Normal, Beta, Gamma x2 ve Weibull Dağılımlarmın İkili Kombinasyonlarından Alınan Değişik Örnek Genişliğindeki Örneklerin Karşılaştırılmasında Testin Gücü

Abstract

Bu çalışmada, Normal, Bcita, Gamma i.2 ve Weibull da ğı l ı m ı gösteren populasyonlar ı n, mümkün olan bütün ikili kombinasyonları ndan mcgsle al ı nan örnekler yard ı mı yla hesaplanan F-Testinin gücü ara ştı r ı lm ıştı r. Bunun için, üzerinde durulan populzsyonlar ı n ikili kombinasyonlar ı n ı n ortalamalan aras ı nda 8=0.5, 8=1.0, 8=1.5, 8=2.0, 8=2.5 ve 8=3.0 standart sapmal ı k fark olacak şekilde, dağı l ı mlardan birisindeki gözlemlere, bütün populasyonlarda olduğundan 8; ilave edilmi ştir. Bu populasyonlar ı n ikili kombinasyonlar ı ndan rasgele olarak al ı nan çeşitli örnek genişliği kombinasyonundaki örnekler yard ı mı yla 100 000 simülasyon denemesi sonunda F-Testinin gücü ampirik olarak belirlenmiştir. F-Testinin istenilen güce %80 veya daha yüksek ula şmas ı nda, dağı l ı m şeklinden ziyade, populasyon ortalamalan aras ı ndaki fark ı n büyüklüğüne bağl ı olarak, bu populasyonlardan rasgele al ı nan örneklerdeki deney ünitesi say ı sı n ı n ve bunlar ı n örneklerde eşit veya dengeli olarak bulunup bulunmad ığı n ı n etkili olduğu sonucuna var ı lmışt ı r.

Keywords

• Testin gücü
• Normal da ğı l ım
• Beta dağı l ı mı
• Gamma Da ğı l ı m ı
• Weibull Dağı l ı mı
• örnek genişliği

The Power of the Test in the Samples of Various Sample Sizes were Taken from the Binary Combinations of the Normal, Beta, Gamma and Weibull Distributions

Abstract

In this study, we investigated the power of the ANOVA in the samples taken from the binary combinations of the populations which are showing Normal, Beta, Gamma and Weibull distributions. The differences between the means of the binary combinations of these populations were 8=0.5, 8=1.0, 8=1.5, 8=2.0, 8=2.5 and 8=3.0 standard deviations. For that, one of the populations was added a constant that is 8, The samples which are equal or unequal sample sizes taken randomly from the binary combinations of these populations and calculated power of the F-Test empirically with 100 000 simulated experiment. The result showed that the shape of the distributions ware ineffective on the power of F-Test but effected the sample sizes depend on difference between the means of the populations.

Keywords

• Power of test
• Normal distribution
• Beta distribution
• Gamma distribution
• Weibull distribution
• sample size

References

1. Akdeniz, F. 1984. Olas ı l ı k ve istatistik. A.Ü.Fen Fakültesi Yay ı nlar ı No:138. 519 S.
2. Andres, A.M., J.D.L. del Castillo, 1990. Multiple Choice Tests: Power, Length and Optimal Numtıer of Choices per !tem. Brit. Jour. Of Math. And Statist. Psyc. (43), 57-71.
3. Banik, N., K. Kohne and P. Bauer, 1986. On the Power of Fisher's Combination Test for 2-Stage Sampling in the Presence of Nuisance ParametrJrs. Biometrical Jour. 38(1), 25-37.
4. Bulgren, W.G. 1971. On Representations of the Doubly NonCentral F Distribution. Jour. of the American Stat. Assoc. 66(333) 184-186.
5. Buning, H. and W. Kossler, 1997. Power of Some Tests for Urnbrella Atternatives in the Multisample Location Problem. Fisher's Combination Test for 2-Stage Sampling in the r?resence of Nuisance Parameters. Biometrical Jour. 38(1), 25-37.
6. Cade, W. 1998. Sampling Procedures and Type I Error Rates (For Nonnormal Populations) (Nonnormal Distribution). DAI-B 59/03, s. 1186.
7. Duchateau,L., B.Mcdermott and G.J.Rowlands, 1998. Power Evaluation of Small Drug and Vaccine Experiments with Binary Outcomes. Statistics in Medicine, 17(1), 111-120
8. Gillett, R. 1996. Retrospective Power Surveys. The Statistician. 45(2), 231-236.

9. Kavuncu, O. 1995. istatistik Teorisi ve Teorik Da ğı l ı mlar. T.C. Ziraat Bankas ı Matbaast, Ankara. 179 s.
10. Lui, K.J. 1994. A Group Sequential Method for One Standard Control and More than One Experimental Treatment. Biometrical Journal, 36(5) 515-529.
11. O'Gorman, T.W. 1995. The effect of Unequal Estimation of the Power of the Kruskal-Wallis Test. Commun. Statist.-Simula., 24(4), 853-867.
12. Olejnik, S. and W.M.Luh, 1994. Type I Error Rates, Power and Sample Sizes For 2 Stage Solutions to the Behrens-Fisher Problem When Population Distributions are Nonnormal. Computational Statistics & Data Analysis, 17(4) 409-420.
13. Price, R. 1964. Some Non-Central F Distributions Expressed in Closed Form. Biometrica, (51) 107-122.
14. Snedecor, G.W. and W.G. Cochran, 1980. Statistical Methods. Seventh Ed. The lowa State University Press. Ames, lowa, U.S.A. 507 s.
15. Sokal, R.R. and F.J.Rohlf, 1995. Biometry. The Principles and Practice of Statistics in Biological Research. Third Ed. W.H.Freeman and Co. New York, 887 s.
16. Taylor, D.J. and K.E.Muller, 1995. Computing Confidence Bounds for Power and Sample Size of the General Linear Univariate Model. The Amer.Statician. 49(1) 43-47.
17. Wassmer G. 1997. A Technical Note on the Power Determination Fisher's Combination Test. Biometrical Jour. 39(7), 831-838.
18. Wellek, S. 1996. A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic. Biometrical Journal, 38(6), 695-710.
19. Zhang J. and D.D.Boos, 1994. Adjusted Power Estimates in Monte Carlo Experiments. Commun. Statist.-Simula., 23(1), 165-173.