Modelos Estatísticos Fatoriais: Conceitos e Aplicações

Pedro Henrique Mainardi; Ederio Dino Bidoia

doi:10.22481/exon.v11i1.20492

Autores

Pedro Henrique Mainardi Universidade Estadual Paulista Júlio de Mesquita Filho https://orcid.org/0000-0003-0234-0052
Ederio Dino Bidoia Universidade Estadual Paulista Júlio de Mesquita Filho https://orcid.org/0000-0001-7040-1983

DOI:

https://doi.org/10.22481/exon.v11i1.20492

Palavras-chave:

Planejamento experimental, delineamento experimental, desenho experimental, design experimental, modelagem estatística, otimização de processos

Resumo

Modelos estatísticos são fundamentais em investigações científicas, amplamente utilizados na caracterização da relação entre variáveis de interesse, quantificação dos efeitos de fatores, estimação da variabilidade experimental, otimização de processos e também na derivação de funções matemáticas que descrevem e preveem o comportamento dos resultados. A partir da utilização de matrizes com todas as combinações possíveis, modelos estatísticos fatoriais foram vistos por serem capazes de gerar grande quantidade de informação em um número reduzido de ensaios, além de considerar simultaneamente as interações de efeito de experimentos com múltiplas variáveis, ajustar modelos com curvatura e supor com precisão a magnitude de erros experimentais. Modelos fatoriais têm sido fracionalizados com sucesso, reduzindo consideravelmente o número de experimentos sem a perda de informação relevante. As características de ortogonalidade e rotacionalidade (rotatabilidade), adquiridas com a adição de pontos estrelas e centrais aos arranjos dos planejamentos fatoriais, têm possibilitado análises robustas dos efeitos individuais e combinados das variáveis, além de propiciar a construção de equações matemáticas quadráticas de forma eficaz. Relatos publicados na literatura indicaram que modelos estatísticos fatoriais, também chamados de designs, delineamentos e planejamentos fatoriais, foram aplicados em extensivos e compreensivos estudos com múltiplos fatores e utilizados com sucesso na elaboração de equações polinomiais simples ou complexas relacionadas com os resultados experimentais. Foram também vastamente utilizados no desenho de gráficos de superfície de resposta, plotes de contorno e na assunção de importantes características de experimentos científicos e processos industriais, como a influência dos efeitos individuais e das interações entre os fatores, a variação residual, e ótimas condições operacionais. Nesta revisão, foram conceitualizados os aspectos fundamentais dos modelos estatísticos fatorais e descritas aplicações bem sucedidas dessa poderosa ferramenta.

Downloads

Não há dados estatísticos.

Referências

ZANCAN, Glaci T. Educação científica: uma prioridade nacional. São Paulo em perspectiva, v. 14, n. 3, p. 3-7, 2000.

CHRISTENSEN, Ronald. Analysis of variance, design, and regression: applied statistical methods. CRC Press, 1996.

MASON, Robert L.; GUNST, Richard F.; HESS, James L. Statistical design and analysis of experiments: with applications to engineering and science. John Wiley & Sons, 2003.

RODRIGUES, Maria Isabel; IEMMA, Antônio Francisco. Planejamento de experimentos e otimização de processos: uma estratégia sequencial de planejamentos. Casa do Pão Editora, 2005.

HINKELMANN, Klaus; KEMPTHORNE, Oscar. Design and analysis of experiments. New York: Wiley, 1994.

MONTGOMERY, Douglas C. Design and analysis of experiments. John wiley & sons, 2017.

MORETTIN, Pedro Alberto; BUSSAB, Wilton Oliveira. Estatística básica. Saraiva Educação SA, 2017.

YULE, G. Udny. On the theory of correlation. Journal of the Royal Statistical Society, v. 60, n. 4, p. 812-854, 1897.

PEARSON, Karl. Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, v. 185, p. 71-110, 1894.

FISHER, Ronald A. XV.—The correlation between relatives on the supposition of Mendelian inheritance. Earth and Environmental Science Transactions of the Royal Society of Edinburgh, v. 52, n. 2, p. 399-433, 1919.

ZWILLINGER, Daniel; KOKOSKA, Stephen. CRC standard probability and statistics tables and formulae. Crc Press, 1999.

SHESKIN, David J. Handbook of parametric and nonparametric statistical procedures. crc Press, 2003.

ALTMAN, Douglas G.; BLAND, J. Martin. Standard deviations and standard errors. Bmj, v. 331, n. 7521, p. 903, 2005.

TSAI, Chih-Ling; CAI, Zongwu; WU, Xizhi. The examination of residual plots. Statistica Sinica, p. 445-465, 1998.

GALTON, Francis. Natural inheritance. Macmillan and Company, 1894.

WILKS, Mathematical Statistics. Princeton Univ. 1944.

ONYIAH, Leonard C. Design and analysis of experiments: classical and regression approaches with SAS. CRC Press, 2008.

BROWNLEE, Kenneth Alexander. Statistical theory and methodology in science and engineering. New York: Wiley, 1965.

KALINOWSKI, Pawel; FIDLER, Fiona. Interpreting significance: the differences between statistical significance, effect size, and practical importance. Newborn and Infant Nursing Reviews, v. 10, n. 1, p. 50-54, 2010.

BEZERRA, Marcos Almeida et al. Response surface methodology (RSM) as a tool for optimization in analytical chemistry. Talanta, v. 76, n. 5, p. 965-977, 2008.

NAKAGAWA, Shinichi; SCHIELZETH, Holger. A general and simple method for obtaining R2 from generalized linear mixed‐effects models. Methods in ecology and evolution, v. 4, n. 2, p. 133- 142, 2013.

NAKAGAWA, Shinichi; CUTHILL, Innes C. Effect size, confidence interval and statistical significance: a practical guide for biologists. Biological reviews, v. 82, n. 4, p. 591-605, 2007.

PATINO, Cecilia Maria; FERREIRA, Juliana Carvalho. Confidence intervals: a useful statistical tool to estimate effect sizes in the real world. Jornal Brasileiro de Pneumologia, v. 41, n. 6, p. 565- 566, 2015.

COX, David Roxbee; HINKLEY, David Victor. Theoretical statistics. CRC Press, 1979.

HOAGLIN, David C.; MOSTELLER, Frederick; TUKEY, John W. (Ed.). Fundamentals of exploratory analysis of variance. John Wiley & Sons, 1991.

HOOPER, Daire; COUGHLAN, Joseph; MULLEN, Michael R. Structural equation modelling: Guidelines for determining model fit. Electronic journal of business research methods, v. 6, n. 1, p. 53-60, 2008.

D'AGOSTINO, Ralph B. Goodness-of-fit-techniques. CRC press, 1986.

MULAIK, Stanley A. et al. Evaluation of goodness-of-fit indices for structural equation models. Psychological bulletin, v. 105, n. 3, p. 430, 1989.

STEIGER, James H. Structural model evaluation and modification: An interval estimation approach. Multivariate behavioral research, v. 25, n. 2, p. 173-180, 1990.

MILES, Jeremy; SHEVLIN, Mark. A time and a place for incremental fit indices. Personality and individual differences, v. 42, n. 5, p. 869-874, 2007.

GRUBBS, Frank E. et al. Sample criteria for testing outlying observations. The Annals of Mathematical Statistics, v. 21, n. 1, p. 27-58, 1950.

COOK, R. Dennis. Detection of influential observation in linear regression. Technometrics, v. 19, n. 1, p. 15-18, 1977.

NETTER, John; WASSERMAN, William; KUTNER, M. H. Applied linear statistical models: Regression, analysis of variance, and experimental designs. RD Irwin: Homewood, IL, 1985.

TUKEY, John W. The future of data analysis. The annals of mathematical statistics, v. 33, n. 1, p. 1-67, 1962.

DRAPER, Norman R.; SMITH, Harry. Applied regression analysis. John Wiley & Sons, 1998.

PORTER, Stuart C.; VERSEPUT, Richard P.; CUNNINGHAM, Charles R. Process optimization using design of experiments. Pharmaceutical technology, v. 21, n. 10, p. 60-71, 1997.

NOORDIN, Mohd Yusof et al. Application of response surface methodology in describing the performance of coated carbide tools when turning AISI 1045 steel. Journal of materials processing technology, v. 145, n. 1, p. 46-58, 2004.

ANDERSON, Mark J.; WHITCOMB, Patrick J. Design of experiments. Kirk‐Othmer Encyclopedia of Chemical Technology, p. 1-22, 2000.

WEISSMAN, Steven A.; ANDERSON, Neal G. Design of experiments (DoE) and process optimization. A review of recent publications. Organic Process Research & Development, v. 19, n. 11, p. 1605-1633, 2014.

FISHER, Ronald Aylmer. Design of experiments. Br Med J, v. 1, n. 3923, p. 554-554, 1936.

CAMACHO, Lucy Mar; FOX, Jessica A.; AJEDEGBA, Johnson O. Optimization of electrodialysis metathesis (EDM) desalination using factorial design methodology. Desalination, v. 403, p. 136-143, 2017.

ARAUJO, C. M. B. et al. The use of factorial design to evaluate systematic changes in the operating conditions of a three-phase separator using simulation as a tool. Brazilian Journal of Petroleum and Gas, v. 12, n. 1, 2018.

ARA, Behisht et al. Spectrophotometric determination of metribuzin herbicide with pdimethylamino-benzaldehyde using factorial designs for optimization of experimental variables. Journal of Saudi Chemical Society, v. 20, p. S566-S572, 2016.

EJIM, Louisa N. et al. A factorial approach to understanding the effect of inner geometry of baffled meso-scale tubes on solids suspension and aial dispersion in continuous, oscillatory liquid– solid plug flows. Chemical Engineering Journal, v. 308, p. 669-682, 2017.

CHAN, Lydia et al. Optimisation of weld overlay cladding parameters using full-factorial design of experiment. In: Materials Science Forum. Trans Tech Publications, v. 880, p. 54-58, 2017.

BOX, George EP; HUNTER, J. Stuart. The 2 k—p fractional factorial designs. Technometrics, v. 3, n. 3, p. 311-351, 1961.

GUNST, Richard F.; MASON, Robert L. Fractional factorial design. Wiley Interdisciplinary Reviews: Computational Statistics, v. 1, n. 2, p. 234-244, 2009.

REZENDE, Camila A. et al. Optimization of biomass pretreatments using fractional factorial experimental design. Biotechnology for biofuels, v. 11, n. 1, p. 206, 2018.

MA, Hao; KÖKKILIÇ, Ozan; WATERS, Kristian E. The use of the emulsion liquid membrane technique to remove copper ions from aqueous systems using statistical experimental design. Minerals Engineering, v. 107, p. 88-99, 2017.

LE, Giang Thi Truc; GRISDANURAK, Nurak; SREEARUNOTHAI, Paiboon. Seawater Ions Effect on the Adsorption of Cesium by Reduced Graphene Oxide Using Fractional Factorial Design. In: Proceedings of the 7th International Conference on Informatics, Environment, Energy and Applications, p. 77-81, 2018.

SILVA, Marta Sofia et al. Speciation of zinc in fish feed by size exclusion chromatography coupled to inductively coupled plasma mass spectrometry–using fractional factorial design for method optimisation and mild extraction conditions. journal of Chromatography B, v. 1104, p. 262-268, 2019.

ADIO, Salawu Omobayo et al. Arsenic and selenium removal from water using biosynthesized nanoscale zero-valent iron: a factorial design analysis. Process Safety and Environmental Protection, v. 107, p. 518-527, 2017.

BOX, George EP & HUNTER, JS. Multi-factor experimental designs for exploring response surfaces. The Annals of Mathematical Statistics, v. 28, n. 1, p. 195-241, 1957.

CONAGIN, Armando. Delineamento composto central com duas estrelas. Pesquisa Agropecuária Brasileira, v. 17, n. 9, p. 1261-1269, 1982.

GHAEDI, Abdol Mohammad et al. Optimization and modeling of simultaneous ultrasoundassisted adsorption of ternary dyes using copper oxide nanoparticles immobilized on activated carbon using response surface methodology and artificial neural network. Ultrasonics sonochemistry, v. 51, p. 264-280, 2019.

OZ, Umut Can et al. Development and Optimization of Alendronate Sodium Loaded PLGA Nanoparticles by Central Composite Design. Macromolecular Research, p. 1-10, 2019.

PEREIRA, Erlon Lopes et al. Central Composite Rotatable Design for Startup Optimization of Anaerobic Sequencing Batch Reactor Treating Biodiesel Production Wastewater. Journal of Environmental Chemical Engineering, p. 103038, 2019.