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題名 具局部相關結構之貝氏試題反應理論模型
Bayesian Item Response Theory Model with Local Dependence
作者 劉孟展
Liu, Meng-Chan
貢獻者 張育瑋
Chang, Yu-wei
劉孟展
Liu, Meng-Chan
關鍵詞 貝氏統計推論
Ising 模型
試題反應理論模型
擬概似函數
Bayesian inference
Ising model
Item response theory model
Pseudo likelihood
日期 2025
上傳時間 4-Aug-2025 15:11:58 (UTC+8)
摘要 傳統的試題反應理論(Item Response Theory; IRT)模型假設一位受試者在所有試題的反應是給定隨機效應參數下彼此獨立，稱為局部獨立，但是在現實應用上，這項假設很難滿足。文獻上已發展各式各樣更符合現實情況的IRT模型，以處理例如題組、多群體、限時測驗、受試者各異反應...等各種不滿足局部獨立假設時的局部相關。近年更新穎的做法是直接發展更一般化的局部相關之IRT模型，例如Chen et al. (2018) 提出的 FLaG-IRT 模型透過將Ising 模型引入試題反應理論模型中來描述一般化的局部相關。然而，該文獻提出使用proximal gradient-based方法求解最大概似估計量相當複雜，在模型進一步推廣時並不容易隨之推廣。本研究針對FLaG-IRT模型提出一套貝氏統計推論流程，讓模型中的許多參數能透過先驗分配彼此互相借用訊息，並且提供一個更簡潔的估計架構以利未來的模型推廣。然而，該模型具有無法直接處理的正規化函數(intractable normalizing functions) 之議題，本研究使用 variational Bayesian 方法 (Kim et al., 2024)，並搭配適當的擬概似函數解決該議題。透過模擬研究呈現本研究所提出之統計推論方法在各種模擬條件下之估計優勢，最後將該方法應用於兩筆實際資料上。
Traditional Item Response Theory (IRT) models assume that the responses to all items by one respondent are conditionally independent, conditional on random-effect parameters, and this is referred to as local independence. However, it is not easy that this assumption is fulfilled in practical applications. Various extensions of IRT models have been proposed in the literature to deal with local dependence (LD) structures arising from situations such as testlets, time limit tests, and different response strategies of individuals. In recent years, a more innovative approach involves directly developing general IRT models that account for local dependence. For example, Chen, Li, Liu, and Ying (2018) proposed the FLaG-IRT model, which incorporates the Ising model into IRT model to capture local dependence. However, their statistical inference relies on proximal gradient-based algorithm for maximum likelihood estimation, which is computationally complex and is difficult to be further generalized. The current study proposes a Bayesian inference for the FLaG-IRT model. The advantage of the Bayesian FLaG-IRT inference is to allow parameters to borrow information through prior distributions and to provide a more concise estimation structure that facilitates future model extensions. The main challenge of Bayesian FLaG-IRT inference is the intractable normalizing function issues, and we adopt variational Bayesian method (Kim, Bhattacharya & Maiti, 2024) with an appropriate pseudo likelihood to overcome this issue. Simulation studies are conducted to evaluate the performance of the proposed method under various conditions with two real data sets illustrating its practical utility.
參考文獻 Andrieu, C., & Roberts, G.O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. The Annals of Statistics, 37, 697–725. Atchade, Y. F., Lartillot, N., & Robert, C. P. (2008). Bayesian computation for statistical models with intractable normalizing constants. Brazilian Journal of Probability and Statistics, 22, 416–436. Barber, R.F., & Drton, M. (2015). High-dimensional Ising model selection with Bayesian information criteria. Electronic Journal of Statistics, 9, 567–607. Beaumont, M.A. (2003). Estimation of population growth or decline in genetically monitored populations. Genetics, 164, 1139–1160. Besag, J. (1974). Spatial Interaction and the Statistical Analysis of Lattice Systems. Journal of the Royal Statistical Society, Series B, 36, 192–236. Bhattacharya, B. B., & Mukherjee, S. (2018), Inference in Ising Models. Bernoulli, 24, 493–525. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord and M. R. Novick, Statistical theories of mental test scores. Reading, MA: Addison-Wesley. Bolt, D. M., Cohen, A. S., & Wollack, J.A. (2002). Item parameter estimation under conditions of test speededness: application of a mixture Rasch model with ordinal constraints. Journal of Educational Measurement, 39, 331–348. Bradlow, E. T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168. Chang, Y.-W., Tsai, R., & Hsu, N.-J. (2014). A speeded Item Response model: Leave the harder till later. Psychometrika, 79, 255–274. Chang, Y.-W., & Tu, J.-Y. (2022). Bayesian estimation for an item response tree model for nonresponse modeling. Metrika, 85, 1023–1047. Chen, J., & Chen, Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95, 759–771. Chen, W.-H., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265–289. Chen, Y., Li, X., Liu, J., & Ying, Z. (2018). Robust measurement via a fused latent and graphical item response theory model. Psychometrika, 83, 538–562. Cho, S.-J., Cohen, A. S., & Kim, S.-H. (2013). Markov chain Monte Carlo estimation of a mixture item response theory model. Journal of Statistical Computation and Simulation, 83, 215–241. College Entrance Examination Center. (2010). Mathematics B Test, 2010 Advanced Subjects Test. https://www.ceec.edu.tw/xmfile?xsmsid=0J052427633128416650 Eckes, T., & Baghaei, P. (2015). Using testlet response theory to examine local dependence in C-tests. Applied Measurement in Education, 28, 85–98. Foygel, R., & Drton, M. (2010). Extended Bayesian information criteria for Gaussian graphical models. Advances in Neural Information Processing Systems, 604–612. Fox, J.-P. (2010). Bayesian item response modeling-Theory and applications. Springer. Frank, B.B., & Kim, S.-H. (2004). Item response theory: parameter estimation techniques, 2. Gelman, A., Carlin, J. B., Rubin, D. B., & Stern, H. S. (2004). Bayesian data analysis. Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511. Goegebeur, Y., De Boeck, P., Wollack, J. A., & Cohen, A.S. (2008). A speeded item response model with gradual process change. Psychometrika, 73, 65–87. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109. Hughes, J., Haran, M., & Caragea, P. (2011). Autologistic Models for Binary Data on a Lattice. Environmetrics, 22, 857–871. Hunter, D.R., & Handcock, M.S. (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15, 565–583. Hunter, D. R., & Handcock, M.S. (2012). Inference in Curved Exponential Family Models for Networks. Journal of Computational and Graphical Statistics, 15, 565–583. Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei, 31, 253–258. Jeon, M., Jin, I. H., Schweinberger, M., & Baugh, S. (2021). Mapping unobserved item-respondent interactions: A latent space item response model with interaction map. Psychometrika, 86, 378–403. Kim, M., Bhattacharya, S., & Maiti, T. (2024). Statistically valid variational Bayes algorithm for Ising model parameter estimation. Journal of Computational and Graphical Statistics, 33, 75–84. Lenz, W. (1920). Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern. Physikalische Zeitschrift, 21, 613–615. Liang, F. (2010). A double Metropolis–Hastings sampler for spatial models with intractable normalizing constants. Journal of Statistical Computation and Simulation, 80, 1007–1022. Liang, F., Jin, I. H., Song, Q., & Liu, J. S. (2016). An adaptive exchange algorithm for sampling from distributions with intractable normalizing constants. Journal of the American Statistical Association, 111, 377–393. Macdonald, P., & Paunonen, S.V. (2002). A Monte Carlo comparison of item and person statistics based on item response theory versus classical test theory. Educational and Psychological Measurement, 62, 921–943. Møller, J., Pettitt, A. N., Reeves, R., & Berthelsen, K. K. (2006). An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93, 451–458. Murray, I., Ghahramani, Z., & MacKay, D. J. C. (2006). MCMC for doubly-intractable distributions. In R. Dechter & T. S. Richardson (Eds.), Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence (UAI-06) (pp. 359–366). AUAI Press. Parikh, N., & Boyd, S. (2013). Proximal algorithms. Foundations and Trends in Optimization, 1, 123–231. Park, J., & Haran, M. (2018). Bayesian inference in the presence of intractable normalizing functions. Journal of the American Statistical Association, 113, 1372–1390. Park, J., Jin, I. H., & Schweinberger, M. (2022). Bayesian model selection for high-dimensional Ising models, with applications to educational data. Computational Statistics and Data Analysis, 165, Article 107325. R Core Team (2020). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Rasch, G. (1960). Probabilistic models for some intelligence and achievement tests. Nielsen and Lydiche, Copenhagen, Denmark. Robins, G., Pattison, P., Kalish, Y., & Lusher, D. (2007). An Introduction to Exponential Random Graph (p*) Models for Social Networks. Social Networks, 29, 173–191. Royal, K. D. (2016). The impact of item sequence order on local item dependence: An item response theory perspective. Survey Practice, 9, 1–7. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., & van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society. Series B, 64, 583–616. Strauss, D.J. (1975). A model for clustering. Biometrika, 62, 467–475. Swaminathan, H., & Gifford, J. A. (1986). Bayesian estimation in the three-parameter logistic model. Psychometrika, 51, 589–601. van der Linden, W. J. (Ed.). (2016). Handbook of item response theory. CRC Press. Yamamoto, K., & Everson, H. (1997). Modeling the effects of test length and test time on parameter estimation using the hybrid model. In J. Rost & R. Langeheine (Eds.), Applications of latent trait and latent class models in the social sciences (pp. 89–98). Yen, W.M. (1993). Scaling performance assessments: strategies for managing local item dependence. Journal of Educational Measurement, 30, 187–213.
描述 碩士
國立政治大學
統計學系
112354025
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0112354025
資料類型 thesis
dc.contributor.advisor 張育瑋zh_TW
dc.contributor.advisor Chang, Yu-weien_US
dc.contributor.author (Authors) 劉孟展zh_TW
dc.contributor.author (Authors) Liu, Meng-Chanen_US
dc.creator (作者) 劉孟展zh_TW
dc.creator (作者) Liu, Meng-Chanen_US
dc.date (日期) 2025en_US
dc.date.accessioned 4-Aug-2025 15:11:58 (UTC+8)-
dc.date.available 4-Aug-2025 15:11:58 (UTC+8)-
dc.date.issued (上傳時間) 4-Aug-2025 15:11:58 (UTC+8)-
dc.identifier (Other Identifiers) G0112354025en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/158716-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 112354025zh_TW
dc.description.abstract (摘要) 傳統的試題反應理論(Item Response Theory; IRT)模型假設一位受試者在所有試題的反應是給定隨機效應參數下彼此獨立，稱為局部獨立，但是在現實應用上，這項假設很難滿足。文獻上已發展各式各樣更符合現實情況的IRT模型，以處理例如題組、多群體、限時測驗、受試者各異反應...等各種不滿足局部獨立假設時的局部相關。近年更新穎的做法是直接發展更一般化的局部相關之IRT模型，例如Chen et al. (2018) 提出的 FLaG-IRT 模型透過將Ising 模型引入試題反應理論模型中來描述一般化的局部相關。然而，該文獻提出使用proximal gradient-based方法求解最大概似估計量相當複雜，在模型進一步推廣時並不容易隨之推廣。本研究針對FLaG-IRT模型提出一套貝氏統計推論流程，讓模型中的許多參數能透過先驗分配彼此互相借用訊息，並且提供一個更簡潔的估計架構以利未來的模型推廣。然而，該模型具有無法直接處理的正規化函數(intractable normalizing functions) 之議題，本研究使用 variational Bayesian 方法 (Kim et al., 2024)，並搭配適當的擬概似函數解決該議題。透過模擬研究呈現本研究所提出之統計推論方法在各種模擬條件下之估計優勢，最後將該方法應用於兩筆實際資料上。zh_TW
dc.description.abstract (摘要) Traditional Item Response Theory (IRT) models assume that the responses to all items by one respondent are conditionally independent, conditional on random-effect parameters, and this is referred to as local independence. However, it is not easy that this assumption is fulfilled in practical applications. Various extensions of IRT models have been proposed in the literature to deal with local dependence (LD) structures arising from situations such as testlets, time limit tests, and different response strategies of individuals. In recent years, a more innovative approach involves directly developing general IRT models that account for local dependence. For example, Chen, Li, Liu, and Ying (2018) proposed the FLaG-IRT model, which incorporates the Ising model into IRT model to capture local dependence. However, their statistical inference relies on proximal gradient-based algorithm for maximum likelihood estimation, which is computationally complex and is difficult to be further generalized. The current study proposes a Bayesian inference for the FLaG-IRT model. The advantage of the Bayesian FLaG-IRT inference is to allow parameters to borrow information through prior distributions and to provide a more concise estimation structure that facilitates future model extensions. The main challenge of Bayesian FLaG-IRT inference is the intractable normalizing function issues, and we adopt variational Bayesian method (Kim, Bhattacharya & Maiti, 2024) with an appropriate pseudo likelihood to overcome this issue. Simulation studies are conducted to evaluate the performance of the proposed method under various conditions with two real data sets illustrating its practical utility.en_US
dc.description.tableofcontents 第一章 緒論 1 第二章 FLaG-IRT 模型 4 2.1 符號定義與經典的IRT模型 4 2.2 FLaG-IRT模型 8 2.3 FLaG-IRT模型的擬概似函數 11 2.4 FLaG-IRT模型之註解 13 第三章 貝氏估計 16 3.1 先驗分佈設定 17 3.2 貝氏估計流程 20 第四章 模擬研究 26 4.1 模擬設定 26 4.2 模擬結果之摘要方式 30 4.3 模擬結果 31 第五章 資料分析 38 5.1 2010年指考數學乙資料分析 38 5.2 2012年PISA資料分析 45 第六章 結論與討論 48 參考文獻 51zh_TW
dc.format.extent 1418131 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0112354025en_US
dc.subject (關鍵詞) 貝氏統計推論zh_TW
dc.subject (關鍵詞) Ising 模型zh_TW
dc.subject (關鍵詞) 試題反應理論模型zh_TW
dc.subject (關鍵詞) 擬概似函數zh_TW
dc.subject (關鍵詞) Bayesian inferenceen_US
dc.subject (關鍵詞) Ising modelen_US
dc.subject (關鍵詞) Item response theory modelen_US
dc.subject (關鍵詞) Pseudo likelihooden_US
dc.title (題名) 具局部相關結構之貝氏試題反應理論模型zh_TW
dc.title (題名) Bayesian Item Response Theory Model with Local Dependenceen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Andrieu, C., & Roberts, G.O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. The Annals of Statistics, 37, 697–725. Atchade, Y. F., Lartillot, N., & Robert, C. P. (2008). Bayesian computation for statistical models with intractable normalizing constants. Brazilian Journal of Probability and Statistics, 22, 416–436. Barber, R.F., & Drton, M. (2015). High-dimensional Ising model selection with Bayesian information criteria. Electronic Journal of Statistics, 9, 567–607. Beaumont, M.A. (2003). Estimation of population growth or decline in genetically monitored populations. Genetics, 164, 1139–1160. Besag, J. (1974). Spatial Interaction and the Statistical Analysis of Lattice Systems. Journal of the Royal Statistical Society, Series B, 36, 192–236. Bhattacharya, B. B., & Mukherjee, S. (2018), Inference in Ising Models. Bernoulli, 24, 493–525. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord and M. R. Novick, Statistical theories of mental test scores. Reading, MA: Addison-Wesley. Bolt, D. M., Cohen, A. S., & Wollack, J.A. (2002). Item parameter estimation under conditions of test speededness: application of a mixture Rasch model with ordinal constraints. Journal of Educational Measurement, 39, 331–348. Bradlow, E. T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168. Chang, Y.-W., Tsai, R., & Hsu, N.-J. (2014). A speeded Item Response model: Leave the harder till later. Psychometrika, 79, 255–274. Chang, Y.-W., & Tu, J.-Y. (2022). Bayesian estimation for an item response tree model for nonresponse modeling. Metrika, 85, 1023–1047. Chen, J., & Chen, Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95, 759–771. Chen, W.-H., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265–289. Chen, Y., Li, X., Liu, J., & Ying, Z. (2018). Robust measurement via a fused latent and graphical item response theory model. Psychometrika, 83, 538–562. Cho, S.-J., Cohen, A. S., & Kim, S.-H. (2013). Markov chain Monte Carlo estimation of a mixture item response theory model. Journal of Statistical Computation and Simulation, 83, 215–241. College Entrance Examination Center. (2010). Mathematics B Test, 2010 Advanced Subjects Test. https://www.ceec.edu.tw/xmfile?xsmsid=0J052427633128416650 Eckes, T., & Baghaei, P. (2015). Using testlet response theory to examine local dependence in C-tests. Applied Measurement in Education, 28, 85–98. Foygel, R., & Drton, M. (2010). Extended Bayesian information criteria for Gaussian graphical models. Advances in Neural Information Processing Systems, 604–612. Fox, J.-P. (2010). Bayesian item response modeling-Theory and applications. Springer. Frank, B.B., & Kim, S.-H. (2004). Item response theory: parameter estimation techniques, 2. Gelman, A., Carlin, J. B., Rubin, D. B., & Stern, H. S. (2004). Bayesian data analysis. Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511. Goegebeur, Y., De Boeck, P., Wollack, J. A., & Cohen, A.S. (2008). A speeded item response model with gradual process change. Psychometrika, 73, 65–87. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109. Hughes, J., Haran, M., & Caragea, P. (2011). Autologistic Models for Binary Data on a Lattice. Environmetrics, 22, 857–871. Hunter, D.R., & Handcock, M.S. (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15, 565–583. Hunter, D. R., & Handcock, M.S. (2012). Inference in Curved Exponential Family Models for Networks. Journal of Computational and Graphical Statistics, 15, 565–583. Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei, 31, 253–258. Jeon, M., Jin, I. H., Schweinberger, M., & Baugh, S. (2021). Mapping unobserved item-respondent interactions: A latent space item response model with interaction map. Psychometrika, 86, 378–403. Kim, M., Bhattacharya, S., & Maiti, T. (2024). Statistically valid variational Bayes algorithm for Ising model parameter estimation. Journal of Computational and Graphical Statistics, 33, 75–84. Lenz, W. (1920). Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern. Physikalische Zeitschrift, 21, 613–615. Liang, F. (2010). A double Metropolis–Hastings sampler for spatial models with intractable normalizing constants. Journal of Statistical Computation and Simulation, 80, 1007–1022. Liang, F., Jin, I. H., Song, Q., & Liu, J. S. (2016). An adaptive exchange algorithm for sampling from distributions with intractable normalizing constants. Journal of the American Statistical Association, 111, 377–393. Macdonald, P., & Paunonen, S.V. (2002). A Monte Carlo comparison of item and person statistics based on item response theory versus classical test theory. Educational and Psychological Measurement, 62, 921–943. Møller, J., Pettitt, A. N., Reeves, R., & Berthelsen, K. K. (2006). An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93, 451–458. Murray, I., Ghahramani, Z., & MacKay, D. J. C. (2006). MCMC for doubly-intractable distributions. In R. Dechter & T. S. Richardson (Eds.), Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence (UAI-06) (pp. 359–366). AUAI Press. Parikh, N., & Boyd, S. (2013). Proximal algorithms. Foundations and Trends in Optimization, 1, 123–231. Park, J., & Haran, M. (2018). Bayesian inference in the presence of intractable normalizing functions. Journal of the American Statistical Association, 113, 1372–1390. Park, J., Jin, I. H., & Schweinberger, M. (2022). Bayesian model selection for high-dimensional Ising models, with applications to educational data. Computational Statistics and Data Analysis, 165, Article 107325. R Core Team (2020). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Rasch, G. (1960). Probabilistic models for some intelligence and achievement tests. Nielsen and Lydiche, Copenhagen, Denmark. Robins, G., Pattison, P., Kalish, Y., & Lusher, D. (2007). An Introduction to Exponential Random Graph (p*) Models for Social Networks. Social Networks, 29, 173–191. Royal, K. D. (2016). The impact of item sequence order on local item dependence: An item response theory perspective. Survey Practice, 9, 1–7. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., & van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society. Series B, 64, 583–616. Strauss, D.J. (1975). A model for clustering. Biometrika, 62, 467–475. Swaminathan, H., & Gifford, J. A. (1986). Bayesian estimation in the three-parameter logistic model. Psychometrika, 51, 589–601. van der Linden, W. J. (Ed.). (2016). Handbook of item response theory. CRC Press. Yamamoto, K., & Everson, H. (1997). Modeling the effects of test length and test time on parameter estimation using the hybrid model. In J. Rost & R. Langeheine (Eds.), Applications of latent trait and latent class models in the social sciences (pp. 89–98). Yen, W.M. (1993). Scaling performance assessments: strategies for managing local item dependence. Journal of Educational Measurement, 30, 187–213.zh_TW