Source: Ann. Appl. Stat. Volume 4, Number 3
(2010), 1517-1532.
In many medical studies, patients are followed longitudinally and
interest is on assessing the relationship between longitudinal
measurements and time to an event. Recently, various authors
have proposed joint modeling approaches for longitudinal and
time-to-event data for a single longitudinal variable. These
joint modeling approaches become intractable with even a few
longitudinal variables. In this paper we propose a regression
calibration approach for jointly modeling multiple longitudinal
measurements and discrete time-to-event data. Ideally, a
two-stage modeling approach could be applied in which the
multiple longitudinal measurements are modeled in the first
stage and the longitudinal model is related to the time-to-event
data in the second stage. Biased parameter estimation due to
informative dropout makes this direct two-stage modeling
approach problematic. We propose a regression calibration
approach which appropriately accounts for informative dropout.
We approximate the conditional distribution of the multiple
longitudinal measurements given the event time by modeling all
pairwise combinations of the longitudinal measurements using a
bivariate linear mixed model which conditions on the event time.
Complete data are then simulated based on estimates from these
pairwise conditional models, and regression calibration is used
to estimate the relationship between longitudinal data and
time-to-event data using the complete data. We show that this
approach performs well in estimating the relationship between
multivariate longitudinal measurements and the time-to-event
data and in estimating the parameters of the multiple
longitudinal process subject to informative dropout. We
illustrate this methodology with simulations and with an
analysis of primary biliary cirrhosis (PBC) data.
References
Abramowitz, M. and Stegun, I. (1974). Handbook of Mathematical Functions. Dover, New York.
Albert, P. S. and Shih, J. H. (2009). On estimating the relationship between longitudinal measurements and time-to-event data using regression calibration.
Biometrics DOI:
10.1111/j.1541-0420.2009.01324.x.
Allen, C., Duffy., S., Teknos, T. Islam, M., Chen, Z., Albert, P. S., Wolf, G. Y. and Van Waes, C. (2007). A prospective study of serial measurements of NF-αβ related serum cytokines as biomarkers of response and survival in patients with advanced oropharyngeal squamous cell carcinoma receiving chemoradiation therapy. Clinical Cancer Research 13 3182–3190.
Chi, Y. Y. and Ibrahim, J. G. (2006). Joint models for multivariate longitudinal and multivariate survival data. Biometrics 62 432–445.
Brown, E. R., Ibrahim, J. G. and DeGruttola, V. (2005). A flexible B-spline model for multiple longitudinal biomarkers and survival. Biometrics 61 64–73.
Doran, H. C. and Lockwood, J. R. (2006). Fitting value-added models in R. Journal of Educational and Behavioral Statistics 31 205–230.
Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Boostrap. Chapman and Hall, New York.
Fieuws, S. and Verbeke, G. (2005). Pairwise fitting of mixed models for the joint modelling of multivariate longitudinal profiles. Biometrics 62 424–431.
Fieuws, S., Verbeke, G. and Molenberghs, G. (2007). Random-effects models for multivariate repeated measures. Stat. Methods Med. Res. 16 387–397.
Fieuws, S., Verbeke, G., Maes, B. and Vanrenterghem, Y. (2008). Predicting renal graft failure using multivariate longitudinal profiles. Biostatistics 9 419–431.
Henderson, R. Diggle, P. and Dobson, A. (2000). Joint modeling of measurements and event time data. Biostatistics 1 465–480.
Huang, W. H., Zeger, S. L., Anthony, J. C. and Garrett, E. (2001). Latent variable model for joint anlaysis of multiple repeated measures and bivariate event times. Amer. Statist. Assoc. 96 906–914.
Ibrahim, J. G., Chen, M. and Sinha, D. (2004). Bayesian methods for jointly modeling of longitudinal and survival data with applications to cancer vaccine trials. Statist. Sinica 14 863–883.
Laird, N. M. and Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics 38 963–974.
Murtaugh, P. A., Dickson, E. R., Van Dam, G. M., Malincho, M., Grambsch, P. M., Langworthy, A. L. and Gips, C. H. (1994). Primary billary cirrhosis: Prediction of short-term survival based on repeated patient visits. Hepatology 20 126–134.
Song, X., Davidian, M. and Tsiatis, A. S. (2002). An estimator for the proportional hazards model with multiple longitudinal covariates measured with error. Biostatistics 3 511–524.
Tsiatis, A. A. and Davidian, M. (2004). Joint modeling of longitudinal and time-to-event data: An overview. Statist. Sinica 14 809–834.
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. J. Amer. Statist. Assoc. 90 27–37.
Venables, W. N., Smith, D. M. and the R Development Core Team (2008). An Introduction to R. Version 2.8.1 (2008-12-22).
Verbeke, G. and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer, New York.
Wei, G. C. G. and Tanner, M. A. (1990). A Monte-Carlo implementation of the E–M algorithm and the poor man’s data augmentation algorithm. J. Amer. Statist. Assoc. 85 699–704.
Wu, M. C. and Carroll, R. J. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics 45 939–955.
Mathematical Reviews (MathSciNet):
MR931633
Wulfsohn, M. S. and Tsiatis, A. A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics 53 330–339.
Xu, J. and Zeger, S. L. (2001a). Joint analysis of longitudinal data comprising repeated measures and times to events. Appl. Statist. 50 375–387.
Xu, J. and Zeger S. L. (2001b). The evaluation of multiple surrogate endpoints. Biometrics 57 81–87.
