成長曲線模型及Fieller-Creasy問題的精確方法之研究

Abstract

異質性一直是線性模型的主要問題之一。傳統的處理方法並無法在所有情況下得到精確的解，甚至在一些簡單的問題只要牽涉到擾亂變數，傳統的分析就會遇到困難。因此一般只能尋求估計方法來導出結論，即使知道這樣的做法在某些特定的樣本大小會估計的相當差。 根據這些事實，本論文將對著名的Fieller- Creasy問題、成長曲線模型以及多維度的變異數分析模型提出精確的推論，我們所根據的方法為 Tsui 和Weerahandi在 1989年所提出的廣義p值法及Weerahandi在 1993年所提出的廣義信賴區間。 我們先簡單介紹廣義p值法及廣義信賴區間的理論，然後提出Fieller-Creasy問題的廣義信賴區間，我們將針對這個問題提出兩個不同的方法：第一個方法直接根據兩個平均數的比值找出廣義的基準量(pivotal quantiy)；第二個方法則是把Fieller-Creasy問題當成一個假的Behrens-Fisher問題來做雙尾檢定，再根據廣義p值求得平均數比值的信賴區間。接著我們將考慮在成長曲線模型下包括單一處理樣本及多重處理樣本的精確檢定及未知參數的信賴區域，這些成長曲線模型假設具有組內相關結構或幾個組內相關結構的組合。最後，我們將對具異質性的多維度變異數分析提出精確檢定方法，Behrens-Fisher問題也包含在內。 文中亦包含了數值範例以及模擬的研究來說明研究結果的顯著性。根據結果顯示，當異質性很強時，最好能取消變異數相同的假設。利用廣義p值法不假設變異數相同所求的的結果較具實用性。

Heteroscedasticity is one of the major practical problems in linear models. Conventional methods do not always provide exact solutions to even some simple problems involving nuisance parameters. As a result, practitioners often resort to asymptotic results in search of approximate solutions even such approximations are known to perform rather poorly with typical sample sizes. In view of this fact, this dissertation will provide the generalized confidence intervals for the well-known Fieller-Cresay problem, the exact inferences for the the growth curve models and Multivariate ANOVA models under unequal intraclass correlation structures. Our approaches will be based on the notions of the generalized $p$-values and generalized confidence intervals which were proposed by Tsui and Weerahandi (1989) and Weerahandi (1993), respectively.\\ % \hspace*{0.6cm} We will start out with a brief introduction of generalized inferences, including generalized $p$-values and generalized confidence intervals. Next, we will provide generalized confidence interval for the ratio of means of two normal populations which is referred as the Fieller-Creasy problem. In this problem, we use two different procedures to find two potential generalized pivotal quantities. One procedure is to find the generalized pivotal quantity based directly on the ratio of means. The other is to treat the problem as a pseudo Behrens-Fisher problem through testing the two-sided hypothesis, and then to construct the confidence interval as a counterpart of generalized $p$-values. After that, we will consider exact tests for the pre-specified treatment effect in growth curve models with a single treatment and the exact tests for the equality of treatment effects in growth curve models with multiple treatment groups. These growth curve models analyzed are under intraclass correlation structure or combination of several intraclass correlation structures. Exact tests as well as generalized confidence regions using generalized $p$-values are obtained. Finally, we will provide the exact inference in multivariate ANOVA model under heteroscedasticity. The well-known Behren-Fisher problem is also under consideration.\\% \hspace*{0.6cm} Numerical examples and simulation studies are given to illustrate the importance of our results. According to our findings, we would be better off dropping the assumption of identical variance when the heteroscedasticity is serious. Therefore tests based on generalized $p$-values without the assumption of identical variance are much more practical than tests with this assumption.