On the Landmark Survival Model for Dynamic Prediction of Event Occurrence Using Longitudinal Data

  • Yayuan Zhu
  • Liang Li
  • Xuelin Huang
Part of the ICSA Book Series in Statistics book series (ICSABSS)


In longitudinal cohort studies, participants are often monitored through periodic clinical visits until the occurrence of a terminal clinical event. A question of interest to both scientific research and clinical practice is to predict the risk of the terminal event at each visit, using the longitudinal prognostic information collected up to the visit. This problem is called the dynamic prediction: a real-time, personalized prediction of the risk of a future adverse clinical event with longitudinally measured biomarkers and other prognostic information. An important method for dynamic prediction is the landmark Cox model and variants. A fundamental difficulty in the current methodological research of this kind of models is that it is unclear whether there exists a joint distribution of the longitudinal and time-to-event data that satisfies the model assumptions. As a result, this model is often viewed as a working model instead of a probability distribution, and its statistical properties are often studied using data simulated from shared random effect models, where the landmark model works under misspecification. In this paper, we demonstrate that a joint distribution of longitudinal and survival data exists that satisfy the modeling assumptions without additional restrictions, and propose an algorithm to generate data from this joint distribution. We further generalize the results to the more flexible landmark linear transformation models that include the landmark Cox model as a special case. These results facilitate future theoretical and numerical research on landmark survival models for dynamic prediction.



The authors gratefully acknowledge the financial support for this research by the National Institutes of Health (grant 5P30CA016672 and 5U01DK103225) and MD Anderson Cancer Center.


  1. Blanche, P., Proust-Lima, C., Loubère, L., Berr, C., Dartigues, J. F., & Jacqmin-Gadda, H. (2015). Quantifying and comparing dynamic predictive accuracy of joint models for longitudinal marker and time-to-event in presence of censoring and competing risks. Biometrics, 71(1), 102–113.MathSciNetCrossRefGoogle Scholar
  2. Cheng, S. C., Wei, L. J., & Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika, 82(4), 835–845.MathSciNetCrossRefGoogle Scholar
  3. Diggle, P. J., Heagerty, P., Liang, K.-Y., & Zeger, S. L. (2002). Analysis of longitudinal data (2nd ed.). Oxford: Oxford University Press.zbMATHGoogle Scholar
  4. Faderl, S., Talpaz, M., Estrov, Z., O’Brien, S., Kurzrock, R., & Kantarjian, H. M. (1999). The biology of chronic myeloid leukemia. New England Journal of Medicine, 341(3), 164–172.CrossRefGoogle Scholar
  5. Gorre, M. E., & Sawyers, C. L. (2002). Molecular mechanisms of resistance to STI571 in chronic myeloid leukemia. Current Opinion in Hematology, 9(4), 303–307.CrossRefGoogle Scholar
  6. Hochhaus, A., Kantarjian, H. M., Baccarani, M., Lipton, J. H., Apperley, J. F., Druker, B. J., et al. (2007). Dasatinib induces notable hematologic and cytogenetic responses in chronic-phase chronic myeloid leukemia after failure of imatinib therapy. Blood, 109(6), 2303–2309.CrossRefGoogle Scholar
  7. Huang, X., Yan, F., Ning, J., Feng, Z., Choi, S. & Cortes, J. (2005). A two-stage approach for dynamic prediction of time-to-event distributions. Statistics in Medicine, 35(13), 2167–2182.MathSciNetCrossRefGoogle Scholar
  8. Jewell, N. P., & Nielsen, J. P. (1993). A framework for consistent prediction rules based on markers. Biometrika, 80(1), 153–164.MathSciNetCrossRefGoogle Scholar
  9. Kalbfleisch, J. D., & Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data (2nd ed.). Hoboken, NJ: Wiley.CrossRefGoogle Scholar
  10. Li, L., Greene, T., & Hu, B. (2018). A simple method to estimate the time-dependent receiver operating characteristic curve and the area under the curve with right censored data. Stat Methods Med Res. 27(8), 2264–2278.MathSciNetCrossRefGoogle Scholar
  11. Li, L., Luo, S., Hu, B., & Greene, T. (2017). Dynamic prediction of renal failure using longitudinal biomarkers in a cohort study of chronic kidney disease. Stat Biosci. 9(2), 357–378.CrossRefGoogle Scholar
  12. Maziarz, M., Heagerty, P., Cai, T., & Zheng, Y. (2016). On longitudinal prediction with time-to-event outcome: Comparison of modeling options. Biometrics Epub ahead of print, Scholar
  13. Quintás-Cardama, A., Choi, S., Kantarjian, H., Jabbour, E., Huang, X., & Cortes, J. (2014). Predicting outcomes in patients with chronic myeloid leukemia at any time during tyrosine kinase inhibitor therapy. Clinical Lymphoma Myeloma and Leukemia, 14(4), 327–334.CrossRefGoogle Scholar
  14. Rizopoulos, D. (2011). Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics, 67(3), 819–829.MathSciNetCrossRefGoogle Scholar
  15. Rizopoulos, D., Hatfield, L. A., Carlin, B. P., & Takkenberg, J. J. M. (2014). Combining dynamic predictions from joint models for longitudinal and time-to-event data using Bayesian model averaging. Journal of the American Statistical Association, 109(508), 1385–1397.MathSciNetCrossRefGoogle Scholar
  16. Sawyers, C. L. (1999). Chronic myeloid leukemia. New England Journal of Medicine, 340(17), 1330–1340.CrossRefGoogle Scholar
  17. Steyerberg, E. W. (2009). Clinical prediction models: A practical approach to development, validation, and updating. New York: Springer.CrossRefGoogle Scholar
  18. Taylor, J. M., Park, Y., Ankerst, D. P., Proust-Lima, C., Williams, S., Kestin, L., et al. (2013). Real-time individual predictions of prostate cancer recurrence using joint models. Biometrics, 69(1), 206–213.MathSciNetCrossRefGoogle Scholar
  19. van Houwelingen, H. C. (2007). Dynamic prediction by landmarking in event history analysis. Scandinavian Journal of Statistics, 34(1), 70–85.MathSciNetCrossRefGoogle Scholar
  20. van Houwelingen, H. C., & Putter H. (2008). Dynamic predicting by landmarking as an alternative for multi-state modeling: an application to acute lymphoid leukemia data. Lifetime Data Analysis, 14(4), 447–463.MathSciNetCrossRefGoogle Scholar
  21. van Houwelingen, H., & Putter, H. (2011). Dynamic prediction in clinical survival analysis. Boca Raton, FL: Chapman & Hall/CRC.zbMATHGoogle Scholar
  22. Wong, S. F. (2009). New dosing schedules of dasatinib for CML and adverse event management. Journal of Hematology & Oncology, 2(1), 10.CrossRefGoogle Scholar
  23. Yan, F., Lin, X., & Huang, X. (2017). Dynamic prediction of disease progression for leukemia patients by functional principal component analysis of longitudinal expression levels of an oncogene. The Annals of Applied Statistics, 11(3), 1649–1670.MathSciNetCrossRefGoogle Scholar
  24. Zheng, Y. Y., & Heagerty, P. J. (2005). Partly conditional survival models for longitudinal data. Biometrics, 61(2), 379–391.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of BiostatisticsMD Anderson Cancer CenterHoustonUSA

Personalised recommendations