Bayesian implementation of Skew-Normal distributions for experimental error in the description of pepper growth
Main Article Content
Abstract
The purpose of this work is to implement and compare three types of Skew-Normal distributions together with the Normal submodel for the experimental error through a Bayesian nonlinear modeling of pepper phenotype growth. The posterior means of the parameters were shown to be invariant to the experimental error. The Sahu Skew-Normal error showed evidence of null asymmetry for all growth models. In general, all Skew-Normal distributions presented the highest accuracy ( ) in relation to the Normal error. The problematic points stand out for the computational cost in the millions of MCMC iterations and the limitations of the BUGS language. The Skew-Normal distribution of Azzalini and Fernandéz & Steel modeled the asymmetry of the data and provided the best goodness of fit (DIC) and the best precision ( ) for the Gompertz and Von Bertalanffy model in relation to the Normal error.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
References
Azzalini, A. A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics. 12, 171–178 (1985). http://www.jstor.org/stable/4615982.
Cancho, V.G. et al. A nonlinear regression model with skew-normal errors. Stat Papers. v51, 547-558 (2008). https://doi.org/10.1007/s00362-008-0139-y.
Castillo, N. O., Gómez, H. W., Leiva, V., Sanhueza, A. On the Fernández–Steel distribution: Inference and application. Computational Statistics & Data Analysis, 55, 2951-2961 (2011). https://doi.org/10.1016/j.csda.2011.04.023.
Cordeiro, G. M., Demétrio, C. G. B. Modelos lineares generalizados e extensões, 1. ed. Recife, PE: [s.n.], 2013.
De la cruz, R., Branco, M. Bayesian analysis for nonlinear regression model under skewed errors, with application in growth curves. Biometrical Journal, Weinheim, 51, 588-609, (2009). https://doi.org/10.1002/bimj.200800154.
Eloy, R. A. O. et al. Distribuição assimétrica t-Student tipo 3: Uma aplicação a delineamentos inteiramente casualizados. SAJEBTT, 6, 55-77, (2020). https://periodicos.ufac.br/index.php/SAJEBTT/article/view/2445 .
Faccin, M.Z. and Rossi, R.M. Bayesian modeling of the Gompertz curve for meat quails growth data considering different error distributions. Brazilian Journal of Biometrics. v.42, 3 260–271, (2024) https://doi.org/10.28951/bjb.v42i3.699.
Fernandes, T. J., Muniz, J. A., Pereira, A. A., Muniz, F. R.; Muianga, C. A. Parameterization effects in nonlinear models to describe growth curves. Acta Scientiarum. Technology, 37, 397-402 (2015). https://doi.org/10.4025/actascitechnol.v37i4.27855.
Fernandes, F. A., Silva, Édipo M., Lima, K. P., Jane, S. A., Fernandes, T. J., Muniz, J. A. Parameterizations of the von bertalanffy model for description of growth curves. Brazilian Journal of Biometrics, 38, 369–384, (2020). https://doi.org/10.28951/rbb.v38i3.457.
Firat, MZ et al. Bayesian Analysis for the Comparison of Nonlinear Regression Model Parameters: An Application to the Growth of Japanese Quail. Revista Brasileira de Ciência Avícola, 18, 19-26, (2016). https://doi.org/10.1590/1806-9061-2015-0066.
Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., & Rubin, D.B. Bayesian Data Analysis. (Chapman and Hall/CRC, 2013).
Guedes, T.A. et al. Applying regression models with skew-normal errors to the height of bedding plants of Stevia rebaudiana (Bert) Bertoni. Acta Scientiarum Technology, 36, 463-468, (2014). https://doi.org/10.4025/actascitechnol.v36i3.21191.
Ghaderinezhad F, et al. Bayesian Inference for Skew-Symmetric Distributions. Symmetry, 12, (2020). https://doi.org/10.3390/sym12040491.
Gujarati, D. N., Porter, D.C. Econometria básica. (AMGH, 2012).
Henze, N. A Probabilistic Representation of the 'Skew-normal' Distribution. Scandinavian Journal of Statistics, 13, 271-275, (1986). http://www.jstor.org/stable/4616036.
Koya, P.R., Goshu, A.T. Generalized Mathematical Model for Biological Growths. Open Journal of Modelling and Simulation, 1, 42-53 (2013). https://www.scirp.org/journal/paperinformation?paperid=38842.
Louis, Thomas A., Carlin, Bradley P. Bayesian Methods for Data Analysis. 2 ed. (United States: CRC Press, 2008).
Luna, J.G., Olinda, R.A. Introdução a modelos lineares. (São Paulo. Livraria da física. 2014).
Macedo, L. R. et al. Bayesian inference for the fitting of dry matter accumulation curves in garlic plants. Pesquisa Agropecuária Brasileira, 52, 572-581, (2017). https://doi.org/10.1590/S0100-204X2017000800002.
Mangueira, R. A. F., Savian, T. V., Muniz, J. A., Sermarini, R. A., Crosariol N, J. Logistic model considering different error distributions applied in maize height data. Brazilian Journal of Biometrics. 34, 317–333, (2016). https://biometria.ufla.br/index.php/BBJ/article/view/143.
Martins Filho, S. et al. Abordagem Bayesiana das curvas de crescimento de duas cultivares de feijoeiro. Ciência Rural, 38, 1516-1521, (2008). https://doi.org/10.1590/S0103-84782008000600004.
Nadarajah, S. The skew logistic distribution. AStA Adv Stat Anal, 187-203, (2009). https://doi.org/10.1007/s10182-009-0105-6.
Nelder, J. A., Wedderburn, R. W. M. Generalized linear models. Journal of the Royal Statistical Society, 135, 370-384, (1972). https://doi.org/10.2307/2344614.
Ntzoufras, I. Bayesian Modeling Using WinBUGS. (1st ed. Wiley, 2008). https://onlinelibrary.wiley.com/doi/book/10.1002/9780470434567.
Nunes Ávila, M. de S., Barbosa, J. M. Análise de crescimento de pimenta-biquinho em diferentes níveis de radiação solar. Brazilian Journal of Development, 5, 31985-31997, (2019). https://doi.org/10.34117/bjdv5n12-279 .
Pérez-Rodríguez, P. et al. A Bayesian Genomic Regression Model with Skew Normal Random Errors. G3 Genes|Genomes|Genetics, 8, 1771-1785, (2018). https://doi.org/10.1534/g3.117.300406 .
Pino, F.A. A questão da não normalidade: Uma revisão. Rev. de Economia Agrícola, 61, 17-33, (2014). http://www.iea.sp.gov.br/ftpiea/publicar/rea2014-2/rea2-22014.pdf.
Rosa, M. et al. Linear and nonlinear models to describe the lactation curve of Girolando cows. Pesquisa Agropecuária Brasileira, 57, 2678 (2022). https://doi.org/10.1590/S1678-3921.pab2022.v57.02678.
Rossi. R.M., Santos. L.A. Modelagem Bayesiana para curvas de crescimentos de codornas assumindo assimetria nos erros. Semina: Ciências Agrárias, 35, 1637-1647, (2014). https://doi.org/10.5433/1679-0359.2014v35n3p1637.
Sahu, S.K., Dey, D.K. and Branco, M.D. A new class of multivariate skew distributions with applications to bayesian regression models. Can J Statistics, v.31, 129-150. (2003). https://doi.org/10.2307/3316064.
Souza, G.S. Introdução aos modelos de Regressão linear e não linear. (Brasília. Embrapa, 1998).
Teixeira, F.R.F., Cecon, P.R., Suela, M.M., Nascimento, M. Nonlinear Mixed-Effect Models to Describe Growth Curves of Pepper Fruits in Eight Cultivars Including Group Effects. Agronomy, v.12, (2023). https://www.mdpi.com/2073-4395/13/8/2042h .