Abstract
In this paper we derive the asymptotic expansion of the null distribution of the F-statistic in one-way ANOVA under non-normality. The asymptotic framework is when the number of treatments is moderate but sample size per treatment (replication size) is small. This kind of asymptotics will be relevant, for example, to agricultural screening trials where large number of cultivars are compared with few replications per cultivar. There is also a huge potential for the application of this kind of asymptotics in microarray experiments. Based on the asymptotic expansion we will devise a transformation that speeds up the convergence to the limiting distribution. The results indicate that the approximation based on limiting distribution are unsatisfactory unless number of treatments is very large. Our numerical investigations reveal that our asymptotic expansion performs better than other methods in the literature when there is skewness in the data or even when the data comes from a symmetric distribution with heavy tails.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akritas M.A., Arnold S.F. (2000). Asymptotics for analysis of variance when the number of levels is large. Journal of the American Statistical association 95:212–226
Bathke A. (2002). ANOVA for large number of treatments, Mathematical Methods of Statistics 11:118–132
Bhattacharya R.N., Rao R.R. (1976). Normal approximation and asymptotic expansions. Wiley, New York
Boos D.D., Brownie C. (1995). ANOVA and rank tests when number of treatments is large. Statistics and Probability Letters 23:183–191
Browne M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. The British Journal of Mathematical and Statistical Psychology 37:62–83
Brownie C., Boos D.D. (1994). Type I error robustness of ANOVA and ANOVA on ranks when the number of treatments is large. Bimetrics 50:542–549
Cope L.M., Irizarry R.A., Jaffee H.A., Wu Z., Speed T.P. (2004). A benchmark for affymetrix genechip expression measures. Bioinformatics 20:323–331
David F.N., Kendall M.G., Barton D.E. (1966). Symmetric function and allied tables. Cambridge University Press, London
Donaldson T.S. (1968). Robustness of the F-test to errors of both kinds and the correlation between the neumerator and denominator of the F-ratio. Journal of the American Statistical Association 63:660–676
Fisher N.I., Hall P. (1990). On bootstrap hypothesis testing. Australian Journal of Statistics 32:177–190
Fujikoshi Y., Ohmae M., Yanagihara H. (1999). Asymptotic approximations of the null distribution of the one-way ANOVA test statistic under nonormality. Journal of Japan Statistical Society 29:147–161
Gupta, A. K., Harrar,S., Fujikoshi, Y. (2005). MANOVA for large hypothesis degrees of freedom under non-normality. Test (in press)
Gupta A.K., Harrar S., Fujikoshi Y. (2006). Asymptotics for testing hypothesis in some variance components model under nonnormality. Journal of Multivariate Analysis 97:148–178
Hall P. (1987). Edgeworth expansion for Students’s statistics under minimal moment conditions. Annals of Probability 15:920–931
Hall P. (1992). The bootstrap and edgeworth expansions. Springer, Berlin Heidelberg New York
Huber P.J. (1973) Robust regression: asymptotics, cojectures and montecarlo. Annals of Statistics 1:799–821
Irizarry R.A., Hobbs B., Collin F., Beazer-Barclay Y.D., Antonellis K.J., Scherf U., Speed T.P. (2003). Exploration, normalization, and summaries of high density oligonucleotide array probe level data. Biostatistics 4:249–264
James G.S. (1954). Test for Linear hypothesis in univariate and multivariate analysis when ratios of the population variances are unknown. Biometrika 41:19–43
Mardia K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530
Tan W.Y., Cheng S.S. (1981). Cumulants of bilinear forms and quadratic forms and their application. Communications in Statistics–Theory and Methods A10:283–298
Yanagihara H. (2000). Asymptotic expansion of the null distribution of the one-way ANOVA test statistic for heteroscedastic case under nonnormality. Communications in Statistics–Theory and Methods 29:463–476
Yanagihara H. (2003). Asymptotic expansion of the null distribution of test statistic for linear hypothesis in non-normal linear model. Journal of Multivariate Analysis 84:222–246
Yanagihara, H. (2005). A family of estimators for multivariate kurtosis in a nonnormal linear regression model. Journal of Multivariate Analysis (in press)
Xu J., Gupta A.K. (2005). Confidence intervals for the mean value of response function in generalized linear models. Statistica Sinica 15:1081–1096
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Harrar, S.W., Gupta, A.K. Asymptotic Expansion for the Null Distribution of the F-statistic in One-way ANOVA under Non-normality. AISM 59, 531–556 (2007). https://doi.org/10.1007/s10463-006-0055-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-006-0055-7