A Bayesian Shared Parameter Model for Analysing Longitudinal Skewed Responses with Nonignorable Dropout
DOI:
https://doi.org/10.6000/1929-6029.2014.03.02.4Keywords:
Bayesian approach, Longitudinal data, Markov Chain Monte Carlo, Missingness mechanism, Nonignorable missing data, Random effects modelAbstract
When the nature of a data set comes from a skew distribution, the use of usual Gaussian mixed effect model can be unreliable. In recent years, skew-normal mixed effect models have been used frequently for longitudinal data modeling in many biomedical studies. These models are flexible for considering skewness of the longitudinal data. In this paper, a shared parameter model is considered for simultaneously analysing nonignorable missingness and skew longitudinal outcomes. A Bayesian approach using Markov Chain Monte Carlo is adopted for parameter estimation. Some simulation studies are performed to investigate the performance of the proposed methods. The proposed methods are applied for analyzing an AIDS data set, where CD4 count measurements are gathered as longitudinal outcomes. In these data CD4 counts measurements are severely skew. In application section, different structures of skew-normal distribution assumptions for random effects and errors are considered where deviance information criterion is used for model comparison.
References
Azzalini A. A class of distribution which includes the normal ones. Scand J Statist 1985; 12: 171-78.
Jara A, Quintana F, San Martn E. Linear mixed models with skew-elliptical distributions: A Bayesian approach. Comput Statist Data Anal 2008; 52: 5033-45. http://dx.doi.org/10.1016/j.csda.2008.04.027 DOI: https://doi.org/10.1016/j.csda.2008.04.027
De la Cruz R, Branco M. Bayesian analysis for nonlinear regression model under skewed errors, with application in growth curves. Biometr J 2009; 51: 588-609. http://dx.doi.org/10.1002/bimj.200800154 DOI: https://doi.org/10.1002/bimj.200800154
Huang Y, Dagne G. A Bayesian approach to joint mixed-effects models with a skew-normal distribution and measurement errors in covariates. Biometrics 2011; 67: 260-69. http://dx.doi.org/10.1111/j.1541-0420.2010.01425.x DOI: https://doi.org/10.1111/j.1541-0420.2010.01425.x
Abanto-Valle CA, Bandyopadhyay D, Lachos VH. Robust Bayesian Analysis of Heavy-tailed Stochastic Volatility Models using Scale Mixtures of Normal Distributions. Computat Statist Data Anal 2010; 54: 2883-98. http://dx.doi.org/10.1016/j.csda.2009.06.011 DOI: https://doi.org/10.1016/j.csda.2009.06.011
Abanto-Valle CA, Migon H, Lachos VH. Bayesian Analysis of Heavy-tailed Stochastic Volatility in Mean model using Scale Mixtures of Normal distributions. J Statist Planning Inference 2011; 141: 1875-87. http://dx.doi.org/10.1016/j.jspi.2010.11.039 DOI: https://doi.org/10.1016/j.jspi.2010.11.039
Zeller CB, Lachos VH, Labra F. Local influence analysis for regression models with skew-normal independent distributions. J Appl Statist 2011; 38: 343-68. http://dx.doi.org/10.1080/02664760903406504 DOI: https://doi.org/10.1080/02664760903406504
Ferreira CS, Lachos VH, Bolfarine H. Skew scale mixtures of normal distributions: properties and estimation. Statist Methodol 2011; 8: 154-71. http://dx.doi.org/10.1016/j.stamet.2010.09.001 DOI: https://doi.org/10.1016/j.stamet.2010.09.001
Lachos VH, Garibay V, Ortega E. A nonlinear model with skew-normal errors. Statist Papers 2010; 51: 547-58. http://dx.doi.org/10.1007/s00362-008-0139-y DOI: https://doi.org/10.1007/s00362-008-0139-y
Huang Y. Segmental modeling of viral load changes for HIV longitudinal data with skewness and detection limits. Statist Med 2013; 32(2): 319-34. http://dx.doi.org/10.1002/sim.5527 DOI: https://doi.org/10.1002/sim.5527
Lin I, Ho J, Chen L. Analysis of multivariate skew-normal models with incomplete data. J Multivar Anal 2009; 100: 2337-51. http://dx.doi.org/10.1016/j.jmva.2009.07.005 DOI: https://doi.org/10.1016/j.jmva.2009.07.005
Baghfalaki T, Ganjali M. An EM estimation approach for analyzing bivariate skew-normal data with non-monotone missing values. Commun Statist Theory Methods 2011; 40(9): 1671-86. http://dx.doi.org/10.1080/03610921003637454 DOI: https://doi.org/10.1080/03610921003637454
Baghfalaki T, Ganjali M. An ECM estimation approach for analyzing multivariate skew-normal data with dropout. Commun Statist Simulat Comput 2012; 41: 1970-88. http://dx.doi.org/10.1080/03610918.2011.627099 DOI: https://doi.org/10.1080/03610918.2011.627099
Ganjali M, Baghfalaki T, Khazaei M. A linear mixed model for analyzing longitudinal skew-normal responses with random dropout. J Korean Statist Soc 2012; 42(2): 149-60. http://dx.doi.org/10.1016/j.jkss.2012.06.004 DOI: https://doi.org/10.1016/j.jkss.2012.06.004
Huang Y, Dagne G. Bayesian Semiparametric Nonlinear Mixed- Effects Joint Models for Data with Skewness, Missing Responses, and Measurement Errors in Covariates. Biometrics 2012; 68: 943-53. http://dx.doi.org/10.1111/j.1541-0420.2011.01719.x DOI: https://doi.org/10.1111/j.1541-0420.2011.01719.x
Little RJ, Rubin D. Statistical analysis with missing data. Second edition. New york, Wiley 2002. DOI: https://doi.org/10.1002/9781119013563
Roy J. Modeling longitudinal data with nonignorable dropouts using a latent dropout class model. Biometrics 2003; 59: 829-36. http://dx.doi.org/10.1111/j.0006-341X.2003.00097.x DOI: https://doi.org/10.1111/j.0006-341X.2003.00097.x
Gao S. A shared random effect parameter approach for longitudinal dementia data with nonignorable missing data. Statist Med 2004; 23: 211-19. http://dx.doi.org/10.1002/sim.1710 DOI: https://doi.org/10.1002/sim.1710
Albert PS, Follmann DA. A random effects transition model for longitudinal binary data with informative missingness. Statist Neerlandic 2003; 57: 100-11. http://dx.doi.org/10.1111/1467-9574.00223 DOI: https://doi.org/10.1111/1467-9574.00223
Albert PS, Follmann DA. Shared-parameter models. In G. Fitzmaurice, M. Davidian, G. Verbeke, and G. Molenberghs (Eds.), Longitudinal data analysis. Boca Raton, FL: Chapman and Hall 2009; pp. 433-452. DOI: https://doi.org/10.1201/9781420011579.ch19
Vonesh EF, Greene T, Schluchter MD. Shared parameter models for the joint analysis of longitudinal data with event times. Statist Med 2006; 25: 143-63. http://dx.doi.org/10.1002/sim.2249 DOI: https://doi.org/10.1002/sim.2249
Yuan Y, Little RJA. Meta-Analysis of Studies with Missing Data. Biometrics 2009; 65: 487-96. http://dx.doi.org/10.1111/j.1541-0420.2008.01068.x DOI: https://doi.org/10.1111/j.1541-0420.2008.01068.x
Azzalini A, Dalla-Valle A. The multivariate skew-normal distribution. Biometrika 1996; 83: 715-26. http://dx.doi.org/10.1093/biomet/83.4.715 DOI: https://doi.org/10.1093/biomet/83.4.715
Sahu SK, Dey DK, Branco M. A new class of multivariate skew distributions with applications to Bayesian regression models. Can J Statist 2003; 31(2): 129-50. http://dx.doi.org/10.2307/3316064 DOI: https://doi.org/10.2307/3316064
Lachos VH, Bandyopadhyay D, Dey DK. Linear and nonlinear mixed-effects models for censored HIV viral loads using normal/independent distributions. Biometrics 2011; 67: 1594-604. http://dx.doi.org/10.1111/j.1541-0420.2011.01586.x DOI: https://doi.org/10.1111/j.1541-0420.2011.01586.x
Arellano-Valle RB, Bolfarine H, Lachos VH. Bayesian inference for skew-normal linear mixed models. J Appl Statist 2007; 34: 663-82. http://dx.doi.org/10.1080/02664760701236905 DOI: https://doi.org/10.1080/02664760701236905
Spiegelhalter DJ, Best NG, Carlin BP, Lindevan der A. Bayesian measures of model complexity and fit. J Royal Statist Soc Ser B 2002; 64: 583-16. http://dx.doi.org/10.1111/1467-9868.00353 DOI: https://doi.org/10.1111/1467-9868.00353
Wu MC, Carroll RJ. Estimation and comparison of changes in the presence of informative right censoring by modelling the censoring process. Biometrics 1988; 44: 175-88. http://dx.doi.org/10.2307/2531905 DOI: https://doi.org/10.2307/2531905
Ten Have TR, Kunselman AR, Pulkstenis EP, Landis JR. Mixed effects logistic regression models for longitudinal binary response data with informative drop-out. Biometrics 1998; 54: 367-83. http://dx.doi.org/10.2307/2534023 DOI: https://doi.org/10.2307/2534023
Albert PS, Follmann DA. Modeling repeated count data subject to informative dropout. Biometrics 2000; 56: 667-77. http://dx.doi.org/10.1111/j.0006-341X.2000.00667.x DOI: https://doi.org/10.1111/j.0006-341X.2000.00667.x
Carlin BP, Louis TA. Bayesian Methods for Data Analysis. Boca Raton, FL: Chapman and Hall - CRC Press 2009.
Guo X, Carlin BP. Separate and joint modeling of longitudinal and event time data using standard computer packages. Am Statist 2004; 58: 16-24. http://dx.doi.org/10.1198/0003130042854 DOI: https://doi.org/10.1198/0003130042854
Goldman AI, Carlin BP, Crane LR, Launer C, Korvick JA, Deyton L, Abrams DI. Response of CD4+ and Clinical Consequences to Treatment Using ddI or ddC in Patients with Advanced HIV Infection. J Acquired Immune Deficiency Syndromes Human Retrovirol 1996; 11: 161-69. http://dx.doi.org/10.1097/00042560-199602010-00007 DOI: https://doi.org/10.1097/00042560-199602010-00007
Gelman A, Rubin DB. Inference from iterative simulation using multiple sequences. Statist Sci 1992; 7: 457-11. http://dx.doi.org/10.1214/ss/1177011136 DOI: https://doi.org/10.1214/ss/1177011136
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2014 M. Ganjali, T. Baghfalaki
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Policy for Journals/Articles with Open Access
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are permitted and encouraged to post links to their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work
Policy for Journals / Manuscript with Paid Access
Authors who publish with this journal agree to the following terms:
- Publisher retain copyright .
- Authors are permitted and encouraged to post links to their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work .
