在植牙研究中，研究者欲了解患者在植入假牙後，假牙植體是否發生併發症，並試圖探討患者植入假牙後發生併發症的危險因子。由於患者在進行植牙手術時，植入的牙體數目略有異同，所以蒐集到的實際資料將為集群或多變量資料(clustered 或multivariate data)。此外，由於多數患者的假牙植體不會發生術後併發症，這些個體被稱之為治癒個體(cure 或 non-susceptible)。一般而言，集群內的資料間將具一定的相關程度。然而，考慮治癒率的文獻還僅限於單變量的離散反應變數。
本論文將針對集群離散存活資料，採用Zhao與Zhou (2008)所提出的等比例混合治癒模式，並利用Yu與Peng (2008)處理集群連續資料之想法，即先忽略集群內相關性而建構所謂邊際概似函數 (marginal likelihood function)，用以求模式中的參數估計方程式，再由此估計方程式求參數估計量。接著利用模擬的方式來驗證參數估計量的一致性，並在不同設限比例與不同集群數下探討參數估計量之表現。最後將此推廣方法應用於實際植牙資料，以協助探討影響植牙併發症的危險因子。

One of the goals of a dental implant study (Wen et al., 2008) is to identify the risk factors associated with dental implant failure based on a 7-years follow up. Since Wen et al. applied life-table method to estimate failure rate, the failure times are only available in yearly interval which can be regard as clustered discrete-time survival data.
Therefore, this thesis applied a marginal regression approach applied by many authors, such as Lipsitz et al. (1994), and Yu and Peng (2008), to estimate the parameters in proportional cure model developed in Zhao and Zhou (2008). The accuracy of the estimators is investigated by a simulation study. When the number of clusters are large enough, the bias of the estimators of regression parameters become smaller. Finally, the proportional cure model is applied to the dental implant data set to identify the risk factor associated with dental implant failure.

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