문항반응모형 모수 추정에서 2-요소 정규혼합분포의 잠재분포로서의 활용 = Using a Two-Component Normal Mixture Distribution as a Latent Distribution in Estimating Parameters of Item Response Models

국문 초록 (Abstract)

검사 점수가 이봉분포를 따르는 경우 Mislevy(1984)는 2-요소 정규혼합분포(2NM)를 문항반응모형의 잠재분포로 활용하는 방법을 제안했다. 본 연구의 목적은 실제 잠재분포가 분포의 모수 값에 따라 형성된 다양한 모양의 2NM일 때 2NM-가정-문항반응모형(2NM-IRM)의 기능을 모의실험을 통해 검토하는 것이다. 본 연구에서는 2NM의 모양의 특징을 나타내고 모양을 결정하는 3개의 모양 모수(shape parameter)를 정의하였으며, 이 3개의 모양 모수 값들에 의해 결정된 총 15가지 모양의 2NM을 모의실험 조건으로 활용하였다. 2NM-IRM의 기능은 정규성-가정-문항반응모형(normal-IRM), 경험적 히스토그램 방법(EHM), Davidian-curve IRT(DC-IRT)와의 비교를 통해 검토되었다. 연구 결과, 2NM-IRM은 실제 잠재분포가 표준정규분포와 매우 유사한 경우를 제외하고는 다른 방법들보다 더 정확한 추정 결과를 산출하는 것으로 나타났다. 모양 모수 추정치에 편의가 존재하는 경우에도 2NM-IRM은 실제 잠재분포의 전반적인 모양을 일관적으로 정확하게 추정하고, 이에 따라 정확한 문항모수 및 능력모수 추정치를 제공하는 것으로 나타났다.

다국어 초록 (Multilingual Abstract)

A two-component normal mixture distribution (2NM) could be used as the latent distribution of item response models when test scores follow a bimodal distribution, an idea that was originally proposed by Mislevy (1984). The purpose of this study is to examine the performance of this aforementioned 2NM-assumption item response model (2NM-IRM) under various shapes of the true latent distributions. Computer simulation techniques were used for this purpose. A reparameterization was enacted to characterize the shape of the 2NM using only three shape parameters, through which 15 total shapes of 2NM were used as simulation conditions. The performance of 2NM-IRM was compared with the normality-assumption, empirical histogram, and Davidian-curve methods. 2NM-IRM produced the most accurate results except when the true latent distribution was practically equivalent to the normal distribution. Even when some biases were present in the shape parameter estimates, 2NM-IRM was consistent at accurately identifying the overall shape of the true latent distributions and thus provided accurate item and ability parameter estimates.

목차 (Table of Contents)

• Ⅰ. Introduction Ⅱ. Estimating Distribution Parameters of 2NM-IRM Ⅲ. The Three Shape Parameters of 2NM Ⅳ. Methods Ⅴ. Results Ⅵ. Discussions and Conclusions

참고문헌 (Reference)

1 강혜진, "잠재변수의 비정규성, 검사 문항 수, 피험자 수가 Davidian 곡선을 적용한 IRT 모수 추정에 미치는 영향" 한국교육평가학회 33 (33): 533-559, 2020

2 Chalmers, R. P., "mirt : A multidimensional item response theory package for the R environment" 48 (48): 1-29, 2012

3 Cai, L., "flexMIRT® version 3.62: Flexible multilevel multidimensional item analysis and test scoring [Computer software]" Vector Psychometric Group. 2020

4 Houts, C. R., "flexMIRT® user’s manual version 3.6: Flexible multilevel multidimensional item analysis and test scoring" Vector Psychometric Group 2020

5 Yadin, A., "Using unique assignments for reducing the bimodal grade distribution" 4 (4): 38-42, 2013

6 De Ayala, R. J., "The theory and practice of item response theory" Guilford Publications 2009

7 Bauer, D. J., "The integration of continuous and discrete latent variable models : potential problems and promising opportunities" 9 (9): 3-29, 2004

8 Hastie, T., "The elements of statistical learning: Data mining, inference, and prediction" Springer Science & Business Media. 2009

9 Hannan, E. J., "The determination of the order of an autoregression" 41 (41): 190-195, 1979

10 Mellenbergh, G. J., "The Rasch model as a loglinear model" 5 (5): 369-376, 1981

11 Harpaz, R., "The EM algorithm as a lower bound optimization technique. CUNY Ph. D. Program in Computer Science Technical Reports" 1-14, 2006

12 Titterington, D. M., "Statistical analysis of finite mixture distributions" Wiley 1985

13 Seong, T. J., "Sensitivity of marginal maximum likelihood estimation of item and ability parameters to the characteristics of the prior ability distributions" 14 (14): 299-311, 1990

14 Finch, H., "Rasch model parameter estimation in the presence of a nonnormal latent trait using a nonparametric Bayesian approach" 76 (76): 662-684, 2016

15 R Core Team, "R: A language and environment for statistical computing" R Foundation for Statistical Computing

16 Mislevy, R. J., "Proceedings of the 1982 item response theory and computerized adaptive testing conference" University of Minnesota, Department of Psychology, Computerized Adaptive Testing Conference. 189-202, 1985

17 Rossnan, S., "Overcoming math anxiety" 1 (1): 1-4, 2006

18 Meyer, J., "Overcoming barriers to student understanding: Threshold concepts and troublesome knowledge" Routledge 2006

19 Behboodian, J., "On the modes of a mixture of two normal distributions" 12 (12): 131-139, 1970

20 Sibbald, T., "Occurrence of bimodal classroom achievement in Ontario" 60 (60): 221-225, 2014

21 Geist, E., "Math anxiety and the"math gap" : How attitudes toward mathematics disadvantages students as early as preschool" 135 (135): 328-336, 2015

22 Bock, R. D., "Marginal maximum likelihood estimation of item parameters : Application of an EM algorithm" 46 (46): 443-459, 1981

23 Thissen, D., "Loglinear item response models, with applications to data from social surveys" 19 : 299-330, 1989

24 Kelderman, H., "Loglinear Rasch model tests" 49 (49): 223-245, 1984

25 Baker, F. B., "Item response theory: Parameter estimation techniques" CRC Press 2004

26 Woods, C. M., "Item response theory with estimation of the latent population distribution using spline-based densities" 71 (71): 281-301, 2006

27 Woods, C. M., "Item response theory with estimation of the latent density using Davidian curves" 33 (33): 102-117, 2009

28 Lubke, G. H., "Investigating population heterogeneity with factor mixture models" 10 (10): 21-39, 2005

29 Dodd, A. W., "Insights from a math phobic" 85 (85): 296-298, 1992

30 Woods, C. M., "Handbook of item response theory modeling" Routledge 78-102, 2014

31 von Davier, M., "Handbook of item response theory (Vol. 1)" Chapman and Hall/CRC 421-434, 2016

32 Eisenberger, I., "Genesis of bimodal distributions" 6 (6): 357-363, 1964

33 Hambleton, R. K., "Fundamentals of item response theory (Vol. 2)" Sage 1991

34 Andersen, E., "Estimating the parameters of the latent population distribution" 42 (42): 357-374, 1977

35 Mislevy, R. J., "Estimating latent distributions" 49 (49): 359-381, 1984

36 Bock, R. D., "Estimating item parameters and latent ability when responses are scored in two or more latent categories" 37 (37): 29-51, 1972

37 Pearl, J., "Detecting latent heterogeneity" 46 (46): 370-389, 2017

38 Girelli, S. A., "Bipolarity in Jungian type theory and the Myers-Briggs type indicator" 60 (60): 290-301, 1993

39 Bess, T. L., "Bimodal score distributions and the Myers-Briggs Type Indicator : fact or artifact?" 78 (78): 176-186, 2002

40 Rytting, M., "Bimodal distributions in a sample of CEOs : Validating evidence for the MBTI" 31 : 16-23, 1994

41 김성훈, "BILOG-MG를 사용한 IRT 모형 모수의 추정에서 기저 능력분포의 경험적 추정이 문항모수 추정에 미치는 영향" 한국교육평가학회 25 (25): 317-336, 2012

42 Baker, F. B., "Analysis of test results via log-linear models" 5 (5): 503-515, 1981

43 Pastor, D. A., "A latent profile analysis of college students’ achievement goal orientation" 32 (32): 8-47, 2007