Bayesian Analysis

Bayesian Quantile Regression with Mixed Discrete and Nonignorable Missing Covariates

Zhi-Qiang Wang and Nian-Sheng Tang

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

Bayesian inference on quantile regression (QR) model with mixed discrete and non-ignorable missing covariates is conducted by reformulating QR model as a hierarchical structure model. A probit regression model is adopted to specify missing covariate mechanism. A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm is developed to simultaneously produce Bayesian estimates of unknown parameters and latent variables as well as their corresponding standard errors. Bayesian variable selection method is proposed to recognize significant covariates. A Bayesian local influence procedure is presented to assess the effect of minor perturbations to the data, priors and sampling distributions on posterior quantities of interest. Several simulation studies and an example are presented to illustrate the proposed methodologies.

Article information

Source
Bayesian Anal., Advance publication (2018), 26 pages.

Dates
First available in Project Euclid: 19 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ba/1560909811

Digital Object Identifier
doi:10.1214/19-BA1165

Subjects
Primary: 62F15: Bayesian inference 62H12: Estimation
Secondary: 62J20: Diagnostics

Keywords
Bayesian analysis local influence analysis non-ignorable missing data quantile regression variable selection

Rights
Creative Commons Attribution 4.0 International License.

Citation

Wang, Zhi-Qiang; Tang, Nian-Sheng. Bayesian Quantile Regression with Mixed Discrete and Nonignorable Missing Covariates. Bayesian Anal., advance publication, 19 June 2019. doi:10.1214/19-BA1165. https://projecteuclid.org/euclid.ba/1560909811


Export citation

References

  • Albert, J. H. and Chib, S. (1993). “Bayesian analysis of binary and polychotomous response data.” Journal of the American Statistical Association, 88(422): 669–679.
  • Alhamzawi, R., Yu, K., and Benoit, D. F. (2012). “Bayesian adaptive Lasso quantile regression.” Statistical Modelling, 12(3): 279–297.
  • Andrews, D. F. and Mallows, C. L. (1974). “Scale mixtures of normal distributions.” Journal of the Royal Statistical Society B, 99–102.
  • Brooks, S. and Andrew, G. (1998). “General methods for monitoring convergence of iterative simulations.” Journal of Computational and Graphical Statistics, 7(4): 434–455.
  • Cade, B. S. and Noon, B. R. (2003). “A gentle introduction to quantile regression for ecologists.” Frontiers in Ecology and the Environment, 1(8): 412–420.
  • Canay, I. A. (2011). “A simple approach to quantile regression for panel data.” The Econometrics Journal, 14(3): 368–386.
  • Casella, G. (2001). “Empirical Bayes Gibbs sampling.” Biostatistics, 2(4): 485–500.
  • Chen, X., Wan, A. T., and Zhou, Y. (2015). “Efficient quantile regression analysis with missing observations.” Journal of the American Statistical Association, 110(510): 723–741.
  • Chernozhukov, V. (2005). “Extremal quantile regression.” Annals of Statistics, 33(2): 806–839.
  • Clarke, B. and Gustafson, P. (1998). “On the overall sensitivity of the posterior distribution to its inputs.” Journal of Statistical Planning and Inference, 71(1): 137–150.
  • Cook, R. D. (1986). “Assessment of local influence.” Journal of the Royal Statistical Society B, 48(1): 133–169.
  • Gaglianone, W. P., Lima, L. R., Linton, O., and Smith, D. R. (2011). “Evaluating value-at-risk models via quantile regression.” Journal of Business and Economic Statistics, 29(1): 150–160.
  • Gelman, A. (1996). Inference and monitoring convergence in Markov chain Monte Carlo in practice. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, eds.
  • Geraci, M. and Bottai, M. (2006). “Quantile regression for longitudinal data using the asymmetric Laplace distribution.” Biostatistics, 8(1): 140–154.
  • Gustafson, P. and Clarke, B. (2004). “A decomposition for the posterior variance.” Journal of Statistical Planning and Inference, 119: 311–327.
  • Gustafson, P. and Wasserman, L. (1995). “Local sensitivity diagnostics for Bayesian inference.” Annals of Statistics, 23(6): 2153–2167.
  • Hallock, K. F. and Koenker, R. W. (2001). “Quantile regression.” Journal of Economic Perspectives, 15(4): 143–156.
  • Hastie, T. J. and Tibshirani, R. J. (1990). Generalized additive models. New York: Chapman and Hall.
  • Hendricks, W. and Koenker, R. (1992). “Hierarchical spline models for conditional quantiles and the demand for electricity.” Journal of the American Statistical Association, 87(417): 58–68.
  • Huang, Y. (2016). “Quantile regression-based Bayesian semiparametric mixed-effects models for longitudinal data with non-normal, missing and mismeasured covariate.” Journal of Statistical Computation and Simulation, 86(6): 1183–1202.
  • Huang, Y. and Chen, J. (2016). “Bayesian quantile regression-based nonlinear mixed-effects joint models for time-to-event and longitudinal data with multiple features.” Statistics in Medicine, 35(30): 5666–5685.
  • Huang, Y., Chen, J., and Qiu, H. (2017). “Bayesian quantile regression for nonlinear mixed-effects joint models for longitudinal data in the presence of mismeasured covariate errors.” Journal of Biopharmaceutical Statistics, 1–15.
  • Ibrahim, J. G., Lipsitz, S. R., and Chen, M. H. (1999). “Missing covariates in generalized linear models when the missing data mechanism is non-ignorable.” Journal of the Royal Statistical Society B, 61(1): 173–190.
  • Ju, Y., Tang, N., and Li, X. (2018). “Bayesian local influence analysis of skew-normal spatial dynamic panel data models.” Journal of Statistical Computation and Simulation, 88(12): 2342–2364.
  • Koenker, R. (2004). “Quantile regression for longitudinal data.” Journal of Multivariate Analysis, 91(1): 74–89.
  • Koenker, R. (2005). Quantile regression. Cambridge University Press.
  • Kottas, A. and Krnjajić, M. (2009). “Bayesian semiparametric modelling in quantile regression.” Scandinavian Journal of Statistics, 36(2): 297–319.
  • Kozumi, H. and Kobayashi, G. (2011). “Gibbs sampling methods for Bayesian quantile regression.” Journal of Statistical Computation and Simulation, 81(11): 1565–1578.
  • Lancaster, T. and Sung, J. (2010). “Bayesian quantile regression methods.” Journal of Applied Econometrics, 25(2): 287–307.
  • Lee, S. Y. and Tang, N. S. (2006). “Bayesian analysis of nonlinear structural equation models with nonignorable missing data.” Psychometrika, 71(3): 541–564.
  • Leng, C., Tran, M. N., and Nott, D. (2014). “Bayesian adaptive lasso.” Annals of the Institute of Statistical Mathematics, 66(2): 221–244.
  • Li, Q., Xi, R., and Lin, N. (2010). “Bayesian regularized quantile regression.” Bayesian Analysis, 1(1): 1–26.
  • Liu, C. (2004). “Robit regression: a simple robust alternative to logistic and probit regression.” In: A. Gelman, and X. L. Meng (Eds.), Applied Bayesian modeling and causal inference from incomplete data perspectives (pp. 227–238). West Sussex: Wiley.
  • Park, T. and Casella, G. (2008). “The Bayesian lasso.” Journal of the American Statistical Association, 103(482): 681–686.
  • Poon, W. and Poon, Y. (1999). “Conformal normal curvature and assessment of local influence.” Journal of the Royal Statistical Society B, 61(1): 51–61.
  • Pregibon, D. (1982). “Resistant fits for some commonly used logistic models with medical applications.” Biometrics, 38: 485–498.
  • Reich, B. J., Bondell, H. D., and Wang, H. J. (2009). “Flexible Bayesian quantile regression for independent and clustered data.” Biostatistics, 11(2): 337–352.
  • Richardson, S. and Green, P. J. (1997). “On Bayesian analysis of mixtures with an unknown number of components.” Journal of the Royal Statistical Society B, 59(4): 731–792.
  • Schmid, M. and Hothorn, T. (2008). “Boosting additive models using component-wise P-splines.” Computational Statistics and Data Analysis, 53(2): 298–311.
  • Tang, N., Chow, S., Ibrahim, J., and Zhu, H. (2017). “Bayesian sensitivity analysis of a nonlinear dynamic factor analysis model with nonparametric prior and possible nonignorable missingness.” Psychometrika, 82(4): 875–903.
  • Tang, N. and Zhao, H. (2014). “Bayesian analysis of nonlinear reproductive dispersion mixed models for longitudinal data with nonignorable missing covariates.” Communications in Statistics – Simulation and Computation, 43(6): 1265–1287.
  • Tanner, M. A. and Wong, W. H. (1987). “The calculation of posterior distributions by data augmentation.” Journal of the American Statistical Association, 82(398): 528–540.
  • Tibshirani, R. (1996). “Regression shrinkage and selection via the lasso.” Journal of the Royal Statistical Society B, 267–288.
  • Wang, Z.-Q. and Tang, N.-S. (2019). “Supplementary Material of “Bayesian Quantile regression with mixed discrete and nonignorable missing covariates”.” Bayesian Analysis.
  • Wei, Y., Ma, Y., and Carroll, R. J. (2012). “Multiple imputation in quantile regression.” Biometrika, 99(2): 423.
  • Wei, Y. and Yang, Y. (2014). “Quantile regression with covariates missing at random.” Statistica Sinica, 24(3): 1277–1299.
  • Wu, Y. and Liu, Y. (2009). “Variable selection in quantile regression.” Statistica Sinica, 19(1): 801–817.
  • Yang, Y. and He, X. (2012). “Bayesian empirical likelihood for quantile regression.” Annals of Statistics, 40(4): 1102–1131.
  • Yi, G. and He, W. (2009). “Median regression models for longitudinal data with dropouts.” Biometrics, 65(2): 618–626.
  • Yu, K. and Moye, R. A. (2001). “Bayesian quantile regression.” Statistics and Probability Letters, 54(4): 437–447.
  • Yuan, Y. and Yin, G. (2010). “Bayesian quantile regression for longitudinal studies with nonignorable missing data.” Biometrics, 66(1): 105–114.
  • Zhang, H., Huang, Y., Wang, W., Chen, H., and Langlandorban, B. (2017). “Bayesian quantile regression-based partially linear mixed-effects joint models for longitudinal data with multiple features.” Statistical Methods in Medical Research, 962280217730852.
  • Zhang, Y. and Tang, N. (2017). “Bayesian local influence analysis of general estimating equations with nonignorable missing data.” Computational Statistics and Data Analysis, 105: 184–200.
  • Zhu, H., Ibrahim, J. G., and Tang, N. (2011). “Bayesian influence analysis: a geometric approach.” Biometrika, 98(2): 307–323.
  • Zhu, H., Ibrahim, J. G., and Tang, N. (2014). “Bayesian sensitivity analysis of statistical models with missing data.” Statistica Sinica, 24(2): 871–896.
  • Zou, H. (2006). “The adaptive lasso and its oracle properties.” Journal of the American Statistical Association, 101(476): 1418–1429.

Supplemental materials