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U-tests for variance components in one-way random effects models
We consider a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null. This test is based on a decomposition of a U-statistic. Its asymptotic null distribution is derived under the mild regularity condition that the second moment of the random effects and the fourth moment of the within-treatment errors are finite. Under the additional assumption that the fourth moment of the random effect is finite, we also derive the distribution of the proposed U-test statistic under a sequence of local alternative hypotheses. We report the results of a simulation study conducted to compare the performance of the U-test with that of the usual F-test. The main conclusions of the simulation study are that (i) under normality or under moderate degrees of imbalance in the design, the F-test behaves well when compared to the U-test, and (ii) when the distribution of the random effects and within-treatment errors are nonnormal, the U-test is preferable even when the number of treatments is small.
First available in Project Euclid: 1 April 2008
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Nobre, Juvêncio S.; Singer, Julio M.; Silvapulle, Mervyn J. U -tests for variance components in one-way random effects models. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 197--210, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000149. https://projecteuclid.org/euclid.imsc/1207058274
-  Akritas, M. and Arnold, S. (2000). Asymptotics for analysis of variance when the number of levels is large. J. Amer. Statist. Assoc. 95 212–226.
-  Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 367–389.
-  Boos, D. D. and Brownie, C. (1995). ANOVA and rank tests when the number of treatments is large. Statist. Probab. Lett. 23 183–191.
-  Brownie, D. D. and Boos, D. D. (1994). Type I error robustness of ANOVA and ANOVA on ranks when the number of treatments is large. Biometrics 50 542–549.
-  Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and their Application. Cambridge Univ. Press.
-  Demidenko, E. (2004). Mixed Models: Theory and Applications. Wiley, Hoboken, NJ.
-  Donner, A. and Koval, J. J. (1989). The effect of imbalance on significance-testing in one-way model II analysis of variance. Comm. Statist. Theory Methods 18 1239–1250.
-  Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) II. Probability Theory 513–535. Univ. California Press, Berkeley.
-  Giampaoli, V. and Singer, J. M. (2007). Generalized likelihood ratio tests for variance components in linear mixed models. Submitted.
-  Hall, D. B. and Præstgaard, J. T. (2001). Order-restricted score tests for homogeneity in generalised linear and nonlinear mixed models. Biometrika 88 739–751.
-  Halmos, P. R. (1946). The theory of unbiased estimation. Ann. Math. Statist. 17 34–43.
-  Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325.
-  Khuri, A. I., Mathew, T. and Sinha, B. K. (1998). Statistical Tests for Mixed Linear Models. Wiley, New York.
-  Lee, A. J. (1990). U-Statistics: Theory and Practice. Dekker, New York.
-  Lee, J. (2003). The effect of design imbalance on the power of the F-test of a variance component in the one-way random model. Biometrical J. 45 238–248.
-  Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
-  Lin, X. (1997). Variance component testing in generalised linear models with random effects. Biometrika 84 309–326.
-  Nobre, J. S. (2007). Tests for variance components using U-statistics. Ph.D. thesis (in Portuguese), Dept. de Estatistica, Univ. de São Paulo, Brazil.
-  Pinheiro, A. S., Sen, P. K. and Pinheiro, H. P. (2007). Decomposability of high-dimensional diversity measures: Quasi U-statistics, martingales and nonstandard asymptotics. Submitted.
-  Savalli, C., Paula, G. A. and Cysneiros, F. J. A. (2006). Assessment of variance components in elliptical linear mixed models. Stat. Model. 6 59–76.
-  Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York.
-  Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc. 82 605–610.
-  Sen, P. K. (1960). On some convergence properties of U-statistics. Calcutta Statist. Assoc. Bull. 10 1–18.
-  Sen, P. K. (1963). On the properties of U-statistics when the observations are not independent. I. Estimation of non-serial parameters in some stationary stochastic process. Calcutta Statist. Assoc. Bull. 12 69–92.
-  Sen, P. K. (1965). On some permutation tests based on U-statistics. Calcutta Statist. Assoc. Bull. 14 106–126.
-  Sen, P. K. (1967). On some multisample permutation tests based on a class of U-statistics. J. Amer. Statist. Assoc. 62 1201–1213.
-  Sen, P. K. (1969). On a robustness property of a class of nonparametric tests based on U-statistics. Calcutta Statist. Assoc. Bull. 18 51–60.
-  Sen, P. K. (1974a). Almost sure behaviour of U-statistics and von Mises’ differentiable statistical functions. Ann. Statist. 2 387–395.
-  Sen, P. K. (1974b). Weak convergence of generalized U-statistics. Ann. Probab. 2 90–102.
-  Sen, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. Wiley, New York.
-  Sen, P. K. (1984). Invariance principles for U-statistics and von Mises’ functionals in the non-I.D. case. Sankhyā Ser. A 46 416–425.
-  Sen, P. K. and Ghosh, M. (1981). Sequential point estimation of estimable parameters based on U-statistics. Sankhyā Ser. A 43 331–344.
-  Sen, P. K. and Singer, J. M. (1993). Large Sample Methods in Statistics: An Introduction with Applications. Chapman and Hall, New York.
-  Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
-  Silvapulle, M. J. and Silvapulle, P. (1995). A score test against one-sided alternatives. J. Amer. Statist. Assoc. 90 342–349.
-  Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference: Inequality, Order, and Shape Restrictions. Wiley, Hoboken, NJ.
-  Stram, D. O. and Lee, J. W. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics 50 1171–1177.
-  Verbeke, G. and Molenberghs, G. (2003). The use of score tests for inference on variance components. Biometrics 59 254–262.
-  Zhu, Z. and Fung, W. K. (2004). Variance component testing in semiparametric mixed models. J. Multivariate Anal. 91 107–118.