Hsieh, F ve Kruskal-Wallis Testlerinin Monte Carlo Simülasyon Tekniği Kullanılarak I. Tip Hata Olasılıkları Bakımından Değerlendirilmesi

Year 2024, Volume: 6 Issue: 2, 132 - 137

Malik Ergin , Özgür Koşkan

https://doi.org/10.55979/tjse.1539525

Abstract

Bu çalışmada, iki parametreli üstel dağılımdan farklı parametrelerle ve χ2(2) dağılımından üretilen örnekler kullanılarak Hsieh, F ve Kruskal-Wallis testleri, I.tip hata olasılıkları açısından karşılaştırılmıştır. Çalışmada incelenen testler arasında, Hsieh testinin I.tip hata olasılıklarının, Klasik F ve Kruskal-Wallis testlerine göre daha yüksek olduğu bulunmuş ve bu durum, I.tip hatayı %5 seviyesinde tutamadığını göstermektedir. Bu durum özellikle küçük örnek genişliklerinde ve varyansların homojenliği varsayımının sağlanmadığı durumlarda daha belirgin hale gelmektedir. Ayrıca, Levene testi tarafından göz ardı edilen heterojen varyans oranlarında bile, Hsieh testinin I.tip hata oranı %13.3'e ulaşmıştır. Sonuç olarak, Hsieh testindeki yüksek I.tip hata olasılığı, özellikle küçük örnek genişliklerinde ve grup varyanslarının homojen olmadığı durumlarda kullanımını sınırlamaktadır.

Keywords

Hsieh testi, Varyans analizi, Kruskal-Wallis testi, I.tip hata olasılığı

Evaluating The Type I Error Rate Performances of Hsieh, F, and Kruskal-Wallis Tests Using Monte Carlo Simulation Technique

Abstract

In this study, the Hsieh, Classical F, and Kruskal-Wallis tests were compared in terms of Type I error probabilities using samples generated from a two-parameter exponential distribution with various parameters and a χ2(2) distribution. Among the tests examined in the study, the Hsieh test was found to have higher Type I error probabilities compared to the Classical F and Kruskal-Wallis tests, indicating that it could not maintain the Type I error at the 5.0% level. The effect of this finding is more pronounced in small sample sizes and when the assumption of homogeneity of variance is not met. In addition, even in heterogeneous variance ratios, that Levene's test neglects, the Type I error rate of the Hsieh test reached 13.3%. In conclusion, the high probability of Type I error in the Hsieh test, especially in small sample sizes and when group variances are not homogeneous, restricts its usage.

Keywords

Hsieh test, Analysis of variance, Kruskal-Wallis test, Type I error rate

References

• Arıcı, K. Y., Özkan, M. M., & Kocabaş, Z. (2011). Comparison of Kruskal-Wallis test and transformed variance analysis in heterogeneous variance groups. 7th National Zootechnical Student Congress. September 14-16, Adana, 14-16.
• Babacan, E. K., & Kaya, S. (2020). Comparison of parameter estimation methods in Weibull Distribution. Sigma Journal of Engineering and Natural Sciences, 38(3), 1609-1621.
• Blanca Mena, M. J., Alarcón Postigo, R., Arnau Gras, J., Bono Cabré, R., & Bendayan, R. (2017). Non-normal data: Is ANOVA still a valid option? Psicothema, 29(4), 552-557. doi.org/10.7334/psicothema2016.383
• Bradley, J. V. (1978). Robustness? British Journal of Mathematical and Statistical Psychology, 31(2), 144-152. doi.org/10.1111/j.2044-8317.1978.tb00581.x
• Cavus, M. (2021). Testing the equality of normal distributed and independent groups' means under unequal variances by doex package. The R Journal, 12(2), 134-154.
• Cavuş, M., & Yazıcı, B. (2020). Comparison of Hsieh test and ANOVA for logtransformed on income data. 20th International Symposium on Econometrics, Operational Research and Statistics. February 12-14, Ankara, 152-158.
• Cessie, S., Goeman, J. J., & Dekkers, O.M. (2020). Who is afraid of non-normal data? Choosing between parametric and non-parametric tests. European Journal of Endocrinology, 182(2), 1-3. doi.org/10.1530/EJE-19-0922
• Fan, W., & Hancock, G. R. (2012). Robust means modeling: An alternative for hypothesis testing of independent means under variance heterogeneity and nonnormality. Journal of Educational and Behavioral Statistics, 37(1), 137-156. doi.org/10.3102/1076998610396897
• Ghosh, M., & Razmpour, A. (1984). Estimation of the common location parameter of several exponentials. Sankhya: The Indian Journal of Statistics, Series A, 46(3), 383-394.
• Hammouri, H. M., Sabo, R. T., Alsaadawi, R., & Kheirallah, K. A. (2020). Handling skewed data: A comparison of two popular methods. Applied Sciences, 10(18), 6247. doi.org/10.3390/app10186247
• Hsieh, H. K. (1986). An exact test for comparing location parameters of k exponential distributions with unequal scales based on type II censored data. Technometrics, 28(2), 157-164. doi.org/10.1080/00401706.1986.10488117
• Kim, B. S., Park, S. G., You, Y. K., & Jung, S. I. (2011). Probability & statistics for engineers & scientists. New York, Pearson.
• Koskan, Ö., & Gürbüz F. (2009). Comparison of F test and resampling approach for type I error rate and test power by simulation method. Journal of Agricultural Science, 15(1), 105-111.
• Krishna, H., & Goel, N. (2018). Classical and Bayesian inference in two parameter exponential distribution with randomly censored data. Computational Statistics, 33, 249-275. doi.org/10.1007/s00180-017-0725-3
• Krishnamoorthy, K., Nguyen, T., & Sang, Y. (2020). Tests for comparing several two-parameter exponential distributions based on uncensored/censored samples. Journal of Statistical Theory and Applications, 19(2), 248-260. doi.org/10.2991/jsta.d.200512.001
• Lantz, B. (2013). The impact of sample non‐normality on ANOVA and alternative methods. British Journal of Mathematical and Statistical Psychology, 66(2), 224-244. doi.org/10.1111/j.2044-8317.2012.02047.x
• Li, J., Song, W., & Shi, J. (2015). Parametric bootstrap simultaneous confidence intervals for differences of means from several two-parameter exponential distributions. Statistics & Probability Letters, 106, 39-45. doi.org/10.1016/j.spl.2015.07.002
• Lix, L. M., Keselman, J. C., & Keselman, H. J. (1996). Consequences of assumption violations revisited: A quantitative review of alternatives to the one-way analysis of variance F test. Review of Educational Research, 66(4), 579-619. doi.org/10.3102/0034654306600457
• Malekzadeh, A., & Jafari, A. A. (2020). Inference on the equality means of several two-parameter exponential distributions under progressively Type II censoring. Communications in Statistics - Simulation and Computation, 49(12), 3196-3211. doi.org/10.1080/03610918.2018.1538452
• Maurya, V., Goyal, A., & Gill, A. N. (2011). Simultaneous testing for the successive differences of exponential location parameters under heteroscedasticity. Statistics & Probability Letters, 81(10), 1507-1517. doi.org/10.1016/j.spl.2011.05.010
• Mendeş, M. (2002). The Comparison of some alternative parametric tests to one - way analysis of variance about Type I error rates and power of test under non - normality and heterogeneity of variance. (PhD thesis, Ankara University).
• Mendeş, M., & Yiğit, S. (2013). Comparison of ANOVA-F and ANOM tests with regard to type I error rate and test power. Journal of Statistical Computation and Simulation, 83(11), 2093-2104. doi.org/10.1080/00949655.2012.679942
• Nwobi, F. N., & Akanno, F. C. (2021). Power comparison of ANOVA and Kruskal–Wallis tests when error assumptions are violated. Advances in Methodology and Statistics / Metodološki zvezki, 18(2), 53-71. doi.org/10.51936/ltgt2135
• Pawlitschko, J. (2001). Robust estimation of the location parameter from a two-parameter exponential distribution (No. 2001, 36). Technical Report.
• R Core Team (2023) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. URL https://www.R-project.org/
• Wilcox, R. R. (2002). Understanding the practical advantages of modern ANOVA methods. Journal of Clinical Child & Adolescent Psychology, 31(3), 399-412.
• Young, D. S. (2010). Tolerance: an R package for estimating tolerance intervals. Journal of Statistical Software, 36, 1-39. doi.org/10.18637/jss.v036.i05
• Zhuang, Y., & Bapat, S.R. (2022). On comparing locations of two-parameter exponential distributions using sequential sampling with applications in cancer research. Communications in Statistics - Simulation and Computation, 51(10), 6114-6135. doi.org/10.1080/03610918.2020.1794007