Testing equality of means in one-way ANOVA using three and four moment approximations

Year 2023, , 587 - 605, 30.09.2023

Gamze Guven

https://doi.org/10.31801/cfsuasmas.1252070

Abstract

In this study, we focus on two test statistics for testing the equality of treatment means in one-way analysis of variance (ANOVA). The first one is the well known Cochran ($C_{LS}$) test statistic based on least squares (LS) estimators and the second one is robust version of it ($RC_{MML}$) based on modified maximum likelihood (MML) estimators. These two test statistics are asymptotically distributed as chi-square. However, distributions of them are unknown for small samples. Therefore, three-moment chi-square and four moment $F$ approximations to the null distributions of $C_{LS}$ and $RC_{MML}$ are derived inspired by Tiku and Wong [19]. To investigate the small and moderate sample properties of these tests based on the mentioned approximations, an extensive Monte-Carlo simulation study is performed when the underlying distribution is long-tailed symmetric (LTS). Simulation results show that four-moment $F$ approximation provides better approximation than the three-moment chi-square approximation for both $C_{LS}$ and $RC_{MML}$ tests. Therefore, the simulated Type I error rates and powers of the $C_{LS}$ and $RC_{MML}$ test statistics are calculated using four-moment $F$ approximation. According to simulation results, $RC_{MML}$ test is more powerful than the corresponding $C_{LS}$ test.

Keywords

Cochran test statistic, three moment chi-square approximation, four-moment F approximation, Monte Carlo simulation

References

• Aydoğdu, H., Senoğlu, B., Kara, M., Parameter estimation in geometric process with Weibull distribution. Appl. Math. Comput., 217(6) (2010), 2657-2665. https://doi.org/10.1016/j.amc.2010.08.003
• Brown, M. B., Forsythe, A. B., The small sample behavior of some statistics which test the equality of several means. Technometrics, 16(1) (1974), 129-132. https://www.tandfonline.com/doi/abs/10.1080/00401706.1974.10489158.
• Cochran, W. G., Problems arising in the analysis of a series of similar experiments. Suppl. J. R. Stat. Soc, 4(1) (1937), 102-118. https://www.jstor.org/stable/2984123
• Gamage, J., Weerahandi, S., Size performance of some tests in one way ANOVA, Comm. Statist. Simulation Comput., 27(3) (1998), 625-640. https://www.tandfonline.com/doi/abs/10.1080/03610919808813500
• Güven, G., Gürer, Ö., Şamkar, H., Şenoglu, B., A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances. J. Stat. Comput. Simul., 89(9) (2019), 1715-1729. https://doi.org/10.1080/00949655.2019.1593985
• Hampel, F. R., Robust estimation: A condensed partial survey, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 27(2) (1973), 87-104. https://link.springer.com/article/10.1007/BF00536619
• Hartung, J., Knapp, G., Sinha, B. K., Statistical meta-analysis with applications. John Wiley and Sons (2011).
• James, G. S., The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika, 38(3/4) (1951), 324-329. https://doi.org/10.2307/2332578
• Krishnamoorthy, K., Lu, F., Mathew, T., A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models. Comput. Stat. Data Anal., 51(12) (2007), 5731-5742. https://doi.org/10.1016/j.csda.2006.09.039
• Li, X.,Wang, J., Liang, H., Comparison of several means: A fiducial based approach. Comput. Stat. Data Anal., 55(5) (2011), 1993-2002. https://doi.org/10.1016/j.csda.2010.12.009
• Mehrotra, D. V., Improving the Brown-Forsythe solution to the generalized Behrens-Fisher problem. Commun. Stat. Simul. Comput., 26(3) (1997), 1139-1145. https://doi.org/10.1080/03610919708813431
• Purutcuoğlu, V., Unit root problems in time series analysis, Master Thesis, Middle East Technical University, 2004.
• Sürücü, B., Sazak, H. S., Monitoring reliability for a three-parameter Weibull distribution. Reliab. Eng. Syst. Saf., 94(2) (2009), 503-508. https://doi.org/10.1016/j.ress.2008.06.001
• Schrader, R. M., Hettmansperger, T. P., Robust analysis of variance based upon a likelihood ratio criterion, Biometrika, 67(1) (1980), 93-101. https://doi.org/10.1093/biomet/67.1.93
• Şenoğlu, B., Tiku, M. L., Analysis of variance in experimental design with nonnormal error distributions. Commun. Stat. Theory Methods, 30(7) (2001), 1335-1352. https://www.tandfonline.com/doi/full/10.1081/STA-100104748
• Tiku, M. L., Estimating the mean and standard deviation from a censored normal sample. Biometrika, 54(1-2) (1967), 155-165. https://doi.org/10.1093/biomet/54.1-2.155
• Tiku, M. L., Estimating the parameters of log-normal distribution from censored samples. J. Am. Stat. Assoc, 63(321) (1968), 134-140. https://doi.org/10.1080/01621459.1968.11009228
• Tiku, M. L., Kumra, S., Expected values and variances and covariances of order statistics for a family of symmetric distributions (Student’st). Selected tables in mathematical statistics, 8 (1981), 141-270.
• Tiku, M. L., Wong, W. K., Testing for a unit root in an AR (1) model using three and four moment approximations: symmetric distributions, Commun. Stat. Simul. Comput., 27(1) (1998), 185-198. https://www.tandfonline.com/doi/abs/10.1080/03610919808813474
• Tiku, M. L., Wong, W. K., Bian, G., Estimating parameters in autoregressive models in nonnormal situations: Symmetric innovations. Commun. Stat. Theory Methods, 28(2) (1999), 315-341. https://doi.org/10.1080/03610929908832300
• Tiku, M. L., Yip, D. Y. N., A four-moment approximation based on the F distribution. Austrian J. Stat., 20(3) (1978), 257-261. https://doi.org/10.1111/j.1467-842X.1978.tb01108.x
• Weerahandi, S., ANOVA under unequal error variances.Biometrics, 51(2) (1995), 589-599. https://doi.org/10.2307/2532947
• Welch, B. L., On the comparison of several mean values: an alternative approach. Biometrika, 38(3/4) (1951) , 330-336. https://doi.org/10.2307/2332579