Generalized score test of homogeneity for mixed effects models
Hongtu Zhu and Heping Zhang
Source: Ann. Statist. Volume 34, Number 3 (2006), 1545-1569.
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.
Primary Subjects: 62F05
Secondary Subjects: 62F40
Keywords: Functional central limit theorem; latent variable; random quadratic form; score test; variance component
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1152540758
Digital Object Identifier: doi:10.1214/009053606000000380
Mathematical Reviews number (MathSciNet):
MR2278367
Zentralblatt MATH identifier:
1113.62018
References
Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69 683--734.
Mathematical Reviews (MathSciNet):
MR1828540
Digital Object Identifier: doi:10.1111/1468-0262.00210
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR1977730
Zentralblatt MATH:
1015.62015
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.
Mathematical Reviews (MathSciNet):
MR1811988
Digital Object Identifier: doi:10.1111/1467-9868.00273
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR2035761
Digital Object Identifier: doi:10.1111/j.1467-9868.2004.00434.x
Chen, Z. and Dunson, D. B. (2003). Random effect selection in linear mixed models. Biometrics 59 762--769.
Mathematical Reviews (MathSciNet):
MR2025100
Digital Object Identifier: doi:10.1111/j.0006-341X.2003.00089.x
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.
Mathematical Reviews (MathSciNet):
MR1436561
Digital Object Identifier: doi:10.1111/1467-9868.00061
JSTOR: links.jstor.org
Cox, D. R. (1983). Some remarks on overdispersion. Biometrika 70 269--274.
Mathematical Reviews (MathSciNet):
MR0742997
Zentralblatt MATH:
0511.62007
Digital Object Identifier: doi:10.1093/biomet/70.1.269
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR2035765
Digital Object Identifier: doi:10.1111/j.1467-9868.2004.00438.x
Daniels, M. J. and Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89 553--566.
Mathematical Reviews (MathSciNet):
MR1929162
Zentralblatt MATH:
1036.62019
Digital Object Identifier: doi:10.1093/biomet/89.3.553
JSTOR: links.jstor.org
de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261--277.
Mathematical Reviews (MathSciNet):
MR0885466
Digital Object Identifier: doi:10.1007/BF00354037
Delhing, H., Mikosch, T. and Sørensen, M., eds. (2002). Empirical Process Techniques for Dependent Data. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1958776
Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman's truncation. J. Amer. Statist. Assoc. 91 674--688.
Mathematical Reviews (MathSciNet):
MR1395735
Digital Object Identifier: doi:10.2307/2291663
JSTOR: links.jstor.org
Guttorp, P. and Lockhart, R. A. (1988). On the asymptotic distribution of quadratic forms in uniform order statistics. Ann. Statist. 16 433--449.
Mathematical Reviews (MathSciNet):
MR0924879
Digital Object Identifier: doi:10.1214/aos/1176350713
Project Euclid: euclid.aos/1176350713
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.
Mathematical Reviews (MathSciNet):
MR1859406
Digital Object Identifier: doi:10.1093/biomet/88.3.739
JSTOR: links.jstor.org
Jacqmin-Gadda, H. and Commenges, D. (1995). Tests of homogeneity for generalized linear models. J. Amer. Statist. Assoc. 90 1237--1246.
Mathematical Reviews (MathSciNet):
MR1379466
Digital Object Identifier: doi:10.2307/2291514
JSTOR: links.jstor.org
Jiang, J. (1996). REML estimation: Asymptotic behavior and related topics. Ann. Statist. 24 255--286.
Mathematical Reviews (MathSciNet):
MR1389890
Digital Object Identifier: doi:10.1214/aos/1033066209
Project Euclid: euclid.aos/1033066209
Jørgensen, B. (1987). Exponential dispersion models (with discussion). J. Roy. Statist. Soc. Ser. B 49 127--162.
Mathematical Reviews (MathSciNet):
MR0905186
JSTOR: links.jstor.org
Kosorok, M. R. (2003). Bootstraps of sums of independent but not identically distributed stochastic processes. J. Multivariate Anal. 84 299--318.
Mathematical Reviews (MathSciNet):
MR1965224
Digital Object Identifier: doi:10.1016/S0047-259X(02)00040-4
Liang, K. (1987). A locally most powerful test for homogeneity with many strata. Biometrika 74 259--264.
Mathematical Reviews (MathSciNet):
MR0903126
Zentralblatt MATH:
0621.62025
Digital Object Identifier: doi:10.1093/biomet/74.2.259
JSTOR: links.jstor.org
Lin, X. (1997). Variance component testing in generalised linear models with random effects. Biometrika 84 309--326.
Mathematical Reviews (MathSciNet):
MR1467049
Zentralblatt MATH:
0881.62074
Digital Object Identifier: doi:10.1093/biomet/84.2.309
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet):
MR1091696
Digital Object Identifier: doi:10.1016/0304-4149(91)90062-H
Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
Mathematical Reviews (MathSciNet):
MR1089429
Zentralblatt MATH:
0741.60001
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.
Mathematical Reviews (MathSciNet):
MR1927753
Digital Object Identifier: doi:10.1016/S0378-3758(02)00242-2
Silvapulle, M. J. and Silvapulle, P. (1995). A score test against one-sided alternatives. J. Amer. Statist. Assoc. 90 342--349.
Mathematical Reviews (MathSciNet):
MR1325141
Digital Object Identifier: doi:10.2307/2291159
JSTOR: links.jstor.org
Tsai, M. (1992). On the power superiority of likelihood ratio tests for restricted alternatives. J. Multivariate Anal. 42 102--109.
Mathematical Reviews (MathSciNet):
MR1177520
Digital Object Identifier: doi:10.1016/0047-259X(92)90081-P
van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1385671
Zentralblatt MATH:
0862.60002
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.
Mathematical Reviews (MathSciNet):
MR0230359
Project Euclid: euclid.aoms/1177698131
Verbeke, G. and Molenberghs, G. (2003). The use of score tests for inference on variance components. Biometrics 59 254--262.
Mathematical Reviews (MathSciNet):
MR1987392
Digital Object Identifier: doi:10.1111/1541-0420.00032
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.
Mathematical Reviews (MathSciNet):
MR2041490
Digital Object Identifier: doi:10.1198/016214503000000981
Zhang, H. and Merikangas, K. (2000). A frailty model of segregation analysis: Understanding the familial transmission of alcoholism. Biometrics 56 815--823.
Zentralblatt MATH:
0981.68692
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.
Mathematical Reviews (MathSciNet):
MR2135868
Digital Object Identifier: doi:10.1111/j.0006-341X.2005.030531.x
Zhu, H. and Zhang, H. (2004). Hypothesis testing in mixture regression models. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 3--16.
Mathematical Reviews (MathSciNet):
MR2035755
Digital Object Identifier: doi:10.1046/j.1369-7412.2003.05379.x
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.
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