Review Articles

Robust dynamic risk prediction with longitudinal studies

Qian M. Zhou ,

Department of Mathematics and Statistics, Mississippi State University, Mississippi, USA

Wei Dai ,

Department of Biostatistics, Harvard School of Public Health, Boston, MA, USA

Yingye Zheng ,

Department of Biostatistics and Biomathematics, Fred Hutchinson Cancer Research Center, Seattle, WA, USA

Tianxi Cai

Department of Mathematics and Statistics, Mississippi State University, Mississippi, USA

Pages 159-170 | Received 21 Mar. 2017, Accepted 31 Oct. 2017, Published online: 27 Nov. 2017,
  • Abstract
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Providing accurate and dynamic age-specific risk prediction is a crucial step in precision medicine. In this manuscript, we introduce an approach for estimating the τ-year age-specific absolute risk directly via a flexible varying coefficient model. The approach facilitates the utilisation of predictors varying over an individual's lifetime. By using a nonparametric inverse probability weighted kernel estimating equation, the age-specific effects of risk factors are estimated without requiring the specification of the functional form. The approach allows borrowing information across individuals of similar ages, and therefore provides a practical solution for situations where the longitudinal information is only measured sparsely. We evaluate the performance of the proposed estimation and inference procedures with numerical studies, and make comparisons with existing methods in the literature. We illustrate the performance of our proposed approach by developing a dynamic prediction model using data from the Framingham Study.


  1. Blanche, P., Dartigues, J.-F., & Jacqmin-Gadda, H. (2013). Estimating and comparing time-dependent areas under receiver operating characteristic curves for censored event times with competing risks. Statistics in Medicine, 32(30), 53815397[Google Scholar]
  2. Cai, T., Tian, L., Uno, H., Solomon, S. D., & Wei, L. J. (2010). Calibrating parametric subject-specific risk estimation. Biometrika, 97(2), 389404[Google Scholar]
  3. Eguchi, S., & Copas, J. (2002). A class of logistic-type discriminant functions. Biometrika, 89(1), 122[Google Scholar]
  4. Gail, M. H., Brinton, L. A., Byar, D. P., Corle, D. K., Green, S. B., Schairer, C., & Mulvihill, J. J. (1989). Projecting individualized probabilities of developing breast cancer for white females who are being examined annually. Journal of the National Cancer Institute, 81(24), 1879[Google Scholar]
  5. Gerds, T. A., Cai, T., & Schumacher, M. (2008). The performance of risk prediction models. Biometrical Journal, 50(4), 457479[Google Scholar]
  6. Li, K.-C., & Duan, N. (1989). Regression analysis under link violation. The Annals of Statistics, 17(3), 10091052[Crossref], [Web of Science ®], [Google Scholar]
  7. Liu, D., Zheng, Y., Prentice, R. L., & Hsu, L. (2014). Estimating risk with time-to-event data: An application to the women's health initiative. Journal of the American Statistical Association, 109(506), 514524[Taylor & Francis Online], [Google Scholar]
  8. Lloyd-Jones, D. M. (2010). Cardiovascular risk prediction basic concepts, current status, and future directions. Circulation, 121(15), 17681777[Google Scholar]
  9. Mosca, L., Appel, L. J., Benjamin, E. J., Berra, K., Chandra-Strobos, N., Fabunmi, R. P., … Williams, C.L. (2004). Evidence-based guidelines for cardiovascular disease prevention in women 1. Journal of the American College of Cardiology, 43(5), 900921[Google Scholar]
  10. Neumann, M. H., & Polzehl, J. (1998). Simultaneous bootstrap confidence bands in nonparametric regression. Journal of Nonparametric Statistics, 9(4), 307333[Taylor & Francis Online], [Google Scholar]
  11. Parast, L., Cheng, S.-C., & Cai, T. (2012). Landmark prediction of long-term survival incorporating short-term event time information. Journal of the American Statistical Association, 107(500), 14921501[Taylor & Francis Online], [Google Scholar]
  12. Park, Y., & Wei, L. J. (2003). Estimating subject-specific survival functions under the accelerated failure time model. Biometrika, 90(3), 717723[Google Scholar]
  13. Ridker, P. M., Buring, J. E., Rifai, N., & Cook, N. R. (2007). Development and validation of improved algorithms for the assessment of global cardiovascular risk in women: The Reynolds risk score. Jama, 297(6), 611619[Crossref], [Web of Science ®], [Google Scholar]
  14. Rizopoulos, D. (2010). Jm: An r package for the joint modelling of longitudinal and time-to-event data. Journal of Statistical Software, 35(9), 133[Google Scholar]
  15. Tian, L., Zucker, D., & Wei, L. J. (2005). On the cox model with time-varying regression coefficients. Journal of the American Statistical Association, 100(469), 172183[Taylor & Francis Online], [Google Scholar]
  16. Tsiatis, A., DeGruttola, V., & 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(429), 27[Taylor & Francis Online], [Google Scholar]
  17. Tsiatis, A. A., & Davidian, M. (2004). Joint modeling of longitudinal and time-to-event data: An overview. Statistica Sinica, 14(3), 809834[Google Scholar]
  18. Uno, H., Cai, T., Tian, L., & Wei, L. J. (2007). Evaluating prediction rules for t-year survivors with censored regression models. Journal of the American Statistical Association, 102(478), 527–537. [Google Scholar]
  19. Wang, Y., & 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(455), 895905[Taylor & Francis Online], [Google Scholar]
  20. Wolf, P. A., D’Agostino, R. B., Belanger, A. J., and Kannel, W. B. (1991). Probability of stroke: A risk profile from the framingham study. Stroke, 22(3), 312318[Google Scholar]
  21. Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics, 14(4), 12611295[Google Scholar]
  22. Ye, W., Lin, X., & Taylor, J. M. G. (2008). Semiparametric modeling of longitudinal measurements and time-to-event data – A two-stage regression calibration approach. Biometrics, 64(4), 12381246[Google Scholar]
  23. Zheng, Y., Cai, T., & Feng, Z. (2006). Application of the time-dependent roc curves for prognostic accuracy with multiple biomarkers. Biometrics, 62(1), 279287[Google Scholar]
  24. Zheng, Y., & Heagerty, P. J. (2004). Semiparametric estimation of time-dependent ROC curves for longitudinal marker data. Biostatistics, 5(4), 615632[Google Scholar]
  25. Zheng, Y., & Heagerty, P. J. (2005). Partly conditional survival models for longitudinal data. Biometrics, 61(2), 379391[Google Scholar]