# Differential Functioning of Items and Tests framework

### Description

`DFIT`

provides functions for calculating the differential item and test functioning
proposed by Raju et al. (1995).

### Details

DFIT provides a set of functions to calculate the noncompensatory (NCDIF), compensatory (CDIF) and test level (DTF) differential functioning indices for items and tests under Raju's (Raju, et al. 1995) DFIT framework. It also provides functions for obtaining cut-off points for identifying differential functioning for these indices following the Monte Carlo Item Parameter Replication approach proposed by Oshima et al. (2006).

This package also improves upon available DFIT software by allowing the covariance matrices for both focal and reference groups to be used. This improves the obtained cut-off points, which result in type I error rates at the nominal level, and increased power, when compared to the cut-off points obtained when using only the focal group item parameter estimates and their estimate covariances (Cervantes, 2012). Furthermore, this package includes functions for obtaining the asymptotic covariance matrices of item parameter estimates (currently only for dichotomous IRT models) and for calculating the DFIT indices base on the focal group distribution as well as ability estimates for a sample from the focal population are included; these enable ad hoc and a priori power calculations for given item parameters and sample sizes to be possible with this package.

### References

de Ayala, R. J., (2009). The theory and practice of item response theory. New York: The Guildford Press

Cervantes, V. H. (2012). On using the Item Parameter Replication (IPR) approach for power calculation of the noncompensatory differential item functioning (NCDIF) index (pp. 206-207). Proceedings of the V European Congress of Methodology. Santiago de Compostela, Spain: Universidade de Santiago de Compostela.

Cohen, A., Kim, S-H and Baker , F. (1993). Detection of differential item functioning in the Graded Response Moodel. Applied psychological measurement, 17(4), 335-350

Holland, P.W., and Thayer, D.T. (1988). Differential Item Performance and the Mantel-Haenszel Procedure. In H. Wainer and H.I. Braun (Eds.), Test Validity. Hillsdale, NJ: Erlbaum.

Li, Y. & Lissitz, R. (2004). Applications of the analytically derived standard errors of Item Response Theory item parameter estimates. Journal of educational measurement, 41(2), 85–117.

Oshima, T. & Morris, S. (2008). Raju's Differential Functioning of Items and Tests (DFIT). Educational Measurement: Issues and Practice, 27(3), 43–50.

Oshima, T., Raju, N. & Nanda, A. (2006). A new method for assessing the statistical significance in the Differential Functioning of Items and Tests (DFIT) framework. Journal of educational measurement, 43(1), 1–17.

Raju, N. (1988). The area between two item characteristic cureves. Psychometricka, 53(4), 495–502.

Raju, N., Fortmann-Johnson, K., Kim, W., Morris, S., Nering, M. & Oshima, T. (2009). The item parameter replication method for detecting differential functioning in the polytomous DFIT framework. Applied psychological measurement, 33(2), 133–147.

Raju, N. S., van der Linden, W. J., & Fleer, P. F. (1995). An IRT-based internal measure of test bias with applications for differential item functioning. Applied Psychological Measurement, 19, 353–368.

Roussos, L., Schnipke, D. & Pashley, P. (1999). A generalized formula for the Mantel-Haenszel Differential Item Functioning parameter. Journal of educational and behavioral statistics, 24(3), 293–322.

Wright, K. (2011). Improvements for Differential Funtioning of Items and Tests (DFIT): Investigating the addition of reporting an effect size measure and power (Unpublished doctoral dissertation). Georgia State University, USA.