We propose a flexible joint longitudinal-survival
framework to examine the association between longitudinally collected
biomarkers and a time-to-event endpoint. More specifically, we use our method
for analyzing the survival outcome of end-stage renal disease patients with
time-varying serum albumin measurements. Our proposed method is robust to
common parametric assumptions in that it avoids explicit specification of the
distribution of longitudinal responses and allows for a subject-specific
baseline hazard in the survival component. Fully joint estimation is performed
to account for uncertainty in the estimated longitudinal biomarkers that are
included in the survival model.
References
[1]
Fung, F., Sherrard, D.J., Gillen, D.L., Wong, C., Kestenbaum, B., Seliger, S., Ball, A. and Stehman-Breen, C. (2002) Increased Risk for Cardiovascular Mortality among Malnourished End-Stage Renal Disease Patients. American Journal of Kidney Diseases, 40, 307-314. https://doi.org/10.1053/ajkd.2002.34509
[2]
Wong, C.S., Hingorani, S., Gillen, D.L., Sherrard, D.J., Watkins, S.L., Brandt, J.R., Ball, A. and Stehman-Breen, C.O. (2002) Hypoalbuminemia and Risk of Death in Pediatric Patients with End-Stage Renal Disease. Kidney International, 61, 630-637.
https://doi.org/10.1046/j.1523-1755.2002.00169.x
[3]
Prentice, R.L. (1982) Covariate Measurement Errors and Parameter Estimation in a Failure Time Regression Model. Biometrika, 69, 331-342.
https://doi.org/10.1093/biomet/69.2.331
[4]
Dafni, U.G. and Tsiatis, A.A. (1998) Evaluating Surrogate Markers of Clinical Outcome When Measured with Error. Biometrics, 54, 1445-1462.
https://doi.org/10.2307/2533670
[5]
Tsiatis, A.A., Degruttola, V. and Wulfsohn, M.S. (1995) Modeling the Relationship of Survival to Longitudinal Data Measured with Error. Applications to Survival and cd4 Counts in Patients with Aids. Journal of the American Statistical Association, 90, 27-37. https://doi.org/10.1080/01621459.1995.10476485
[6]
Verbeke, G. and Molenberghs, G. (2000) Linear Mixed Models for Longitudinal Data. Springer, New York. https://doi.org/10.1007/978-1-4419-0300-6
[7]
Pinheiro, J.C. and Bates, D.M. (2000) Mixed Effects Models in S and S-Plus. Springer, New York. https://doi.org/10.1007/b98882
[8]
Christensen, R., Johnson, W., Branscum, A. and Hanson, T.E. (2010) Bayesian Ideas and Data Analysis: An Introduction for Scientists and Statisticians. CRC Texts in Statistical Science, CRC Press, Boca Raton.
[9]
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (2003) Bayesian Data Analysis. Chapman and Hall, London. https://doi.org/10.1201/9780429258480
[10]
Shi, J.Q., Wang, B., Murray-Smith, R. and Titterington, D.M. (2007) Gaussian Process Functional Regression Modeling for Batch Data. Biometrics, 63, 714-723.
https://doi.org/10.1111/j.1541-0420.2007.00758.x
[11]
Liu, Q., Lin, K.K., Andersen, B., Smyth, P. and Ihler, A. (2010) Estimating Replicate Time Shifts Using Gaussian Process Regression. Bioinformatics, 26, 770-776.
https://doi.org/10.1093/bioinformatics/btq022
[12]
Bycott, P. and Taylor, J. (1998) A Comparison of Smoothing Techniques for cd4 Data Measured with Error in a Time-Dependent Cox Proportional Hazards Model. Statistics in Medicine, 17, 2061-2077.
https://doi.org/10.1002/(SICI)1097-0258(19980930)17:18%3C2061::AID-SIM896%3E3.0.CO;2-O
[13]
Hanson, T.E., Branscum, A.J. and Johnson, W.O. (2011) Predictive Comparison of Joint Longitudinal-Survival Modeling: A Case Study Illustrating Competing Approaches. Lifetime Data Analysis, 17, 3-28.
https://doi.org/10.1007/s10985-010-9162-0
[14]
Wang, Y. and Taylor, J.M.G. (2001) Jointly Modeling Longitudinal and Event Time Data with Application to Acquired Immunodeficiency Syndrome. Journal of the American Statistical Association, 96, 895-905.
https://doi.org/10.1198/016214501753208591
[15]
Faucett, C.L. and Thomas, D.C. (1996) Simultaneously Modelling Censored Survival Data and Repeatedly Measured Covariates: A Gibbs Sampling Approach. Statistics in Medicine, 15, 1663-1685.
https://doi.org/10.1002/(SICI)1097-0258(19960815)15:15%3C1663::AID-SIM294%3E3.0.CO;2-1
[16]
Brown, E. and Ibrahim, J. (2003) A Bayesian Semiparametric Joint Hierarchical Model for Longitudinal and Survival Data. Biometrics, 59, 221-228.
https://doi.org/10.1111/1541-0420.00028
[17]
Wulfsohn, M.S. and Tsiatis, A.A. (1997) A Joint Model for Survival and Longitudinal Data Measured with Error. Biometrics, 53, 330-339.
https://doi.org/10.2307/2533118
[18]
Song, X., Davidian, M. and Tsiatis, A.A. (2002) An Estimator for the Proportional Hazards Model with Multiple Longitudinal Covariates Measured with Error. Biostatistics, 3, 511-528. https://doi.org/10.1093/biostatistics/3.4.511
[19]
Law, N.J., Taylor, J.M.G. and Sandler, H. (2002) The Joint Modeling of a Longitudinal Disease Progression Marker and the Failure Time Process in the Presence of cure. Biostatistics, 3, 547-563. https://doi.org/10.1093/biostatistics/3.4.547
[20]
Erango, M.A. and Goshu, A.T. (2019) Bayesian Joint Modelling of Survival Time and Longitudinal CD4 Cell Counts Using Accelerated Failure Time and Generalized Error Distributions. Open Journal of Modelling and Simulation, 7, 79-95.
https://doi.org/10.4236/ojmsi.2019.71004
[21]
USRD (U S Renal Data System) (2010) USRDS 2010 Annual Data Report: Atlas of Chronic Kidney Disease and End-Stage Renal Disease in the United States. National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases, Bethesda.
[22]
Banerjee, S., Carlin, B.P. and Gelfand, A.E. (2014) Hierarchical Modeling and Analysis for Spatial Data. CRC Press, New York. https://doi.org/10.1201/b17115
[23]
Diggle, P.J. and Ribeiro, P.J. (2007) Model-Based Geostatistics. 1st Edition, Springer Series in Statistics, Springer, New York. https://doi.org/10.1007/978-0-387-48536-2
[24]
Neal, R.M. (2011) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC Press, Boca Raton.
[25]
Flaxman, S, Gelman, A., Neill, D., Smola, A., Vehtari, A. and Gordon Wilson, A. (2015) Fast Hierarchical Gaussian Processes. Manuscript in Preparation.
[26]
Struthers, C. and Kalbiesch, J. (1986) An Estimator for the Proportional Hazards Model with Multiple Longitudinal Covariates Measured with Error. Biometrika, 73, 363-369. https://doi.org/10.1093/biomet/73.2.363
[27]
Martinussen, T. and Vansteelandt, S. (2013) On Collapsibility and Confounding Bias in Cox and Aalen Regression Models. Lifetime Data Analysis, 19, 279-296.
https://doi.org/10.1007/s10985-013-9242-z