Published

2015-01-01

Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal

Estimación y pruebas de hipótesis en ANOVA a una vía cuando los errores se distribuyen como normal sesgados

DOI:

https://doi.org/10.15446/rce.v38n1.48802

Keywords:

ANOVA, Modified Likelihood, Iteratively Reweighting Algorithm, Skew-Normal, Monte Carlo Simulation, Robustness (en)
ANOVA, Estimación, Normal sesgada, Pruebas de hipótesis, Robustez. (es)

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Authors

  • Nuri Celik Bartin University, Bartin, Turkey
  • Birdal Senoglu Ankara University, Ankara, Turkey
  • Olcay Arslan Ankara University, Ankara, Turkey

We consider one-way analysis of variance (ANOVA) model when the error terms have skew- normal distribution. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies (see, Tiku 1967). In the ML method, iteratively reweighting algorithm (IRA) is used to solve the likelihood equations. The MML approach is a non-iterative method used to obtain the explicit estimators of model parameters. We also propose new test statistics based on these estimators for testing the equality of treatment effects. Simulation results show that the proposed estimators and the tests based on them are more efficient and robust than the corresponding normal theory solutions. Also, real data is analysed to show the performance of the proposed estimators and the tests.

Se considera el modelo de análisis de varianza a una vía (ANOVA) cuando los términos de error siguen una distribución normal sesgada. Se obtienen estimadores de los parámetros desconocidos mediante el uso de la metodología de máxima verosimilitud (ML). Se proponen nuevos estadísticos de prueba basados en estos estimadores. Los resultados de la simulación muestran que los estimadores propuestos y los tests basados en ellos son más eficientes y robustos que los correspondientes a las soluciones de la teoría normal. Un conjunto de datos real es analizado con el fin de mostrar el desempeño de los estimadores propuestos y sus tests relacionados.

https://doi.org/10.15446/rce.v38n1.488020

Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal

Estimación y pruebas de hipótesis en ANOVA a una vía cuando los errores se distribuyen como normal sesgados

NURI CELIK1, BIRDAL SENOGLU2, OLCAY ARSLAN3

1Bartin University, Science Faculty, Department of Statistics, Bartin, Turkey. Professor. Email: ncelik@bartin.edu.tr
2Ankara University, Science Faculty, Department of Statistics, Ankara, Turkey. Professor. Email: senoglu@science.ankara.edu.tr
3Ankara University, Science Faculty, Department of Statistics, Ankara, Turkey. Professor. Email: oarslan@ankara.edu.tr

Abstract

We consider one-way analysis of variance (ANOVA) model when the error terms have skew- normal distribution. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies (see, Tiku 1967). In the ML method, iteratively reweighting algorithm (IRA) is used to solve the likelihood equations. The MML approach is a non-iterative method used to obtain the explicit estimators of model parameters. We also propose new test statistics based on these estimators for testing the equality of treatment effects. Simulation results show that the proposed estimators and the tests based on them are more efficient and robust than the corresponding normal theory solutions. Also, real data is analysed to show the performance of the proposed estimators and the tests.

Key words: ANOVA, Modified Likelihood, Iteratively Reweighting Algorithm, Skew-Normal, Monte Carlo Simulation, Robustness.

Resumen

Se considera el modelo de análisis de varianza a una vía (ANOVA) cuando los términos de error siguen una distribución normal sesgada. Se obtienen estimadores de los parámetros desconocidos mediante el uso de la metodología de máxima verosimilitud (ML). Se proponen nuevos estadísticos de prueba basados en estos estimadores. Los resultados de la simulación muestran que los estimadores propuestos y los tests basados en ellos son más eficientes y robustos que los correspondientes a las soluciones de la teoría normal. Un conjunto de datos real es analizado con el fin de mostrar el desempeño de los estimadores propuestos y sus tests relacionados.

Palabras clave: ANOVA, estimación, normal sesgada, pruebas de hipótesis, robustez.

Texto completo disponible en PDF

References

1. Acitas, S., Kasap, S., Senoglu, B. & Arslan, O. (2013), 'Robust estimation with the skew t_2 distribution', Pakistan Journal of Statistics 29(4), 409-430.

2. Arrellano-Valle, R., Bolfarine, H. & Lachos, V. (2005), 'Skew-normal linear mixed models', Journal of Data Science 3, 415-438.

3. Arslan, O. & Genc, .. (2009), 'The skew generalized t (sgt) distribution as the scale mixture of a skew exponential distribution and its application in robust estimation', Statistics 43, 481-498.

4. Azzalini, A. (1985), 'A class of distributions which includes the normal ones', Scandinavian Journal of Statistics 12, 171-178.

5. Azzalini, A. (1986), 'Further results on a class of distributions which includes the normal ones', Statistica 46, 199-208.

6. Azzalini, A. (2005), 'The skew-normal distribution and related multivariate families (with discussion)', Scandinavian Journal of Statistics 32, 159-188.

7. Bowman, K. & Shenton, L. (2001), 'Weibull distributions when the shape parameter is defined', Computational Statistics and Data Analysis 36, 299-310.

8. Box, G. (1953), 'Non-normality and test of variances', Biometrika 40, 336-346.

9. Box, G. & Tiao, G. (1964), 'A Bayesian approach to the importance of assumptions applied to the comparison of variances', Biometrika 51, 153-167.

10. Donaldson, T. (1968), 'Robustness of the f-test to errors of both kinds and the correlation between the numerator and denominator of the F ratio', Journal of American Statistical Association 63(322), 660- 667.

11. Garay, A., Lachos, V. & Abanto-Valle, C. (2011), 'Nonlinear regression models based on scale mixtures of skew-normal distributions', Journal of Korean Statistical Society 40, 115-124.

12. Garay, A., Lachos, V., Labra, F. & Ortega, E. (2013), 'Statistical diagnostics for nonlinear regression models based on scale mixtures of skew normal distributions', Journal of Statistical Computation and Simulation 84, 1761-1778.

13. Geary, R. (1947), 'Testing for normality', Biometrika 34, 209-242.

14. Gupta, A. & Huang, W. (2002), 'Quadratic forms in skew normal variates', Journal of Mathematical Analysis and Applications 273, 558-564.

15. Huber, P. (1981), Robust Statistics, Jonh Wiley, New York.

16. Islam, M. & Tiku, M. (2004), 'Multiple linear regression model under nonnormality', Communication Statistics -Theory Methods 33, 2443-2467.

17. Kantar, Y. & Senoglu, B. (2008), 'A comparative study for the location and scale parameters of the weibull distribution with given shape parameter', Computers and Geosciences 34, 1900-1909.

18. Lachos, V., Bandyopadhyay, D. & Garay, A. (2011), 'Heteroscedastic non linear regression models based on scale mixtures of skew-normal distributions', Statistics and Probability Letters 81, 1208-1217.

19. Lachos, V., Bolfarine, H., Arellano-Valle, R. B. & Montenegro, L. (2007), 'Likelihood based inference for multivariate skew normal regression models', Communications in Statistics 36(9), 1769-1786.

20. Lachos, V., Ghosh, P. & Arellano-Valle, R. (2010), 'Likelihood based inference for skew-normal independent linear mixed models', Statistica Sinica 20, 303-322.

21. Martínez-Flórez, G., Vergara-Cardozo, S. & González, L. M. (2013), 'The family of Log-Skew-Normal Alpha-power distributions using precipitation data', Revista Colombiana de Estadística 36(1), 43-57.

22. Montgomery, D. (2005), Design and Analysis of Experiments, John Wiley & Sons Inc., United States of America.

23. Pearson, E. (1932), 'The analysis of variance in cases of nonnormal variation', Biometrika 23, 114-133.

24. Pereira, J. R., Marques, L. A. & da Costa, J. M. (2012), 'An empirical comparison of EM initialization methods and model choice criteria for mixtures of Skew-Normal distributions', Revista Colombiana de Estadística 35(3), 457-478.

25. Senoglu, B. & Tiku, M. (2001), 'Analysis of variance in experimental design with nonnormal error distributions', Communication Statistical Theory Methods 30, 1335-1352.

26. Senoglu, B. & Tiku, M. (2002), 'Linear contrasts in experimental design with non-identical error distributions', Biometrical Journal 44(3), 359-374.

27. Spjotvoll, E. & Aastveit, H. (1980), 'Comparison of robust estimators on some data from field experiments', Scandinavian Journal of Statistics 7, 1-13.

28. Srivastava, A. (1959), 'Effect of nonnormality on the power of the analysis of variance test', Biometrika 46, 114-122.

29. Tan, W. & Tiku, M. (1999), Sampling Distributions in Terms of Laguerre Polynomials with Applications, New Age International (formerly, Wiley Eastern), New Delhi.

30. Tiku, M. (1967), 'Estimating the mean and standard deviation from censored normal samples', Biometrika 54, 155-165.

31. Tiku, M. & Akkaya, A. (2004), Robust Estimation and Hypothesis Testing, New Age International, New Delhi.

32. Tiku, M., Tan, W. & Balakrishnan, N. (1986), Robust Inference, Marcel Dekker, New York.

33. Tukey, J. (1960), A survey of sampling from contaminated distributions, 'Contributions to Probability and Statistics', Stanford University Press, Stanford.

34. Xie, F., Wei, B. & Lin, J. (2009), 'Homogeneity diagnostics for skew-normal nonlinear regression models', Statistics and Probability Letters 79, 821-827.

[Recibido en noviembre de 2013. Aceptado en junio de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n1a04,
    AUTHOR  = {Celik, Nuri and Senoglu, Birdal and Arslan, Olcay},
    TITLE   = {{Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {1},
    pages   = {75-91}
}

References

Acitas, S., Kasap, S., Senoglu, B. & Arslan, O. (2013), ‘Robust estimation with the skew t2 distribution’, Pakistan Journal of Statistics 29(4), 409–430.

Arrellano-Valle, R., Bolfarine, H. & Lachos, V. (2005), ‘Skew-normal linear mixed models’, Journal of Data Science 3, 415–438.

Arslan, O. & Genc, A. (2009), ‘The skew generalized t (sgt) distribution as the scale mixture of a skew exponential distribution and its application in robust estimation’, Statistics 43, 481–498.

Azzalini, A. (1985), ‘A class of distributions which includes the normal ones’, Scandinavian Journal of Statistics 12, 171–178.

Azzalini, A. (1986), ‘Further results on a class of distributions which includes the normal ones’, Statistica 46, 199–208.

Azzalini, A. (2005), ‘The skew-normal distribution and related multivariate families (with discussion)’, Scandinavian Journal of Statistics 32, 159–188.

Bowman, K. & Shenton, L. (2001), ‘Weibull distributions when the shape parameter is defined’, Computational Statistics and Data Analysis 36, 299–310.

Box, G. (1953), ‘Non-normality and test of variances’, Biometrika 40, 336–346.

Box, G. & Tiao, G. (1964), ‘A Bayesian approach to the importance of assumptions applied to the comparison of variances’, Biometrika 51, 153–167.

Donaldson, T. (1968), ‘Robustness of the f-test to errors of both kinds and the correlation between the numerator and denominator of the F ratio’, Journal of American Statistical Association 63(322), 660– 667.

Garay, A., Lachos, V. & Abanto-Valle, C. (2011), ‘Nonlinear regression models based on scale mixtures of skew-normal distributions’, Journal of Korean Statistical Society 40, 115–124.

Garay, A., Lachos, V., Labra, F. & Ortega, E. (2013), ‘Statistical diagnostics for nonlinear regression models based on scale mixtures of skew normal distributions’, Journal of Statistical Computation and Simulation 84, 1761–1778.

Geary, R. (1947), ‘Testing for normality’, Biometrika 34, 209–242.

Gupta, A. & Huang, W. (2002), ‘Quadratic forms in skew normal variates’, Journal of Mathematical Analysis and Applications 273, 558–564.

Huber, P. (1981), Robust Statistics, Jonh Wiley, New York.

Islam, M. & Tiku, M. (2004), ‘Multiple linear regression model under nonnormality’, Communication Statistics -Theory Methods 33, 2443–2467.

Kantar, Y. & Senoglu, B. (2008), ‘A comparative study for the location and scale parameters of the weibull distribution with given shape parameter’, Computers and Geosciences 34, 1900–1909.

Lachos, V., Bandyopadhyay, D. & Garay, A. (2011), ‘Heteroscedastic non linear regression models based on scale mixtures of skew-normal distributions’, Statistics and Probability Letters 81, 1208–1217.

Lachos, V., Bolfarine, H., Arellano-Valle, R. B. & Montenegro, L. (2007), ‘Likelihood based inference for multivariate skew normal regression models’, Communications in Statistics 36(9), 1769–1786.

Lachos, V., Ghosh, P. & Arellano-Valle, R. (2010), ‘Likelihood based inference for skew-normal independent linear mixed models’, Statistica Sinica 20, 303–322.

Martínez-Flórez, G., Vergara-Cardozo, S. & González, L. M. (2013), ‘The family of Log-Skew-Normal Alpha-power distributions using precipitation data’, Revista Colombiana de Estadística 36(1), 43–57.

Montgomery, D. (2005), Design and Analysis of Experiments, John Wiley & Sons Inc., United States of America.

Pearson, E. (1932), ‘The analysis of variance in cases of nonnormal variation’, Biometrika 23, 114–133.

Pereira, J. R., Marques, L. A. & da Costa, J. M. (2012), ‘An empirical comparison of EM initialization methods and model choice criteria for mixtures of Skew-Normal distributions’, Revista Colombiana de Estadística 35(3), 457–478.

Senoglu, B. & Tiku, M. (2001), ‘Analysis of variance in experimental design with nonnormal error distributions’, Communication Statistical Theory Methods 30, 1335–1352.

Senoglu, B. & Tiku, M. (2002), ‘Linear contrasts in experimental design with non-identical error distributions’, Biometrical Journal 44(3), 359–374.

Spjotvoll, E. & Aastveit, H. (1980), ‘Comparison of robust estimators on some data from field experiments’, Scandinavian Journal of Statistics 7, 1–13.

Srivastava, A. (1959), ‘Effect of nonnormality on the power of the analysis of variance test’, Biometrika 46, 114–122.

Tan, W. & Tiku, M. (1999), Sampling Distributions in Terms of Laguerre Polynomials with Applications, New Age International (formerly, Wiley Eastern), New Delhi.

Tiku, M. (1967), ‘Estimating the mean and standard deviation from censored normal samples’, Biometrika 54, 155–165.

Tiku, M. & Akkaya, A. (2004), Robust Estimation and Hypothesis Testing, New Age International, New Delhi.

Tiku, M., Tan, W. & Balakrishnan, N. (1986), Robust Inference, Marcel Dekker, New York.

Tukey, J. (1960), A survey of sampling from contaminated distributions, in I. Olkin, ed., ‘Contributions to Probability and Statistics’, Stanford University Press, Stanford.

Xie, F., Wei, B. & Lin, J. (2009), ‘Homogeneity diagnostics for skew-normal nonlinear regression models’, Statistics and Probability Letters 79, 821–827.

How to Cite

APA

Celik, N., Senoglu, B. and Arslan, O. (2015). Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal. Revista Colombiana de Estadística, 38(1), 75–91. https://doi.org/10.15446/rce.v38n1.48802

ACM

[1]
Celik, N., Senoglu, B. and Arslan, O. 2015. Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal. Revista Colombiana de Estadística. 38, 1 (Jan. 2015), 75–91. DOI:https://doi.org/10.15446/rce.v38n1.48802.

ACS

(1)
Celik, N.; Senoglu, B.; Arslan, O. Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal. Rev. colomb. estad. 2015, 38, 75-91.

ABNT

CELIK, N.; SENOGLU, B.; ARSLAN, O. Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal. Revista Colombiana de Estadística, [S. l.], v. 38, n. 1, p. 75–91, 2015. DOI: 10.15446/rce.v38n1.48802. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/48802. Acesso em: 24 apr. 2024.

Chicago

Celik, Nuri, Birdal Senoglu, and Olcay Arslan. 2015. “Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal”. Revista Colombiana De Estadística 38 (1):75-91. https://doi.org/10.15446/rce.v38n1.48802.

Harvard

Celik, N., Senoglu, B. and Arslan, O. (2015) “Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal”, Revista Colombiana de Estadística, 38(1), pp. 75–91. doi: 10.15446/rce.v38n1.48802.

IEEE

[1]
N. Celik, B. Senoglu, and O. Arslan, “Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal”, Rev. colomb. estad., vol. 38, no. 1, pp. 75–91, Jan. 2015.

MLA

Celik, N., B. Senoglu, and O. Arslan. “Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal”. Revista Colombiana de Estadística, vol. 38, no. 1, Jan. 2015, pp. 75-91, doi:10.15446/rce.v38n1.48802.

Turabian

Celik, Nuri, Birdal Senoglu, and Olcay Arslan. “Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal”. Revista Colombiana de Estadística 38, no. 1 (January 1, 2015): 75–91. Accessed April 24, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/48802.

Vancouver

1.
Celik N, Senoglu B, Arslan O. Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal. Rev. colomb. estad. [Internet]. 2015 Jan. 1 [cited 2024 Apr. 24];38(1):75-91. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/48802

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CrossRef citations8

1. Abdullah Yalçınkaya, Ufuk Yolcu, Birdal Şenoǧlu. (2021). Maximum likelihood and maximum product of spacings estimations for the parameters of skew-normal distribution under doubly type II censoring using genetic algorithm. Expert Systems with Applications, 168, p.114407. https://doi.org/10.1016/j.eswa.2020.114407.

2. Abdullah Yalçınkaya, İklim Gedik Balay, Birdal Şenoǧlu. (2021). A new approach using the genetic algorithm for parameter estimation in multiple linear regression with long-tailed symmetric distributed error terms: An application to the Covid-19 data. Chemometrics and Intelligent Laboratory Systems, 216, p.104372. https://doi.org/10.1016/j.chemolab.2021.104372.

3. Gamze GUVEN. (2022). Testing the equality of treatment means in one-way ANOVA: Short-tailed symmetric error terms with heterogeneous variances. Hacettepe Journal of Mathematics and Statistics, 51(6), p.1736. https://doi.org/10.15672/hujms.1055277.

4. Abdullah Yalçınkaya, Birdal Şenoğlu, Ufuk Yolcu. (2018). Maximum likelihood estimation for the parameters of skew normal distribution using genetic algorithm. Swarm and Evolutionary Computation, 38, p.127. https://doi.org/10.1016/j.swevo.2017.07.007.

5. Mehmet Niyazi Çankaya, Yakup Murat Bulut, Fatma Zehra Dogru, Olcay Arslan. (2015). A Bimodal Extension of the Generalized Gamma Distribution. Revista Colombiana de Estadística, 38(2), p.353. https://doi.org/10.15446/rce.v38n2.51666.

6. Sukru Acitas, Birdal Senoglu, Olcay Arslan. (2015). Alpha-Skew Generalized t Distribution. Revista Colombiana de Estadística, 38(2), p.371. https://doi.org/10.15446/rce.v38n2.51665.

7. N. Celik. (2022). Welch’s ANOVA: Heteroskedastic skew-t error terms. Communications in Statistics - Theory and Methods, 51(9), p.3065. https://doi.org/10.1080/03610926.2020.1788084.

8. Murtadha Alaa Alkhafaji, Rabab Abdulrida Saleh. (2021). A Statistical Study on the Parameters of the Skew Normal Distribution Depending on the Use of the Genetic Algorithm Using the Simulation Method. Journal of Physics: Conference Series, 1879(3), p.032017. https://doi.org/10.1088/1742-6596/1879/3/032017.

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