政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/33913

政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/33913

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政大機構典藏 > 商學院 > 統計學系 > 學位論文 > Item 140.119/33913

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題名:	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 Sequential Designs with Measurement Errors in Logistic Models with Applications to Educational Testing
作者:	盧宏益 Lu, Hung-Yi
貢獻者:	張源俊 Chang, Yuan-Chin 盧宏益 Lu, Hung-Yi
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 Item Response Theory Computerized Adaptive Testing online calibration measurement error sequential design sequential estimation stopping time variable length item calibration
日期:	2005
上傳時間:	2009-09-17 18:47:44 (UTC+8)
摘要:	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.
參考文獻:	[1] Abdelbasit, K. M. and Plackett, R. L. (1983). Experimental Design for Binary Data. Journal of the American Statistical Association, 78, 90-98. [2] Berger, M. P. F. (1991). On the e_ciency of IRT models when applied to di_erent sampling designs. Applied Psychological Measurement, 15, 293-306. [3] Berger, M. P. F. (1992). Sequential Sampling Designs for the Two-parameter Item Response Theory Model. Psychometrika, 57, 521-538. [4] Berger, M. P. F. (1994). D-optimal Sequential Sampling Designs for Item Response Theory Models. Journal of Educational Statistics, 19, 43-56. [5] Buyske, S. G. (1998). Optimal Design for Item Calibration in Computerized Adaptive Testing, unpublished Ph.D. dissertation. [6] Chang, H. H. and Ying, Z. (1996). A Global Information Approach to Computerized Adaptive Testing. Applied Psychological Measurement, 20, 213-229. [7] Chang, H. H. and Ying, Z. (1999). _-strati_ed Multistage Computerized Adaptive Testing. Applied Psychological Measurement, 23, 211-222. [8] Chang, Y.-c. I. and Martinsek, A. (1992). Fixed Size Con_dence Regions for Parameters of a Logistic Regression Model. The Annals of Statistics, 20(4), 1953- 1969. [9] Chang, Y.-c. I. (1999). Strong Consistency of maximum Quasi-likelihood Estimate in Generalized Linear Models Via a Last Time. Statistics and Probability Letters, 45, 237-246. [10] Chang, Y.-c. I. (2001). Sequential Con_dence Regions of Generalized Linear Models with Adaptive Designs. Journal of Statistical Planning and Inference, 93, 277-293. [11] Chang, Y.-c. I. and Ying, Z. (2004). Sequential Estimate in Variable Length Computerized Adaptive Testing. Journal of Statistical Planning and Inference, 121, 249-264. [12] Chang, Y.-c. I. (2006). Maximum Quasi-likelihood Estimate in Generalized Linear Models with Measurement Errors in Fixed and Adaptive Designs. Technical Report C-2006-01, Institute of Statistical Science, Academia Sinica. [13] Chiang, J. (1990). Sequential Designs for the Linear Logistic Model, unpublished Ph.D. dissertation. The Pennsylvania State University, Department of Statistics. [14] Chow, Y. S. and H. Teicher (1998). Probability Theory (2nd ed.). New York, USA:Springer. [15] Hambleton, R. K. and Swaminathan, H. (1985). Item Response Theory : Prin- ciples and Applications. Kluwer. [16] Jones, D. H. and Jin, Z. (1994). Optimal Sequential Designs for On-line Item Estimation. Psychometrika, 59, 59-75. [17] Jones, D. H., Chiang, J. and Jin, Z. (1997). Optimal Designs for Simultaneous Item Estimation. Nonlinear Analysis, Theory, Methods and Applications, 30, 4051-4058. [18] Jones, D. H., Nediak, M. andWang, X. B. (1999). Sequential Optimal Designs for On-line Item Calibration. Technical Report. Rutgers University. Rutcor Research Report 2-99. [19] Kalish, L. A. and Rosenberger, J. L. (1978). Optimal Designs for the Estimation of the Logistic Function. Technical Report 33. The Pennsylvania State University, Department of Statistics. [20] Minkin, S. (1987). Optimal Designs for Binary Data. Journal of the American Statistical Association, 82, 1098-1103. [21] Ortega, J. A. and W. C. Rheinboldt (1970). Iterative solution of nonlinear equa- tions in several variables. San Diago, CA: Academia Press, Inc. [22] Silvey, S. D. (1980). Optimal Design. London: Chapman and Hall. [23] Sitter, R. R. and Forbes, B. E. (1997). Optimal Two-Stage Designs for Binary Response Experiments. Statistica Sinica, 7, 941-955. [24] Stefanski, L. A. and Carroll, R. J. (1985). Covariate measurement error in logistic regression. The Annals of Statistics, 13(4), 1335-1351. [25] van der Linden and W. J. (2000). Capitalization on item calibration error in adaptive testing. Applied Measurement in Education, 13(1), 35-53. [26] Wu, C. F. J. (1985). E_cient sequential designs with binary data. Journal of American Statistical Association, 80, 974-984.
描述:	博士國立政治大學統計研究所 90354501 94
資料來源:	http://thesis.lib.nccu.edu.tw/record/#G0903545011
資料類型:	thesis
顯示於類別:	[統計學系] 學位論文

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