Alternative methods in presence of violation of assumptions in factorial experiment designs
DOI:
https://doi.org/10.21704/ac.v82i2.1795Keywords:
Multiple comparisons, Factorial design, non-parametric methods, robust methods, permutations, R, transformationsAbstract
The objective of this research was to compare and test different methods and alternatives to approach factorial designs with fixed effects, when the assumptions of normality or homogeneity of variances are not met. Twenty methods investigated in the literature as an alternative to classical ANOVA (variance analysis) were described and tested, including non-parametric techniques, robust methods, permutations, methods for heterogeneous variances and transformations; which are currently available and implemented in R software. The methods were tested in a 3A2B factorial design, where factor A was varieties of pineapple (Golden, Cayenne Lisa and Hawaiian), factor B was type of crop management (conventional and organic), and the response variable was the average percentage of Brix degrees. Among the proposed methods, 15 rejected the interaction hypothesis, and when comparing the type I error rates through simulations it was found that the permutations methods, the robust methods, the ART, van der Waerder and the BDM yielded error rates by below nominal value. When selecting ART as an alternative to perform the post hoc test, the best combination of treatments was the Lisa Cayena variety in organic management and the Hawaiian variety with conventional management, obtaining with these combinations percentages of Brix degrees above the average.
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• Adam, L., & Bejda, P. (2018). Robust estimators based on generalization of trimmed mean. Communications in Statistics - Simulation and Computation, 47(7), 2139-2151. https://doi.org/10.1080/03610918.2017.1337136
• Aho, K. A. (2013). Robust ANOVA. ln Chapman & Hall/CRC (Eds.). Foundational and Applied Statistics for Biologists using R (1st ed., pp. 493). Taylor & Francis Group. Disponible en: https://libgen.is/book/index.php?md5=FC7D4BB1 EF0423899FFE8E20DED19E09
• Aho, K. A. (2020). asbio: A collection of statistical tools for biologists. R package version 1.6-7. https://rdrr.io/cran/asbio/
• Akritas, M. G., Arnold, S. F., & Brunner, E. (1997). Nonparametric hypotheses and rank statistics for unbalanced factorial designs. Journal of the American Statistical Association, 92(437), 258- 265. https://doi.org/10.1080/01621459.1997.10473623
• Afonso, A., & Pereira, D. G. (2019). Comparação entre métodos não paramétricos para a análise de variância com dois fatores: um estudo de simulação. Instituto Nacional de Estatística, 147- 158.
• Algina, J., & Olejnik, S. F. (1984). Implementing the Welch-James procedure with factorial designs. Educational and psychological measurement, 44(1), 39-48. https://doi.org/10.1177/0013164484441004
• Andriani, S. (2017). Uji Park Dan Uji Breusch Pagan Godfrey Dalam pendeteksian heteroskedastisitas pada analisis regresi. Al-Jabar. Jurnal Pendidikan Matematika, 8(1), 63-72.https://doi.org/10.24042/ajpm.v8i1.1014
• Beasley, T. M., & Zumbo, B. D. (2009). Aligned rank tests for interactions in Split- Plot Designs: distributional assumptions and stochastic heterogeneity. Journal of Modern Applied Statistical Methods, 8(1), 16-50.
http://dx.doi.org/10.22237/jmasm/1241136180
• Box, G. E., & Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26(2), 211- 252. https://doi.org/10.1111/j.2517- 6161.1964.tb00553.x
• Brown, M. B., & Forsythe, A. B. (1974). The anova and multiple comparisons for data with heterogeneous variances. Biometrics, 30(4), 719- 724. https://doi.org/10.2307/2529238
• Brunner, E., Dette, H., & Munk, A. (1997). Box- Type approximations in nonparametric factorial designs. Journal of the American Statistical Association, 92(440), 1494-1502. https://doi.org/10.1080/01621459.1997.10473671
• Brunner, E., Konietschke, F., Pauly, M., & Puri, M. (2016). Rank-Based procedures in factorial designs: Hypotheses about nonparametric treatment effects. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(5), 1463-1485. https://doi.org/10.1111/rssb.12222
• Brunner, E.; Bathke, A.; Konietschke, F. (2018). Two-Factor Crossed Designs. Rank and Pseudo - Rank Procedures for Independent Observations in Factorial Designs Using R and SAS. (1st ed, pp. 287-292). Editorial Springer, suiza. url: http://library.lol/main/96D22A7D31F896490321 AEBE84933872
• Conover, W. J., & Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. American Statistician 35(3), 124-129. https://doi.org/10.2307/2683975
• Delgado, J. (1992). Algunos problemas básicos del análisis de varianza. (1era ed.). Editorial Ediciones Universidad de Salamanca, España.
• De Neve, J., & Thas, O. (2017). A Mann–Whitney type effect measure of interaction for factorial designs. Communications in Statistics - Theory and Methods, 46(22), 11243-11260. https://doi.org/10.1080/03610926.2016.1263739
• Feys, J. (2016). Nonparametric tests for the interaction in two-way factorial designs using R. The R Journal 8(1), 367-378. https://doi.org/10.32614/RJ-2016-027
• Friedrich, S., Konietschke, F., & Pauly, M. (2017). GFD: An R Package for the analysis of general factorial designs. Journal of Statistical Software, 79(1), 1-18. https://doi.org/10.18637/jss.v079.c01
• Frossard, J., & Renaud, O. (2019). permuco: Permutation tests for regression, (Repeated Measures) ANOVA/ANCOVA and Comparison of Signals. R package version 1.1.0. https://cran.r- project.org/web/packages/permuco/index.html
• Gao, X., & Alvo, M. (2005). A nonparametric test for interaction in two‐way layouts. Canadian Journal of Statistics, 33(4), 529-543. doi:10.1002/cjs.5550330405
• Higgins, J., & Tashtoush, S. (1994). An aligned rank transform test for interaction. Nonlinear World, 1(2), 201-211.
• Hocking, R. (1985). The Analysis of Linear Models (1st ed.). Editorial Brooks/Cole, Monterey, California.
• Howell, D. (2013). Statistical Methods for Psychology. Wadsworth (9th ed.). https://www.uvm.edu/~statdhtx/StatPages/
• Jimenes, C.; Pérez, J. (1989). Diseños experimentales en ciencias de la conducta: un método de análisis de varianza de libre distribución (no paramétrico). Anuario de psicología, 42, 31-48.
• Kassambara, A. (2019). Homogeneity of variance (1st ed. pp. 14). Practical Statistics in R for Comparing Groups: Numerical Variables. Editorial Datanovia, Francia.
• Kay, M., & Wobbrock, J. (2020). ARTool: Aligned rank transform. R package version 0.10.7. https://cran.r- project.org/web/packages/ARTool/index.html.
• Keselman, H. J., Carriere, K. C., & Lix, L. M. (1996). Robust and powerful nonorthogonal analyses. Psychometrika, 60(3), 395-418. https://doi.org/10.1007/BF02294383
• Kloke, J., & McKean, J. (2020). Rfit: Rank-Based estimation for linear models. R package version 0.24.2. Disponible en: https://cran.r- project.org/web/packages/Rfit/index.html
• Konietschke, F.; Friedrich, S.; Brunner, E.; Pauly,
M. (2020). rankFD: Rank-Based Tests for General Factorial Designs. R package version 0.0.5. Diponible en: https://cran.r- project.org/web/packages/rankFD/index.html
• Kreutzmann, A., Medina, & Rojas, N. (2018). trafo: Estimation, comparison and selection of transformations. R package version 1.0.1. Disponible en: https://cran.r- project.org/web/packages/trafo/index.html
• Lawrence, M. (2016). ez: Easy analysis and visualization of factorial experiments. R package version 4.4.0. Disponible en: https://cran.r- project.org/web/packages/ez/index.html
• Lawson, J. (2015). Completely randomized designs with one factor (1st ed. Pp. 31-36). Design and Analysis of Experiments with R. Editorial CHAPMAN & HALL/CRC, Brigham Young University Provo, Utah, USA. http://library.lol/main/BE472FED84FD3B4E5D5 D53D6C1719C30
• Luepsen, H. (2018). Comparison of nonparametric analysis of variance methods: A vote for van der Waerden. Journal Communications in Statistics - Simulation and Computation, 47(9), 2547-2576. https://doi.org/10.1080/03610918.2017.1353613
• Luepsen, H. (2020). R Functions for the analysis of variance. Disponible en: http://www.uni- koeln.de/~a0032/R/
• Mair, P., & Wilcox, R. (2020). WRS2: A collection of robust statistical methods. R package version 1.1-0. Disponible en: https://cran.r- project.org/web/packages/WRS2/
• Mangiafico, S. (2016). Summary and Analysis of Extension Program Evaluation. R package version 1.18.1. Disponible en: rcompanion.org/documents/RHandbookProgramE valuation.pdf
• Manly, B. (1997). Randomization, bootstrap, and Monte Carlo methods in biology (2nd ed.). Editorial Chapman Hall, London.
• Mansouri, H., & Chang, G. (1995). A comparative study of some rank tests for interaction. Computational Statistics & Data Analysis, 19(1), 85-96. https://doi.org/10.1016/0167- 9473(93)E0045-6
• Melo, O. O., López, L. A, & Melo, S. (2020). Comparaciones múltiples y validación de supuestos (2da ed.). Diseño de experimentos Métodos y aplicaciones. Editorial Coordinación de publicaciones - Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, D. C., Colombia.
• Mewhort, D. J., Brendan, B. T., & Matthew, k. (2010). Applying the permutation test to factorial designs. Behavior Research Methods, 42(2), 366- 372.
• Puri, M., & Sen, P. (1985). Nonparametric Methods in General Linear Models. (1st ed. pp. 235). Editorial Wiley, New York.
• Pelea, L. (2018). ¿Cómo proceder ante el incumplimiento de las premisas de los métodos paramétricos? o ¿cómo trabajar con variables biológicas no normales? Revista Del Jardín Botánico Nacional, 39, 1-12.
• Quinglong, L. (2015). StatMethRank: Statistical methods for ranking data. R package version 1.3. Disponible en: https://cran.r- project.org/web/packages/StatMethRank
• Ribeiro-Oliveira, J. P., Garcia, D., Pereira, V. & Machado, C. (2018). Data transformation: an underestimated tool by inappropriate use. Acta Scientiarum-agronomy, 40, 35-300. https://doi.org/10.4025/actasciagron.v40i1.35300
• Ripley, B., Venables, B., Bates, D., Hornik, K., Gebhardt, A., & Firth, D. (2020). MASS: Support Functions and Datasets for Venables and Ripley's MASS. R package version 7.3-53. Disponible en: https://cran.r- project.org/web/packages/MASS/index.html
• Saste, S., Sananse, S., & Sonar, C. (2016). On parametric and nonparametric analysis of two factor factorial experiment. International Journal of Applied Research, 2(7), 653-656.
• Sawilowsky, S. S (1990). Nonparametric Tests of Interaction in Experimental Design. Review of Educational Research, 60(1), 91-126. https://doi.org/10.2307/1170226
• Scheirer, C. J., Ray, W. S., & Hare, N. (1976). The analysis of ranked data derived from completely randomized factorial designs. Biometrics, 32(2), 429-434. https://doi.org/10.2307/2529511
• Shoemaker, L. H. (1986). A
• Wobbrock, J. O., Findlater, L., Gergle, D., & Higgins, J. J. (2011). The Aligned Rank Transform for Nonparametric Factorial Analyses Using Only ANOVA Procedures. In Proceedings of the ACM Conference on Human Factors in Computing Systems, Vancouver, British Columbia, New York, 7-12. https://doi.org/10.1145/1978942.1978963
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