Bayesian Analysis of Transition Model for Longitudinal Ordinal Response Data: Application to Insomnia Data

Authors

  • S. Noorian Department of Statistics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
  • M. Ganjali Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

DOI:

https://doi.org/10.6000/1929-6029.2012.01.02.08

Keywords:

Bayesian Analysis, Bayes Factor, Conditional Predictive Ordinate, Logistic Regression, Markov Model

Abstract

In this paper, we present a Bayesian framework for analyzing longitudinal ordinal response data. In analyzing longitudinal data, the possibility of correlations between responses given by the same individual needs to be taken into account. Various models can be used to handle such correlations such as marginal modeling, random effect modeling and transition (Markov) modeling. Here a transition modeling is used and a Bayesian approach is presented for analyzing longitudinal data. A cumulative logistic regression model and the Bayesian method, using MCMC, are implemented for obtaining the parameters estimates. Our approach is applied on a two-period longitudinal Insomnia data where the Bayesian estimate for measure of association, , between the initial and follow-up ordinal responses is obtained in each level of a treatment variable. Then, the sensitivity of posterior summaries to changes of prior hyperparameters is investigated. We also use Bayes factor criterion for testing some important hypotheses

Author Biographies

S. Noorian, Department of Statistics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran

Department of Statistics, Faculty of Mathematical Sciences

M. Ganjali, Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

Department of Statistics, Faculty of Mathematical Sciences

References

Agresti A. Modelling ordered categorical data: recent advances and future challenges. Stat Med 1999; 18: 2197-207. http://dx.doi.org/10.1002/(SICI)1097-0258(19990915/30)18:17/18<2191::AID-SIM249>3.0.CO;2-M DOI: https://doi.org/10.1002/(SICI)1097-0258(19990915/30)18:17/18<2191::AID-SIM249>3.0.CO;2-M

Lall R, Campbell MJ, Walters SJ, Morgan K. A review of ordinal regression models applied on health-related quality of life assessments. Stat Methods Med Res 2002; 11: 49-67. http://dx.doi.org/10.1191/0962280202sm271ra DOI: https://doi.org/10.1191/0962280202sm271ra

Ten Have TR, Landis JR, Hartzel J. Population-averaged and cluster-specific models for clustered ordinal response data. Stat Med 1996; 15: 2573-88. http://dx.doi.org/10.1002/(SICI)1097-0258(19961215)15:23<2573::AID-SIM389>3.0.CO;2-O DOI: https://doi.org/10.1002/(SICI)1097-0258(19961215)15:23<2573::AID-SIM389>3.0.CO;2-O

Kim K. A bivariate cumulative probit regression model for ordered categorical data. Stat Med 1995; 14: 1341-52. http://dx.doi.org/10.1002/sim.4780141207 DOI: https://doi.org/10.1002/sim.4780141207

Liang KY, Zeger SL, Qaqish BF. Multivariate regression analyses for categorical data. J R Stat Soc Ser B Stat Methodol 1992; 54(1): 3-40. DOI: https://doi.org/10.1111/j.2517-6161.1992.tb01862.x

Molenberghs G, Lesaffre E. Marginal modeling of correlated ordinal data using a multivariate Plackett distribution. J Am Stat Assoc 1994; 89: 633-44. http://dx.doi.org/10.1080/01621459.1994.10476788 DOI: https://doi.org/10.1080/01621459.1994.10476788

Harville DA, Mee RW. A mixed-model procedure for analyzing ordered categorical data. Biometrics 1984; 40: 393-408. http://dx.doi.org/10.2307/2531393 DOI: https://doi.org/10.2307/2531393

Verbeke G, Lesaffre E. A linear mixed-effects model with heterogeneity in the random- effects population. J Am Stat Assoc 1996; 91: 217-21. http://dx.doi.org/10.1080/01621459.1996.10476679 DOI: https://doi.org/10.1080/01621459.1996.10476679

Tutz G, Hennevogl W. Random effects in ordinal regression models. Comput Stat Data Anal 1996; 22: 537-57. http://dx.doi.org/10.1016/0167-9473(96)00004-7 DOI: https://doi.org/10.1016/0167-9473(96)00004-7

Verbeke G, Molenberghs G. Linear mixed models in practice: a SAS-oriented approach. New York: Spinger 1997. DOI: https://doi.org/10.1007/978-1-4612-2294-1

Diggle PJ, Heagerty PJ, Liang KY, Zeger SL. Analysis of longitudinal data. Oxford: Oxford University Press 2002.

Tutz G. Modelling of repeated ordered measurements by isotonic sequential regression. Stat Model 2005; 5(4): 269-87. http://dx.doi.org/10.1191/1471082X05st101oa DOI: https://doi.org/10.1191/1471082X05st101oa

Garber AM. A discrete-time model of the acquisition of antibiotic-resistant infections in hospitalized patients. Biometrics 1989; 45: 797-16. http://dx.doi.org/10.2307/2531684 DOI: https://doi.org/10.2307/2531684

Francom SF, Chuang-Stein C, Landis JR. A log-linear model for ordinal data to characterize differential change among treatments. Stat Med 1989; 8: 571-82. http://dx.doi.org/10.1002/sim.4780080506 DOI: https://doi.org/10.1002/sim.4780080506

Chung H, Park YS, Lanza ST. Latent transition analysis with covariates: pubertal timing and substance use behaviours in adolescent females. Stat Med 2005; 24: 2895-10. http://dx.doi.org/10.1002/sim.2148 DOI: https://doi.org/10.1002/sim.2148

Rezaei Ghahroodi Z, Ganjali M. Testing homogeneity in Markov models for analyzing longitudinal ordinal response data with random dropout. J Stat Theory Appl 2009; 8(2): 125-39. http://dx.doi.org/10.1080/10543400902964100

Rezaei Ghahroodi Z, Ganjali M, Berridge D. A transition model for ordinal response data with random dropout: an application to the uvoxamine data. J Biopharm Statist 2009; 19(4): 658-71. DOI: https://doi.org/10.1080/10543400902964100

McCullagh P. Regression models for ordinal data. J R Stat Soc Ser B Stat Methodol 1980; 42(2): 109-42. DOI: https://doi.org/10.1111/j.2517-6161.1980.tb01109.x

Agresti A. Analysis of ordinal categorical data. 2nd ed. New York: John Wiley & Sons 2010. http://dx.doi.org/10.1002/9780470594001 DOI: https://doi.org/10.1002/9780470594001

Sung M, Soyer R, Nhan N. Bayesian analysis of non-homogeneous Markov chains: application to mental health data. Stat Med 2007; 26: 3000-17. http://dx.doi.org/10.1002/sim.2775 DOI: https://doi.org/10.1002/sim.2775

Zeghnoun A, Czernichow P, Declercq C. Assessment of short-term association be- tween health outcomes and ozone concentrations using a Markov regression model. Environmetrics 2003; 14: 271-82. http://dx.doi.org/10.1002/env.585 DOI: https://doi.org/10.1002/env.585

Chan JSK, Wan WY. Bayesian approach to analysing longitudinal bivariate binary data with informative dropout.Comput Stat 2011; 26: 121-44. http://dx.doi.org/10.1007/s00180-010-0213-5 DOI: https://doi.org/10.1007/s00180-010-0213-5

Mansouriana M, Kazemnejada A, Kazemi I, Zayeri F, Soheilian M. Bayesian analysis of longitudinal ordered data with flexible random effects using McMC: application to diabetic macular Edema data. J Appl Stat 2012; 39(5): 1087-100. DOI: https://doi.org/10.1080/02664763.2011.638367

Mandel M, Gauthier SA, Guttmann CRG, Weiner HL, Betensky RA. Estimating time to event from longitudinal categorical data: an analysis of multiple sclerosis progression. J Am Stat Assoc 2007; 102: 1254-66. http://dx.doi.org/10.1198/016214507000000059 DOI: https://doi.org/10.1198/016214507000000059

Anderson TW, Goodman LA. Statistical inference about Markov chains. Annals of mathematical statistics 1957; 28(1): 89-10. http://dx.doi.org/10.1214/aoms/1177707039 DOI: https://doi.org/10.1214/aoms/1177707039

Lee TC, Judge GG, Zellner A. Estimating the parameters of the Markov probability model from aggregate time series data. 2nd ed. Amsterdam: North-Holland Publication Company 1970.

Meshkani M. Empirical Bayes estimation of transition probabilities for Markov chains. Ph.D. Dissertation: Florida State University 1978.

Healy BC, Engler D. Modeling disease-state transition heterogeneity through Bayesian variable selection. Stat Med 2009; 28: 1353-68. http://dx.doi.org/10.1002/sim.3545 DOI: https://doi.org/10.1002/sim.3545

Goodman LA, Kruskal WH. Measures of association for cross-classification. J Am Stat Assoc 1954; 49: 732-804. DOI: https://doi.org/10.2307/2281536

Lindley DV. The Bayesian analysis of contingency tables. Annals of mathematical statistics 1964; 35: 1622-43. http://dx.doi.org/10.1214/aoms/1177700386 DOI: https://doi.org/10.1214/aoms/1177700386

Good I. The population frequencies of species and the estimation of population parameters. Biometrika 1953; 40: 237-64. DOI: https://doi.org/10.1093/biomet/40.3-4.237

Agresti A, Hitchcock DB. Bayesian inference for categorical data analysis. Stat Methods Appt 2005; 14(3): 297-30. http://dx.doi.org/10.1007/s10260-005-0121-y DOI: https://doi.org/10.1007/s10260-005-0121-y

Muenz LR, Rubinstein LV. Markov models for covariate dependence of binary sequences. Biometrics 1985; 41(1): 91-101. http://dx.doi.org/10.2307/2530646 DOI: https://doi.org/10.2307/2530646

Zeger S, Qaqish B. Markov regression models for time series: a quasi-likelihood approach. Biometrics 1988; 44: 1019-31. http://dx.doi.org/10.2307/2531732 DOI: https://doi.org/10.2307/2531732

Ganjali M, Rezaee Z. A transition model for analysis of repeated measure ordinal response data to identify the effects of different treatments. Drug Inf J 2007; 41: 527-34. DOI: https://doi.org/10.1177/009286150704100411

Newton MA, Raftery AE. Approximate Bayesian inference by the weighted likelihood bootstrap. J R Stat Soc Ser B Stat Methodol 1994; 56(1): 3-48. DOI: https://doi.org/10.1111/j.2517-6161.1994.tb01956.x

Geisser S, Eddy WF. A predictive approach to model selection. J Am Stat Assoc 1979; 74: 153-60. http://dx.doi.org/10.1080/01621459.1979.10481632 DOI: https://doi.org/10.1080/01621459.1979.10481632

Gelfand AE, Dey DK. Bayesian model choice, asymptotics and exact calculations. J R Stat Soc Ser B Stat Methodol 1994; 56(3): 50-514. DOI: https://doi.org/10.1111/j.2517-6161.1994.tb01996.x

Jeffreys H. Theory of probability. 3rd ed. Oxford: Oxford University Press 1961.

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Published

2012-12-20

How to Cite

Noorian, S., & Ganjali, M. (2012). Bayesian Analysis of Transition Model for Longitudinal Ordinal Response Data: Application to Insomnia Data. International Journal of Statistics in Medical Research, 1(2), 148–161. https://doi.org/10.6000/1929-6029.2012.01.02.08

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General Articles