¹Medtronic Bakken Research Center, Cardiac Rhythm Disease Management, Clinical Outcomes, Research and Biometry, Maastricht, Netherlands. Biostatistician. Email: javier.castaneda@medtronic.com
²Medtronic Bakken Research Center, Cardiac Rhythm Disease Management, Clinical Outcomes, Research and Biometry, Maastricht, Netherlands. Principal Statistician. Email: bart.gerritse@medtronic.com

During the disease-recovery process of many diseases, such as in Heart Failure (HF), often more than one type of event plays a role. Some clinical trials use the combined endpoint of death and a secondary event; for instance, HF-related hospitalizations. This is often analyzed with time-to-first-event survival analysis which ignores possible subsequent events, such as several HF-related hospitalizations. Accounting for multiple events provides more detailed information on the disease-control process, and allows a more precise understanding of the prognosis of patients.
In this paper we explore and illustrate several modelling strategies to study time to repeated events of disease-related hospitalizations and death per subject. Marginal models are revised in order to account for intra-subject correlation and competing risks. Finally, we recommend a Multi-state model which allows a flexible modelling strategy that incorporates important features in the analysis of HF-related hospitalizations and death, and at the same time extends relevant characteristics of the Andersen & Gill (1982), Wei et al. (1989) and Prentice et al. (1981) models.

Key words: Survival analysis, Competing risks, Correlated observations, Marginal models.

Algunos ensayos clínicos para estudiar el efecto de nuevos tratamientos en pacientes con insuficiencia cardiaca (IC) se basan en la evaluación de hospitalizaciones relacionadas con IC y muerte. Frecuentemente el análisis se enfoca en el tiempo a la primera ocurrencia de alguno de estos dos desenlaces. Este tipo de análisis ignora importantes eventos como nuevas hospitalizaciones relacionadas con IC, que permiten una mejor descripción y compresión del proceso de recuperación de estos pacientes.
En este trabajo se describen y exploran varias estrategias para el análisis de tiempo a repetidas hospitalizaciones relacionadas con IC y tiempo a la muerte. Se estudian modelos marginales para incorporar la correlación intra-sujeto y riesgos competitivos propios de este tipo de ensayos clínicos. Finalmente, se recomienda un modelo multi-estado como una estrategia sencilla y flexible que incorpora elementos importantes en el análisis de hospitalizaciones relacionadas con IC y muerte, y a la vez extiende características relevantes de los modelos de Andersen & Gill (1982), Wei et al. (1989) and Prentice et al. (1981).

Palabras clave: análisis de sobrevida, riesgos competitivos, observaciones correlacionadas, modelos marginales.

References

1. Adamson, P. B., Smith, A. L., Abraham, W. T., Kleckner, K. J., Stadler, R. W., Shih, A. & Rhodes, M. M. (2004), 'Continuous Autonomic Assessment in Patients with Symptomatic Heart Failure. Prognostic Value of Heart Rate Variability Measured by an Implanted Cardiac Resynchronization Device', Circulation 110, 2389-2394.         [ Links ]

2. Andersen, P. K., Borgan, O., Gill, R. D. & Keiding, N. (1993), Statistical Models Based on Counting Processes, Springer, New York, United States.         [ Links ]

3. Andersen, P. K. & Gill, R. D. (1982), 'Cox's Regression Model for Counting Pprocesses: A large Sample Study', Annals of Statistics 10, 1100-1120.         [ Links ]

4. Andersen, P. K. & Keiding, N. (2002), 'Multi-State Models for Event History Analysis', Statistical Methods in Medical Research 11, 91-115.         [ Links ]

5. Chung, E. S., Leon, A. R., Tavazzi, L., Sun, J. P., Nihoyannopoulos, P., Merlino, J., Abraham, W. T., Ghio, S., Leclercq, C., Bax, J. J., Yu, C. M., Gorcsan, J., Sutton, S. M., De Sutter, J. & Murillo, J. (2008), 'Results of the Predictors of Response to CRT (PROSPECT) Trial', Circulation 117, 2608-2616.         [ Links ]

6. Commenges, D. (2002), 'Inference for Multi-State Models From Interval-Censored Data', Statistical Methods in Medical Research 11, 167-182.         [ Links ]

7. Cook, R. J. & Lawless, J. F. (2002), 'Analysis of Repeated Events', Statistical Methods in Medical Research 11, 141-166.         [ Links ]

8. Cox, R. D. (1972), 'Regression Models and Life-Tables (with Discussion)', Journal on the Royal Statistical Society. Series B.         [ Links ]

9. Fleming, T. R. & Harrington, D. P. (1991), Counting Processes and Survival Analysis, John Wiley & Sons, Inc..         [ Links ]

10. Gheorghiade, M., Zannad, F., Sopko, G., Klein, L., Piña, I. L., Konstam, M. A., Massie, B. M., Roland, E., Targum, S., Collins, S. P., Filippatos, G. & Tavazzi, L. (2005), 'Acute Heart Failure Syndromes-Current State and Framework for Future Research', Circulation 112, 3958-3968.         [ Links ]

11. Grambsch, P. M. & Therneau, T. M. (1994), 'Proportional Hazards Tests and Diagnostics Based on Weighted Residuals', Biometrika 81, 515-526.         [ Links ]

12. Johnson, C. J., Boyce, M. S., Schwartz, C. C. & Haroldson, M. A. (2004), 'Modeling Survival: Application of the Andersen-Gill Model to Yellowstone Grizzly Bears', The Journal of Wildlife Management 68, 966-978.         [ Links ]

13. Kelly, P. J. & Lim, L. L. (2000), 'Survival Analysis for Recurrent Event Data: an Application to Childhood Infectious Diseases', Statistics in Medicine 19, 13-33.         [ Links ]

14. Klein, J. P. & Shu, Y. (2002), 'Multi-State Models for Bonemarrow Transplantation Studies', Statistical Methods in Medical Research 11, 117-139.         [ Links ]

15. Liang, K. Y. & Zeger, S. L. (1986), 'Longitudinal Data Analysis Using Generalized Linear Models', Biometrika 73(1), 13-22.         [ Links ]

16. Lipsitz, S. R., Laird, N. M. & Harrington, D. P. (1990), 'Using the Jackknife to Estimate the Variance of Regression Estimators from Repeated Measures Studies', Communication in Statistics. Theory and Methods 19, 821-845.         [ Links ]

17. Metcalfe, C. & Thompson, S. G. (2007), 'Wei, Lin and Weissfeld's Marginal Analysis of Multivariate Failure Time Data', Statistical Methods in Medical Research 16, 103-122.         [ Links ]

18. Prentice, R. L., Williams, B. J. & Peterson, A. V. (1981), 'On the Regression Analysis of Multivariate Failure Time Data', Biometrika 68, 373-379.         [ Links ]

19. Putter, H., Fiocco, M. & Geskus, R. B. (2007), 'Tutorial in Biostatistics: Competing Risks and Multi-State Models', Statistics in Medicine 26, 2389-2430.         [ Links ]

20. Solomon, S. D., Dobson, J., Pocock, S., Skali, H., McMurray, J. J., Granger, C. B., Tusuf, S., Swedberg, K., Young, J. B., Michelson, E. L. & Pfeffer, M. A. (2007), 'Influence of Nonfatal Hospitalization for Heart Failure on Subsequent Mortality in Patients with Chronic Heart Failure', Circulation 116, 1482-1487.         [ Links ]

21. Therneau, T. M. & Grambsch, P. M. (2000), Modelling Survival Data: Extending the Cox Model, Springer, New York, United States.         [ Links ]

22. Wei, L. J. & Glidden, D. V. (1997), 'An Overview of Statistical Methods for Multiple Failure Time Data in Clinical Trials', Statistics in Medicine 16, 833-839.         [ Links ]

23. Wei, L. J., Lin, D. Y. & Weissfeld, L. (1989), 'Regression Analysis of Multivariate Incomplete Failure Time Data by Modeling Marginal Distributions', Journal of the American Statistical Association 84, 1065-1073.         [ Links ]

24. Yu, C. M., Abraham, W. T., Bax, J. J., Chung, E. S., Fedewa, M., Ghio, S., Leclercq, C., Leon, A. R., Merlino, J., Nihoyannopoulos, P., Notabartolo, D., Sun, J. & Tavazzi, L. (2005), 'Predictors of Response to Cardiac Resynchronization Therapy (PROSPECT): study design', American Heart Journal 149, 600-605.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX: