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研究生:盧宏益
研究生(外文):Lu, Hung-Yi
論文名稱:自變數有測量誤差的羅吉斯迴歸模型之序貫設計探討及其在教育測驗上的應用
論文名稱(外文):Sequential Designs with Measurement Errors in Logistic Models with Applications to Educational Testing
指導教授:張源俊張源俊引用關係
指導教授(外文):Chang, Yuan-Chin
學位類別:博士
校院名稱:國立政治大學
系所名稱:統計研究所
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:英文
論文頁數:74
中文關鍵詞:電腦化適性測驗線上校準測量誤差序貫設計變動長度試題反應理論試題校準
外文關鍵詞:Item Response TheoryComputerized Adaptive Testingonline calibrationmeasurement errorsequential designsequential estimationstopping timevariable lengthitem calibration
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本論文探討當自變數存在測量誤差時,羅吉斯迴歸模型的估計問題,並將此結果應用在電腦化適性測驗中的線上校準問題。在變動長度電腦化測驗的假設下,我們證明了估計量的強收斂性。試題反應理論被廣泛地使用在電腦化適性測驗上,其假設受試者在試題的表現情形與本身的能力,可以透過試題特徵曲線加以詮釋,羅吉斯迴歸模式是最常見的試題反應模式。藉由適性測驗的施行,考題的選取可以依據不同受試者,選擇最適合的題目。因此,相較於傳統測驗而言,在適性測驗中,題目的消耗量更為快速。在題庫的維護與管理上,新試題的補充與試題校準便為非常重要的工作。線上試題校準意指在線上測驗進行中,同時進行試題校準。因此,受試者的能力估計會存在測量誤差。從統計的觀點,線上校準面臨的困難,可以解釋為在非線性模型下,當自變數有測量誤差時的實驗設計問題。我們利用序貫設計降低測量誤差,得到更精確的估計,相較於傳統的試題校準,可以節省更多的時間及成本。我們利用處理測量誤差的技巧,進一步應用序貫設計的方法,處理在線上校準中,受試者能力存在測量誤差的問題。
In this dissertation, we focus on the estimate in logistic
regression models when the independent variables are subject to some measurement errors. The problem of this dissertation is motivated by online calibration in Computerized Adaptive Testing (CAT). We apply the measurement error model techniques and adaptive sequential design methodology to the online calibration problem of CAT. We prove that the estimates of item parameters are strongly consistent under the variable length CAT setup. In an adaptive testing scheme, examinees are presented with different sets of items chosen from a
pre-calibrated item pool. Thus the speed of attrition in items will be very fast, and replenishing of item pool is essential for CAT. The online calibration scheme in CAT refers to estimating the item parameters of new, un-calibrated items by presenting them to examinees during the course of their ability testing together with previously calibrated items. Therefore, the estimated latent trait levels of examinees are used as the design points for estimating the parameter of the new items, and naturally these designs, the estimated latent trait levels, are subject to some estimating errors. Thus the problem of the online calibration under CAT setup can be formulated as a sequential estimation problem with measurement errors in the independent variables, which are also chosen sequentially. Item Response Theory (IRT) is the most commonly used psychometric model in CAT, and the logistic type models are the most popular models used in IRT based tests. That's why the nonlinear design problem and the nonlinear measurement error models are involved. Sequential design procedures proposed here can provide more accurate estimates of parameters, and are more efficient in terms of sample size (number of examinees used in calibration). In traditional calibration process in paper-and-pencil tests, we usually have to pay for the examinees
joining the pre-test calibration process. In online calibration,
there will be less cost, since we are able to assign new items to the examinees during the operational test. Therefore, the proposed procedures will be cost-effective as well as time-effective.
1 Introduction 1
2 Experimental Design in Regression Models 6
2.1 Designs in linear regression models 6
2.2 Designs in logistic models 7
2.2.1 Multiple-stage designs 7
2.2.2 Sequential sample size for logistic models 8
3 Optimal Designs for Item Calibration in Computerized Adaptive Testing 9
3.1 Designs for online calibration 11
3.2 Sequential sample size for two parameter logistic models12
4 Estimation of Logistic Regression Model with Measurement Error 15
4.1 Online calibration in two parameter logistic model 16
4.2 Estimate of logistic regression with measurement error in designs 18
5 Empirical Study 25
5.1 D-optimal designs in two parameter logistic models 25
5.2 Synthesized data 26
5.2.1 Initial stage 26
5.2.2 Design stage 27
5.3 Empirical studies based on The Basic Competence Test for Junior High School Students41
6 Discussion and Further Research 50
6.1 Future work 51
A Designs in Two Parameter Logistic Models 55
A.1 Design 1 : Kalish and Rosenberger's design 55
A.2 Design 2 : Abdelbasit and Plackett's design 55
A.3 Design 3 : Multiple stage design 56
A.4 Design 4 : Minkin's design 56
A.5 Design 5 : Sitter and Forbes's design 56
B Estimates of Latent Trait Levels in CAT 60
C Estimates of Other Exams 62
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