1Universidade de São Paulo, Instituto de Matemática e Estatística, São Paulo, Brasil. Mestre em Estatística. Email: erilaniaalmeida@yahoo.com.br
2Universidade de São Paulo, Instituto de Matemática e Estatística, São Paulo, Brasil. Professor doutor. Email: selian@ime.usp.br
3Universidade Federal do Ceará, Departamento de Estatística e Matemática Aplicada, Fortaleza, Brasil. Professor adjunto I. Email: juvencio@ufc.br ]]>
Os testes usuais para comparar variâncias e médias, teste de Bartlett e teste F, supõem que as amostras sejam provenientes de populações com distribuições normais. Para o teste de igualdade de médias, a suposição de homogeneidade de variâncias também é necessária. Alguns problemas se destacam quando tais suposições básicas são violadas, como tamanho excessivo e baixo poder. Neste trabalho descrevemos inicialmente o teste de Levene para igualdade de variâncias, que é robusto à não normalidade, e o teste de Brown e Forsythe para igualdade de médias quando existe desigualdade de variâncias. Apresentamos várias modificações do teste de Levene e do teste de Brown e Forsythe, propostas por diferentes autores. Analisamos e aplicamos uma forma do teste modificado de Brown e Forsythe a um conjunto de dados reais. Este teste é uma alternativa robusta com relação a desvios de normalidade e homocedasticidade e também na presença de observações discrepantes. Na comparação de variâncias, destaca-se o teste de Levene com centralização na mediana.
Palavras chave: teste de Levene, teste de Brown e Forsythe, médias aparadas, variâncias winsorizadas, bootstrap.
The usual tests to compare variances and means (e.g. Bartletts test and F-test) assume that the sample comes from a normal distribution. In addition, the test for equality of means requires the assumption of homogeneity of variances. In some situation those assumptions are not satisfied, hence we may face problems like excessive size and low power. In this paper, we describe two tests, namely the Levenes test for equality of variances, which is robust under nonnormality; and the Brown and Forsythes test for equality of means. We also present some modifications of the Levenes test and Brown and Forsythes test, proposed by different authors. We analyzed and applied one modified form of Brown and Forsythes test to a real data set. This test is a robust alternative under nonnormality, heteroscedasticity and also when the data set has influential observations. The equality of variance can be well tested by Levenes test with centering at the sample median.
Key words: Levene's test, Brown and Forsythe's test, Trimmed Means, Winsorized variances, bootstrap.
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1. Bickel, P. J. (2005), `One-step Haber Estimates in the Linear Model´, Journal of the American Statistical Association 70, 428-434. [ Links ]
2. Brown, M. B. & Forsythe, A. B. (1974a), `Robust Tests for the Equality of Variances´, Journal of the American Statistical Association 69, 364-367. [ Links ]
3. Brown, M. B. & Forsythe, A. B. (1974b), `The Small Sample Behavior of Some Statistics which Test the Equality of Several Means´, Technometrics 16, 129-132. [ Links ]
4. Carrol, R. J. & Ruppert, D. (1982), `Robust Estimation in Heteroscedastic Linear Models´, Annals of Statistics 10, 429-441. [ Links ]
5. Carrol, R. J. & Schneider, H. (1985), `A Note on Levene's Test for Equality of Variances´, Statistics and Probability Letters 3, 191-194. [ Links ]
6. Conover, W. J., Johnson, M. E. & Johnson, M. M. (1981), `A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data´, Technometrics 23, 351-361. [ Links ]
7. Davison, A. C. & Hinkley, D. V. (1997), Bootstrap Methods and Their Applications, Cambridge University Press, Cambridge, United States. [ Links ]
8. Elian, S. N. & Santos, L. D. (2003), Relatório de Análise Estatística Sobre o Projeto: ``Tipos Psicológicos Associados a Variáveis Estratégicas Em Empreendedores de Pequena e Micro Empresa'', São Paulo, IME-USP. [ Links ]
9. Francis, R. I. C. C. & Manly, B. F. J. (2001), `Bootstrap Calibration to Improve the Reliability of Tests to Compare Means and Variances´, Environmetrics 12, 713-729. [ Links ]
10. Hines, W. G. S. & O'Hara Hines, R. J. (2000), `Increased Power with Modified Forms of the Levene (Med) Test for Heterogeneity of Variance´, Biometrics 56, 451-454. [ Links ]
11. James, G. S. (1954), `Tests of Linear Hypotheses in Univariate and Multivariate Analysis when the Ratios of the Population Variances are Unknown´, Biometrika 41, 19-43. [ Links ]
12. Keselman, H. J. & Wilcox, R. R. (1999), `The Improved Brown and Forsythe Test for Mean Equality: Some Things Can't be Fixed´, Communications in Statistics-Simulation 28, 687-698. [ Links ]
13. Levene, H. (1960), Robust Test for Equality of Variances, `Contributions to Probability and Statistics: Essays in Honor of Harold Hotteling´, Stanford University Press, California, United States, p. 278-292. [ Links ]
14. Manly, B. F. J. (1995), `Randomization Tests to Compare Means with Unequal Variation´, Sankhyã 57, 200-222. [ Links ]
15. Manly, B. F. J. (2004), `One-sided Tests of Bioequivalence with Nonnormal Distributions and Unequal Variances´, Journal of Agricultural, Biological and Environmental Statistics 9, 270-283. [ Links ]
16. Manly, B. F. J. & Francis, R. I. C. C. (2002), `Testing for Mean and Variance Differences with Samples from Distributions that May Be Non-Normal with Unequal Variances´, Journal of Statistical Computation and Simulation 72, 633-646. [ Links ]
17. Mehrotra, D. V. (1997), `Improving the Brown-Forsythe Solution to the Generalized Behrens-Fisher Problem´, Communications in Statistics-Simulation and Computation 26, 1139-1145. [ Links ]
18. O'Neill, M. E. & Mathews, K. (2000), `A Weighted Least Squares Approach to Levene's Test of Homogeneity of Variance´, Austral. & New Zealand J. Statist. 42, 81-100. [ Links ]
19. Pereira, C. A. B. & Stern, J. M. (2003), `Evidence and Credibility: Full Bayesian Significance Test for Precise Hypothesis´, Entropy 1, 99-110. [ Links ]
20. R Development Core Team, (2007), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *http://www.R-project.org [ Links ]
21. Sattherthwaite, F. E. (1941), `Synthesis of Variance´, Psychometrika 6, 309-316. [ Links ]
22. Welch, B. L. (1951), `On the Comparison of Several Mean Values: An Alternative Approach´, Biometrika 38, 330-336. [ Links ]
23. Wilcox, R. R. (1996), Statistics for the Social Sciences, Academic Press, New York, United States. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv31n2a08,
AUTHOR = {de Almeida, Antônia and Elian, Silvia and Nobre, Juvêncio},
TITLE = {{Modificações e alternativas aos testes de Levene e de Brown e Forsythe para igualdade de variâncias e médias}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2008}, ]]>
volume = {31},
number = {2},
pages = {241-260}
}