Abstract
In this paper, we present a novel nonparametric approach for multivariate analysis of two-way crossed factorial design based on nonparametric combination applied to synchronized permutation tests. This nonparametric hypothesis testing procedure not only allows to overcome the shortcomings of MANOVA test like violation of assumptions such as multivariate normality or covariance homogeneity, but, in an extensive simulation study, reveals to be a powerful instrument both in case of small sample size and many response variables. We contextualize its application in the field of industrial experiments and we assume a linear additive model for the data set analysis. Indeed, the linear additive model interpretation well adapts to the industrial production environment because of the way control of production machineries is implemented. The case of small sample size reflects the frequent needs of practitioners in the industrial environment where there are constraints or limited resources for the experimental design. Furthermore, an increase in rejection rate can be observed under alternative hypothesis when the number of response variables increases with fixed number of observed units. This could lead to a strategical benefit considering that in many real problems it could be easier to collect more information on a single experimental unit than adding a new unit to the experimental design. An application to industrial thermoforming processes is useful to illustrate and highlight the benefits of the adoption of the herein presented nonparametric approach.
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References
Basso D, Chiarandini M, Salmaso L (2007) Synchronized permutation tests in replicated i\(\times \) j designs. J Stat Plan Inference 137(8):2564–2578
Basso D, Pesarin F, Salmaso L, Solari A (2009) Permutation tests for stochastic ordering and ANOVA: theory and applications with R. Springer, New York
Bathke AC, Harrar SW (2008) Nonparametric methods in multivariate factorial designs for large number of factor levels. J Stat Plan Inference 138(3):588–610
Bathke AC, Harrar SW, Madden LV (2008) How to compare small multivariate samples using nonparametric tests. Comput Stat Data Anal 52(11):4951–4965
Brunner E, Munzel U, Puri ML (2002) The multivariate nonparametric behrens-fisher problem. J Stat Plan Inference 108(1–2):37–53
Corain L, Salmaso L (2013) Nonparametric permutation and combination-based multivariate control charts with applications in microelectronics. Appl Stoch Model Bus Ind 29(4):334–349
Corain L, Salmaso L (2015) Improving power of multivariate combination-based permutation tests. Stat Comput 25(2):203–214
Giancristofaro RA, Corain L, Ragazzi S (2012a) A comparison among combination-based permutation statistics for randomized complete block design. Commun Stat Simul Comput 41(7):964–979
Giancristofaro RA, Corain L, Ragazzi S (2012b) The multivariate randomized complete block design: a novel permutation solution in case of ordered categorical variables. Commun Stat Theory Methods 41(16–17):3094–3109
Giancristofaro RA, Bonnini S, Corain L, Salmaso L (2016) Dependency and truncated forms of combinations in multivariate combination-based permutation tests and ordered categorical variables. J Stat Comput Simul 86(18):3608–3619
Good PI (2005) Permutation, parametric, and bootstrap tests of hypotheses. Springer Series in Statistics. Springer, New York
Goutis C, Casella G, Wells MT (1996) Assessing evidence in multiple hypotheses. J Am Stat Assoc 91(435):1268–1277
Harrar SW, Bathke AC (2008a) A nonparametric version of the bartlett-nanda-pillai multivariate test. asymptotics, approximations, and applications. Am J Math Manag Sci 28(3–4):309–335
Harrar SW, Bathke AC (2008b) Nonparametric methods for unbalanced multivariate data and many factor levels. J Multivar Anal 99(8):1635–1664
Harrar SW, Bathke AC (2012) A modified two-factor multivariate analysis of variance: asymptotics and small sample approximations. Ann Inst Stat Math 64(1):135–165
Hoeffding W (1952) The large-sample power of tests based on permutations of observations. Ann Math Stat pp. 169–192
Jarrett R (2017) Does theory work in practice? two case studies. Qual Eng 29(1):141–159
Khaleel K, Mousa A, Qashlaq A, Hassan B, Ramadan A, Raslan B (2017) Using design of experiment to predict concert compressive strength using fly ash and ggbfs as cement alternative. In: Proceedings of the international conference on industrial engineering and operations management, pp 741–749
Khreis H, Deflorio A, Lee W.L, De Larramendi M.R, Schmuelling B (2017) Sensitivity analysis for induction machine manufacturing tolerances: Modeling of electrical parameters deviation. In: IEEE twelfth international conference on ecological vehicles and renewable energies (EVER), pp 1–9
Kruskal WH (1952) A nonparametric test for the several sample problem. Ann Math Stat pp 525–540
Kruskal WH, Wallis WA (1952) Use of ranks in one-criterion variance analysis. J Am Stat Assoc 47(260):583–621
Lakens D (2013) Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and anovas. Front Psychol 4:863
Liptak T (1958) On the combination of independent tests. Magyar Tud Akad Mat Kutato Int Kozl 3:171–197
Liu Q, Chen X, Gindy N (2007) Evaluation of superalloy heavy-duty grinding based on multivariate tests. Proc Inst Mech Eng Part B 221(9):1421–1430
Melas VB, Pepelyshev A, Shpilev P, Salmaso L, Corain L, Arboretti R (2015) On the optimal choice of the number of empirical fourier coefficients for comparison of regression curves. Stat Pap 56(4):981–997
Munzel U, Brunner E (2000) Nonparametric methods in multivariate factorial designs. J Stat Plann Inference 88(1):117–132
Pesarin F (1992) A resampling procedure for nonparametric combination of several dependent tests. J Ital Stat Soc 1(1):87–101
Pesarin F (1999) Permutation testing of multidimensional hypotheses by nonparametric combination of dependent tests. Cleup
Pesarin F (2001) Multivariate permutation tests: with applications in biostatistics, vol 240. Wiley, Chichester
Pesarin F, Salmaso L (2010a) Permutation tests for complex data: theory, applications and software. Wiley series in probability and statistics. Wiley, New York
Pesarin F, Salmaso L (2010b) Finite-sample consistency of combination-based permutation tests with application to repeated measures designs. J Nonparametr Stat 22(5):669–684
Ronchi F, Salmaso L, De Dominicis M, Illert J, Krajewski K, Link R, Monti L (2017) Optimal designs to develop and support an experimental strategy on innovation of thermoforming production process. Statistica 77(2):109
Salmaso L (2003) Synchronized permutation tests in 2k factorial designs. Commun Stat Theory Methods 32(7):1419–1437
Sangnuan K, Laosiritaworn WS (2016) Determining the optimal parameter of coordinate measuring machine with design of experiment. In: MATEC Web of Conferences, vol. 68, p. 06009. EDP Sciences
Wiemer H, Schwarzenberger M, Dietz G, Juhrisch M, Ihlenfeldt S (2017) A holistic and doe-based approach to developing and putting into operation complex manufacturing process chains of composite components. Procedia CIRP 66:147–152
Zhang Q, Zhang T, Zhang Z, Zhou X, Wang Y, He C (2017) Experiment of integrated fermentation hydrogen production by photosynthetic bacteria cooperating with enterobacter aerogenes. Trans Chin Soc Agric Eng 33(9):243–249
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Authors would like to thank the Editor and Referees for useful comments and suggestions which helped improving the manuscript. The research was supported by UNIPD BIRD185315/18.
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Arboretti, R., Ceccato, R., Corain, L. et al. Multivariate small sample tests for two-way designs with applications to industrial statistics. Stat Papers 59, 1483–1503 (2018). https://doi.org/10.1007/s00362-018-1032-y
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DOI: https://doi.org/10.1007/s00362-018-1032-y