A general skew-\(t\) mixed model that allows different degrees of freedom for random effects and error distributions. (English) Zbl 1278.62077
Summary: This paper develops a robust mixed model that assumes a multivariate skew-\(t\) distribution for random effects and an independent multivariate \(t\)-distribution for the errors. It simultaneously captures skewness and heavy tailedness in data, while allowing the random effects and error distributions to have different degrees of freedom. It is fit using an EM-type algorithm. Simulations show that its efficiency for estimating mean response is comparable to that of the recent skew-\(t\) mixed model. But it may be considerably more efficient than the latter for estimating variance-covariance parameters when at least one of the random effects distributions or the error distributions have heavy tails, possibly due to outliers. The proposed model is used to analyze a data set consisting of lengths of claws of fiddler crabs (Uca mjoebergi).
MSC:
| 62H12 | Estimation in multivariate analysis |
| 62H10 | Multivariate distribution of statistics |
| 65C60 | Computational problems in statistics (MSC2010) |
Keywords:
EM algorithm; heavy tailed distributions; method comparisons; outliers; robust mixed-effects models; skew-\(t\) distributionsReferences:
| [1] | Arellano-Valle, R. B.; Bolfarine, H.; Lachos, V. H., Skew-normal linear mixed models, J. Data Sci., 3, 415-438 (2005) |
| [2] | Azzalini, A., A class of distributions which includes the normal ones, Scand. J. Statist., 12, 171-178 (1985) · Zbl 0581.62014 |
| [3] | Azzalini, A.; Capitanio, A., Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution, J. Roy. Statist. Soc. Ser. B, 65, 367-389 (2003) · Zbl 1065.62094 |
| [4] | Box, G. E.P., Sampling and Bayes’ inference in scientific modeling and robustness, J. Roy. Statist. Soc. Ser. A, 143, 383-430 (1980) · Zbl 0471.62036 |
| [5] | Branco, M. D.; Dey, D. K., A general class of multivariate skew-elliptical distributions, J. Multivariate Anal., 79, 99-113 (2001) · Zbl 0992.62047 |
| [6] | Butler, S. M.; Louis, T. A., Random effects models with nonparametric priors, Statist. Med., 11, 1981-2000 (1992) |
| [7] | Choudhary, P. K.; Yin, K., Bayesian and frequentist methodologies for analyzing method comparison studies with multiple methods, Statist. Biopharmaceutical Res., 2, 122-132 (2010) |
| [8] | Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B, 39, 1-38 (1977) · Zbl 0364.62022 |
| [10] | Heritier, S.; Cantoni, E.; Copt, S.; Victoria-Feser, M.-P., Robust Methods in Biostatistics (2009), John Wiley: John Wiley New York · Zbl 1163.62085 |
| [11] | Ho, H. J.; Lin, T. I., Robust linear mixed models using the skew t distribution with application to schizophrenia data, Biom. J., 52, 449-469 (2010) · Zbl 1197.62055 |
| [12] | Jiang, J., Conditional inference about generalized linear mixed models, Ann. Statist., 27, 1974-2007 (1999) · Zbl 0961.62062 |
| [13] | Jiang, J., Linear and Generalized Linear Mixed Models and Their Applications (2007), Springer: Springer New York · Zbl 1152.62040 |
| [14] | Jiang, J., The subset argument and consistency of MLE in GLMManswer to an open problem and beyond, Ann. Statist., 41, 177-195 (2013) · Zbl 1347.62042 |
| [15] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions, 2nd ed., vol. 1 (1994), John Wiley: John Wiley New York · Zbl 0811.62001 |
| [16] | Lachos, V. H.; Ghosh, P.; Arellano-Valle, R. B., Likelihood based inference for skew-normal independent linear mixed models, Statist. Sinica, 20, 303-322 (2010) · Zbl 1186.62071 |
| [17] | Lailvaux, S. P.; Reaney, L. T.; Backwell, P. R.Y., Dishonest signalling of fighting ability and multiple performance traits in the fiddler crab Uca mjoebergi, Funct. Ecology, 23, 359-366 (2009) |
| [18] | Lange, K. L.; Little, R. J.; Taylor, J. M.G., Robust statistical modeling using the \(t\) distribution, J. Amer. Statist. Assoc., 84, 881-896 (1989) |
| [19] | Lehmann, E. L., Elements of Large-Sample Theory (1998), Springer: Springer New York · Zbl 0916.62017 |
| [20] | Liang, K.-Y.; Zeger, S., Longitudinal data analysis using generalized linear models, Biometrika, 73, 13-22 (1986) · Zbl 0595.62110 |
| [21] | Liu, C.; Rubin, D. B., The ECME algorithma simple extension of EM and ECM with faster monotone convergence, Biometrika, 81, 4, 633-648 (1994) · Zbl 0812.62028 |
| [22] | McLachlan, G. J.; Krishnan, T., The EM Algorithm and Extensions (2007), John Wiley: John Wiley New York |
| [23] | Meng, X.-L.; Rubin, D. B., Maximum likelihood estimation via the ECM algorithma general framework, Biometrika, 80, 267-278 (1993) · Zbl 0778.62022 |
| [24] | Pinheiro, J. C.; Bates, D. M., Mixed-Effects Models in S and S-PLUS (2000), Springer: Springer New York · Zbl 0953.62065 |
| [26] | Pinheiro, J. C.; Liu, C.; Wu, Y. N., Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate \(t\) distribution, J. Comput. Graph. Statist., 10, 249-276 (2001) |
| [27] | R: A Language and Environment for Statistical Computing (2012), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria, 〈http://www.R-project.org�� |
| [28] | Richardson, A. M., Bounded influence estimation in the mixed linear model, J. Amer. Statist. Assoc., 92, 437, 154-161 (1997) · Zbl 1090.62545 |
| [29] | Seber, G. A.F.; Lee, A. J., Linear Regression Analysis (2003), John Wiley: John Wiley New York · Zbl 1029.62059 |
| [32] | Song, P. X.-K.; Zhang, P.; Qu, A., Maximum likelihood inference in robust linear mixed-effects models using multivariate t distribution, Statist. Sinica, 17, 929-943 (2007) · Zbl 1133.62013 |
| [33] | Stahel, W. A.; Welsh, A. H., Approaches to robust estimation in the simplest variance components model, J. Statist. Plann. Inference, 57, 295-319 (1997) · Zbl 1003.62528 |
| [34] | Verbeke, G.; Lesaffre, E., A linear mixed-effects model with heterogeneity in the random-effects population, J. Amer. Statist. Assoc., 91, 217-221 (1996) · Zbl 0870.62057 |
| [35] | Wang, P.; Tsai, G.-F.; Qu, A., Conditional inference functions for mixed-effects models with unspecified random-effects distribution, J. Amer. Statist. Assoc., 107, 725-736 (2012) · Zbl 1261.62068 |
| [36] | Zhang, D.; Davidian, M., Linear mixed models with flexible distributions of random effects for longitudinal data, Biometrics, 57, 3, 795-802 (2001) · Zbl 1209.62087 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
