Testing for Random Effect in the Mixed Analysis of Variance Model

Abstract

We consider the mixed analysis of variance (ANOVA) model. It is assumed that random effects in the model are from non-normal universes. Approximate test for random effect is established. The proposed approximate test is based on the asymptotic distribution of the F-ratio in this model and its robustness is given.

Keywords

• The Mixed ANOVA Model
• Asymptotic Distribution
• Approximate F Test
• Robustness of Test

Karışık Varyans Analizi Modelinde Etki Faktorünün Rassallık Testi

Abstract

Bu çalışmada bazı karışık varyans modelleri genelleştirildi. Bu modellerin rassal etki faktörleri ve hata terimleri normal olmayan bir kitleden geldiği varsayıldı. Etki faktörünün rassallığını test eden bir test elde edildi ve bu testin sağlamlığı çalışıldı.

Keywords

• Karışık ANOVA Modeli
• Asimptotik Dağılım
• Yaklaşık F-Testi
• Testin Güçlülüğü

References

1. Akritas, M., Arnold, S., 2000. "Asymptotics for Analysis of Variance When the Number of Levels is Large", Journal of the American Statistical Association, 95, 212-226.
2. Akritas, M. G., Papadatos, N., 2004. "Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests", Journal of the American Statistical Association, 99, 368-382.
3. Arnold, S. F., 1980. "Asymptotic Validity of F Tests for the Ordinary Linear Model and the Multiple Correlation Model", Journal of the American Statistical Association, 75, 890-894.
4. Arnold, S. F., 1981. The Theory of Linear Models and Multivariate Analysis, New York: Wiley.
5. Bathke, A., 2004. "ANOVA F Test can Still be Used in Some Balanced Designs with Unequal Variances and Nonnomal Data", Journal of Statistical Planning and Inference, 126, 413-422.
6. Bathke, A., Harrar, S. W., Madden, L. W., 2008. "How to Compare Small Multivariate Samples Using Nonparametric Test", Computational Statistics and Data Analysis, 52, 4951-4965.
7. Boos, D. D., Brownie, C., 1995. "ANOVA and Rank Tests When the Number of Treatments is Large", Statistics and Probability Letters, 23, 183-191.
8. Brunner, E., Dette, H., Munk, A., 1997. "Box-Type Approximations in Nonparametric Factorial Design", Journal of the American Statistical Association, 92, 1494-1502.

9. Fuller, W. A., Battese, G. E., 1973. "Transformations for Estimation of Linear Models with Nested-Error Structure", Journal of the American Statistical Association, 68, 626-632.
10. Güven, B., 1995. "Maximum Likelihood Estimation in Simple Linear Regression with One Fold Nested Error", Communications in Statistics-Theory and Methods, 24, 121-130.
11. Güven, B., 2006. "The Limiting Distribution of the F-Statistic From Nonnormal Universes", Statistics, 40, 545-557.
12. Güven, B., 2012. "Testing for Random Effect in Fuller-Battesse Model", Accepted for publication in Statistics.
13. Harrar, S. W., Gupta, A. K., 2007. "Asymtotic Expansion for the Null Distribution of the F-Statistic in One-Way ANOVA Under Non-Normality", Annuals of the Institute of Statistical Mathematics, 59, 531-556.
14. Horn, S. D., Horn, R. A., 1975. "Comparison of Estimators Heteroscedastic Variances in Linear Models", Journal of the American Statistical Association, 70, 872-879.
15. Khuri, A. I., Mathew, T. and Sinha, B. K., 1998. Statistical Tests for Mixed Linear Models, New York, Wiley.
16. Park, D. J., Burdick, R. K., 2005. "Performance of Confidence Intervals in Regression Models with Unbalanced One-Fold Nested Error Structure", Communications in Statistics-Computation and Simulation, 32, 717-732.
17. Rudin, W., 1964. Principles of Mathematical Analysis, New York, McGraw-Hill.
18. Scheffé, H., 1959. The Analysis of Variance, New York: Wiley.
19. Van der Vaart, A. W., 1998. Asymptotic Statistics, New York, Cambridge.
20. Wang, L., Akritas M. G., 2006. "Two-Way Heteroscedastic ANOVA When the Number of Levels is Large", Statistica Sinica, 16, 1387-1408.
21. Vonesh, F. E., Carter, R. L., 1987. "Efficient Inference for Random Coefficient Growth Models with Unbalanced Data", Biometrics, 43, 617-628.
22. Westfall, P. H., 1986. "Asymptotic Normality of the Anova Estimates of Components of Variance in the Nonnormal Unbalanced Hierarchal Mixed Model", The Annals of Statistics, 14, 1572-1582.
23. Westfall, P. H., 1987. "A Comparison of Variance Component Estimates for Arbitrary Underlying Distributions", Journal of the American Statistical Association, 82, 866-874.