Probabilidade do erro do tipo I nas cartas X e S de Shewhart sob não normalidade
Probability of type I error in X and S Shewhart control charts under non-normality
Korzenowski, André Luis; Werner, Liane
http://dx.doi.org/10.1590/S0103-65132012005000059
Prod, vol.22, n4, p.807-816, 2012
Resumo
O objetivo deste artigo é verificar o comportamento das cartas de média e desvio padrão de Shewhart em relação à probabilidade do erro do tipo I quando da violação da suposição de normalidade. Foi realizada a simulação de uma série de 500.000 amostras (subgrupos) de tamanho n = 3, 5, 7, 10, 15, 20 e 25. As amostras foram simuladas a partir das distribuições normal, t de Student, exponencial, qui-quadrado, gamma e Weibull. Verificou-se em dados não normais o aumento na probabilidade de erro do tipo I na carta de médias em todas as distribuições simuladas. O mínimo tamanho da amostra necessário está relacionado ao grau de assimetria da distribuição dos dados, sendo que, em alguns casos, nem mesmo n = 25 apresentou resultados satisfatórios. No gráfico S, o aumento da probabilidade de erro do tipo I é significativamente superior em quase todas as distribuições simuladas e seu comportamento é influenciado não só pelo tipo de distribuição, mas também pelo tamanho da amostra.
Palavras-chave
Gráficos de controle de Shewhart. Suposição de normalidade. Erro do tipo I. Robustez.
Abstract
The purpose of this article was to examine the performance of Shewhart average and standard deviation control charts in relation to the probability of type I error at the occurrence of normality assumption violation. A simulation of a series of 500,000 samples (subgroups) of sizes n=3, 5, 7, 10, 15, 20 and 25 was performed. The samples were simulated from the following distributions: Normal, Binomial, Poisson, Exponential, Chi-square, Gamma and Weibull. It was possible to find that, in non-normal data, there was an increase in the probability of type I error on the average chart of all simulated distributions. The minimum sample size required is related to the degree of asymmetry of the data distribution and, in some cases, not even n=25 presented satisfactory results. In the S chart, the increased probability of type I error is significantly higher for almost all distributions simulated and its performance is influenced not only by the type of distribution but also by the sample size.
Keywords
Shewhart control charts. Normality assumption. Type I error. Robustness.
References
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CAULCUTT, R. The Rights and Wrongs of Control Charts. Applied Statistics, v. 44, n. 3, p. 279-288, 1995.
CHAN, L. K.; HAPUARACHCHI, K. P.; MACPHERSON, B. D. Robustness of X and R chart. IEEE Transactions on reliability, v. 37, n. 1, p. 117-123, 1998.
COSTA, A. F. B.; EPPRECHT, E. K.; CARPINETTI, L. C. R. Controle estatístico de qualidade. São Paulo: Atlas, 2005.
COX, D. R. The mean and coefficient of variation of range in small samples from non-normal populations. Biometrika, v. 41, n. 3-4, p. 469-48, 1954.
DUNCAN, A. J. Quality control and industrial statistics. 4th ed. Chicago: Irwin Professional Publishing, 1974.
JAMES, B. J. Probabilidade: um curso em nível intermediário. Rio de Janeiro: IMPA; CNPq, 1996.
JARRET, J. E.; PAN, X. The quality control chart for monitoring multivariate autocorrelated processes. Computational Statistics & Data Analysis, v. 51, p. 3862-3870, 2007. http://dx.doi.org/10.1016/j.csda.2006.01.020
LIN, Y.-C.; CHOU, C.-Y. On the design of variable sample size and sampling intervals charts X under non-normality. Internacional Journal of Production Economics, v. 96, p. 249-261, 2005a. http://dx.doi.org/10.1016/j.ijpe.2004.05.001
LIN, Y.-C.; CHOU, C.-Y. Non-normality and the variable parameters X control charts. European Journal of Operational Research, v. 176, p. 361-373, 2005b.
MEYER, P. L. Probabilidade: Aplicação à Estatística. Rio de Janeiro: LTC, 1983.
MULLIKEN, G. A.; JOHNSON, D. E. Analysis of Messy Data. Boca Ratón: Chapman & Hall/CRC, 1992. v. 1: Design of Experiments.
MONTGOMERY, D. Design and Analysis of Experiments. Hoboken: John Willey & Sons, 2001.
MONTGOMERY, D. Introdução ao Controle Estatístico da Qualidade. Rio de Janeiro: LTC, 2004.
MOORE, P. Normality in Quality Control Charts. Applied Statistics, v. 6, n. 3, p. 171-179, 1957.
PEDRINI, D. C.; TEN CATEN, C. S.; MOREIRA JUNIOR, F. J. Proposal of an Alternative to Control Chart Based on Residuals. Praha: ISBIS, 2008. Book of Abstracts.
NAGENDRA, Y.; RAI, G. Optimum sample size and sampling interval for controlling the mean of non-normal variables. Journal of the American Statistical Association, v. 66, n. 335, p. 637-640, 1971.
STOWMBOS, Z. G. et al. The state of statistical process control as we proceed into the 21st century. Journal of the American Statistical Association, v. 95, n. 451, p. 992-998, 2000.
STOWMBOS, Z. G.; REYNOLDS, M. R. Robustness to non-normality and autocorrelation of individuals control charts. Journal of Statistical Computation and Simulation, v. 66, p. 145-187, 2000.
WHEELER, D. J.; CHAMBERS, D. S. Understanding Statistical Process Control. Knoxville: Statistical Process Control Inc., 1986.
YOURSTONE, S. A.; ZIMMER, W. J. Non-normality and the design of control chart averages. Decisions Sciences, v. 23, n. 5, p. 1099-1113, 1992. http://dx.doi.org/10.1111/j.1540-5915.1992.tb00437.x
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