The Annals of Statistics

Generalized score test of homogeneity for mixed effects models

Hongtu Zhu and Heping Zhang

Full-text: Open access

Abstract

Many important problems in psychology and biomedical studies require testing for overdispersion, correlation and heterogeneity in mixed effects and latent variable models, and score tests are particularly useful for this purpose. But the existing testing procedures depend on restrictive assumptions. In this paper we propose a class of test statistics based on a general mixed effects model to test the homogeneity hypothesis that all of the variance components are zero. Under some mild conditions, not only do we derive asymptotic distributions of the test statistics, but also propose a resampling procedure for approximating their asymptotic distributions conditional on the observed data. To overcome the technical challenge, we establish an invariance principle for random quadratic forms indexed by a parameter. A simulation study is conducted to investigate the empirical performance of the test statistics. A real data set is analyzed to illustrate the application of our theoretical results.

Article information

Source
Ann. Statist., Volume 34, Number 3 (2006), 1545-1569.

Dates
First available in Project Euclid: 10 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1152540758

Digital Object Identifier
doi:10.1214/009053606000000380

Mathematical Reviews number (MathSciNet)
MR2278367

Zentralblatt MATH identifier
1113.62018

Subjects
Primary: 62F05: Asymptotic properties of tests
Secondary: 62F40: Bootstrap, jackknife and other resampling methods

Keywords
Functional central limit theorem latent variable random quadratic form score test variance component

Citation

Zhu, Hongtu; Zhang, Heping. Generalized score test of homogeneity for mixed effects models. Ann. Statist. 34 (2006), no. 3, 1545--1569. doi:10.1214/009053606000000380. https://projecteuclid.org/euclid.aos/1152540758


Export citation

References

  • Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69 683--734.
  • Bentler, P. M. and Dudgeon, P. (1996). Covariance structure analysis: Statistical practice, theory and directions. Ann. Review Psychology 47 563--592.
  • Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. J. Amer. Statist. Assoc. 88 9--25.
  • Chen, H. and Chen, J. (2003). Tests for homogeneity in normal mixtures in the presence of a structural parameter. Statist. Sinica 13 351--365.
  • Chen, H., Chen, J. and Kalbfleisch, J. D. (2001). A modified likelihood ratio test for homogeneity in finite mixture models. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 19--29.
  • Chen, H., Chen, J. and Kalbfleisch, J. D. (2004). Testing for a finite mixture model with two components. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 95--115.
  • Chen, Z. and Dunson, D. B. (2003). Random effect selection in linear mixed models. Biometrics 59 762--769.
  • Commenges, D. and Jacqmin-Gadda, H. (1997). Generalized score test of homogeneity based on correlated random effects models. J. Roy. Statist. Soc. Ser. B 59 157--171.
  • Cox, D. R. (1983). Some remarks on overdispersion. Biometrika 70 269--274.
  • Crainiceanu, C. M. and Ruppert, D. (2004). Likelihood ratio tests in linear mixed models with one variance component. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 165--185.
  • Daniels, M. J. and Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89 553--566.
  • de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261--277.
  • Delhing, H., Mikosch, T. and Sørensen, M., eds. (2002). Empirical Process Techniques for Dependent Data. Birkhäuser, Boston.
  • Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman's truncation. J. Amer. Statist. Assoc. 91 674--688.
  • Guttorp, P. and Lockhart, R. A. (1988). On the asymptotic distribution of quadratic forms in uniform order statistics. Ann. Statist. 16 433--449.
  • Hall, D. B. and Præ stgaard, J. T. (2001). Order-restricted score tests for homogeneity in generalised linear and nonlinear mixed models. Biometrika 88 739--751.
  • Jacqmin-Gadda, H. and Commenges, D. (1995). Tests of homogeneity for generalized linear models. J. Amer. Statist. Assoc. 90 1237--1246.
  • Jiang, J. (1996). REML estimation: Asymptotic behavior and related topics. Ann. Statist. 24 255--286.
  • Jørgensen, B. (1987). Exponential dispersion models (with discussion). J. Roy. Statist. Soc. Ser. B 49 127--162.
  • Kosorok, M. R. (2003). Bootstraps of sums of independent but not identically distributed stochastic processes. J. Multivariate Anal. 84 299--318.
  • Liang, K. (1987). A locally most powerful test for homogeneity with many strata. Biometrika 74 259--264.
  • Lin, X. (1997). Variance component testing in generalised linear models with random effects. Biometrika 84 309--326.
  • Merikangas, K. R., Stevens, D. E., Fenton, B., Stolar, M., O'Malley, S., Woods, S. and Risch, N. (1998). Co-morbidity and familial aggregation of alcoholism and anxiety disorders. Psychological Medicine 28 773--788.
  • Mikosch, T. (1991). Functional limit theorems for random quadratic forms. Stochastic Process. Appl. 37 81--98.
  • Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
  • Sen, P. K. and Silvapulle, M. J. (2002). An appraisal of some aspects of statistical inference under inequality constraints. J. Statist. Plann. Inference 107 3--43.
  • Silvapulle, M. J. and Silvapulle, P. (1995). A score test against one-sided alternatives. J. Amer. Statist. Assoc. 90 342--349.
  • Tsai, M. (1992). On the power superiority of likelihood ratio tests for restricted alternatives. J. Multivariate Anal. 42 102--109.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
  • Vaida, F. and Xu, R. (2000). Proportional hazards model with random effects. Statistics in Medicine 19 3309--3324.
  • Varberg, D. E. (1968). Almost sure convergence of quadratic forms in independent random variables. Ann. Math. Statist. 39 1502--1506.
  • Verbeke, G. and Molenberghs, G. (2003). The use of score tests for inference on variance components. Biometrics 59 254--262.
  • Wei, B. (1998). Exponential Family Nonlinear Models. Springer, Berlin.
  • Zhang, H., Feng, R. and Zhu, H. (2003). A latent variable model of segregation analysis for ordinal traits. J. Amer. Statist. Assoc. 98 1023--1034.
  • Zhang, H. and Merikangas, K. (2000). A frailty model of segregation analysis: Understanding the familial transmission of alcoholism. Biometrics 56 815--823.
  • Zheng, G. and Chen, Z. (2005). Comparison of maximum statistics for hypothesis testing when a nuisance parameter is present only under the alternative. Biometrics 61 254--258.
  • Zhu, H. and Zhang, H. (2004). Hypothesis testing in mixture regression models. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 3--16.
  • Zhu, H. and Zhang, H. (2005). Generalized score test of homogeneity for mixed effects models: Supplement. Technical report, Yale Univ. School of Medicine. Available at peace.med.yale.edu.