因為估計量的一致性和半母數模型的有效性，聯合概似模型已廣泛的被使用在同時處理長期和存活資料上。當加速失效模型被運用於聯合概似模型存活資料的部分時，沒有特定形式的基礎風險函數通常被假設為分段函數，而這種不平滑的階層函數假設會導致劇烈跳動的概似函數，並且造成得到的參數估計值不是對應到概似函數的極大值。同時，因為滑概似函數不平滑的情形，直接尋找估計值的方法被用來得到參數的最大概似估計，但此一方法對於參數收斂的速度是相當緩慢且耗時。為了解決這些困難，我們提出平滑擬似-概似函數的方法來取代原本不平滑的階層平滑假設。除此之外，為了增加參數估計的效及減少計算所耗費時間，我們使用Gaussian Quadrature取代蒙地卡羅方法來計算EM演算法中M-step的積分，藉以得到更加正確的積分結果。另一方面，平滑擬似-概似函數也推廣到多重存活時間與脆弱模型的分析上。為了實務的方便，我們發展一MATLAB程式，程式中包含存活部分的半母數風險迴歸模型 (Cox 比例風除和加速失效模)以及長期資料部分的混合效用有母數和無母數基底的資料分析。 Joint likelihood approaches have been widely used to handle longitudinal and survival data at the same time because the estimation is consistent and semi-parametrically efficient. When accelerated failure time (AFT) model is employed as the survival component of the joint likelihood, the unspecified baseline hazard function is usually assumed to be a piecewise constant function. The non-smooth step function leads to very spiky likelihood function which is very hard to find the globe maximum. Moreover, due to non-smoothness of the likelihood function, direct search methods are used for maximization, which is very slow for parameter convergence and time consuming. Thus, to overcome the difficulties, we proposed a kernel smooth pseudo-likelihood function to replace the non-smooth step function. Besides, we replace MC integration method by the Gaussian quadrature approximation to obtain a more accurate numerical integration. The kernel smooth pseudo-likelihood approach can be extended to multivariate survival time cases by incorporating with frailty. For practical purpose, we use MATLAB to develop a program for popular join model approaches in the recent literature. The program includes semi-parametric hazard regression models, the Cox model and the AFT model, for the survival component, and parametric basis as well as nonparametric basis of mixed effects model for the longitudinal component.
