Kernel equating: A framework of observed score equating

doi:10.3724/SP.J.1042.2020.00855

Abstract

Abstract:

Kernel equating procedures include pre-smoothing, estimation of the score probabilities, continuization, equating, and evaluation of equating performance. By incorporating linear equating and equipercentile equating methods, kernel equating is more extensible and comprehensive. Pre-smoothing and continuization are distinctive features in kernel equating to reduce the standard error of equating. Standard error of the difference between equating functions are calculated as criterion for evaluating the performances of different kernel equatings. Continuization methods, bandwidth selection methods, etc., can affect the performance of kernel equating. New equating methods based on kernel equating provide an innovative perspective for researchers. Further researchers could focus on extending kernel equating framework by integrating other methods, updating smoothing procedures, and comparative studies.

Key words: kernel equating, continuization, bandwidth selection, new equating methods

CLC Number:

B841

WANG Shaojie, ZHANG Minqiang, LI Tuoyu, LIANG Zhengyan. Kernel equating: A framework of observed score equating[J]. Advances in Psychological Science, 2020, 28(5): 855-870.

Figures/Tables 1

References 84

1	陈俊丽 . ( 2008). 核等值与其它等值方法的比较研究 (硕士学位论文). 北京语言大学.
2	关丹丹, 景春丽 . ( 2018). 新高考改革背景下不分文理的数学成绩差异研究. 数学教育学报, 27( 4), 31-34.
3	罗莲 . (2008a). 基于HSK数据对核等值法与其他等值方法的比较研究 (博士学位论文). 北京语言大学.
4	罗莲 . (2008b). 一种新的等值方法:核等值法. 心理学探新, 28( 2), 69-74.
5	张敏强, 胡晖 . ( 1988). 略论测验等值的理论、方法和应用. 华南师范大学学报(社会科学版), ( 4), 113-118.
6	Andersson B. ( 2016). Asymptotic standard errors of observed-score equating with polytomous IRT models. Journal of Educational Measurement, 53( 4), 459-477.
7	Andersson B., Bränberg K., & Wiberg M . ( 2013). Performing the kernel method of test equating with the package kequate. Journal of Statistical Software, 55( 6), 1-25.
8	Andersson B. & von Davier, A. A . ( 2014). Improving the bandwidth selection in kernel equating. Journal of Educational Measurement, 51( 3), 223-238.
9	Andersson B. &Wiberg M. , ( 2017). Item response theory observed-score kernel equating. Psychometrika, 82( 1), 48-66.
10	Arıkan Ç. A., &Gelbal S. , ( 2018). A comparison of traditional and kernel equating methods. International Journal of Assessment Tools in Education, 5( 3), 417-427.
11	Chen H. ( 2012). A comparison between linear IRT observed- score equating and Levine observed-score equating under the generalized kernel equating framework. Journal of Educational Measurement, 49( 3), 269-284.
12	Chen H. &Holland P. , ( 2009). Construction of chained true score equipercentile equatings under the kernel equating (KE) framework and their relationship to Levine true score equating. ETS Research Report Series, 2009( 1), i-15.
13	Chen H. &Holland P. , ( 2010). New equating methods and their relationships with Levine observed score linear equating under the kernel equating framework. Psychometrika, 75( 3), 542-557.
14	Chen H. H., Livingston S. A., & Holland P. W . ( 2009). Generalized equating functions for NEAT designs. In S. E. Fienberg, & W. J. van der Linden (Series Eds.) & A. A. von Davier (Vol. Ed.), Statistics for social and behavioral sciences: Statistical models for test equating, scaling, and linking( pp. 185-200). New York City, NY: Springer.
15	Choi S.I . ( 2009). A comparison of kernel equating and traditional equipercentile equating methods and the parametric bootstrap methods for estimating standard errors in equipercentile equating (Unpublished doctorial dissertation). University of Illinois at Urbana-Champaign.
16	Cid J. A., & von Davier, A. A . ( 2015). Examining potential boundary bias effects in kernel smoothing on equating: An introduction for the adaptive and Epanechnikov kernels. Applied Psychological Measurement, 39( 3), 208-222.
17	de Ayala R. J., Smith B., & Norman Dvorak R . ( 2018). A comparative evaluation of kernel equating and test characteristic curve equating. Applied Psychological Measurement, 42( 2), 155-168.
18	Dorans N. J., Liu J., & Hammond S . ( 2008). Anchor test type and population invariance: An exploration across subpopulations and test administrations. Applied Psychological Measurement, 32( 1), 81-97.
19	Dorans N. J., &Puhan G. , ( 2017). Contributions to score linking theory and practice. In B. Veldkamp, & M. von Davier (Series Eds.) & R. E. Bennett, & M. von Davier (Vol. Eds.), Methodology of educational measurement and assessment: Advancing human assessment: The methodological, psychological and policy contributions of ETS( pp. 79-132). Cham, Zug, Switzerland: Springer.
20	Duong M. & von Davier, A. A . (2008,March). Kernel equating with observed mixture distributions in a single- group design. Paper presented at the annual meeting of the National Council on Measurement in Education, New York, NY.
21	ETS. ( 2007a). GENASYS [Computer software]. Princeton, NJ: Author.
22	ETS. ( 2007b). KE Software [Computer software]. Princeton, NJ: Author.
23	Godfrey K.E . ( 2007). A comparison of kernel equating and IRT true score equating methods (Unpublished doctorial dissertation). The University of North Carolina at Greensboro.
24	González, J. ( 2014). SNSequate: Standard and nonstandard statistical models and methods for test equating. Journal of Statistical Software, 59( 7), 1-30.
25	González J., Barrientos A. F., & Quintana F. A . ( 2015). Bayesian nonparametric estimation of test equating functions with covariates. Computational Statistics & Data Analysis, 89, 222-244.
26	González J. & von Davier, A. A . ( 2016). An illustration of the Epanechnikov and adaptive continuization methods in kernel equating.In L. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W. -C. Wang (Vol. Eds), Springer proceedings in mathematics & statistics: Vol. 196. Quantitative psychology: The 81st annual meeting of the Psychometric Society, Asheville, North Carolina, 2016 .(Cham, Zug, Switzerland: Springer.
27	Grant M. C., Zhang L., & Damiano I . ( 2009). An evaluation of kernel equating: Parallel equating with classical methods in the SAT subject tests™ program. ETS Research Report Series, 2009(1), i-25.
28	Haberman S.J . ( 1984). Adjustment by minimum discriminant information. The Annals of Statistics, 12( 3), 971-988.
29	Haberman S.J . ( 2015). Pseudo-equivalent groups and linking. Journal of Educational and Behavioral Statistics, 40( 3), 254-273.
30	Häggström J. &Wiberg M. , ( 2014). Optimal bandwidth selection in observed‐score kernel equating. Journal of Educational Measurement, 51( 2), 201-211.
31	Holland P. W., &Thayer D. T . ( 2000). Univariate and bivariate loglinear models for discrete test score distributions. Journal of Educational and Behavioral Statistics, 25( 2), 133-183.
32	Holland P. W., von Davier A. A., Sinharay S., & Han N . ( 2006). Testing the untestable assumptions of the chain and poststratification equating methods for the NEAT design. ETS Research Report Series, 2006(( 1), i-38.
33	Jiang Y., von Davier A. A., & Chen H . ( 2012). Evaluating equating results: Percent relative error for chained kernel equating. Journal of Educational Measurement, 49( 1), 39-58.
34	Jones M. C., Marron J. S., & Sheather S. J . ( 1996). A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association, 91( 433), 401-407.
35	Kim H.Y . ( 2014). A comparison of smoothing methods for the common item nonequivalent groups design (Unpublished doctorial dissertation). University of Iowa, Iowa City.
36	Kim S. &Lu R. , ( 2018). The pseudo-equivalent groups approach as an alternative to common-item equating. ETS Research Report Series, 2018( 1), 1-13.
37	Kolen M. J., &Brennan R. L . ( 2014). Test equating, scaling, and linking: methods and practices. New York City, NY: Springer Science & Business Media.
38	Lee Y. H., & von Davier, A. A . ( 2008). Comparing alternative kernels for the kernel method of test equating: Gaussian, logistic, and uniform kernels. ETS Research Report Series, 2008( 1), i-26.
39	Lee Y. H., & von Davier, A. A . ( 2011). Equating through alternative kernels. In S. E. Fienberg, & W. J. van der Linden (Series Eds.) & A. A. von Davier (Vol. Ed.), Statistics for social and behavioral sciences: Statistical models for test equating, scaling, and linkingpp. 159-273). New York City, NY: Springer.
40	Leôncio W. &Wiberg M. , ( 2017). Evaluating equating transformations from different frameworks. In M. Wiberg, S. Culpepper, R. Janssen, J. González, & D. Molenaar (Vol. Eds), Springer proceedings in mathematics & statistics: Vol. 233. Quantitative psychology: The 82nd annual meeting of the Psychometric Society, Zurich, Switzerland, 2017(pp. 101-110). Cham, Zug, Switzerland: Springer.
41	Liang T. & von Davier, A. A . ( 2014). Cross-validation: An alternative bandwidth-selection method in kernel equating. Applied Psychological Measurement, 38( 4), 281-295.
42	Liu J. &Low A. C . ( 2007). An exploration of kernel equating using SAT® data: Equating to a similar population and to a distant population. ETS Research Report Series, 2007( 1), i-22.
43	Liu J. &Low A. C . ( 2008). A comparison of the kernel equating method with traditional equating methods using SAT® data. Journal of Educational Measurement, 45( 4), 309-323.
44	Longford N.T . ( 2015). Equating without an anchor for nonequivalent groups of examinees. Journal of Educational and Behavioral Statistics, 40( 3), 227-253.
45	Lord F.M . ( 1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.
46	Lu R. &Guo H. , ( 2018). A simulation study to compare nonequivalent groups with anchor test equating and pseudo-equivalent group linking. ETS Research Report Series, 2018( 1), 1-16.
47	Mao X. ( 2006). An investigation of the accuracy of the estimates of standard errors for the kernel equating functions (Unpublished doctorial dissertation). University of Iowa, Iowa City.
48	Meng Y. ( 2012). Comparison of kernel equating and item response theory equating methods (Unpublished doctorial dissertation). University of Massachusetts Amherst.
49	Moses T. &Holland P. , ( 2007). Kernel and traditional equipercentile equating with degrees of presmoothing. ETS Research Report Series, 2007( 1), 1-39.
50	Moses T. &Holland P. , ( 2008). Notes on a general framework for observed score equating. ETS Research Report Series, 2008( 2), i-34.
51	Moses T., Yang W. L., & Wilson C . ( 2007). Using kernel equating to assess item order effects on test scores. Journal of Educational Measurement, 44( 2), 157-178.
52	Norman Dvorak, R. L . ( 2009). A comparison of kernel equating to the test characteristic curve method (Unpublished doctorial dissertation). University of Nebraska, Lincoln.
53	Puhan G., von Davier A., & Gupta S . ( 2008). Impossible scores resulting in zero frequencies in the anchor test: Impact on smoothing and equating. ETS Research Report Series, 2008( 1), i-26.
54	R Core Team . ( 2017). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing.
55	Sansivieri V. &Wiberg M. , ( 2016). IRT observed-score equating with the nonequivalent groups with covariates design. In L. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W. -C. Wang (Vol. Eds), Springer proceedings in mathematics & statistics: Vol. 196. Quantitative psychology: The 81st annual meeting of the Psychometric Society, Asheville, North Carolina, 2016 (pp. 275-285). Cham, Zug, Switzerland: Springer.
56	Sansivieri V., Wiberg M., & Matteucci M . ( 2017). A review of test equating methods with a special focus on IRT-based approaches. Statistica, 77( 4), 329-352.
57	Sinharay S. &Holland P. W . ( 2010). A new approach to comparing several equating methods in the context of the NEAT design. Journal of Educational Measurement, 47( 3), 261-285.
58	Underhill J.L . ( 2017). The robustness of kernel equating as non-normality occurs under the equivalent groups design (Unpublished doctorial dissertation). University of Florida, Gainesville.
59	van der Linden, W. J . ( 2010). On bias in linear observed-score equating. Measurement: Interdisciplinary Research & Perspective, 8( 1), 21-26.
60	van der Linden, W. J . ( 2013). Some conceptual issues in observed-score equating. Journal of Educational Measurement, 50( 3), 249-285.
61	van der Linden, W. J., &Wiberg M. , ( 2010). Local observed-score equating with anchor-test designs. Applied Psychological Measurement, 34( 8), 620-640.
62	von Davier, A. A . ( 2011a). An observed-score equating framework.In P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, S. Zeger (Series. Eds.) & N. J. Dorans, & S. Sinharay (Vol. Eds.), Lecture notes in statistics: Proceedings: Vol 202. Looking back: proceedings of a conference in honor of Paul W. Holland(pp. 221-238). New York City, NY: Springer.
63	von Davier, A. A . ( 2011 b). A statistical perspective on equating test scores. In S. E. Fienberg, & W. J. van der Linden (Series Eds.) & A. A. von Davier (Vol. Ed.), Statistics for social and behavioral sciences: Statistical models for test equating, scaling, and linking( pp. 1-17). New York City, NY: Springer.
64	von Davier, A. A . ( 2013). Observed-score equating: An overview. Psychometrika, 78( 4), 605-623.
65	von Davier, A. A., &Chen H. , ( 2013). The kernel levine equipercentile observed-score equating function. ETS Research Report Series,( 2), i-27.
66	von Davier A. A., Fournier-Zajac S., & Holland P. W . ( 2007). An equipercentile version of the Levine linear observed-score equating function using the methods of kernel equating. ETS Research Report Series,( 1), i-19.
67	von Davier A. A., Holland P. W., Livingston S. A., Casabianca J., Grant M. C., & Martin K . ( 2006). An evaluation of the kernel equating method: A special study with pseudotests constructed from real test data. ETS Research Report Series,( 1), i-31.
68	von Davier A. A., Holland P. W., & Thayer D. T . ( 2004). The kernel method of test equating. New York City, NY: Springer-Verlag.
69	von Davier, A. A., &Kong N. , ( 2005). A unified approach to linear equating for the nonequivalent groups design. Journal of Educational and Behavioral Statistics, 30( 3), 313-342.
70	Wallin G., Häggström J., & Wiberg M . ( 2017). How to select the bandwidth in kernel equating-An evaluation of five different methods. In M. Wiberg, S. Culpepper, R. Janssen, J. González, & D. Molenaar (Vol. Eds), Springer proceedings in mathematics & statistics: Vol. 233. Quantitative psychology: The 82nd annual meeting of the Psychometric Society, Zurich, Switzerland, 2017 (pp. 91-100). Cham, Zug, Switzerland: Springer.
71	Wallin G. &Wiberg M. , ( 2016). Nonequivalent groups with covariates design using propensity scores for kernel equating. In L. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W. -C. Wang (Vol. Eds), Springer proceedings in mathematics & statistics: Vol. 196. Quantitative psychology: The 81st annual meeting of the Psychometric Society, Asheville, North Carolina, 2016 (pp. 309-319). Cham, Zug, Switzerland: Springer.
72	Wallin G. &Wiberg M. , ( 2019). Kernel equating using propensity scores for nonequivalent groups. Journal of Educational and Behavioral Statistics, 44( 4), 390-414.
73	Wang, T. ( 2007). An alternative continuization method to the kernel method in von Davier, Holland and Thayer’s (2004) test equating framework (No.11).. Retrieved Jan 8, 2020, from
74	Wang T. ( 2011). An alternative continuization method: The continuized log-linear method. In S. E. Fienberg, & W. J. van der Linden (Series Eds.) & A. A. von Davier (Vol. Ed.), Statistics for Social and Behavioral Sciences: Statistical models for test equating, scaling, and linking( pp. 141-157). New York City, NY: Springer.
75	Wang T., Lee W. C., Brennan R. L., & Kolen M. J . ( 2008). A comparison of the frequency estimation and chained equipercentile methods under the common-item nonequivalent groups design. Applied Psychological Measurement, 32( 8), 632-651.
76	Wedman J. ( 2017). Theory and validity evidence for a large-scale test for selection to higher education (Unpublished doctorial dissertation). Umeå University.
77	Wiberg M. ( 2016 a). Alternative linear item response theory observed-score equating methods. Applied Psychological Measurement, 40( 3), 180-199.
78	Wiberg M. ( 2016b). Ensuring test quality over time by monitoring the equating transformations. In L. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W. -C. Wang (Vol. Eds), Springer proceedings in mathematics & statistics: Vol. 196. Quantitative psychology: The 81st annual meeting of the Psychometric Society, Asheville, North Carolina, 2016 (pp. 239-251). Cham, Zug, Switzerland: Springer.
79	Wiberg M. &Bränberg K. , ( 2015). Kernel equating under the non-equivalent groups with covariates design. Applied Psychological Measurement, 39( 5), 349-361.
80	Wiberg M. &González J. , ( 2016). Statistical assessment of estimated transformations in observed-score equating. Journal of Educational Measurement, 53( 1), 106-125.
81	Wiberg M. & van der Linden, W. J . ( 2011). Local linear observed-score equating. Journal of Educational Measurement, 48( 3), 229-254.
82	Wiberg M ., van der Linden, W. J., & von Davier, A. A. ( 2014). Local observed-score kernel equating. Journal of Educational Measurement, 51( 1), 57-74.
83	Wiberg M. & von Davier, A. A . ( 2017). Examining the impact of covariates on anchor tests to ascertain quality over time in a college admissions test. International Journal of Testing, 17( 2), 105-126.
84	Xin T. &Zhang J. , ( 2015). Local equating of cognitively diagnostic modeled observed scores. Applied Psychological Measurement, 39( 1), 44-61.

等值设计	CTT等值	核等值
EG	等百分位等值	核等值(最优带宽)
EG	线性等值	核等值(较大带宽, ${{h}_{X}}>$ $10{{\sigma }_{X}}$, 下同)
NEAT	等百分位链等值	核链等值(最优带宽)
	等百分位后分层等值	核后分层等值(最优带宽)
	线性链等值	核链等值(较大带宽)
	Tucker等值	核后分层等值(较大带宽, 特定条件下)
	Levine观察分数等值	-

等值设计	CTT等值	核等值
EG	等百分位等值	核等值(最优带宽)
EG	线性等值	核等值(较大带宽, ${{h}_{X}}>$ $10{{\sigma }_{X}}$, 下同)
NEAT	等百分位链等值	核链等值(最优带宽)
	等百分位后分层等值	核后分层等值(最优带宽)
	线性链等值	核链等值(较大带宽)
	Tucker等值	核后分层等值(较大带宽, 特定条件下)
	Levine观察分数等值	-

[1]	SHANG Junjie, SHI Zhu, SHEN Kejie. Video game-based assessment of spatial ability [J]. Advances in Psychological Science, 2025, 33(1): 11-24.
[2]	WEN Congcong. A new measurement invariance test method: Penalized alignment [J]. Advances in Psychological Science, 2025, 33(1): 176-190.
[3]	SHUI Xinyu, XIAO Yaheng, CHEN Jingjing, HU Xin, ZHANG Dan. A profile-perspective on daily-life multi-situational individual differences assessment [J]. Advances in Psychological Science, 2024, 32(12): 2050-2066.
[4]	GAO Xuliang, LI Ning. Application of machine learning methods in test security [J]. Advances in Psychological Science, 2024, 32(11): 1814-1828.
[5]	HAN Ming, KUAI Shu-Guang. Toolkits for virtual reality research in psychology [J]. Advances in Psychological Science, 2024, 32(11): 1800-1813.
[6]	GUO Mingqian, PAN Wanke, HU Chuanpeng. Model comparison in cognitive modeling [J]. Advances in Psychological Science, 2024, 32(10): 1736-1756.
[7]	XIAO Yue, LIU Hongyun, XU Yongze. Model construction for intensive longitudinal dyadic data analysis [J]. Advances in Psychological Science, 2024, 32(9): 1450-1462.
[8]	SONG Lihong, WANG Wenyi, DING Shuliang. Q-matrix theory and its applications in cognitive diagnostic assessment [J]. Advances in Psychological Science, 2024, 32(6): 1010-1033.
[9]	ZHAO Jiawei, XIA Tao, HU Chuanpeng. The status quo, challenges, and recommendations of pre-registration in psychological science [J]. Advances in Psychological Science, 2024, 32(5): 715-727.
[10]	LUO Xiaohui, LIU Hongyun. Estimating test reliability of intensive longitudinal studies: Perspectives on multilevel structure and dynamic nature [J]. Advances in Psychological Science, 2024, 32(4): 700-714.
[11]	WANG Su-Jing, WANG Yan, Li Jingting, DONG Zizhao, ZHANG Jianhang, LIU Ye. Cross-modal analysis of facial EMG in micro-expressions and data annotation algorithm [J]. Advances in Psychological Science, 2024, 32(1): 1-13.
[12]	CHEN Ping, DAI Yi, HUANG Yingshi. Test mode effect: Sources, detection, and applications [J]. Advances in Psychological Science, 2023, 31(10): 1966-1980.
[13]	BAO Han-Wu-Shuang, WANG Zi-Xi, CHENG Xi, SU Zhan, YANG Ying, ZHANG Guang-Yao, WANG Bo, CAI Hua-Jian. Using word embeddings to investigate human psychology: Methods and applications [J]. Advances in Psychological Science, 2023, 31(6): 887-904.
[14]	LIU Yue, FANG Fan, LIU Hongyun, LEI Yi. Model construction and sample size planning for mixed-effects location-scale models [J]. Advances in Psychological Science, 2023, 31(6): 958-969.
[15]	FANG Junyan, WEN Zhonglin. The endogeneity issue in longitudinal research: Sources and solutions [J]. Advances in Psychological Science, 2023, 31(4): 507-518.