競爭風險之下對邊際分配的估計，是許多領域中常遇到的問題。由於主要事件及次要事件互相競爭，只要一種事件先發生即終止對另一事件的觀察，在兩事件同時發生的機率為0之下，連一筆完整的資料我們都無法蒐集到。除非兩事件互為獨立或加上其它條件，否則會有邊際分配無法識別的問題。但是獨立的條件在有些情況下並不合理，為解決相依競爭風險之邊際分配無法識別的問題，可先假定兩事件發生時間之間的關係。 由於關聯結構定義出兩變數間的結合關係，我們可利用關聯結構解釋兩事件發生時間之間的關係。假定兩變數之相關性參數為已知，且採用機率積分轉換的觀念，本論文討論了Zheng 與 Klein提出的關聯結構-圖形估計量，是否會依設限程度、相關性強度和關聯結構形式的不同，以致估計能力有別。 / The problem of estimating marginal distributions in a competing risks study is often met in scientific fields. Because main event and secondary event compete with each other, and a first occurring event prevents us from observing another event promptly, the intact lifetimes or survival times are unable to be collected in the circumstances that the probability of both lifetimes coinciding is 0. Unless lifetimes being independent or adding other conditions, there is a problem that the marginal distributions are non-identifiable. But the condition of independence is not always reasonable, we may assume the relation between lifetimes has some special form Because the copula defines the association between two variables, it can be employed to explain relation between lifetimes. Assuming that the dependence parameter in the copula framework is known, and adopting the concept of the probability integral transformations, this thesis has demonstrated whether the estimating abilities of the copula-graphic estimator, that Zheng and Klein put forward, are different in rates of censoring, intensities of dependence, and forms of the copula.

進行調查時，最後的抽樣單位常是從不同的群集取得的，而同一群集內的樣本對象，因背景類似而對於某些問題常會傾向相同或類似的反應，研究者若忽略這種群內相關性，仍以獨立性樣本進行分析時，因其共變異數矩陣通常會與多項模式的共變異數矩陣相差懸殊，而造成所謂的額外變異或不足變異的現象。本文在不同的情況下，提出了Dirichlet-Multinomial模式(簡稱DM模式)、擴展的DM模式、以及兩種平均數-共變異數矩陣模式，以適當的彙整所有的群集資料。並討論DM與EDM模式中相關之參數及格機率之最大概似估計法，且分別對此兩種平均數-共變異數矩陣模式，提出求導廣義最小平方估計的程序。此外，也針對幾種特殊的二維表及三維表結構，探討對應的參數及格機率之估計方法。並提出計算簡易的Score統計檢定量以判斷群內相關(intra-cluster correlation)之存在性，及判斷資料集具有額外或不足變異，而對於不同母體的群內相關同質性檢定亦提出討論。 / This paper presents a modelling method of analyzing categorical data with overdispersion or underdispersion.　In many studies, data are collected from differ clusters, and members within the same cluster behave similary.　Thus, the responses of members within the same cluster are not independent and the multinomial distribution is not the correct distribution for the observed counts.　Therefore, the covariance matrix of the sample proportion vector tends to be much different from that of the multinomial model.　We discuss four different models to fit counts data with overdispersion or underdispersion feature, witch include Dirichlet-Multinomial model (DM model), extended DM model (EDM model), and two mean-covariance models.　Method of maximum-likelihood estimation is discussed for DM and EDM models.　Procedures to derive generalized least squares estimates are proposed for the two mean-covariance models respectively. As to the cell probabilities, we also discuss how to estimate them under several special structures of　two-way and three-way tables.　More easily evaluated Score test statistics are derived for the DM and EDM models to test the existence of　the intra-cluster correlation.　And the test of homogeneity of intra-cluster correlation among several populations is also derived.