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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 17-Sep-2009 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
dc.contributor.advisor 張源俊zh_TW
dc.contributor.advisor Chang, Yuan-Chinen_US
dc.contributor.author (Authors) 盧宏益zh_TW
dc.contributor.author (Authors) Lu, Hung-Yien_US
dc.creator (作者) 盧宏益zh_TW
dc.creator (作者) Lu, Hung-Yien_US
dc.date (日期) 2005en_US
dc.date.accessioned 17-Sep-2009 18:47:44 (UTC+8)-
dc.date.available 17-Sep-2009 18:47:44 (UTC+8)-
dc.date.issued (上傳時間) 17-Sep-2009 18:47:44 (UTC+8)-
dc.identifier (Other Identifiers) G0903545011en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/33913-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 90354501zh_TW
dc.description (描述) 94zh_TW
dc.description.abstract (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。zh_TW
dc.description.abstract (摘要) 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.		en_US
dc.description.tableofcontents 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		zh_TW
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dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0903545011en_US
dc.subject (關鍵詞) 電腦化適性測驗zh_TW
dc.subject (關鍵詞) 線上校準zh_TW
dc.subject (關鍵詞) 測量誤差zh_TW
dc.subject (關鍵詞) 序貫設計zh_TW
dc.subject (關鍵詞) 變動長度zh_TW
dc.subject (關鍵詞) 試題反應理論zh_TW
dc.subject (關鍵詞) 試題校準zh_TW
dc.subject (關鍵詞) Item Response Theoryen_US
dc.subject (關鍵詞) Computerized Adaptive Testingen_US
dc.subject (關鍵詞) online calibrationen_US
dc.subject (關鍵詞) measurement erroren_US
dc.subject (關鍵詞) sequential designen_US
dc.subject (關鍵詞) sequential estimationen_US
dc.subject (關鍵詞) stopping timeen_US
dc.subject (關鍵詞) variable lengthen_US
dc.subject (關鍵詞) item calibrationen_US
dc.title (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zh_TW
dc.title (題名) Sequential Designs with Measurement Errors in Logistic Models with Applications to Educational Testingen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] Abdelbasit, K. M. and Plackett, R. L. (1983). Experimental Design for Binaryzh_TW
dc.relation.reference (參考文獻) Data. Journal of the American Statistical Association, 78, 90-98.zh_TW
dc.relation.reference (參考文獻) [2] Berger, M. P. F. (1991). On the e_ciency of IRT models when applied to di_erentzh_TW
dc.relation.reference (參考文獻) sampling designs. Applied Psychological Measurement, 15, 293-306.zh_TW
dc.relation.reference (參考文獻) [3] Berger, M. P. F. (1992). Sequential Sampling Designs for the Two-parameterzh_TW
dc.relation.reference (參考文獻) Item Response Theory Model. Psychometrika, 57, 521-538.zh_TW
dc.relation.reference (參考文獻) [4] Berger, M. P. F. (1994). D-optimal Sequential Sampling Designs for Item Responsezh_TW
dc.relation.reference (參考文獻) Theory Models. Journal of Educational Statistics, 19, 43-56.zh_TW
dc.relation.reference (參考文獻) [5] Buyske, S. G. (1998). Optimal Design for Item Calibration in Computerizedzh_TW
dc.relation.reference (參考文獻) Adaptive Testing, unpublished Ph.D. dissertation.zh_TW
dc.relation.reference (參考文獻) [6] Chang, H. H. and Ying, Z. (1996). A Global Information Approach to Computerizedzh_TW
dc.relation.reference (參考文獻) Adaptive Testing. Applied Psychological Measurement, 20, 213-229.zh_TW
dc.relation.reference (參考文獻) [7] Chang, H. H. and Ying, Z. (1999). _-strati_ed Multistage Computerized Adaptivezh_TW
dc.relation.reference (參考文獻) Testing. Applied Psychological Measurement, 23, 211-222.zh_TW
dc.relation.reference (參考文獻) [8] Chang, Y.-c. I. and Martinsek, A. (1992). Fixed Size Con_dence Regions forzh_TW
dc.relation.reference (參考文獻) Parameters of a Logistic Regression Model. The Annals of Statistics, 20(4), 1953-zh_TW
dc.relation.reference (參考文獻) 1969.zh_TW
dc.relation.reference (參考文獻) [9] Chang, Y.-c. I. (1999). Strong Consistency of maximum Quasi-likelihood Estimatezh_TW
dc.relation.reference (參考文獻) in Generalized Linear Models Via a Last Time. Statistics and Probabilityzh_TW
dc.relation.reference (參考文獻) Letters, 45, 237-246.zh_TW
dc.relation.reference (參考文獻) [10] Chang, Y.-c. I. (2001). Sequential Con_dence Regions of Generalized Linearzh_TW
dc.relation.reference (參考文獻) Models with Adaptive Designs. Journal of Statistical Planning and Inference,zh_TW
dc.relation.reference (參考文獻) 93, 277-293.zh_TW
dc.relation.reference (參考文獻) [11] Chang, Y.-c. I. and Ying, Z. (2004). Sequential Estimate in Variable Lengthzh_TW
dc.relation.reference (參考文獻) Computerized Adaptive Testing. Journal of Statistical Planning and Inference,zh_TW
dc.relation.reference (參考文獻) 121, 249-264.zh_TW
dc.relation.reference (參考文獻) [12] Chang, Y.-c. I. (2006). Maximum Quasi-likelihood Estimate in Generalized Linearzh_TW
dc.relation.reference (參考文獻) Models with Measurement Errors in Fixed and Adaptive Designs. Technicalzh_TW
dc.relation.reference (參考文獻) Report C-2006-01, Institute of Statistical Science, Academia Sinica.zh_TW
dc.relation.reference (參考文獻) [13] Chiang, J. (1990). Sequential Designs for the Linear Logistic Model, unpublishedzh_TW
dc.relation.reference (參考文獻) Ph.D. dissertation. The Pennsylvania State University, Department of Statistics.zh_TW
dc.relation.reference (參考文獻) [14] Chow, Y. S. and H. Teicher (1998). Probability Theory (2nd ed.). New York,zh_TW
dc.relation.reference (參考文獻) USA:Springer.zh_TW
dc.relation.reference (參考文獻) [15] Hambleton, R. K. and Swaminathan, H. (1985). Item Response Theory : Prin-zh_TW
dc.relation.reference (參考文獻) ciples and Applications. Kluwer.zh_TW
dc.relation.reference (參考文獻) [16] Jones, D. H. and Jin, Z. (1994). Optimal Sequential Designs for On-line Itemzh_TW
dc.relation.reference (參考文獻) Estimation. Psychometrika, 59, 59-75.zh_TW
dc.relation.reference (參考文獻) [17] Jones, D. H., Chiang, J. and Jin, Z. (1997). Optimal Designs for Simultaneouszh_TW
dc.relation.reference (參考文獻) Item Estimation. Nonlinear Analysis, Theory, Methods and Applications, 30,zh_TW
dc.relation.reference (參考文獻) 4051-4058.zh_TW
dc.relation.reference (參考文獻) [18] Jones, D. H., Nediak, M. andWang, X. B. (1999). Sequential Optimal Designs forzh_TW
dc.relation.reference (參考文獻) On-line Item Calibration. Technical Report. Rutgers University. Rutcor Researchzh_TW
dc.relation.reference (參考文獻) Report 2-99.zh_TW
dc.relation.reference (參考文獻) [19] Kalish, L. A. and Rosenberger, J. L. (1978). Optimal Designs for the Estimationzh_TW
dc.relation.reference (參考文獻) of the Logistic Function. Technical Report 33. The Pennsylvania State University,zh_TW
dc.relation.reference (參考文獻) Department of Statistics.zh_TW
dc.relation.reference (參考文獻) [20] Minkin, S. (1987). Optimal Designs for Binary Data. Journal of the Americanzh_TW
dc.relation.reference (參考文獻) Statistical Association, 82, 1098-1103.zh_TW
dc.relation.reference (參考文獻) [21] Ortega, J. A. and W. C. Rheinboldt (1970). Iterative solution of nonlinear equa-zh_TW
dc.relation.reference (參考文獻) tions in several variables. San Diago, CA: Academia Press, Inc.zh_TW
dc.relation.reference (參考文獻) [22] Silvey, S. D. (1980). Optimal Design. London: Chapman and Hall.zh_TW
dc.relation.reference (參考文獻) [23] Sitter, R. R. and Forbes, B. E. (1997). Optimal Two-Stage Designs for Binaryzh_TW
dc.relation.reference (參考文獻) Response Experiments. Statistica Sinica, 7, 941-955.zh_TW
dc.relation.reference (參考文獻) [24] Stefanski, L. A. and Carroll, R. J. (1985). Covariate measurement error in logisticzh_TW
dc.relation.reference (參考文獻) regression. The Annals of Statistics, 13(4), 1335-1351.zh_TW
dc.relation.reference (參考文獻) [25] van der Linden and W. J. (2000). Capitalization on item calibration error inzh_TW
dc.relation.reference (參考文獻) adaptive testing. Applied Measurement in Education, 13(1), 35-53.zh_TW
dc.relation.reference (參考文獻) [26] Wu, C. F. J. (1985). E_cient sequential designs with binary data. Journal ofzh_TW
dc.relation.reference (參考文獻) American Statistical Association, 80, 974-984.zh_TW