Sains Malaysiana 45(10)(2016): 1565–1572

 

A Bayesian Approach to the One Way ANOVA under Unequal Variance

(Pendekatan Bayesian kepada ANOVA Sehala di bawah Varians tak Sama)

 

NOPPAKUN TONGMOL¹, WUTTICHAI SRISODAPHOL²* & ANGKANA BOONYUED¹

 

¹Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002

Thailand

 

²Department of Statistics, Faculty of Science, Khon Kaen University, Khon Kaen 40002

Thailand

 

Diserahkan: 8 Mei 2015/Diterima: 15 Februari 2016

 

ABSTRACT

This study involves testing the equality of several normal means under unequal variances, which is the setup of one-way analysis of variances (one-way ANOVA). Several tests are available in the literature, however, most of them perform poorly in terms of type I error rate under unequal variances. In fact, Type I errors can be highly inflated for some of the commonly used tests, a serious issue that seems to have been overlooked. Even though several tests have been proposed to overcome the problem, most of them show difficulty in calculation. Accordingly, the test for ANOVA with estimation of parameters using Bayesian approach is proposed as an alternative to such tests. The proposed test is compared with four existing tests such as the original test, James’s test, Welch’s test and the parametric bootstrap (PB) test. Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our results indicated that the performance of the proposed test is superior to the original test and is comparable to James’s test, Welch’s test and the PB test, controlling Type I error rate quite well and showing high power of the test. Our study suggested that the proposed test has high performance and should be used as an alternative to the four existing tests due to its simple formula.

 

Keywords: Bayesian approach; power of the test; Type I error rate; unequal variance

 

ABSTRAK

Kajian ini melibatkan ujian kesamaan dalam beberapa cara yang biasa di bawah varians tak sama yang merupakan persediaan varians analisis sehala (ANOVA sehala). Beberapa ujian telah sedia ada dalam penulisan ilmiah, walau bagaimanapun, tidak menunjukkan keputusan memberangsangkan daripada segi kadar ralat Jenis I di bawah varians tak sama. Malah, ralat Jenis I boleh melambung tinggi bagi sesetengah ujian yang biasa digunakan, suatu isu yang serius yang seolah-olah telah diabaikan. Walaupun beberapa ujian telah dicadangkan untuk mengatasi masalah ini, sebahagian besar menunjukkan kesukaran dalam pengiraan. Sehubungan dengan itu, ujian bagi ANOVA dengan parameter anggaran menggunakan pendekatan Bayesian dicadangkan sebagai alternatif kepada ujian tersebut. Ujian yang dicadangkan dibandingkan dengan empat ujian sedia ada seperti ujian asal, ujian James, ujian Welch dan ujian butstrap berparameter (PB). Kadar ralat Jenis I dan kuasa ujian dinilai menggunakan simulasi Monte Carlo. Keputusan kajian kami menunjukkan bahawa prestasi ujian cadangan itu lebih cemerlang berbanding ujian asal dan setanding dengan ujian James, Welch dan PB, mengawal kadar ralat Jenis I dengan baik dan menunjukkan kuasa tinggi ujian tersebut. Kajian kami menyarankan bahawa ujian cadangan mempunyai prestasi yang tinggi dan harus digunakan sebagai suatu alternatif kepada empat ujian sedia ada kerana formula yang mudah.

 

Kata kunci: Kadar ralat Jenis I; kuasa ujian; pendekatan Bayesian; varians tak sama

 

RUJUKAN

 

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James, G.S. 1951. The comparison of several groups of observations when the ratios of population variances are unknown. Biometrika 38: 324-329.

Krishnamoorthy, K., Lu, F. & Mathew, T. 2007. A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models. Computational Statistics and Data Analysis 51: 5731-5742.

Robert, C.P. 2007. The Bayesian Choice from Decision-Theoretic Foundations to Computational Implementation. 2nd ed. New York: Springer Verlag.

Seber, G.A.F. 1977. Linear Regression Analysis. New York: John Wiley and Sons.

Welch, B.L. 1951. On the comparison of several mean values: An alternative approach. Biometrika 38: 330-336.

Welch, B.L. 1937. The significance of the difference between two means when the population variance are unequal. Biometrika 29: 350-362.

 

 

*Pengarang untuk surat-menyurat; email: wuttsr@kku.ac.th