## Electronic Journal of Statistics

### Flexible linear mixed models with improper priors for longitudinal and survival data

#### Abstract

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.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 572-598.

Dates
First available in Project Euclid: 27 February 2018

https://projecteuclid.org/euclid.ejs/1519700495

Digital Object Identifier
doi:10.1214/18-EJS1401

Mathematical Reviews number (MathSciNet)
MR3769189

Zentralblatt MATH identifier
06864470

#### Citation

Rubio, F. J.; Steel, M. F. J. Flexible linear mixed models with improper priors for longitudinal and survival data. Electron. J. Statist. 12 (2018), no. 1, 572--598. doi:10.1214/18-EJS1401. https://projecteuclid.org/euclid.ejs/1519700495

#### References

• 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.
• Z. W. Birnbaum and S. C. Saunders. A new family of life distributions., Journal of Applied Probability, 6:319–327, 1969.
• 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.
• 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.
• 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.
• C. Fernández and M. F. J. Steel. Bayesian regression analysis with scale mixtures of normals., Econometric Theory, 16(1):80–101, 2000.
• 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.
• 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.
• O. Güler., Foundations of Optimization. Springer Science & Business Media, New York, USA, 2010.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• K. J. Lee and S. G. Thompson. Flexible parametric models for random-effects distributions., Statistics in Medicine, 27(3):418–434, 2008.
• Y. Maruyama and W. E. Strawderman. Robust Bayesian variable selection in linear models with spherically symmetric errors., Biometrika, 101(4):992–998, 2014.
• 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.
• J. Osiewalski and M. F. J. Steel. Robust Bayesian inference in elliptical regression models., Journal of Econometrics, 57:345–363, 1993.
• N. G. Polson and J. G. Scott. On the half-Cauchy prior for a global scale parameter., Bayesian Analysis, 7(4):887–902, 2012.
• G. O. Roberts and J. S. Rosenthal. Examples of adaptive MCMC., Journal of Computational and Graphical Statistics, 18(2):349–367, 2009.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• M. West. Outlier models and prior distributions in Bayesian linear regression., Journal of the Royal Statistical Society. Series B, 46:431–439, 1984.
• D. Zhang and M. Davidian. Linear mixed models with flexible distributions of random effects for longitudinal data., Biometrics, 57(3):795–802, 2001.