最佳降維分數之估計及其延伸至設限資料

Abstract

排序指標— 用來評估反應值y 及其對應p 個解釋變數的一致性, 由於它在存活分析的應用層面非常廣泛，所以長期以來一直被關注。傳統上，多個解釋變數的線性結合被用來當作一個分數來看y 關係，在本論文中，我們發展一個新的排序指標，將線性結合延伸至多維多項式結合，並且能充分解釋y 所包含x 的最大訊息。這新的指標幫助我們更容易用圖形來分析迴歸關係。為了引進這樣的概念，一個廣義的遞增模型—描述y 與一個真實多項式分數有遞增關係，是需要的。 而真實多項式分數是由最佳的解釋變數降維子空間所建立出來的最小次數多項式。由這個模型，C 指標被定出來，並且對於這個指標，我們引進了所謂的「最佳SDR 分數」— 具有最大性，唯一性和最佳性。善用多項式可展成廣義性線的特質，我們發展排序基礎的貝式訊息判定，來估計未知的多項式次數以及降維空間的維度。當p 很大的時候，我們也提供顯著性變數選擇法將不重要的變數排除。此外，我們發展一個有效的演算法，來計算這數量不小的廣義線性所對應的係數值。更進一步，利用外積微分法來估計最佳SDR 分數以及C 指標。我們也提出另一個方法— 半母數參數化方式計算C 指標最大值，來得到估計式。存活分析當中，許多資料因為被設限而無法完全觀測到。在這種情況之下，我們發現，用部分排序法的概念，處理完整資料的估計程序可以直接套用。就C 指標來說，利用可觀測資訊來補足不可觀測的二項式計數過程，使我們得到估計式。最後，我們設計一系列的模擬和實務資料來驗證我們方法論在分析上優勢和廣泛應用。

Rank-based measures, which is used to access the concordance between the univariate response variable y and a linear composite score of its p-dimensional explanatory variable z, has been studied because of its applicability to a wide variety of survival data. In this article, a new rank-based measure is developed for extending a linear score to a multivariate polynomial one, which captures the most information of z with respect to y. This new measure explores the simplicity of the graphical view of regression; that is, we can regress y against this multivariate polynomial score based on dimension reduction framework. To introduce this concept, a general semiparametric model, which characterizes the dependence of a response on covariates through a multivariate polynomial transformation of the central subspace (CS) directions with unknown structural degree and dimension, is proposed. In light of the monotonic model structure and defined concordance index (C-index) function, such a composite score, which is referred to as the optimal sufficient dimension reduction (SDR) score, is shown to enjoy the existence, optimality, and uniqueness up to scale and location. By means of these properties and the generalized single-index (SI) representation of any multivariate polynomial function, the concordance-based generalized Bayesian information criterion (BIC) is proposed to estimate the optimal SDR score and its corresponding C-index, say Cmax. Meanwhile, effective computational algorithms are offered to carry out the presented estimation procedure. With estimated structural degree and dimension from this BIC, an alternative approach is further developed to estimate the optimal SDR score and Cmax. In addition, we establish the consistency of structural degree and dimension estimators and the asymptotic normality of optimal SDR score and Cmax estimators. As for significant covariates, a variable selection is proposed to retain important confounding variables when p is large. In general, survival data is partially observed due to right-censoring. In this case, a partial rank-based approach allows us to follow the similar estimation procedure with the completed data. Further, we adopt an imputation method to recover unobserved counting process for estimating Cmax. The performance and practicality of our proposal are also investigated by a series of simulations and illustrated examples.