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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 Received:
8 May 2015/Accepted: 15 February 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 karya 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 REFERENCES Algina, J.,
Oshima, T.C. & Lin, W. 1994. Type
I error rates for Welch’s test and James’ second-order test
under nonnormality and inequality of variance when there are two
groups. Journal of Educational and Behavioral Statistics
19: 275-291. Box, G.E.P. 1954. Some theorems
on quadratic forms in the study of analysis of variance problems.
I. Effect of inequality of variance in the one-way classification.
Annals of Mathematical Statistics 25: 290-302. 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. *Corresponding author; email: wuttsr@kku.ac.th
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