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# Assessing Item Fit for Unidimensional Item Response Theory Models Using Residuals from Estimated Item Response Functions

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## Abstract

Residual analysis (e.g. Hambleton & Swaminathan, Item response theory: principles and applications, Kluwer Academic, Boston, 1985; Hambleton, Swaminathan, & Rogers, Fundamentals of item response theory, Sage, Newbury Park, 1991) is a popular method to assess fit of item response theory (IRT) models. We suggest a form of residual analysis that may be applied to assess item fit for unidimensional IRT models. The residual analysis consists of a comparison of the maximum-likelihood estimate of the item characteristic curve with an alternative ratio estimate of the item characteristic curve. The large sample distribution of the residual is proved to be standardized normal when the IRT model fits the data. We compare the performance of our suggested residual to the standardized residual of Hambleton et al. (Fundamentals of item response theory, Sage, Newbury Park, 1991) in a detailed simulation study. We then calculate our suggested residuals using data from an operational test. The residuals appear to be useful in assessing the item fit for unidimensional IRT models.

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## Notes

1. 1.

By standardization, we refer to dividing the difference of a variable and its expectation by the standard deviation of the difference.

2. 2.

The program is available on request from us.

3. 3.

We also computed our suggested residuals for 15 equispaced values between −2.8 and 2.8 to make these values the same as the midpoints of the intervals used to compute the standardized residuals (Hambleton et al. 1991). The results were virtually unchanged.

4. 4.

Sinharay (2010) reported the average disattenuated correlations among subtest scores from 20+ operational tests. The lowest value reported was 0.69.

5. 5.

We imposed this condition because we noticed that for some easy items, the values of both $$\hat{F}_{j}(\theta)$$ and $$\bar {F}_{j}(\theta)$$ are larger than 0.99 for 0<θ<2 so that the corresponding residual should not be practically significant, but it is statistically significant.

6. 6.

Note that this reordering was done for convenience. Operationally, the anchor items are interspersed with the operational items.

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## Author information

Correspondence to Sandip Sinharay.

Note: Any opinions expressed in this publication are those of the authors and not necessarily of Educational Testing Service. Sandip Sinharay conducted this study and wrote this report while on staff at Educational Testing Service. He is currently at CTB/McGraw-Hill.

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Haberman, S.J., Sinharay, S. & Chon, K.H. Assessing Item Fit for Unidimensional Item Response Theory Models Using Residuals from Estimated Item Response Functions. Psychometrika 78, 417–440 (2013). https://doi.org/10.1007/s11336-012-9305-1