie, V. Weakly informative prior for
point estimation of covariance matrices in hierarchical models. Journal of Educational and Behavioral Statistics 40, 136–157 (2015).
12. Cruz, M., Carvalho, D.,Colombo, R.,Yokota, L., Silva, A.,Neto, H. & Roberto, S. Exploratory
analysis of the sensory attributes of american grape juice blends. Agronomy Science and Biotechnology 4, 79–79 (2018).
13. Di Brisco, A. M., Bongiorno, E. G., Goia, A. & Migliorati, S. Bayesian flexible beta regression
model with functional covariate. Computational Statistics 38, 623–645 (2023).
14. Dunson, D. B. Commentary: practical advantages of Bayesian analysis of epidemiologic data.
American journal of Epidemiology 153, 1222–1226 (2001).
15. Fernández-Vázquez, R., Hewson, L., Fisk, I., Vila, D. H., Mira, F. J. H., Vicario, I. M. & Hort,
J. Colour influences sensory perception and liking of orange juice. Flavour 3, 1 (2014).
16. Ferrari, S. & Cribari-Neto, F. Beta regression for modelling rates and proportions. Journal of
applied statistics 31, 799–815 (2004).
17. Figueroa-Zúñiga, J. I., Arellano-Valle, R. B. & Ferrari, S. L. Mixed beta regression: A Bayesian
perspective. Computational Statistics & Data Analysis 61, 137–147 (2013).
18. Flores, R. D., Sanders, C. A., Duan, S. X., Bishop-Chrzanowski, B. M., Oyler, D. L., Shim,
H., Clocksin, H. E., Miller, A. P. & Merkle, E. C. Before/after Bayes: A comparison of frequentist
and Bayesian mixed-effects models in applied psychological research. British Journal of
Psychology 113, 1164–1194 (2022).
19. Gadrich, T., Pennecchi, F. R., Kuselman, I., Hibbert, D. B., Semenova, A. A. & Cheow, P. S.
Ordinal analysis of variation of sensory responses in combination with multinomial ordered
logistic regression vs. chemical composition: a case study of the quality of a sausage from different
producers. Journal of Food Quality 2022, 4181460 (2022).
20. Gelman, A, Carlin, J, Stern, H, Dunson, D, Aki, V. & Rubin, D. Bayesian data analysis Gelman.
J Chem Inf Model 53, 1689–99 (2013).
21. Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. Bayesian data analysis (Chapman and
Hall/CRC, 1995).
22. Gilks, W. R., Richardson, S. & Spiegelhalter, D. Markov Chain Monte Carlo in Practice (Chapman
and Hall/CRC, 1995).
23. Gous, A. G., Almli, V. L., Coetzee, V. & de Kock, H. L. Effects of varying the color, aroma,
bitter, and sweet levels of a grapefruit-like model beverage on the sensory properties and liking
of the consumer. Nutrients 11, 464 (2019).
24. Huang, K. The development of sensory analysis techniques in the food industry and the research progress. Theoretical and Natural Science 71, 164–169 (2024).
25. Kardas, M., Rakuła, M., Kołodziejczyk, A. & Staśkiewicz-Bartecka, W. Consumer Preferences,
Sensory Evaluation, and Color Analysis of Beetroot and Tomato Juices: Implications for
Product Development and Marketing in Health-Promoting Beverages. Foods 13, 4059 (2024).
26. Kass, R. E. &Wasserman, L. The selection of prior distributions by formal rules. Journal of the
American statistical Association 91, 1343–1370 (1996).
27. Kruschke, J. Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan 2nd (Academic
Press, 2014).
28. Matsuura, F. C. A., CARDOSO, R. L. & RIBEIRO, D. E. Qualidade sensorial de frutos de
híbridos de bananeira cultivar Pacovan. Revista Brasileira de Fruticultura 24, 263–266 (2002).
29. McCullagh, P. & Nelder, J. A. Generalized Linear Models (Chapman and Hall, 1989).
30. Meiselman, H., Jaeger, S., Carr, B. & Churchill, A. Approaching 100 years of sensory and
consumer science: Developments and ongoing issues. Food Quality and Preference 100, 104614
(2022).
31. Moskowitz, H. R. The Role of Sensory Science in the Coming Decade. Viewpoints and Controversies in Sensory Science and Consumer Product Testing, 1 (2008).
32. Paulino, C. D. M., Turkman, M. A. A. & Murteira, B. Estatística Bayesiana (Fundação Calouste
Gulbenkian, 2018).
33. Plummer, M. rjags: Bayesian Graphical Models using MCMC R package version 4-17 (2025).
https://CRAN.R-project.org/package=rjags.
34. R Core Team. R: A Language and Environment for Statistical Computing R Foundation for Statistical Computing (Vienna, Austria, 2024). https://www.R-project.org/.
35. Revuelta, J., Hidalgo, B. & Alcazar-Córcoles, M. Á. Bayesian estimation and testing of a
beta factor model for bounded continuous variables. Multivariate Behavioral Research 57, 57–78
(2022).
36. Robert, C. P. et al. The Bayesian choice: from decision-theoretic foundations to computational implementation (Springer, 2007).
37. Smithson, M. & Verkuilen, J. A better lemon squeezer? Maximum-likelihood regression with
beta-distributed dependent variables. Psychological methods 11, 54 (2006).
38. Spence, C. On the relationship (s) between color and taste/flavor. Experimental psychology
(2019).
39. Ugba, E. R., Mörlein, D. & Gertheiss, J. Smoothing in ordinal regression: An application to
sensory data. Stats 4, 616–633 (2021).
40. Vehtari, A., Gelman, A. & Gabry, J. Practical Bayesian model evaluation using leave-one-out
cross-validation andWAIC. Statistics and Computing 27, 1413–1432 (2017).