在生物醫學研究的過程當中，除了事件時間之外還經常收集到長期追蹤資料 (longitudinal data)。一般最常使用比例風險模型 (proportional hazard model, Cox model) 推估時間相依共變數 (time-dependent covariate) 與存活時間的關聯性。然而，由於此方法需要病人的完整資訊，造成了資料收集之困難，所以將利用聯合模型 (joint model) 的概念來對資料做分析。此模型包含兩大部分：其一為長期追蹤資料，其二為存活資訊。在第一部分使用線性隨機效應模型 (linear random effect model) 處理長期追蹤資料，第二部分則是使用Cox模型描述共變數與存活時間之間的關係。結合這兩部分建構出聯合模型且利用EM演算法 (expectation maximize algorithm) 求得參數之最大概似估計值 (maximum likelihood estimate, MLE)。因在EM演算法中，許多積分並無封閉解，故必須使用數值積分，本研究之目的即是比較不同數值積分方法對於聯合模型參數估計值之影響。最後則藉由愛滋病和地中海果蠅資料驗證所估計參數之變化。

Time-dependent covariates along with survival information are very common to be collected at the same time in many medical researches. For such kind of data, it is very popular to use Cox model to study the relationship between time-dependent covariates and the survival time. However, the partial likelihood requires the complete covariate information from the patients, which is usually not available in many medical researches. Joint model approach is a solution to analyze such kind of data. The longitudinal data is described by a linear random effects model, and the survival time is fitted by the Cox model. To derive the parameter estimates, EM algorithm is implemented to search for the maximum likelihood estimates. However, the expectation part in the EM algorithm involves multiple integration which has no closed-form and must be solved by the numerical integration. The purpose of this research is to compare different numerical integration methods for parameter estimation of the joint model. We demonstrate the properties of the estimation through studies of AIDS and medfly data.

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