In this study, the delta method was applied to estimate the standard errors of the true score equating when using the characteristic curve methods with the generalized partial credit model in test equating under the context of the common-item nonequivalent groups equating design. Simulation studies were further conducted to compare the performance of the delta method with that of the bootstrap method and the multiple imputation method. The results indicated that the standard errors produced by the delta method were very close to the criterion empirical standard errors as well as those yielded by the bootstrap method and the multiple imputation method under all the manipulated conditions.
American Educational Research Association, American Psychological Association, & National Council on Measurement in Education. (2014). Standards for educational and psychological testing. American Educational Research Association.
2.
AnderssonB. (2018). Asymptotic variance of linking coefficient estimators for polytomous IRT models. Applied Psychological Measurement, 42(3), 192–205.
3.
BakerF. B. (1992). Equating tests under the graded response model. Applied Psychological Measurement, 16(1), 87–96.
4.
BakerF. B.Al-KarniA. (1991). A comparison of two procedures for computing IRT equating coefficients. Journal of Educational Measurement, 28(2), 147–162.
5.
BattauzM. (2013). IRT test equating in complex linkage plans. Psychometrika, 78(3), 464–480.
6.
BattauzM. (2015). Factors affecting the variability of IRT equating coefficients. Statistica Neerlandica, 69(2), 85–101.
7.
BenichouJ.GailM. H. (1989). A delta method for implicitly defined random variables. The American Statistician, 43(1), 41–44.
8.
BockR. D.AitkinM. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443–459.
9.
BockR. D.LiebermanM. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35(2), 179–197.
10.
ChalmersR. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1–29.
11.
ChalmersR. P. (2018). Numerical approximation of the observed information matrix with Oakes’ identity. British Journal of Mathematical and Statistical Psychology, 71(3), 415–436.
12.
CohenA. S.KimS.-H. (1998). An investigation of linking methods under the graded response model. Applied Psychological Measurement, 22(2), 116–130.
13.
CookL. L.EignorD. R. (1991). IRT equating methods. Educational Measurement: Issues and Practice, 10(3), 37–45.
14.
EfronB.TibshiraniR. J. (1993). An introduction to the bootstrap. Chapman & Hall.
15.
HaebaraT. (1980). Equating logistic ability scales by a weighted leas squares method. Japanese Psychological Research, 22(3), 144–149.
16.
HansonB. A.BéguinA. A. (2002). Obtaining a common scale for item response theory item parameters using separate versus concurrent estimation in the common-item equating design. Applied Psychological Measurement, 26(1), 3–24.
17.
KimS. H.CohenA. S. (1998). A comparison of linking and concurrent calibration under item response theory. Applied Psychological Measurement, 22(2), 131–143.
18.
KimS. H.CohenA. S. (2002). A comparison of linking and concurrent calibration under the graded response model. Applied Psychological Measurement, 26(1), 25–41.
19.
KimJ.HansonB. A. (2002). Test equating under the multiple-choice model. Applied Psychological Measurement, 26(3), 255–270.
20.
KimS.KolenM. J. (2007). Effects of scale linking of different definitions of criterion functions for the IRT characteristic curve methods. Journal of Educational and Behavioral Statistics, 32(4), 371–397.
21.
KimS.LeeW.-C. (2006). An extension of four IRT linking methods for mixed-format test. Journal of Educational Measurement, 43(1), 53–76.
22.
KolenM.BrennanR. (2004). Test equating, scaling, and linking: Methods and practices (2nd ed.). Springer-Verlag.
23.
LoydB. H.HooverH. D. (1980). Vertical equating using the Rasch model. Journal of Educational Measurement, 17(3), 179–193.
24.
MarcoG. L. (1977). Item characteristic curve solutions to three intractable testing problems. Journal of Educational Measurement, 14, 139–160.
25.
MurakiE. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.
26.
OgasawaraH. (2000). Asymptotic standard errors of IRT equating coefficients using moments. Economic Review (Otaru University of Commerce), 51(1), 1–23.
27.
OgasawaraH. (2001a). Item response theory true score equating and their standard errors. Journal of Educational and Behavioral Statistics, 26(1), 31–50.
28.
OgasawaraH. (2001b). Standard errors of item response theory equating/linking by response function methods. Applied Psychological Measurement, 25(1), 53–67.
29.
PaekI.CaiL. (2014). A comparison of item parameter standard error estimation procedures for unidimensional and multidimensional item response theory modelling. Educational and Psychological Measurement, 74(1), 58–76.
30.
R Development Core Team. (2016). R: A language and environment for statistical computing. R Foundation for Statistical Computing.
31.
SamejimaF. (1969). Estimation of latent ability using a response pattern of graded scores (Psychometrika Monograph No. 17). Psychometric Society.
32.
StockingM. L.LordF. M. (1983). Developing a common metric in item response theory. Applied Psychological Measurement, 7(2), 201–210.
33.
TsaiT. H.HansonB. A.KolenM. J.ForsythA. R. A. (2001). A comparison of bootstrap standard errors of IRT equating methods for the common-item nonequivalent groups design. Applied Measurement in Education, 14(1), 17–30.
34.
WongC. C. (2015). Asymptotic standard errors for item response theory true score equating of polytomous items. Journal of Educational Measurement, 52(1), 106–120.
35.
ZhangZ. H. (2020a). Asymptotic standard errors of equating coefficients using the characteristic curve methods for the graded response model. Applied Measurement in Education, 1–22.
36.
ZhangZ. H. (2020b). Estimating standard errors of IRT true score equating coefficients using imputed item parameters. The Journal of Experimental Education, 1–23.
37.
ZhangZ. H. (2021). Asymptotic standard errors of parameter scale transformation coefficients in test equating under the nominal response model. Applied Psychological Measurement, 45, 134–138.
38.
ZhangZ. H.ZhaoM. R. (2019). Standard errors of IRT parameter scale transformation coefficients: Comparison of bootstrap method, delta method, and multiple imputation method. Journal of Educational Measurement, 56(2), 302–330.
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