## Abstract

Linkage plans can be rather complex, including many forms, several links, and the connection of forms through different paths. This article studies item response theory equating methods for complex linkage plans when the common-item nonequivalent group design is used. An efficient way to average equating coefficients that link the same two forms through different paths will be presented and the asymptotic standard errors of indirect and average equating coefficients are derived. The methodology is illustrated using simulations studies and a real data example.

This is a preview of subscription content, log in to check access.

## References

Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: application of an EM algorithm.

*Psychometrika*,*46*, 443–459.Braun, H.I., & Holland, P.W. (1982). Observed-score test equating: a mathematical analysis of some ETS equating procedures. In P.W. Holland & D.B. Rubin (Eds.),

*Test equating*(pp. 9–49). New York: Academic Press.Guo, H. (2010). Accumulative equating error after a chain of linear equatings.

*Psychometrika*,*75*, 438–453.Guo, H., Liu, J., Dorans, N., & Feigenbaum, M. (2011).

*Multiple linking in equating and random scale drift*. Princeton: Educational Testing Service. (ETS RR-11-46).Haberman, S.J. (2009).

*Linking parameter estimates derived from an item response model through separate calibrations*. Princeton: Educational Testing Service. (ETS RR-09-40).Holland, P.W., & Strawderman, W.E. (2011). How to average equating functions if you must. In A.A. von Davier (Ed.),

*Statistical models for test equating, scaling, and linking*(pp. 89–107). New York: Springer.Kolen, M.J., & Brennan, R.L. (2004).

*Test equating, scaling, and linking: methods and practices*(2nd ed.). New York: Springer.Li, D., Jiang, Y., & von Davier, A.A. (2012). The accuracy and consistency of a series of IRT true-score equatings.

*Journal of Educational Measurement*,*49*, 167–189.Li, D., Li, S., & von Davier, A.A. (2011). Applying time-series analysis to detect scale drift. In A.A. von Davier (Ed.),

*Statistical models for test equating, scaling, and linking*, New York: Springer.Ogasawara, H. (2000). Asymptotic standard errors of IRT equating coefficients using moments.

*Economic Review (Otaru University of Commerce)*,*51*, 1–23.Ogasawara, H. (2001a). Item response theory true score equatings and their standard errors.

*Journal of Educational and Behavioral Statistics*,*26*, 31–50.Ogasawara, H. (2001b). Standard errors of item response theory equating/linking by response function methods.

*Applied Psychological Measurement*,*25*, 53–67.Ogasawara, H. (2003). Asymptotic standard errors of IRT observed-score equating methods.

*Psychometrika*,*68*, 193–211.Ogasawara, H. (2011). Applications of asymptotic expansion in item response theory linking. In A.A. von Davier (Ed.),

*Statistical models for test equating, scaling, and linking*, New York: Springer.Puhan, G. (2009). Detecting and correcting scale drift in test equating: an illustration from a large scale testing program.

*Applied Measurement in Education*,*22*, 79–103.R Development Core Team (2012).

*R: a language and environment for statistical computing*. R Foundation for Statistical Computing, Vienna, Austria.Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses.

*Journal of Statistical Software*,*17*, 1–25.van der Linden, W.J., & Hambleton, R.K. (1997).

*Handbook of modern item response theory*. Berlin: Springer.von Davier, A.A. (2011). Quality control and data mining techniques applied to monitoring scaled scores. In M. Pechenizkiy, T. Calders, C. Conati, S. Ventura, C. Romero, & J. Stamper (Eds.),

*Proceedings of the 4th international conference on educational data mining*, Eindhoven, The Netherlands, July 6–8, 2011. Eindhoven: University of Technology Library.

## Acknowledgements

This work was supported by grants of the Italian Ministry for Education, University, and Research (MIUR).

The author thanks the editor, the associate editor, and three anonymous reviewers for their comments that contributed to improve the quality of the paper. The author is grateful to Professor R. Bellio for his helpful suggestions.

## Author information

## Appendices

### Appendix A. Partial Derivatives of the Equating Coefficients with Respect to the Item Parameters

### A.1 Indirect Coefficients

Irrespective of the method used to obtain direct equating coefficients, the partial derivatives of indirect equating coefficients used in Equation (5) are as follows:

Note that \(\frac{\partial A_{{g-1, g}}}{\partial a_{gj}}\) and \(\frac{\partial A_{{g, g+1}}}{\partial a_{gj}}\) are both different from zero only if item *j* of form *g* is present in both form *g*−1 and in form *g*+1. Similarly,

while

where \(\frac{\partial B_{h-1, h}}{\partial a_{gj}}\) is equal to zero if *g*≠*h*−1 and *g*≠*h* and \(\frac{\partial A_{h, \dots, l}}{\partial a_{gj}}\) is equal to zero if *g*<*h*. Finally,

### A.2 Bisector Equating Coefficients

The partial derivatives of the bisector coefficients with respect to direct and indirect equating coefficients relative to one of the paths that link to forms used in Equation (7) are as follows:

### Appendix B. Equating Coefficients for Common-Item Equating to a Calibrated Pool

It is assumed that *a*
_{
gj
}=1 for all *g* and *j*. Then the equating coefficient for converting the parameters of Form 2 on the scale of Form 1 is given by

where *I*
_{12} is the set containing the items in common between forms 1 and 2. *I*
_{13} and *I*
_{23} are defined similarly. We denote by *I*
_{
p3}=*I*
_{13}∪*I*
_{23} the set of items in common between the pool and Form 3, by *n*
_{
p3}=*n*
_{13}+*n*
_{23} (assuming that *I*
_{13}∩*I*
_{23}=⊘) the cardinality of *I*
_{
p3} and by *b*
_{
pj
} the *j*-th difficulty item parameter of the pool. The equating coefficient for converting the parameters of Form 3 on the scale of the pool is given by

## Rights and permissions

## About this article

### Cite this article

Battauz, M. IRT Test Equating in Complex Linkage Plans.
*Psychometrika* **78, **464–480 (2013). https://doi.org/10.1007/s11336-012-9316-y

Received:

Revised:

Published:

Issue Date:

### Key words

- asymptotic standard errors
- chain equating
- double equating
- equating coefficients
- item response theory
- multiple equating
- weighted bisector