Analysis of the Proportional Hazards Model With Sparse Longitudinal Covariates

Abstract

Regression analysis of censored failure observations via the proportional hazards model permits time-varying covariates that are observed at death times. In practice, such longitudinal covariates are typically sparse and only measured at infrequent and irregularly spaced follow-up times. Full likelihood analyses of joint models for longitudinal and survival data impose stringent modeling assumptions that are difficult to verify in practice and that are complicated both inferentially and computationally. In this article, a simple kernel weighted score function is proposed with minimal assumptions. Two scenarios are considered: half kernel estimation in which observation ceases at the time of the event and full kernel estimation for data where observation may continue after the event, as with recurrent events data. It is established that these estimators are consistent and asymptotically normal. However, they converge at rates that are slower than the parametric rates that may be achieved with fully observed covariates, with the full kernel method achieving an optimal convergence rate that is superior to that of the half kernel method. Simulation results demonstrate that the large sample approximations are adequate for practical use and may yield improved performance relative to last value carried forward approach and joint modeling method. The analysis of the data from a cardiac arrest study demonstrates the utility of the proposed methods. Supplementary materials for this article are available online.

• Convergence rates
• Cox model
• Kernel weighted estimation.

Additional information

Notes on contributors

Hongyuan Cao

Hongyuan Cao is Assistant Professor, Department of Statistics, University of Missouri-Columbia, Columbia, MO 65201 (E-mail: [email protected]). Mathew M. Churpek is Assistant Professor, Department of Health Studies and Department of Medicine, University of Chicago, Chicago, IL 60637 (E-mail: [email protected]). Donglin Zeng is Professor (E-mail: [email protected]) and Jason P. Fine is Professor (E-mail: [email protected]), Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514. Donglin Zeng is partially supported by NIH grant R01CA082659 and P01 CA142538; Jason P. Fine is partially supported by NIH grant 5R01CA09483-07 and P01 CA142538.We thank Zhuang Ma for help with simulation and Trevor C Yuen for help with data preparation.

Mathew M. Churpek

Hongyuan Cao is Assistant Professor, Department of Statistics, University of Missouri-Columbia, Columbia, MO 65201 (E-mail: [email protected]). Mathew M. Churpek is Assistant Professor, Department of Health Studies and Department of Medicine, University of Chicago, Chicago, IL 60637 (E-mail: [email protected]). Donglin Zeng is Professor (E-mail: [email protected]) and Jason P. Fine is Professor (E-mail: [email protected]), Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514. Donglin Zeng is partially supported by NIH grant R01CA082659 and P01 CA142538; Jason P. Fine is partially supported by NIH grant 5R01CA09483-07 and P01 CA142538.We thank Zhuang Ma for help with simulation and Trevor C Yuen for help with data preparation.

Donglin Zeng

Hongyuan Cao is Assistant Professor, Department of Statistics, University of Missouri-Columbia, Columbia, MO 65201 (E-mail: [email protected]). Mathew M. Churpek is Assistant Professor, Department of Health Studies and Department of Medicine, University of Chicago, Chicago, IL 60637 (E-mail: [email protected]). Donglin Zeng is Professor (E-mail: [email protected]) and Jason P. Fine is Professor (E-mail: [email protected]), Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514. Donglin Zeng is partially supported by NIH grant R01CA082659 and P01 CA142538; Jason P. Fine is partially supported by NIH grant 5R01CA09483-07 and P01 CA142538.We thank Zhuang Ma for help with simulation and Trevor C Yuen for help with data preparation.

Jason P. Fine

Hongyuan Cao is Assistant Professor, Department of Statistics, University of Missouri-Columbia, Columbia, MO 65201 (E-mail: [email protected]). Mathew M. Churpek is Assistant Professor, Department of Health Studies and Department of Medicine, University of Chicago, Chicago, IL 60637 (E-mail: [email protected]). Donglin Zeng is Professor (E-mail: [email protected]) and Jason P. Fine is Professor (E-mail: [email protected]), Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514. Donglin Zeng is partially supported by NIH grant R01CA082659 and P01 CA142538; Jason P. Fine is partially supported by NIH grant 5R01CA09483-07 and P01 CA142538.We thank Zhuang Ma for help with simulation and Trevor C Yuen for help with data preparation.