Bayesian Joint Modeling of Skew-Positive Longitudinal-Survival Data Using Birnbaum-Saunders Distribution
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Abstract
Background: There has been a great interest in joint modeling of longitudinal and survival data in recent two decades. Joint models have less restrictive assumptions in multivariate modeling and could address various research questions. This has led to their wide applications in practice. However, earlier models had normality assumption on the distribution in longitudinal part that is usually violated in real data. Hence, recent research have focused on circumventing this issue. Using various skewed distributions has been proposed and applied in the literature. Nevertheless, the flexibility of the proposed methods is limited especially when the data are skew positive.
Methods: In this paper, we introduce the use of Birnbaum-Saunders (BS) distribution in joint modeling context. BS distribution is more flexible and could cover a wide range of skew, kurtotic or bimodal data.
Results: We analyzed publicly available ddI/ddC data both with normal and BS distributions in Bayesian setting and compared their fit by Widely Applicable Information Criterion (WAIC). The joint BS model showed a better fit to the data.
Conclusion: We introduced and applied BS distribution in joint modeling of longitudinal-survival data. Using multi-parameter distributions such as BS in Bayesian setting could improve the fit of models without limitations that arise in transformation of data from original scale.
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20.Birnbaum Z, Saunders SC. A new family of life distributions. Journal of Applied Probability. 1969:319-27.
21.Rieck JR, Nedelman JR. A Log-Linear Model for the Birnbaum-Saunders Distribution. Technometrics. 1991;33(1):51-60.
22.Leiva V, Vilca F, Balakrishnan N, Sanhueza A. A skewed sinh-normal distribution and its properties and application to air pollution. Communications in Statistics-Theory and Methods. 2010;39(3):426-43.
23.Vilca F, Sanhueza A, Leiva V, Christakos G. An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile. Stochastic Environmental Research and Risk Assessment. 2010;24(5):771-82.
24.Paula GA, Leiva V, Barros M, Liu S. Robust statistical modeling using the Birnbaum‐Saunders‐t distribution applied to insurance. Applied Stochastic Models in Business and Industry. 2012;28(1):16-34.
25.Lemonte AJ. A log-Birnbaum-Saunders regression model with asymmetric errors. Journal of Statistical Computation and Simulation. 2012;82(12):1775-87.
26.Sanhueza A, Leiva V, Balakrishnan N. The generalized Birnbaum–Saunders distribution and its theory, methodology, and application. Communications in Statistics-Theory and Methods. 2008;37(5):645-70.
27.Leiva V. The Birnbaum-Saunders Distribution: Academic Press; 2015.
28.Stan Development Team. Stan Modeling Language Users Guide and Reference Manual, Version 2.12.0. 2016.
29.Guo X, Carlin BP. Separate and joint modeling of longitudinal and event time data using standard computer packages. Americal Statistician. 2004;58(1):16-24.
30.Spiegelhalter DJ, Best NG, Carlin BP, Linde A. The deviance information criterion: 12 years on. Journal of Royal Statistical Society, Series B. 2014;76(3):485-93.
31.Plummer M. Penalized loss functions for Bayesian model comparison. Biostatistics. 2008;9:523–39.
32.van der Linde A. DIC in variable selection. Statistica Neerlandica. 2005;1:45–56.
33.Geisser S. IN DISCUSSION OF Box, George EP. Sampling and Bayes' inference in scientific modelling and robustness. Journal of Royal Statistical Society, Series A. 1980;143(4):383-430.
34.Gelfand AE, Dey DK, Chang H. Model determination using predictive distributions with implementation via sampling-based methods. DTIC Document, 1992.
35.Ibrahim JG, Chen M-H, Sinha D. Bayesian Survival Analysis: Springer-Verlag New York; 2001.
36.Watanabe S. Asymptotic equivalence of Bayes cross validation and widely application information criterion in singular learning theory. Journal of Machine Learning Research. 2010;11:3571-94.
37.Gelman A, Carlin JB, Stern HS, Rubin DB. Bayesian data analysis: Chapman & Hall/CRC Boca Raton, FL, USA; 2014.
38.Goldman A, Carlin B, Crane L, Launer C, Korvick J, Deyton L, et al. Response of CD4+ and Clinical Consequences to Treatment Using ddI or ddC in Patients with Advanced HIV Infection. Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology. 1996;11:161–9
2.Fitzmaurice GM, Laird NM, Ware JH. Applied longitudinal analysis. Hoboken, New Jersey: John Wiley & Sons; 2012.
3.Chi YY, Ibrahim JG. Joint models for multivariate longitudinal and multivariate survival data. Biomet. 2006;62(2):432-45.
4.Henderson R, Diggle P, Dobson A. Joint modelling of longitudinal measurements and event time data. Biostatistics. 2000;1(4):465-80.
5.Ibrahim JG, Chu H, Chen LM. Basic concepts and methods for joint models of longitudinal and survival data. Journal of Clinical Oncology,. 2010;28(16):2796-801.
6.Wulfsohn MS, Tsiatis AA. A joint model for survival and longitudinal data measured with error. Biomet. 1997:330-9.
7.Faucett CL, Thomas DC. Simultaneously modelling censored survival data and repeatedly measured covariates: a Gibbs sampling approach. Statistics in Medicine,. 1996;15(15):1663-85.
8.Rizopoulos D, Ghosh P. A Bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time-to-event. Stat Med. 2011;30(12):1366-80.
9.Brown ER, Ibrahim JG. A Bayesian semiparametric joint hierarchical model for longitudinal and survival data. Biomet. 2003;59(2):221-8.
10.Yu M, Law NJ, Taylor JM, Sandler HM. Joint longitudinal-survival-cure models and their application to prostate cancer. Statistica Sinica. 2004;14(3):835-62.
11.Song H, Peng Y, Tu D. A new approach for joint modelling of longitudinal measurements and survival times with a cure fraction. Canadian Journal of Statistics. 2012;40(2):207-24.
12.Baghfalaki T, Ganjali M, Berridge D. Robust joint modeling of longitudinal measurements and time to event data using normal/independent distributions: A Bayesia approach. Biometrical Journal. 2013;55(6):844-65.
13.Tang N-S, Tang A-M, Pan D-D. Semiparametric Bayesian joint models of multivariate longitudinal and survival data. Computional Statisitics and Data Analysis. 2014;77:113–29.
14.Chen J, Huang Y. A Bayesian mixture of semiparametric mixed‐effects joint models for skewed‐longitudinal and time‐to‐event data. Statistics in Medicine. 2015;34:2820-43.
15.Huang Y, Yan C, Xing D, Zhang N, Chen H. Jointly Modeling Event Time and Skewed-Longitudinal Data with Missing Response and Mismeasured Covariate for AIDS Studies. Journal of Biopharmaceutical Statistics,. 2015;25:670–94.
16.Baghfalaki T, Ganjali M, Hashemi R. Bayesian joint modeling of longitudinal measurements and time-to-event data using robust distributions. Journal of Biopharmaceutical Statistics. 2014;24(4):834-55.
17.Baghfalaki T, Ganjali M. A Bayesian Approach for Joint Modeling of Skew-Normal Longitudinal Measurements and Time to Event Data. REVSTAT–Statistical Journal. 2015;13(2):169-91.
18.Leiva V, Barros M, Paula GA, Sanhueza A. Generalized Birnbaum‐Saunders distributions applied to air pollutant concentration. Environmet. 2008;19(3):235-49.
19.Balakrishnan N, Leiva V, Sanhueza A, Labra FEV. Estimation in the Birnbaum-Saunders distribution based on scale-mixture of normals and the EM-algorithm. SORT- Statistics and Operations Research Transactions. 2009;33(2):171-92.
20.Birnbaum Z, Saunders SC. A new family of life distributions. Journal of Applied Probability. 1969:319-27.
21.Rieck JR, Nedelman JR. A Log-Linear Model for the Birnbaum-Saunders Distribution. Technometrics. 1991;33(1):51-60.
22.Leiva V, Vilca F, Balakrishnan N, Sanhueza A. A skewed sinh-normal distribution and its properties and application to air pollution. Communications in Statistics-Theory and Methods. 2010;39(3):426-43.
23.Vilca F, Sanhueza A, Leiva V, Christakos G. An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile. Stochastic Environmental Research and Risk Assessment. 2010;24(5):771-82.
24.Paula GA, Leiva V, Barros M, Liu S. Robust statistical modeling using the Birnbaum‐Saunders‐t distribution applied to insurance. Applied Stochastic Models in Business and Industry. 2012;28(1):16-34.
25.Lemonte AJ. A log-Birnbaum-Saunders regression model with asymmetric errors. Journal of Statistical Computation and Simulation. 2012;82(12):1775-87.
26.Sanhueza A, Leiva V, Balakrishnan N. The generalized Birnbaum–Saunders distribution and its theory, methodology, and application. Communications in Statistics-Theory and Methods. 2008;37(5):645-70.
27.Leiva V. The Birnbaum-Saunders Distribution: Academic Press; 2015.
28.Stan Development Team. Stan Modeling Language Users Guide and Reference Manual, Version 2.12.0. 2016.
29.Guo X, Carlin BP. Separate and joint modeling of longitudinal and event time data using standard computer packages. Americal Statistician. 2004;58(1):16-24.
30.Spiegelhalter DJ, Best NG, Carlin BP, Linde A. The deviance information criterion: 12 years on. Journal of Royal Statistical Society, Series B. 2014;76(3):485-93.
31.Plummer M. Penalized loss functions for Bayesian model comparison. Biostatistics. 2008;9:523–39.
32.van der Linde A. DIC in variable selection. Statistica Neerlandica. 2005;1:45–56.
33.Geisser S. IN DISCUSSION OF Box, George EP. Sampling and Bayes' inference in scientific modelling and robustness. Journal of Royal Statistical Society, Series A. 1980;143(4):383-430.
34.Gelfand AE, Dey DK, Chang H. Model determination using predictive distributions with implementation via sampling-based methods. DTIC Document, 1992.
35.Ibrahim JG, Chen M-H, Sinha D. Bayesian Survival Analysis: Springer-Verlag New York; 2001.
36.Watanabe S. Asymptotic equivalence of Bayes cross validation and widely application information criterion in singular learning theory. Journal of Machine Learning Research. 2010;11:3571-94.
37.Gelman A, Carlin JB, Stern HS, Rubin DB. Bayesian data analysis: Chapman & Hall/CRC Boca Raton, FL, USA; 2014.
38.Goldman A, Carlin B, Crane L, Launer C, Korvick J, Deyton L, et al. Response of CD4+ and Clinical Consequences to Treatment Using ddI or ddC in Patients with Advanced HIV Infection. Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology. 1996;11:161–9
| Files | ||
| Issue | Vol 6 No 1 (2020) | |
| Section | Original Article(s) | |
| DOI | https://doi.org/10.18502/jbe.v6i1.4757 | |
| Keywords | ||
| Birnbaum-Saunders distribution Joint model Skew-positive | ||
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How to Cite
1.
Jafari-Koshki T, Hosseini SM, Arsang-Jang S. Bayesian Joint Modeling of Skew-Positive Longitudinal-Survival Data Using Birnbaum-Saunders Distribution. JBE. 2020;6(1):30-39.
