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題名 混合試題與受試者模型於試題差異功能分析之研究
A Mixture Items-and-Examinees Model Analysis on Differential Item Functioning
作者 黃馨瑩
Huang, Hsin Ying
貢獻者 余民寧<br>溫福星
Yu, Min Ning<br>Wen, Fur Hsing
黃馨瑩
Huang, Hsin Ying
關鍵詞 混合試題反應理論
隨機試題
試題差異功能
mixture item response theory
random item
differential item functioning
日期 2013
上傳時間 4-Jun-2014 14:45:48 (UTC+8)
摘要 依據「多層次混合試題反應理論」與「隨機試題混合模型」，本研究提出「混合試題與受試者模型」。本研究旨在評估此模型在不同樣本數、不同試題差異功能的試題數下，偵測試題差異功能的表現，以及其參數回復性情形。研究結果顯示，「混合試題與受試者模型」在樣本數大、試題差異功能試題數較多之情境下，具有正確的參數回復性，能正確判斷出試題是否存在試題差異功能，且具有良好的難度估計值，並能將樣本正確地分群，其也與「隨機試題混合模型」的估計表現頗為相近。建議未來可將「混合試題與受試者模型」應用於大型教育資料庫相關研究上，並加入其他變項後進一步探討。
Drawing upon the framework of the multilevel mixture item response theory model and the random item mixture model, the study attempts to propose one model, called the mixture items and examinees model(MIE model). The purpose of this study was to assess the respective performances of the model on different sample-sizes and differential item functioning (DIF) items. Particularly, the study assessed the model performances in the detection of DIF items, and the accurate parameters recovery. The results of the study revealed that with large sample-sizes and more DIF items, the MIE model had the good parameters recovery, the accurate detection of the DIF items, the good estimate of the item difficulty, and the accurate classifications of the sub-samples. These model performances appeared similar to those of the random item mixture model. The findings suggest that future studies should apply the MIE model to the analyses on large-scale education databases, and should add more variables to the MIE model.
參考文獻 1.Bolt, D. M., Cohen, A. S., & Wollack, J. A. (2001). A mixture item response for multiple choice data. Journal of Educational and Behavioral Statistics, 26, 381-409.
2.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.
3.Camilli, G. (1992). A conceptual analysis of differential item functioning in terms of a multidimensional item response model. Applied Psychological Measurement, 16, 129-147.
4.Chaimongkol, S. (2005). Modeling differential item functioning (DIF) using multilevel logistic regression models: A Bayesian perspective. (Unpublished doctoral dissertation). Florida State University, Tallahassee, FL.
5.Chaimongkol, S., Huffer, F. W., & Kamata, A. (2007). An explanatory differential item functioning (DIF) model by the WinBUG 1.4. Songklanakarin Journal of Science and Technology, 29(2), 449-459.
6.Cheong, Y. F. (2006). Analysis of school context effects on differential item functioning using hierarchical generalized linear models. International Journal of Testing, 6(1), 57-79.
7.Cho, S. J., & Cohen, A. S. (2010). Multilevel mixture IRT model with an application to DIF. Journal of Educational and Behavioral Statistics, 35, 336-370.
8.Cho, S. J., Cohen, A. S., & Kim, S. H. (2006, June). An investigation of priors on the probabilities of mixtures in the mixture Rasch model. Paper presented at the International Meeting of the Psychometric Society: The 71st annual meeting of the Psychometric Society, Montreal, Canada.
9.Cohen, A. S., & Bolt, D. M. (2005). A mixture model analysis of differential item functioning. Journal of Educational Measurement, 42, 133-148.
10.Cohen, A. S., Cho, S. J., & Kim, S. H. (2005, April). A mixture testlet model for educational tests. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Canada.
11.Cohen, A. S., Gregg, N., & Deng, M. (2005). The role of extended time and item content on a high-stakes mathematics test. Learning Disabilities Research & Practice, 20, 225-233.
12.Dai, Y. (2013). A mixture Rasch model with a covariate a simulation study via Bayesian Markov Chain Monte Carlo estimation. Applied Psychological Measurement, 37(5), 375-396.
13.De Boeck, P. (2008). Random item IRT models. Psychometrika, 73, 533–559.
14.De Boeck, P., Cho, S. J., & Wilson, M. (2011). Explanatory secondary dimension modeling of latent differential item functioning. Applied Psychological Measurement, 35, 583-603.
15.DeAyala, R. J., Kim, S. H., Stapleton, L. M., & Dayton, C. M. (2002). Differential item functioning: A mixture distribution conceptualization. International Journal of Testing, 2, 243-276.
16.Demar, C. E., & Lau, A. (2011). Differential item functioning detection with latent classes: How accurately can we detect who is responding differentially? Educational and Psychological Measurement, 71(4), 597-616.
17.Dorans, N. J., & Kulick, E. (1986). Demonstrating the utility of the standardization approach to assessing unexpected differential item performance on the scholastic aptitude test. Journal of Educational Measurement, 23(4), 355-368.
18.Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah, NJ: Lawrence-Erlbaum.
19.Finch, W. H. (2005). The MIMIC model as a method for detecting DIF: Comparison with Mantel-Haenszel, SIBTEST, and the IRT Likelihood Ratio. Applied Psychological Measurement, 29, 278-295
20.Finch, W. H. (2012).The MIMIC model as a tool for differential bundle functioning detection. Applied Psychological Measurement, 36, 40-59.
21.Fox, J. P. (2005). Multilevel IRT using dichotomous and polytomous response data. British Journal of Mathematical and Statistical Psychology, 58, 145-172.
22.Frederickx, S., Tuerlinckx, F., De Boeck, P., & Magis, D. (2010). RIM: A random item mixture model to detect differential item functioning. Journal of Educational Measurement, 47, 432–457.
23.French, B. F., & Finch, W. H. (2010). Hierarchical logistic regression: Accounting for multilevel data in DIF detection. Journal of Educational Measurement, 47(3), 299-317.
24.Frick, H., Strobl, C., & Zeileis, A. (2013). Rasch mixture models for DIF detection: A comparison of old and new score specifications. Retrieved from http://eeecon.uibk.ac.at/wopec2/repec/inn/wpaper/2013-36.pdf
25.Holland, P. W., & Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In H. Wainer & H. I. Braun (Eds.), Test Validity (pp. 129-145). Hillsdale, NJ: Lawrence Erlbaum Associates.
26.Jiao, H., Lissitz, R. W., Macready, G., Wang, S., & Liang, S. (2011). Exploring levels of performance using the mixture Rasch model for standard setting. Psychological Test and Assessment Modeling, 53(4), 499-522.
27.Johnson, V., & Albert, J. (1998). Ordinal data modeling. New York: Springer.
28.Kamata, A. (1998). Some generalizations of the Rasch model: An application of the hierarchical generalized linear model. (Unpublished doctoral dissertation). Michigan State University, East Lansing, MI.
29.Kamata, A. (2001). Item analysis by the hierarchical generalized linear model. Journal of Educational Measurement, 38(1), 79-93.
30.Kang, T., & Cohen, A. S. (2003, April). A mixture model analysis of ethnic group DIF. Paper presented at the annual meeting of the National Council on Measurement in Education, Chicago, IL.
31.Li, F., Cohen, A. S., Kim, S. H., & Cho, S. J. (2006, April). Model selection methods for mixture dichotomous IRT models. Paper presented at the annual meeting of the National Council on Measurement in Education, San Francisco, CA.
32.Lord, F. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.
33.Lu, R., & Jiao, H. (2009, April). Detecting DIF using the mixture Rasch model. Paper presented at the annual meeting of the National Council on Measurement in Education, San Diego, CA.
34.Maier, K. S. (2002). Modeling incomplete scaled questionnaire data with a partial credit hierarchical measurement model. Journal of Educational and Behavioral Statistics, 27, 271-289.
35.Maij-de Meij, A. M., Kelderman, H., & van der Flier, H. (2010). Improvement in detection of differential item functioning using a mixture item response theory model. Multivariate Behavioral Research, 45, 975-999.
36.McLachlan, G., &, Peel, D. (2000). Finite mixture models. New York, NY: Wiley.
37.Mislevy, R. J., & Verhelst, N. (1990). Modeling item responses when different subjects employ different solution strategies. Psychometrika, 55, 195-215.
38.Navas-Ara, M. J., & Gomez-Benito, J. (2002). Effects of ability scale purification on the identification of DIF. European Journal of Psychological Assessment, 19, 9–15.
39.Penfield, R. D. (2010). Modelling DIF effects using distractor-level invariance effects: Implications for understanding the causes of DIF. Applied Psychological Measurement, 34, 151-165.
40.Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized multilevel structural equation modeling. Psychometrika, 69, 167–190.
41.Raju, N. S. (1990). Determining the significance of estimated signed and unsigned areas between two item response functions. Applied Psychological Measurement, 14, 197–207.
42.Rost, J. (1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14, 271-282.
43.Rost, J. (1997). Logistic mixture models. In W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 449-463). New York: Springer.
44.Roussos, L., & Stout, W. (1996). A multidimensionality-based DIF analysis paradigm. Applied Psychological Measurement, 20, 355-371.
45.Rudner, L. M., Getson, P. R., & Knight, D. L. (1980). Biased item detection techniques. Journal of Educational Statistics, 6, 213-233.
46.Samuelsen, K. (2005). Examining differential item functioning from a latent class perspective. (Unpublished doctoral dissertation). University of Maryland, College Park, MD.
47.Samuelsen, K. (2008). Examining differential item functioning from a latent class perspective. In G. R. Hancock & K. M. Samuelsen (Eds.) Advances in latent variable mixture models (pp. 67-113). Charlotte, NC: Information Age.
48.Shealy, R. & Stout, W. F. (1993). A model-based standardization approach that separates true bias/DIF from group differences and detects test bias/DIF as well as item bias/DIF. Psychometrika, 58, 159-194.
49.Shepard, L. A. (1982). Definitions of bias. In R. A. Berk (Ed.), Handbook of methods for detecting test bias. London, UK: The John Hopkins Press.
50.Shepard, L. A., Camilli, G., & Averill, M. (1981). Comparison of six procedures for detecting test item bias using both internal and external ability criteria. Journal of Educational Statistics, 6, 317-375.
51.Shin, C. L., & Wang, W. C. (2009). Differential item functioning detection using the multiple indicators, multiple causes method with a pure short anchor. Applied Psychological Measurement, 33, 184-199.
52.Snijders, T. A. B., & Bosker, R. (2011). Multilevel analysis: An introduction to basic and advanced multilevel modeling. Thousand Oaks, CA: Sage.
53.Soares, T. M., Goncalves, F. B., & Gamerman, D. (2009). An integrated Bayesian model for DIF analysis. Journal of Educational and Behavioral Statistics, 34, 348–377.
54.Swaminathan, H., & Rogers, H. J. (1990). Detecting differential item functioning using logistic regression procedures. Journal of Educational Measurement, 27(4), 361-370.
55.Tay, L., Newman, D. A., & Vermunt, J. K. (2011). Using mixed-measurement item response theory with covariates (MM-IRT-C) to ascertain observed and unobserved measurement equivalence. Organizational Research Methods, 14, 147-176.
56.Thissen, D., Steinberg, L., & Gerrard, M. (1986). Beyond group-mean differences: The concept of item bias. Psychological Bulletin, 99, 118-128.
57.Thissen, D., Steinberg, L., & Wainer, H. (1993). Detection of differential item functioning using the parameters of item response models. In P. W. Holland & H. Wainer. (Eds.), Differential item functioning (pp. 67-113). Hillsdale, NJ: Lawrence Erlbaum Associates.
58.Vermunt, J. K., & Magidson, J. (2005). Technical guide for Latent GOLD 4.0: Basic and advanced. Belmont MA: Statistical Innovations.
59.Von Davier, M., & Yamamoto, K. (2004). Partially observed mixtures of IRT models: An extension of the generalized partial credit model. Applied Psychological Measurement, 28, 389-406.
60.Wang, W. C., & Shin, C. L. (2010). MIMIC methods for assessing differential item functioning in polytomous items. Applied Psychological Measurement, 34, 166-180.
61.Wang, W. C., Shin, C. L., & Yang, C. C. (2009). The MIMIC method with scale purification for detecting differential item functioning. Educational and Psychological Measurement, 69, 713-731.
62.Wiberg, M. (2007). Measuring and detecting differential item functioning in criterion-referenced licensing test. (EM No. 60. Umea, Sweden: Umea University.
63.Wollack, J. A., Cohen, A. S., & Wells, C. S. (2003). A method for maintaining scale stability in the presence of test speededness. Journal of Educational Measurement, 40, 307-330.
64.Woods, C. M., Oltmanns, T. F., & Turkheimer, E. (2009). Illustration of MIMIC-model DIF testing with the schedule for nonadaptive and adaptive personality. Journal of Psychopathology and Behavioral Assessment, 31, 320-330.
65.Zumbo, B. D., & Gelin, M. N. (2005). A matter of test bias in educational policy research: Bringing the context into picture by investigating sociological/community moderated (or mediated) test and item bias. Journal of Educational Research & Policy Studies, 5, 1-23.
66.Zwick, R. (2012). A review of ETS differential item functioning assessment procedures: Flagging rules, minimum sample size requirements, and criterion refinement. Retrieved from http://www.ets.org/Media/Research/pdf/RR-12-08.pdf
描述 博士
國立政治大學
教育研究所
98152501
102
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0098152501
資料類型 thesis
dc.contributor.advisor 余民寧<br>溫福星zh_TW
dc.contributor.advisor Yu, Min Ning<br>Wen, Fur Hsingen_US
dc.contributor.author (Authors) 黃馨瑩zh_TW
dc.contributor.author (Authors) Huang, Hsin Yingen_US
dc.creator (作者) 黃馨瑩zh_TW
dc.creator (作者) Huang, Hsin Yingen_US
dc.date (日期) 2013en_US
dc.date.accessioned 4-Jun-2014 14:45:48 (UTC+8)-
dc.date.available 4-Jun-2014 14:45:48 (UTC+8)-
dc.date.issued (上傳時間) 4-Jun-2014 14:45:48 (UTC+8)-
dc.identifier (Other Identifiers) G0098152501en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/66502-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 教育研究所zh_TW
dc.description (描述) 98152501zh_TW
dc.description (描述) 102zh_TW
dc.description.abstract (摘要) 依據「多層次混合試題反應理論」與「隨機試題混合模型」，本研究提出「混合試題與受試者模型」。本研究旨在評估此模型在不同樣本數、不同試題差異功能的試題數下，偵測試題差異功能的表現，以及其參數回復性情形。研究結果顯示，「混合試題與受試者模型」在樣本數大、試題差異功能試題數較多之情境下，具有正確的參數回復性，能正確判斷出試題是否存在試題差異功能，且具有良好的難度估計值，並能將樣本正確地分群，其也與「隨機試題混合模型」的估計表現頗為相近。建議未來可將「混合試題與受試者模型」應用於大型教育資料庫相關研究上，並加入其他變項後進一步探討。zh_TW
dc.description.abstract (摘要) Drawing upon the framework of the multilevel mixture item response theory model and the random item mixture model, the study attempts to propose one model, called the mixture items and examinees model(MIE model). The purpose of this study was to assess the respective performances of the model on different sample-sizes and differential item functioning (DIF) items. Particularly, the study assessed the model performances in the detection of DIF items, and the accurate parameters recovery. The results of the study revealed that with large sample-sizes and more DIF items, the MIE model had the good parameters recovery, the accurate detection of the DIF items, the good estimate of the item difficulty, and the accurate classifications of the sub-samples. These model performances appeared similar to those of the random item mixture model. The findings suggest that future studies should apply the MIE model to the analyses on large-scale education databases, and should add more variables to the MIE model.en_US
dc.description.tableofcontents 中文摘要 iii
英文摘要 v
目次 vii
表次 viii
圖次 ix
第壹章 緒論 1
第一節 研究緣起與特點 1
第二節 研究特色與待答問題 3
第三節 名詞解釋 7
第四節 研究範圍 8
第貳章 文獻探討 9
第一節 偵測DIF的方法 9
第二節 多層次混合試題反應理論模型 16
第三節 隨機試題混合模型 18
第四節 小結 20
第參章 研究方法 21
第一節 理論基礎與模型 21
第二節 模擬因子與估計精準度 26
第三節 研究工具 32
第四節 實徵資料 33
第肆章 研究結果 35
第一節 模擬研究結果 35
第二節 實徵資料結果 39
第伍章 結論與建議 41
參考文獻 47		zh_TW
dc.format.extent 470668 bytes-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0098152501en_US
dc.subject (關鍵詞) 混合試題反應理論zh_TW
dc.subject (關鍵詞) 隨機試題zh_TW
dc.subject (關鍵詞) 試題差異功能zh_TW
dc.subject (關鍵詞) mixture item response theoryen_US
dc.subject (關鍵詞) random itemen_US
dc.subject (關鍵詞) differential item functioningen_US
dc.title (題名) 混合試題與受試者模型於試題差異功能分析之研究zh_TW
dc.title (題名) A Mixture Items-and-Examinees Model Analysis on Differential Item Functioningen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) 1.Bolt, D. M., Cohen, A. S., & Wollack, J. A. (2001). A mixture item response for multiple choice data. Journal of Educational and Behavioral Statistics, 26, 381-409.
2.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.
3.Camilli, G. (1992). A conceptual analysis of differential item functioning in terms of a multidimensional item response model. Applied Psychological Measurement, 16, 129-147.
4.Chaimongkol, S. (2005). Modeling differential item functioning (DIF) using multilevel logistic regression models: A Bayesian perspective. (Unpublished doctoral dissertation). Florida State University, Tallahassee, FL.
5.Chaimongkol, S., Huffer, F. W., & Kamata, A. (2007). An explanatory differential item functioning (DIF) model by the WinBUG 1.4. Songklanakarin Journal of Science and Technology, 29(2), 449-459.
6.Cheong, Y. F. (2006). Analysis of school context effects on differential item functioning using hierarchical generalized linear models. International Journal of Testing, 6(1), 57-79.
7.Cho, S. J., & Cohen, A. S. (2010). Multilevel mixture IRT model with an application to DIF. Journal of Educational and Behavioral Statistics, 35, 336-370.
8.Cho, S. J., Cohen, A. S., & Kim, S. H. (2006, June). An investigation of priors on the probabilities of mixtures in the mixture Rasch model. Paper presented at the International Meeting of the Psychometric Society: The 71st annual meeting of the Psychometric Society, Montreal, Canada.
9.Cohen, A. S., & Bolt, D. M. (2005). A mixture model analysis of differential item functioning. Journal of Educational Measurement, 42, 133-148.
10.Cohen, A. S., Cho, S. J., & Kim, S. H. (2005, April). A mixture testlet model for educational tests. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Canada.
11.Cohen, A. S., Gregg, N., & Deng, M. (2005). The role of extended time and item content on a high-stakes mathematics test. Learning Disabilities Research & Practice, 20, 225-233.
12.Dai, Y. (2013). A mixture Rasch model with a covariate a simulation study via Bayesian Markov Chain Monte Carlo estimation. Applied Psychological Measurement, 37(5), 375-396.
13.De Boeck, P. (2008). Random item IRT models. Psychometrika, 73, 533–559.
14.De Boeck, P., Cho, S. J., & Wilson, M. (2011). Explanatory secondary dimension modeling of latent differential item functioning. Applied Psychological Measurement, 35, 583-603.
15.DeAyala, R. J., Kim, S. H., Stapleton, L. M., & Dayton, C. M. (2002). Differential item functioning: A mixture distribution conceptualization. International Journal of Testing, 2, 243-276.
16.Demar, C. E., & Lau, A. (2011). Differential item functioning detection with latent classes: How accurately can we detect who is responding differentially? Educational and Psychological Measurement, 71(4), 597-616.
17.Dorans, N. J., & Kulick, E. (1986). Demonstrating the utility of the standardization approach to assessing unexpected differential item performance on the scholastic aptitude test. Journal of Educational Measurement, 23(4), 355-368.
18.Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah, NJ: Lawrence-Erlbaum.
19.Finch, W. H. (2005). The MIMIC model as a method for detecting DIF: Comparison with Mantel-Haenszel, SIBTEST, and the IRT Likelihood Ratio. Applied Psychological Measurement, 29, 278-295
20.Finch, W. H. (2012).The MIMIC model as a tool for differential bundle functioning detection. Applied Psychological Measurement, 36, 40-59.
21.Fox, J. P. (2005). Multilevel IRT using dichotomous and polytomous response data. British Journal of Mathematical and Statistical Psychology, 58, 145-172.
22.Frederickx, S., Tuerlinckx, F., De Boeck, P., & Magis, D. (2010). RIM: A random item mixture model to detect differential item functioning. Journal of Educational Measurement, 47, 432–457.
23.French, B. F., & Finch, W. H. (2010). Hierarchical logistic regression: Accounting for multilevel data in DIF detection. Journal of Educational Measurement, 47(3), 299-317.
24.Frick, H., Strobl, C., & Zeileis, A. (2013). Rasch mixture models for DIF detection: A comparison of old and new score specifications. Retrieved from http://eeecon.uibk.ac.at/wopec2/repec/inn/wpaper/2013-36.pdf
25.Holland, P. W., & Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In H. Wainer & H. I. Braun (Eds.), Test Validity (pp. 129-145). Hillsdale, NJ: Lawrence Erlbaum Associates.
26.Jiao, H., Lissitz, R. W., Macready, G., Wang, S., & Liang, S. (2011). Exploring levels of performance using the mixture Rasch model for standard setting. Psychological Test and Assessment Modeling, 53(4), 499-522.
27.Johnson, V., & Albert, J. (1998). Ordinal data modeling. New York: Springer.
28.Kamata, A. (1998). Some generalizations of the Rasch model: An application of the hierarchical generalized linear model. (Unpublished doctoral dissertation). Michigan State University, East Lansing, MI.
29.Kamata, A. (2001). Item analysis by the hierarchical generalized linear model. Journal of Educational Measurement, 38(1), 79-93.
30.Kang, T., & Cohen, A. S. (2003, April). A mixture model analysis of ethnic group DIF. Paper presented at the annual meeting of the National Council on Measurement in Education, Chicago, IL.
31.Li, F., Cohen, A. S., Kim, S. H., & Cho, S. J. (2006, April). Model selection methods for mixture dichotomous IRT models. Paper presented at the annual meeting of the National Council on Measurement in Education, San Francisco, CA.
32.Lord, F. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.
33.Lu, R., & Jiao, H. (2009, April). Detecting DIF using the mixture Rasch model. Paper presented at the annual meeting of the National Council on Measurement in Education, San Diego, CA.
34.Maier, K. S. (2002). Modeling incomplete scaled questionnaire data with a partial credit hierarchical measurement model. Journal of Educational and Behavioral Statistics, 27, 271-289.
35.Maij-de Meij, A. M., Kelderman, H., & van der Flier, H. (2010). Improvement in detection of differential item functioning using a mixture item response theory model. Multivariate Behavioral Research, 45, 975-999.
36.McLachlan, G., &, Peel, D. (2000). Finite mixture models. New York, NY: Wiley.
37.Mislevy, R. J., & Verhelst, N. (1990). Modeling item responses when different subjects employ different solution strategies. Psychometrika, 55, 195-215.
38.Navas-Ara, M. J., & Gomez-Benito, J. (2002). Effects of ability scale purification on the identification of DIF. European Journal of Psychological Assessment, 19, 9–15.
39.Penfield, R. D. (2010). Modelling DIF effects using distractor-level invariance effects: Implications for understanding the causes of DIF. Applied Psychological Measurement, 34, 151-165.
40.Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized multilevel structural equation modeling. Psychometrika, 69, 167–190.
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