在執行臨床試驗前必須先決定所需的樣本數，才能針對不同類型的疾病進行相關的研究以達到所需的檢定力。而復發型疾病在現今社會中是一種常見的類型，例如：心臟病。由於患者會歷經疾病反復發作，或是伴隨發生死亡的情況，讓人們對於此類型疾病的治療效果越趨重視。因此，隨著醫療的進步，針對不同的復發型疾病，也會相繼發展出許多新的治療方法。在發展出新的治療方法之後，為了瞭解新的治療方法跟原本的治療方法是否有差異，則需要進行非常嚴謹的臨床試驗，以檢定新的治療方法對於抑制復發型疾病的有效性。因此在上述臨床試驗執行前，決定樣本數為試驗的最開始，也是最重要的一步。本研究將採用聯合脆弱模型 （joint frailty models），對疾病的復發率與發生風險建立模型，透過估計模型中的參數，瞭解樣本數與檢定力的關係，最後提供不同參數組合下所建議的樣本數。本研究透過模擬研究計算出特定檢定力下所需的樣本數，並且可將此結果應用於不同復發率或死亡風險之復發型疾病的樣本數計算。

Recurrent events are frequently observed today, such as heart failures. We can conduct clinical trials to compare recurrent rate or mortality rate in different treatment groups. Before performing a clinical trial, we must determine the required sample size to achieve a target power. This study aims to calculate the sample size by using the joint frailty models based on interested the parameters of the models. The joint frailty models describe the rates of recurrent events and terminal event, which can be affected by the frailty term that follows a gamma distribution. This paper present simulation studies to evaluate the performance of sample sizes and power, such as under different settings of parameters or different estimating methods of baseline hazard function. Finally, we provide the sample sizes required for three examples.

[1] Anand, I. S., Win, S., Rector, T. S., Cohn, J. N. and Taylor, A. L. (2014). Effect of Fixed-Dose Combination of Isosorbide Dinitrate and Hydralazine on All Hospitalizations and on 30-Day Readmission Rates in Patients With Heart Failure Results From the African-American Heart Failure Trial. Circulation: Heart Failure, 7(5), 759-765.

[2] Charlson, M. E., Pompei P, Ales K. L. and MacKenzie, C. R. (1987). A new method of classifying prognostic comorbidity in longitudinal studies: development and validation. Journal of Chronic Diseases, 40(5), 373-383.

[3] Gary, R. J. (1992). Flexible Methods for Analyzing Survival Data Using Splines, With Applications to Breast Cancer Prognosis. Journal of the American Statistical Association, 87(420), 942-951.

[4] Ghosh, D., Lin, D.Y. (2000). Nonparametric Analysis of Recurrent Events and Death. Biometrics, 56(2), 554-62.

[5] Gonza ́lez, J. R., Fernandez, E., Moreno, V., Ribes, J., Peris, M., Navarro, M., Cambray, M. and Borra ́s, J. M. (2005). Sex differences in hospital readmission among colorectal cancer patients. Journal of Epidemiology and Community Health, 59, 506–511.

[6] Król, A., Mauguen, A., Mazroui,Z Y., Laurent, A., Michiels, S. and Rondeau, V. (2017). Tutorial in Joint Modeling and Prediction: A Statistical Software for Correlated Longitudinal Outcomes, Recurrent Events and a Terminal Event. Journal of Statistical Software, 81(3).

[7] Liu, L., Wolfe R. A. and Huang, X. (2004). Shared Frailty Models for Recurrent Events and a Terminal Event. Biometrics, 60(3), 747-756.

[8] Librero J., Peiro ́ S. and Ordin ̃ana R. (1999). Chronic comorbidity and outcomes of hospital care: length of stay, mortality, and readmission at 30 and 365 days. Journal of Clinical Epidemiology, 52(3), 171-179.

[9] Mack, M. J., Abraham, W. T., Lindenfeld, J., Bolling, S. F., Feldman, T. E., Graybur, P. A., Kapadia, S. R., McCarthy, P. M., Lim, D. S., Udelson, J. E., Zile, M. R., Gammie, J. S., Gillinov, A. M., Glower, D. D., Heimansohn, D. A., Suri, R. M., Ellis, J. T., Shu, Y., Kar, S., Weissman, N. J. and Stone, G. W. (2018). Cardiovascular Outcomes Assessment of the MitraClip in Patients with Heart Failure and Secondary Mitral Regurgitation: Design and rationale of the COAPT trial. American Heart Journal, 205, 1-11.

[10] Nielsen, G. G., Gill, R. D., Andersen, P. K. and Sorensen, T. I. A. (1992). A Counting Process Approach to Maximum Likelihood Estimation in Frailty Models. Scandinavian Journal of Statistics 19, 25-43.

[11] Ramsay, J. O. (1988). Monotone Regression Splines in Action. Statistical Science, 3, 425-261.

[12] Rondeau, V., Mathoulin-Pélissier, S., Jacqmin-Gadda, H., Brouste, V. and Soubeyran, P. (2007). General joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events. Biostatistics, 8(4), 708-721.