# zbMATH — the first resource for mathematics

Flexible linear mixed models with improper priors for longitudinal and survival data. (English) Zbl 06864470
Summary: We propose a Bayesian approach using improper priors for hierarchical linear mixed models with flexible random effects and residual error distributions. The error distribution is modelled using scale mixtures of normals, which can capture tails heavier than those of the normal distribution. This generalisation is useful to produce models that are robust to the presence of outliers. The case of asymmetric residual errors is also studied. We present general results for the propriety of the posterior that also cover cases with censored observations, allowing for the use of these models in the contexts of popular longitudinal and survival analyses. We consider the use of copulas with flexible marginals for modelling the dependence between the random effects, but our results cover the use of any random effects distribution. Thus, our paper provides a formal justification for Bayesian inference in a very wide class of models (covering virtually all of the literature) under attractive prior structures that limit the amount of required user elicitation.

##### MSC:
 62F15 Bayesian inference 62J05 Linear regression; mixed models 62N01 Censored data models
##### Software:
spBayes; bayesSurv; lmec
Full Text:
##### References:
 [1] D. Bandyopadhyay, V. H. Lachos, L. M. Castro, and D. K. Dey. Skew-normal/independent linear mixed models for censored responses with applications to HIV viral loads., Biometrical Journal, 54(3):405-425, 2012. · Zbl 1244.62076 [2] Z. W. Birnbaum and S. C. Saunders. A new family of life distributions., Journal of Applied Probability, 6:319-327, 1969. · Zbl 0209.49801 [3] T. S. Breusch, J. C. Robertson, and A. H. Welsh. The emperor’s new clothes: a critique of the multivariate $$t$$ regression model., Statistica Neerlandica, 51(3):269-286, 1997. · Zbl 0929.62062 [4] D. Dunson. Nonparametric Bayes applications to biostatistics. In N. L. Hjort, C. Holmes, P. Müller, and S. G. Walker, editors, Bayesian Nonparametrics, pages 223-273. Cambridge University Press, Cambridge, UK, 2010. [5] C. Fernández and M. F. J. Steel. On Bayesian modeling of fat tails and skewness., Journal of the American Statistical Association, 93(441):359-371, 1998. · Zbl 0910.62024 [6] C. Fernández and M. F. J. Steel. Bayesian regression analysis with scale mixtures of normals., Econometric Theory, 16(1):80-101, 2000. · Zbl 0945.62031 [7] C. Fernández, J. Osiewalski, and M. F. J. Steel. On the use of panel data in stochastic frontier models with improper priors., Journal of Econometrics, 79(1):169-193, 1997. · Zbl 0878.68060 [8] A. O. Finley, S. Banerjee, and B. P. Carlin. spBayes: an R package for univariate and multivariate hierarchical point-referenced spatial models., Journal of Statistical Software, 19(4):1, 2007. [9] O. Güler., Foundations of Optimization. Springer Science & Business Media, New York, USA, 2010. [10] J. P. Hobert and G. Casella. The effect of improper priors on Gibbs sampling in hierarchical linear mixed models., Journal of the American Statistical Association, 91(436) :1461-1473, 1996. · Zbl 0882.62020 [11] A. Jara, F. Quintana, and E. San-Martín. Linear mixed models with skew-elliptical distributions: A Bayesian approach., Computational Statistics & Data Analysis, 52(11) :5033-5045, 2008. · Zbl 1452.62223 [12] A. Jara, T. E. Hanson, and E. Lesaffre. Robustifying generalized linear mixed models using a new class of mixtures of multivariate Polya trees., Journal of Computational and Graphical Statistics, 18(4):838-860, 2009. [13] A. Komárek and E. Lesaffre. Bayesian accelerated failure time model for correlated interval-censored data with a normal mixture as error distribution., Statistica Sinica, (17):549-569, 2007. · Zbl 1144.62022 [14] A. Komárek and E. Lesaffre. Bayesian accelerated failure time model with multivariate doubly interval – censored data and flexible distributional assumptions., Journal of the American Statistical Association, 103(482):523-533, 2008. · Zbl 05564507 [15] V. H. Lachos, P. Ghosh, and R. B. Arellano-Valle. Likelihood based inference for skew-normal independent linear mixed models., Statistica Sinica, 20:303-322, 2010. · Zbl 1186.62071 [16] K. J. Lee and S. G. Thompson. Flexible parametric models for random-effects distributions., Statistics in Medicine, 27(3):418-434, 2008. [17] Y. Maruyama and W. E. Strawderman. Robust Bayesian variable selection in linear models with spherically symmetric errors., Biometrika, 101(4):992-998, 2014. · Zbl 1306.62077 [18] G. S. Mudholkar and A. D. Hutson. The epsilon – skew – normal distribution for analyzing near-normal data., Journal of Statistical Planning and Inference, 83(2):291-309, 2000. · Zbl 0943.62012 [19] J. Osiewalski and M. F. J. Steel. Robust Bayesian inference in elliptical regression models., Journal of Econometrics, 57:345-363, 1993. · Zbl 0776.62029 [20] N. G. Polson and J. G. Scott. On the half-Cauchy prior for a global scale parameter., Bayesian Analysis, 7(4):887-902, 2012. · Zbl 1330.62148 [21] G. O. Roberts and J. S. Rosenthal. Examples of adaptive MCMC., Journal of Computational and Graphical Statistics, 18(2):349-367, 2009. [22] G. J. M. Rosa, C. R. Padovani, and D. Gianola. Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation., Biometrical Journal, 45(5):573-590, 2003. [23] F. J. Rubio. On the propriety of the posterior of hierarchical linear mixed models with flexible random effects distributions., Statistics & Probability Letters, 96:154-161, 2015. · Zbl 1396.62045 [24] F. J. Rubio and M. G. Genton. Bayesian linear regression with skew-symmetric error distributions with applications to survival analysis., Statistics in Medicine, 35(4) :2441-2454, 2016. [25] F. J. Rubio and M. F. J Steel. Inference in two-piece location-scale models with Jeffreys priors., Bayesian Analysis, 9(1):1-22, 2014. · Zbl 1327.62157 [26] F. J. Rubio and M. F. J. Steel. Bayesian modelling of skewness and kurtosis with two-piece scale and shape distributions., Electronic Journal of Statistics, 9(2) :1884-1912, 2015. · Zbl 1331.62090 [27] F. J. Rubio and K. Yu. Flexible objective Bayesian linear regression with applications in survival analysis., Journal of Applied Statistics, 44(5):798-810, 2017. [28] F. J. Rubio, E. O. Ogundimu, and J. L. Hutton. On modelling asymmetric data using two-piece sinh – arcsinh distributions., Brazilian Journal of Probability and Statistics, 30(3):485-501, 2016. · Zbl 1381.60042 [29] D. Sun, R. K. Tsutakawa, and Z. He. Propriety of posteriors with improper priors in hierarchical linear mixed models., Statistica Sinica, 11(1):77-95, 2001. · Zbl 1057.62525 [30] G. C. Tiao and W. Y. Tan. Bayesian analysis of random-effect models in the analysis of variance: I. posterior distribution of variance-components., Biometrika, 52(1/2):37-53, 1965. · Zbl 0144.42204 [31] F. Vaida and L. Liu. Fast implementation for normal mixed effects models with censored response., Journal of Computational and Graphical Statistics, 18(4):797-817, 2009. [32] C. A. Vallejos and M. F. J. Steel. Objective Bayesian survival analysis using shape mixtures of log-normal distributions., Journal of the American Statistical Association, 110(510):697-710, 2015. · Zbl 1373.62506 [33] I. Verdinelli and L. Wasserman. Computing Bayes factors using a generalization of the Savage-Dickey density ratio., Journal of the American Statistical Association, 90(430):614-618, 1995. · Zbl 0826.62022 [34] M. West. Outlier models and prior distributions in Bayesian linear regression., Journal of the Royal Statistical Society. Series B, 46:431-439, 1984. · Zbl 0567.62022 [35] D. Zhang and M. Davidian. 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. 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.