# zbMATH — the first resource for mathematics

Improved $$U$$-tests for variance components in one-way random effects models. (English) Zbl 1445.62120
Summary: Based on a decomposition of a $$U$$-statistic, N. Balakrishnan (ed.) et al. [Beyond parametrics in interdisciplinary research: Festschrift in honor of Professor Pranab K. Sen. Beachwood, OH: IMS, Institute of Mathematical Statistics (2008; Zbl 1159.62002)] proposed a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null, specially useful when very mild assumptions are imposed on the underlying distributions. We consider a bootstrap version of that $$U$$-test and evaluate its performance via simulation studies in different scenarios. The bootstrap $$U$$-test has better statistical properties than the original test even in small samples. Furthermore, it is easy to implement and has a low computational cost. We consider two examples with unbalanced small sample datasets, for illustrative purposes.
##### MSC:
 62H15 Hypothesis testing in multivariate analysis 62G30 Order statistics; empirical distribution functions 62P10 Applications of statistics to biology and medical sciences; meta analysis
Full Text:
##### References:
 [1] Alkhamisi, M. (2000). Asymptotic analysis of the one-way random effects models. Ph.D. thesis, University of Toronto, Graduate Department of Statistics. Toronto. [2] Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew $$t$$-distribution. Journal of the Royal Statistical Society, Series B, Statistical Methodology 65, 367-389. · Zbl 1065.62094 [3] Chernick, M. R. and LaBudde, R. A. (2011). Bootstrap Methods with Application to R. New York: John Wiley & Sons. [4] Crainiceanu, C. M. (2008). Likelihood ratio testing for zero variance components in linear mixed models. In Random Effect and Latent Variable Model Selection (D. B. Dunson, ed.). Lecture Notes in Statistics 192, 3-18. New York: Springer. [5] Crainiceanu, C. M. and Ruppert, D. (2004). Likelihood ratio tests in linear mixed models with one variance component. Journal of the Royal Statistical Society, Series B, Statistical Methodology 66, 165-185. · Zbl 1061.62027 [6] Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Applications. Cambridge: Cambridge University Press. · Zbl 0886.62001 [7] Demidenko, E. (2013). Mixed Models: Theory and Applications with R, 2nd ed. New York: John Wiley & Sons. · Zbl 1276.62049 [8] Giampaoli, V. and Singer, J. M. (2009). Generalized likelihood ratio tests for variance components in linear mixed models. Journal of Statistical Planning and Inference 139, 1435-1448. · Zbl 1419.62033 [9] Greven, S., Crainiceanu, C. M., Küchenhoff, H. and Peters, A. (2008). Restricted likelihood ratio testing for zero variance components in linear mixed models. Journal of Computational and Graphical Statistics 17, 870-891. [10] Hall, D. and Praestgaard, J. T. (2001). Order-restricted score tests for homogeneity in generalised linear and nonlinear mixed models. Biometrika 88, 739-751. · Zbl 1009.62057 [11] Halmos, P. R. (1946). The theory of unbiased estimation. The Annals of Mathematical Statistics 17, 34-43. · Zbl 0063.01891 [12] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. The Annals of Mathematical Statistics 19, 293-325. · Zbl 0032.04101 [13] Khuri, A. I., Mathew, T. and Sinha, B. K. (1998). Statistical Tests for Mixed Linear Models. New York: John Wiley & Sons. · Zbl 0893.62009 [14] Kowalski, J., Pagano, M. and DeGruttola, V. (2002). A nonparametric test of gene region heterogeneity associated with phenotype. Journal of the American Statistical Association 97, 398-408. · Zbl 1073.62581 [15] Kowalski, J. and Tu, X. M. (2007). Modern Applied $$U$$-Statistics. New York: John Wiley & Sons. · Zbl 1167.62002 [16] Lahiri, S. N. (2003). Resampling Methods for Dependent Data. New York: Springer. · Zbl 1028.62002 [17] Lee, A. J. (1990). $$U$$-Statistics: Theory and Practice. New York: Marcel Dekker. · Zbl 0771.62001 [18] Lencina, V. B., Singer, J. M. and Stanek, E. J. III (2005). Much ado about nothing: The mixed models controversy revisited. International Statistical Review 73, 9-20. · Zbl 1104.62001 [19] Lin, X. (1997). Variance component testing in generalised linear models with random effects. Biometrika 84, 309-326. · Zbl 0881.62074 [20] McCulloch, C. E., Searle, S. R. and Neuhaus, J. M. (2008). Generalized, Linear, and Mixed Models, 2nd ed. New York: John Wiley & Sons. · Zbl 1165.62050 [21] Nobre, J. S. (2007). Test for variance components using $$U$$-statistics. Unpublished Ph.D. thesis, Departamento de Estatística, Universidade de São Paulo, Brazil (in Portuguese). [22] Nobre, J. S., Singer, J. M. and Sen, P. K. (2013). $$U$$ tests for variance components in linear mixed models. Test 22, 580-605. · Zbl 1283.62096 [23] Nobre, J. S., Singer, J. M. and Silvapulle, M. J. (2008). $$U$$-Tests for variance components in one-way random effects models. In Beyond Parametrics in Interdisciplinary Research, Festschrift to P.K. Sen (E. N. Balakrishnan, E. Pena and M. J. Silvapulle, eds.). IMS Lecture Notes-Monograph Series, 197-210. Hayward, CA: Institute of Mathematical Statistics. [24] Pinheiro, A., Sen, P. K. and Pinheiro, H. P. (2009). Decomposability of high-dimensional diversity measures: Quasi $$U$$-statistics, martingales and nonstandard asymptotics. Journal of Multivariate Analysis 100, 1645-1656. · Zbl 1190.62130 [25] Savalli, C., Paula, G. A. and Cysneiros, F. J. A. (2006). Assesment of variance components in elliptical linear mixed models. Statistical Modelling 6, 59-76. [26] Sen, P. K., Singer, J. M. and Pedroso de Lima, A. C. (2010). From Finite Sample to Asymptotic Methods in Statistics. New York: Cambridge University Press. · Zbl 1210.62001 [27] Self, S.G. and Liang, K.Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association 82, 605-610. · Zbl 0639.62020 [28] Serfling, R. J. (1980). Approximation theorems of mathematical statistics. New York: John Wiley & Sons. · Zbl 0538.62002 [29] Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference. New York: John Wiley & Sons. · Zbl 1077.62019 [30] Silvapulle, M. J. and Silvapulle, P. (1995). A score test against one-sided alternatives. Journal of the American Statistical Association 90, 342-349. · Zbl 0818.62022 [31] Sinha, S. K. (2009). Bootstrap tests for variance components in generalized linear mixed models. Canadian Journal of Statistics 37, 219-234. · Zbl 1176.62013 [32] Snedecor, G. W. and Cochran, W. G. (1980). Statistical Methods, 7th ed. Iowa: Iowa State College Press. · Zbl 0727.62003 [33] Stram, D. O. and Lee, J. W. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics 50, 1171-1177. · Zbl 0826.62054 [34] Verbeke, G. and Molenberghs, G. (2003). The use of score tests for inference on variance components. Biometrics 59, 254-262. · Zbl 1210.62013 [35] Zhang, D. and Lin, X. (2008). Variance components testing in generalized linear mixed models for longitudinal/clustered data and other related topics. In Random Effect and Latent Variable Model Selection (D. B. Dunson, ed.). Lecture Notes in Statistics 192, 19-36. New York: Springer. [36] Zhu, Z. · Zbl 1051.62021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.