本篇論文主要考慮成群資料的存活分析，其特點為群內個體間具有相關性，並假定群內個體具有相同但無法觀測到的事故傾向。首先，探討事故傾向服從任一連續分配時混合Weibull迴歸模式的特性，接著，推導出事故傾向服從血Inverse Gaussian吧時之混合Weibull模式，並介紹參數的估計問題。然後，推導出群內個體是否獨立之分數檢定統計量，以分別就兩種最常見的存活資料型態一完整型態與右設限型態：檢定模式中事故傾向的效應是否存在。最後，並以實例說明分數檢定之程序。 / In this paper, we study survival analysis for grouped data, where the within group correlations are considered. It is also assumed that individuals within the same group share a common but unobservable random frailty. First, we discuss the properties of the Weibull regression model mixed by any continuous distribution. Next, we derive an Inverse Gaussan mixture of Weibull regression model, and discuss the estimation problem. Then, we derive the score test for testing independence between components within the same group, where the two most common cases are discussed the complete data case and the right censoring case. Finally, the testing procedures are illustrated by two examples.

國中基測實施迄今已十年，但是各種多元管道仍以國中基測量尺分數作為分發篩選之重要參據，多元能力評量參採比重偏低，國中學生升學競爭壓力未得緩解。本研究透過數學與統計分析的工具，尋找採用學生在校成績的方法，希望能找出更好的方式來代表學生在校三年的學習現況與學習成果，以做為免試升學採計在校成績的參考與依據。 本研究主要目的是要探討如何取決各科在校成績的權重(也就是在每個科目的分數之前乘上一個加權比重係數)，以求得一個新的合成變量(由數個科目分數組成的線性組合)，並用這個新合成變量做為學生在校的“綜合學科能力表現分數”，代表學生在校三年的基本學習能力及程度。 研究方法運用主成份分析與典型相關分析的觀念，但因限制條件設定的範圍與傳統主成分分析及典型相關分析的要求不一致，因此，我們便將所用的研究方法命名為「類主成分分析」與「類典型相關分析」。 研究中，方法主要在比較「類主成分分析」、「主成分分析」、「類典型相關分析」、以及「典型相關分析」四種方法與一般學校常用的「等加權比重」算平均成績的方法之分別；了解這些不同加權機制對同一所學校內學生的學科加權平均分數之成績排名百分比結果，以及與基測排名結果的差異。 「類主成份分析」研究結果發現，各科學科成績中變異數大的科目將獲得較大的權重比例，成為主導學生加權平均成績中舉足輕重的科目。另外；運用「類典型相關分析法」所求得的典型相關係數，其結果與傳統典型相關分析法以及使用最佳數值分析軟體(GAMS)所得的典型相關係數完全相同。 本研究最重要的貢獻之一，是我們在「類典型相關分析法」中證明並推導出一個求得各科權重的公式，只要使用此公式代入簡單的MATLAB程式，其所得的權重結果與最佳化數值分析軟體(GAMS)所得的結果完全相同，但花費的計算時間及成本卻遠少於GAMS所需，是一個求權重極便捷的方法，讀者可以在本論文附錄7.5.2或政大應數系網站上下載此程式。本研究最後結論也發現，類主成份分析的變異解釋率是所有方法中較高的；與基測總分結果較相近的則是類典型相關分析所得的權重機制；而等加權方法所得的排名結果則與基測排名結果差異最小。 / The BCTEST (Basic Competence Test) for junior high school students has been implemented for ten years, however, the screenings for a variety of entrance programs are still based on the scale scores of the BCTEST with a low proportion of multi-intelligence. Hence, the competitive entrance pressure for junior high school students remains un-relieved. In view of this, via mathematics and statistics, this study is to explore an alternative approach which can not only reflect students' in-school grade, their learning situations and achievements but also represent a reference for entering senior high schools and junior colleges without entrance examinations. The purpose of this study is to determine the different weightings of five learning subjects (that is, multiply the score of each subject by a weighted coefficient) and acquire a new composite variable from the linear combinations of five learning subjects. Then, use this new composite variable as the synthetic score of students' in-school academic performance. Principal Component Analysis and Canonical Correlation Analysis are used in this study. Due to inconsistent restraints, the other two approaches we use are based on the concept of previously mentioned methodologies and denominated Principal Component Type of Analysis and Canonical Correlation Type of Analysis. In the study, we compare with the different results of Principal Component Analysis, Principal Component Type of Analysis, Canonical Correlation Analysis, Canonical Correlation Type of Analysis and identical weighted method to realize how these different weighted schemes affect the rankings of students from the same school on both their weighted in-school grade and scores of the BCTEST (Basic Competence Test). The outcomes of Principal Component Type of Analysis show that subjects with greater variance acquire larger weightings and play a dominant role in weighted in-school grade. Moreover, the correlation coefficients of Canonical Correlation Type of Analysis are completely the same as the ones of Canonical Correlation Analysis and GAMS. One of the most important contributions in this study is we have proven and derived a formula to acquire different weightings of five learning subjects by using Canonical Correlation Type of Analysis. The acquired weightings are completely the same as the ones of GAMS with less time consuming. Readers can download this program in appendix 7.5.2 or from the website of Department of Mathematical Sciences, National Chengchi University(NCCU). We have also found that, the explanation rate of variance obtained from Principal Component Type of Analysis is the highest; the weighted scheme of Canonical Correlation Type of Analysis is more similar to the scores of the BCTEST; the difference of the rankings between identical weighted method and the BCTEST is the smallest.

本研究目的，乃在運用試題反應理論中最大測驗訊息量的概念於精熟標準設定上作為探討的主軸，透過其歷史的演進與發展，衍生出詮釋本研究最大測驗訊息量法的三個面向，分別為：元素的搭配組合與調整、廣義測驗建構流程、多元效度等，並以此概念賦予解釋運用最大測驗訊息量於精熟標準設定時的合理性與適切性。同時，確立最大測驗訊息量法於公式意涵、試題選擇與統計考驗力面向的合理性，建立其於精熟標準上的理論基礎，而後，再輔以精熟/未精熟者分類一致性信度值以期提供多元效度證據。最後，探討測驗分數的轉換方法、差異能力描述，期能同時獲得量與質的測驗結果解釋。 綜整分析，可發現以下幾點結論： 一、運用最大測驗訊息量法於精熟標準設定時，在分類的信度指標上，顯示由此求得精熟標準，經交叉驗證後，大致可獲得滿意的結果，皆有高達九成以上的精確分類水準，且藉由區間的概念亦能充分顯現出，以最大測驗訊息量法求得之標準，可作為專家設定精熟標準時參考、判斷的優勢。而在分數轉換上，不論搭配換算古典測驗分數法或測驗特徵曲線構圖法時，其分類精熟/未精熟者的一致性表現，大致可獲得滿意的結果，乃是值得參照的組合策略。 二、在運用定錨點以解釋由最大測驗訊息量法於國中基本學力測驗求得之精熟標準時，可發現未精熟者乃僅需具備學科基礎知識與簡易圖示理解能力，而對於精熟者而言，則需進一步擁有對於廣泛學科知識的了解；複雜問題、資料與圖表詮釋；邏輯推理、分析實驗結果以獲得相關論點等能力，或者更高階之具備進階學科知識；綜合、評鑑資料、情境傳遞之訊息的能力。 三、探討測驗長度因素時，分析結果顯示不論採行最大測驗訊息量法、換算古典測驗分數法或是測驗特徵曲線構圖法，皆受此因素的影響，顯示測驗長度愈長，分類一致性愈高，此項結果乃與過去大多數的研究一致。另，由本資料分析結果乃建議測驗長度20題時，會是必備的基本題數要求值。此外，若從細部精確錯誤分類人數角度分析時，於實務用途上，可發現對於影響轉換分數時，產生差異分數的因素，決策者並不容易掌握與控制，但卻可藉由增加測驗長度，分散分數點的人數，以彌平錯誤分類的影響。 四、探討測驗異質性因素時，最大測驗訊息量法因具有因試題參數而調整估計受試者能力的特性，使得在異質測驗時，分類一致性仍能維持在不錯的水準之上。反觀換算古典測驗分數法與測驗特徵曲線構圖法，在固定精熟標準下，則有明顯的錯誤分類比率，此現象也反應出現行以固定60分作為及格(精熟)標準的缺失。 五、探討採用簡易測驗、困難測驗或常態測驗間於轉換分數上之效果時，由換算古典測驗分數法或測驗特徵曲線構圖法轉換來自最大測驗訊息量法之精熟標準時，資料分析結果顯示，不論於何種測驗難度類型中，採用何種轉換方式，並不會嚴重影響轉換分數間一致性分類的效果。另，若從細部精確錯誤分類人數角度分析時，本研究所採之最大測驗訊息量法，因具備隨測驗難易程度來決定門檻的特性，於簡易測驗中求得之精熟標準較低，而於困難測驗中求得之精熟標準相對較高，使得於轉換分數上，即使有較大的差異分數，亦不會造成嚴重的錯誤分類人數。 六、在探討測驗長度、測驗異質性因素與定錨點題目篩選間互動關係時，分析結果顯示，測驗長度與測驗異質性，並非是絕對影響定錨點題目篩選的因素，更重要的在於最大試題訊息量所對應之最適能力值是否能與定錨點相搭配。 綜整之，本研究所採最大測驗訊息量法，經檢驗後，於分類一致性上乃具有不錯的表現，且搭配相對強韌、嚴謹的理論支持與適切測驗結果解釋方法等，是最適合用於大型考試上使用。因此，乃建議未來政府單位或實務工作者於進行大型證照、資格檢定考試時，可考慮使用本策略。 / The purpose of this study is to adopt the concepts of IRT maximum test information to standard setting. At first, we are trying to discover three facets of interpretation in using the maximum test information to standard setting through the historical movement of standard setting. The three facets are component combination and adjustment, generalized test construction processes and multiple validities. Depending on these three concepts, we can easily explain the reasonableness and appropriateness of maximum test information approach. After that, we further investigate the reasonableness from the dimensions of definition of formula, item selections and statistical power to establish the basic theory of the maximum information approach in standard setting. In addition, we also examine the effects on exact classification of master/non-master in expectation to provide multiple evidences for validity. Finally, the method of classical test scores transformation and difference ability description are discussed to provide quantitative and qualitative test result interpretation simultaneously. In sum, some conclusions are proposed. 1.In applying the maximum test information approach to standard setting, the effect on exact classification of master/ non-master may come to a satisfying result. We may have at least 90% exact classification performance. At the same time, we also find that the mastery standard deriving from the maximum test information approach may have some advantages being a starting reference point for experts to adjust on the basis of the view of confidence interval. In the aspect of classical test scores transformation, no matter what approach you take, the transformed classical test scores approach or test characteristic curve mapping method, the consistency of exact classification of master/ non-master may hold. We may suggest the combination strategy is really worthy to take into consideration in standard setting. 2.In applying the anchor point to interpret Basic Competency Test result, we may find non-master only has basic academic knowledge and simple graph understanding ability, but for the master, he may need extensive academic knowledge; ability of complicated problems、data and graph interpretation; logic reasoning、analyzing experimental result to get related issues. Moreover, advanced academic knowledge; ability of synthesizing and evaluating information from data and surroundings are also included. 3.In the aspect of test length, the result of this research shows no matter what approach you take, maximum test information approach、transformed classical test scores approach or test characteristic curve mapping method, they are all influenced. It shows the longer test length, the higher consistency of exact classification of master/non-master. This result is consistent to most of the studies in the past. On the other hand, we suggest the 20 items is a fundamental value. Moreover, from the view of exact number of error classification, we can find that the real factor affecting the difference scores in transforming classical test score is unable to control in practical usage, but we can just disperse the numbers of people in each test score point to reduce the influence of error classification by increasing test length. 4.In the aspect of diverse test difficulty, because the maximum test information approach possesses the characteristic of examinees’ ability adjustment depending on item parameters, it is less influenced to maintain a acceptable level of consistent classification. In contrast with the maximum test information approach, the transformed classical test scores approach and test characteristic curve mapping method may have obvious high ratio of error classification under the fixed mastery standard. This also reflects the deficiency of current fixed 60 points passing scores. 5.In the aspect of analyzing the effect of score transformation between easy、hard and normal test, this research shows no matter what approach you take in any type of test difficulty, they may not severely influenced. Furthermore, from the view of exact number of error classification, because the maximum test information approach possesses the characteristic of deciding passing level depending on the degree of test difficulty (the lower mastery standard in easy test and the higher in hard test), it may not lead to a severe error classification even if there exists a large difference score in classical test score transformation. 6.In the aspect of interaction between test length、diverse test difficulty and anchor items selection, this research shows that test length and diverse test difficulty are not the real factors affecting anchor items selection. The more accurate cause is if the mastery standard deriving from the maximum test information approach may coordinate with the anchor point or not. In sum, the maximum test information approach may not only lead to a satisfying exact classification performance after analysis, but also be supported by strong and strict theory and accompany proper test result interpretation method. It is the most proper method in standard setting for large-sized test. Finally, we suggest the government or practitioners may consider adopting this strategy for future usage.

本研究使用「臺灣教育長期追蹤資料庫」第二波（2003）國三與第三波（2005）高二的CP追蹤數據，採用反事實模型框架下的傾向分數配對法估計單一性別環境對臺灣高中女生選組行爲（自然組／社會組）的因果效應。使用學生進入高中前的國三資料，本研究在學生因素，家庭因素和學校、班級因素三個方面充分平衡處置組（進入女校）和控制組（進入男女合校）的樣本，使配對樣本之間除了進入女校或進入男女合校之外，在以上三個方面儘可能相等，嘗試克服觀察性數據因選擇性偏誤問題對因果效應的估計帶來的阻礙。結果顯示，在進行傾向分數配對後，處置組和控制組之間達到了很好的平衡，進入女校的女生與進入男女合班的男女合校中的女生相比，女校顯著地促進了高中女生選自然組的機率，然而該效應在數學成績水平不同的女生中存在異質性，女校顯著地促進了數學成績處於高水平的女生的選擇自然組的行爲。作爲教育分流的重要組成部分，臺灣高中生選組行爲的性別隔離長期存在，選組行爲與學生未來大學科系選擇和職業選擇關係密切，自然組中低比例的女生組成顯現出隱藏的教育機會不平等。該現象長期存在的背後因素，除了學生個人因素和學生家庭背景因素帶來的影響，本研究關注學校這一角色對高中女生選組行爲的影響。 / Based on the Taiwan Educational Panel Survey’s core panel data from wave 2 (junior high school/9th grade) and wave 3 (high school/11th grade), researcher applies the propensity score analysis in counterfactual framework to study the causal effect of single-sex schooling on high school girls’ curriculum tracking (science track or humanity track) selection. Observational data always be the obstacles of making the causal analysis because it's lacking random assignment and being under threat of selection bias and unobserved variables. Using the pretreatment variables from students in 9th grade, students in the treatment group were matched with those in control groups by the individual factors, family factors and school, classroom environment factors. After matching, students come from different groups are supposed to be relatively equivalent on all the matching variables except for attending single-sex school or not. The researcher can make suitable comparison and prediction between well-matched samples. The result shows that more girls in single-sex high schools tend to choose science track. Effect of single-sex schooling is not the same among girls with different level of math scores. It benefits girls with top math performance significantly. As an important part of the educational tracking system, alternative curriculum tracking selection in Taiwan high school is worth noteworthy. The long lasting gender segregation is closely associated with students college major as well as occupation selection and it reveals the hidden educational inequality between gender. Other than focusing on the individual and family background effect on high school girls’ curriculum tracking selection, this study highlights the school effect on it.

在迴歸分析中，當資料中存在很多離群值時，偵測的工作變得非常不容易。 在此狀況下，我們無法使用傳統的殘差分析正確地偵測出其是否存在，此現象稱為遮蔽效應(The Masking Effect)。 而為了避免此效應的發生，我們利用最小中位數穩健迴歸估計值(Least Median Squares Estimator)正確地找出這些群集離群值，此估計值擁有最大即50﹪的容離值 (Breakdown point)。 在這篇論文中，用來求出最小中位數穩健迴歸估計值的演算法稱為步進搜尋演算法 (the Forward Search Algorithm)。 結果顯示，我們可以利用此演算法得到的穩健迴歸估計值，很快並有效率的找出資料中的群集離群值；另外，更進一步的結果顯示，我們只需從資料中隨機選取一百次子集，並進行步進搜尋，即可得到概似的穩健迴歸估計值並正確的找出那些群集離群值。 最後，我們利用鐘乳石圖(Stalactite Plot)列出所有被偵測到的離群值。 在多變量資料中，我們若使用Mahalanobis距離也會遭遇到同樣的屏蔽效應。 而此一問題，隨著另一高度穩健估計值的採用，亦可迎刃而解。 此估計值稱為最小體積橢圓體估計值 (Minimum Volume Ellipsoid)，其亦擁有最大即50﹪的容離值。 在此，我們也利用步進搜尋法求出此估計值，並利用鐘乳石圖列出所有被偵測到的離群值。 這篇論文的第二部分則利用變數轉換的技巧將迴歸資料中的殘差項常態化並且加強其等變異的特性以利後續的資料分析。 在步進搜尋進行的過程中，我們觀察分數統計量(Score Statistic)和其他相關診斷統計量的變化。 結果顯示，這些統計量一起提供了有關轉換參數選取豐富的資訊，並且我們亦可從步進搜尋進行的過程中觀察出某些離群值對參數選取的影響。 / Detecting regression outliers is not trivial when there are many of them. The methods of using classical diagnostic plots sometimes fail to detect them. This phenomenon is known as the masking effect. To avoid this, we propose to find out those multiple outliers by using a highly robust regression estimator called the least median squares (LMS) estimator which has maximal breakdown point. The algorithm in search of the LMS estimator is called the forward search algorithm. The estimator found by the forward search is shown to lead to the rapid detection of multiple outliers. Furthermore, the result reveals that 100 repeats of a simple forward search from a random starting subset are shown to provide sufficiently robust parameter estimators to reveal multiple outliers. Finally, those detected outliers are exhibited by the stalactite plot that shows greatly stable pattern of them. Referring to multivariate data, the Mahalanobis distance also suffers from the masking effect that can be remedied by using a highly robust estimator called the minimum volume ellipsoid (MVE) estimator. It can also be found by using the forward search algorithm and it also has maximal breakdown point. The detected outliers are then displayed in the stalactite plot. The second part of this dissertation is the transformation of regression data so that the approximate normality and the homogeneity of the residuals can be achieved. During the process of the forward search, we monitor the quantity of interest called score statistic and some other diagnostic plots. They jointly provide a wealth of information about transformation along with the effect of individual observation on this statistic.