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Fully exponential Laplace approximations for the joint modelling of survival and longitudinal data

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  • Dimitris Rizopoulos
  • Geert Verbeke
  • Emmanuel Lesaffre

Abstract

Summary. A common objective in longitudinal studies is the joint modelling of a longitudinal response with a time‐to‐event outcome. Random effects are typically used in the joint modelling framework to explain the interrelationships between these two processes. However, estimation in the presence of random effects involves intractable integrals requiring numerical integration. We propose a new computational approach for fitting such models that is based on the Laplace method for integrals that makes the consideration of high dimensional random‐effects structures feasible. Contrary to the standard Laplace approximation, our method requires much fewer repeated measurements per individual to produce reliable results.

Suggested Citation

  • Dimitris Rizopoulos & Geert Verbeke & Emmanuel Lesaffre, 2009. "Fully exponential Laplace approximations for the joint modelling of survival and longitudinal data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 637-654, June.
    • Handle: RePEc:bla:jorssb:v:71:y:2009:i:3:p:637-654
    DOI: 10.1111/j.1467-9868.2008.00704.x
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    1. Fushing Hsieh & Yi-Kuan Tseng & Jane-Ling Wang, 2006. "Joint Modeling of Survival and Longitudinal Data: Likelihood Approach Revisited," Biometrics, The International Biometric Society, vol. 62(4), pages 1037-1043, December.
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    3. Kauermann, Goran & Xu, Ronghui & Vaida, Florin, 2008. "Stacked Laplace-EM algorithm for duration models with time-varying and random effects," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2514-2528, January.
    4. Abrahantes, Jose Cortinas & Legrand, Catherine & Burzykowski, Tomasz & Janssen, Paul & Ducrocq, Vincent & Duchateau, Luc, 2007. "Comparison of different estimation procedures for proportional hazards model with random effects," Computational Statistics & Data Analysis, Elsevier, vol. 51(8), pages 3913-3930, May.
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    Cited by:

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    3. Philipson, Pete & Hickey, Graeme L. & Crowther, Michael J. & Kolamunnage-Dona, Ruwanthi, 2020. "Faster Monte Carlo estimation of joint models for time-to-event and multivariate longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 151(C).
    4. Bernhardt, Paul W. & Zhang, Daowen & Wang, Huixia Judy, 2015. "A fast EM algorithm for fitting joint models of a binary response and multiple longitudinal covariates subject to detection limits," Computational Statistics & Data Analysis, Elsevier, vol. 85(C), pages 37-53.
    5. Bianconcini, Silvia & Cagnone, Silvia, 2012. "Estimation of generalized linear latent variable models via fully exponential Laplace approximation," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 183-193.
    6. Atanu B & Gajendra V & Jesna J & Ramesh V, 2017. "Multiple Imputations for Determining an Optimum Biological Dose of a Metronomic Chemotherapy," Biostatistics and Biometrics Open Access Journal, Juniper Publishers Inc., vol. 3(5), pages 129-140, October.
    7. Tang, Nian-Sheng & Tang, An-Min & Pan, Dong-Dong, 2014. "Semiparametric Bayesian joint models of multivariate longitudinal and survival data," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 113-129.
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    9. Dimitris Rizopoulos, 2011. "Dynamic Predictions and Prospective Accuracy in Joint Models for Longitudinal and Time-to-Event Data," Biometrics, The International Biometric Society, vol. 67(3), pages 819-829, September.
    10. Rong Fu & Peter B. Gilbert, 2017. "Joint modeling of longitudinal and survival data with the Cox model and two-phase sampling," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(1), pages 136-159, January.
    11. Dimitris Rizopoulos & Geert Verbeke & Geert Molenberghs, 2010. "Multiple-Imputation-Based Residuals and Diagnostic Plots for Joint Models of Longitudinal and Survival Outcomes," Biometrics, The International Biometric Society, vol. 66(1), pages 20-29, March.
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    13. Murray, James & Philipson, Pete, 2022. "A fast approximate EM algorithm for joint models of survival and multivariate longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 170(C).
    14. Karl, Andrew T. & Yang, Yan & Lohr, Sharon L., 2014. "Computation of maximum likelihood estimates for multiresponse generalized linear mixed models with non-nested, correlated random effects," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 146-162.
    15. Bikram Karmakar & Peng Liu & Gourab Mukherjee & Hai Che & Shantanu Dutta, 2022. "Improved retention analysis in freemium role‐playing games by jointly modelling players’ motivation, progression and churn," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(1), pages 102-133, January.
    16. Zhang, Cuihong & Ning, Jing & Cai, Jianwen & Squires, James E. & Belle, Steven H. & Li, Ruosha, 2024. "Dynamic risk score modeling for multiple longitudinal risk factors and survival," Computational Statistics & Data Analysis, Elsevier, vol. 189(C).
    17. Karl Andrew T., 2012. "The Sensitivity of College Football Rankings to Several Modeling Choices," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 8(3), pages 1-44, October.
    18. Silvia Cagnone & Paola Monari, 2013. "Latent variable models for ordinal data by using the adaptive quadrature approximation," Computational Statistics, Springer, vol. 28(2), pages 597-619, April.
    19. Yihao Li & Danh V. Nguyen & Esra Kürüm & Connie M. Rhee & Yanjun Chen & Kamyar Kalantar‐Zadeh & Damla Şentürk, 2020. "A multilevel mixed effects varying coefficient model with multilevel predictors and random effects for modeling hospitalization risk in patients on dialysis," Biometrics, The International Biometric Society, vol. 76(3), pages 924-938, September.
    20. Hui Song & Yingwei Peng & Dongsheng Tu, 2017. "Jointly modeling longitudinal proportional data and survival times with an application to the quality of life data in a breast cancer trial," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(2), pages 183-206, April.
    21. Liang Li & Sheng Luo & Bo Hu & Tom Greene, 2017. "Dynamic Prediction of Renal Failure Using Longitudinal Biomarkers in a Cohort Study of Chronic Kidney Disease," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 9(2), pages 357-378, December.
    22. Rui Martins, 2022. "A flexible link for joint modelling longitudinal and survival data accounting for individual longitudinal heterogeneity," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(1), pages 41-61, March.
    23. Bernardi, Mauro & Costola, Michele, 2019. "High-dimensional sparse financial networks through a regularised regression model," SAFE Working Paper Series 244, Leibniz Institute for Financial Research SAFE.
    24. Zhang, Zili & Charalambous, Christiana & Foster, Peter, 2023. "A Gaussian copula joint model for longitudinal and time-to-event data with random effects," Computational Statistics & Data Analysis, Elsevier, vol. 181(C).

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