Evaluation of Third-Order Method for the Tests of Variance Component in Linear Mixed Models
Affiliation(s)
1 Department of Public Health Sciences, University of Hawaii at Manoa, Honolulu, HI, USA. 2 Department of Mathematics and Statistics, York University, Toronto, ON, Canada. 3 Lunenfeld-Taunenbaum Research Institute, Mount Sinai Hospital, Toronto, ON, Canada.
1 Department of Public Health Sciences, University of Hawaii at Manoa, Honolulu, HI, USA. 2 Department of Mathematics and Statistics, York University, Toronto, ON, Canada. 3 Lunenfeld-Taunenbaum Research Institute, Mount Sinai Hospital, Toronto, ON, Canada.
ABSTRACT
Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boundary conditions. It is well documented in literature that the traditional first-order methods: likelihood ratio statistic, Wald statistic and score statistic, provide an excessively conservative approximation to the null distribution. However, the magnitude of the conservativeness has not been thoroughly explored. In this paper, we propose a likelihood-based third-order method to the mixed models for testing the null hypothesis of zero and non-zero variance component. The proposed method dramatically improved the accuracy of the tests. Extensive simulations were carried out to demonstrate the accuracy of the proposed method in comparison with the standard first-order methods. The results show the conservativeness of the first order methods and the accuracy of the proposed method in approximating the p-values and confidence intervals even when the sample size is small.
Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boundary conditions. It is well documented in literature that the traditional first-order methods: likelihood ratio statistic, Wald statistic and score statistic, provide an excessively conservative approximation to the null distribution. However, the magnitude of the conservativeness has not been thoroughly explored. In this paper, we propose a likelihood-based third-order method to the mixed models for testing the null hypothesis of zero and non-zero variance component. The proposed method dramatically improved the accuracy of the tests. Extensive simulations were carried out to demonstrate the accuracy of the proposed method in comparison with the standard first-order methods. The results show the conservativeness of the first order methods and the accuracy of the proposed method in approximating the p-values and confidence intervals even when the sample size is small.
KEYWORDS
Family Data, Genetic Variant, Likelihood Ratio Test, Random Effects, Third-Order Method, Variance Component
Family Data, Genetic Variant, Likelihood Ratio Test, Random Effects, Third-Order Method, Variance Component
Cite this paper
Wu, Y. , Wong, A. , Monette, G. and Briollais, L. (2015) Evaluation of Third-Order Method for the Tests of Variance Component in Linear Mixed Models. Open Journal of Statistics, 5, 233-244. doi: 10.4236/ojs.2015.54025.
Wu, Y. , Wong, A. , Monette, G. and Briollais, L. (2015) Evaluation of Third-Order Method for the Tests of Variance Component in Linear Mixed Models. Open Journal of Statistics, 5, 233-244. doi: 10.4236/ojs.2015.54025.
References
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http://dx.doi.org/10.2307/2533100 [2] Liang, K.Y. (1987) A Locally Most Powerful Test for Homogeneity with Many Strata. Biometrics, 51, 259-264.
http://dx.doi.org/10.1093/biomet/74.2.259 [3] Chernoff, H. (1954) On the Distribution of the Likelihood Ratio. Annals of Mathematical Statistics, 25, 573-578.
http://dx.doi.org/10.1214/aoms/1177728725 [4] Self, S.F. and Liang, K.Y. (1987) Asymptotic Properties of Maximum Likelihood Estimator and Likelihood Ratio Tests under Nonstandard Conditions. Journal of the American Statistical Association, 82, 605-610.
http://dx.doi.org/10.1080/01621459.1987.10478472 [5] Stram, D.O. and Lee, J.W. (1994) Variance Components Testing in the Longitudinal Mixed Effects Model. Biometrics, 50, 71171-1177.
http://dx.doi.org/10.2307/2533455 [6] Patterson, H.D. and Thompson, R. (1971) Recovery of Interblock Information When Block Sizes Are Unequal. Biometrika, 58, 545-554.
http://dx.doi.org/10.1093/biomet/58.3.545 [7] Crainiceanu, D.M. and Ruppert, D. (2004) Likelihood Ratio Tests in Linear Mixed Models with One Variance Component. Journal of the Royal Statistical Society: Series B, 66, 165-185.
http://dx.doi.org/10.1111/j.1467-9868.2004.00438.x [8] Fraser, D.A.S., Reid, N. and Wu. J. (1999) A Simple General Formula for Tail Probabilities for Frequentist and Bayesian Inference. Biometrika, 86, 249-264.
http://dx.doi.org/10.1093/biomet/86.2.249 [9] Barndorff-Nielsen, O.E. (1986) Inference on Full and Partial Parameters, Based on the Standardized Signed Log-Like- lihood Ratio. Biometrika, 73, 307-322.
http://dx.doi.org/10.2307/2336207 [10] Barndorff-Nielsen, O.E. (1991) Modified Signed Log-Likelihood Ratio Statistic. Biometrika, 78, 557-563.
http://dx.doi.org/10.1093/biomet/78.3.557 [11] Fraser, D.A.S. and Reid. M. (1995) Ancillaries and Third-Order Significance. Utilitas Mathematica, 47, 33-53. [12] Verbeke, G. and Molenberghs, G. (2003) The Use of Score Tests for Inference on Variance Components. Biometrics, 59, 254-262.
http://dx.doi.org/10.1111/1541-0420.00032 [13] Almasy, L., Dyer, T.D., Peralta, J.M., Jun, G., Fuchsberger, C., Almeida, M.A., Kent, J.W., Fowler, S., Duggirala, R. and Blangero. J. (2014) Data for Genetic Analysis Workshop 18: Human Whole Genome Sequence, Blood Pressure, and Simulated Phenotypes in Extended Pedigrees. BMC Proceedings, 8, S2. [14] Wu, Y.Y. and Briollais, L. (2014) Mixed-Effects Models for Joint Modeling of Sequence Data in Longitudinal Studies. BMC Proceedings, 8, S92.
[1] Britton. T. (1997) Tests to Detect Clustering of Infected Individuals within Families. Biometrics, 53, 98-109.
http://dx.doi.org/10.2307/2533100 [2] Liang, K.Y. (1987) A Locally Most Powerful Test for Homogeneity with Many Strata. Biometrics, 51, 259-264.
http://dx.doi.org/10.1093/biomet/74.2.259 [3] Chernoff, H. (1954) On the Distribution of the Likelihood Ratio. Annals of Mathematical Statistics, 25, 573-578.
http://dx.doi.org/10.1214/aoms/1177728725 [4] Self, S.F. and Liang, K.Y. (1987) Asymptotic Properties of Maximum Likelihood Estimator and Likelihood Ratio Tests under Nonstandard Conditions. Journal of the American Statistical Association, 82, 605-610.
http://dx.doi.org/10.1080/01621459.1987.10478472 [5] Stram, D.O. and Lee, J.W. (1994) Variance Components Testing in the Longitudinal Mixed Effects Model. Biometrics, 50, 71171-1177.
http://dx.doi.org/10.2307/2533455 [6] Patterson, H.D. and Thompson, R. (1971) Recovery of Interblock Information When Block Sizes Are Unequal. Biometrika, 58, 545-554.
http://dx.doi.org/10.1093/biomet/58.3.545 [7] Crainiceanu, D.M. and Ruppert, D. (2004) Likelihood Ratio Tests in Linear Mixed Models with One Variance Component. Journal of the Royal Statistical Society: Series B, 66, 165-185.
http://dx.doi.org/10.1111/j.1467-9868.2004.00438.x [8] Fraser, D.A.S., Reid, N. and Wu. J. (1999) A Simple General Formula for Tail Probabilities for Frequentist and Bayesian Inference. Biometrika, 86, 249-264.
http://dx.doi.org/10.1093/biomet/86.2.249 [9] Barndorff-Nielsen, O.E. (1986) Inference on Full and Partial Parameters, Based on the Standardized Signed Log-Like- lihood Ratio. Biometrika, 73, 307-322.
http://dx.doi.org/10.2307/2336207 [10] Barndorff-Nielsen, O.E. (1991) Modified Signed Log-Likelihood Ratio Statistic. Biometrika, 78, 557-563.
http://dx.doi.org/10.1093/biomet/78.3.557 [11] Fraser, D.A.S. and Reid. M. (1995) Ancillaries and Third-Order Significance. Utilitas Mathematica, 47, 33-53. [12] Verbeke, G. and Molenberghs, G. (2003) The Use of Score Tests for Inference on Variance Components. Biometrics, 59, 254-262.
http://dx.doi.org/10.1111/1541-0420.00032 [13] Almasy, L., Dyer, T.D., Peralta, J.M., Jun, G., Fuchsberger, C., Almeida, M.A., Kent, J.W., Fowler, S., Duggirala, R. and Blangero. J. (2014) Data for Genetic Analysis Workshop 18: Human Whole Genome Sequence, Blood Pressure, and Simulated Phenotypes in Extended Pedigrees. BMC Proceedings, 8, S2. [14] Wu, Y.Y. and Briollais, L. (2014) Mixed-Effects Models for Joint Modeling of Sequence Data in Longitudinal Studies. BMC Proceedings, 8, S92.