Abstract
This study explored potential sources of differential item functioning (DIF) among accommodated and nonaccommodated groups by examining skills and cognitive processes hypothesized to underlie student performance on the National Assessment for Educational Progress (NAEP). Out of 53 released NAEP items in 2007 for grade 8, a total of 25 items were flagged as DIF among the four studied groups (nonaccommodated, accommodated with extra time, accommodated with read aloud, and accommodated with small groups) by a generalized logistic regression method. The Reparameterized Unified Model was fit to the same data using a Q-matrix containing 25 skills that included content-, process-, and item-type attributes. The nonaccommodated group yielded the highest averages of attribute mastery probabilities as well as the largest proportion of mastered examinees among all the groups. The three accommodated groups tended to have similar attribute mastery means, with the group accommodated with small groups yielding a larger proportion of mastery examinees when compared to the other two accommodated groups.
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Notes
The Journal of Educational Measurement (2007, no. 4) had a special issue on the IRT-based CDMs and related methods.
The difGenLogistic function allows for studying uniform, nonuniform, or both types of DIF. It investigates one item at a time and treats the rest of the items as DIF-free, unless item purification is used. The generalized logistic regression DIF model, as presented by Magis et al. (2011), has the following form: \( {\text{logit}}\left( {\pi_{ig} } \right) = \alpha + \beta S_{i} + \alpha_{g} + \beta_{g} S_{i} , \) where \( \pi_{ig} \) is the probability of examinee i from group g correctly responding to an item, logit is the natural log of the odds of correctly answering an item, \( \alpha \) and \( \beta \) are common intercept and slope parameters (i.e., for all groups), \( \alpha_{g} \) and \( \beta_{g} \) are group-specific intercept and slope parameters, and \( S_{i} \) is the total test score for examinee i (a matching variable and a proxy for the ability level of the examinee). An item is said to contain DIF if the probability \( \pi_{ig} \) varies across the groups of examinees, meaning that there is an interaction between group membership and the item response. When all group-specific parameters equal zero, we would conclude the absence of DIF. Specifically, the following three hypotheses can be tested with regard to DIF using the difGenLogistic function: (a) \( H_{0} : \alpha_{1} = \cdots = \alpha_{F} |\beta_{1} = \cdots = \beta_{F} = 0 \), when testing for uniform DIF; (b) \( H_{0} : \beta_{1} = \cdots = \beta_{F} = 0 \), when testing for nonuniform DIF, and (c) \( H_{0} : \alpha_{1} = \cdots = \alpha_{F} = \beta_{1} = \cdots = \beta_{F} = 0 \), when testing for both types of DIF. Using maximum likelihood, the null hypotheses are tested using different methods, such as the Wald test or, as in our study, the likelihood ratio test.
We refer here to the initial Q-matrix used in the RUM analysis. As we explain later, our focus shifts to the DIF items that elicited 24 attributes from the list of the original 25 skills. One processing attribute (applying and evaluating mathematical correctness) was coded as present in the 53 examined items but was not found to be associated with the DIF identified items.
Proportion (percent) correct in NAEP represents the proportion of all US students that would have gotten an item correct on the NAEP assessment had all students received an opportunity to respond to the item. Due to the multistage and stratified random sampling design used by NAEP, proportion correct values and their standard errors are calculated using student sampling weights via the jackknife repeated replication procedure (Brown et al. 2015). Readers are directed to the NAEP Primer (Allen et al. 2001) and other published technical reports (e.g., Beaton et al. 2011) for technical details.
Three SCR items in Panel (c) in Table 1 were scored dichotomously by NAEP, while the remaining two SCR items were scored on three- or four-point scales. In the analysis, all items were dichotomized by researchers.
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Communicated by Russell George Almond.
Appendices
Appendix 1: RUM model fit summary
All RUM analyses were run using Markov chain Monte Carlo procedures via Arpeggio software. For each analysis, one chain of 13,000 iterations was run, after the burn-in phase of 17,000 iterations. The considered iterations were thinned by 20 and the remaining iterations were pooled to yield 1500 draws from the posterior distribution for use of model fit. Following Hartz and Roussos (2008), Henson et al. (2005), and Roussos et al. (2007), fusion model fit was examined in three ways, by: (1) visual evaluation of thinned chain plots, estimated posterior distributions, and autocorrelations of the chain estimates; (2) checking item mastery statistics; and (3) comparing observed scores with fusion model predicted scores at both item and examinee levels. Results of the aforementioned procedures showed that analyses of all four booklets have acceptable model fit. Selected plots are included below.
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1.
Visual examination of selected posterior distribution parameter estimates (after thinning) across booklets
-
2.
Bias and RMSEs between observed scores and model-estimated scores for both items and examinees
Summary statistics
Booklet | Bias | RMSE | Correlation | |
|---|---|---|---|---|
Summary of fusion model statistics | ||||
Examinee level | B110 | −0.5857 | 2.4847 | 0.9607 |
B133 | −0.1128 | 2.3851 | 0.9597 | |
B145 | −0.4380 | 1.9907 | 0.9603 | |
B150 | −0.5885 | 2.0509 | 0.9622 | |
Item level | B110 | −0.0154 | 0.0209 | 0.9967 |
B133 | −0.0027 | 0.0108 | 0.9987 | |
B145 | −0.0118 | 0.0206 | 0.9967 | |
B150 | −0.0153 | 0.0256 | 0.9951 | |
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3.
Observed and model implied scores for both persons and items (similarity between the lines in the plots below suggest adequate model fit).
Appendix 2
This table specifies which attributes are associated with each item, and the performance of the various groups across these attributes. Within each item, a more complete attribute mastery for the groups can be seen
Content-based attributes | Process-based attributes | Item-type attributes |
|---|---|---|
C1 Whole numbers and integers C2 Fractions and decimals C3 Elementary algebra C4 Two-dimensional geometry C5 Data and basic statistics C6 Measuring and estimating | P1 Translate/formulate equations P2 Computation application P3 Judgmental application P4 Rule application in algebra P5 Logical reasoning P6 Solution search P7 Visual figures and graphs P9 Data management P10 Quantitative reading | S1 Unit conversion S2 Number sense S3 Figures, tables, and graphs S4 Approximation and estimation S5 Evaluate/verify options S6 Pattern recognition S7 Proportional reasoning S10 Open-ended items S11 Word problems |
Item level skill mastery attribute probability means
Item ID | Group | C1 | C2 | C3 | C4 | C5 | C6 | P1 | P2 | P3 | P4 | P5 | P6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
M143601 | NonA | 0.77 | 0.43 | 0.73 | 0.65 | ||||||||
A_ET | 0.47 | 0.21 | 0.43 | 0.38 | |||||||||
A_RA | 0.43 | 0.12 | 0.42 | 0.35 | |||||||||
A_SG | 0.46 | 0.18 | 0.47 | 0.39 | |||||||||
M091701 | NonA | 0.43 | 0.73 | ||||||||||
A_ET | 0.21 | 0.43 | |||||||||||
A_RA | 0.12 | 0.42 | |||||||||||
A_SG | 0.18 | 0.47 | |||||||||||
M144901 | NonA | 0.77 | 0.43 | 0.42 | 0.54 | 0.45 | |||||||
A_ET | 0.47 | 0.21 | 0.22 | 0.31 | 0.34 | ||||||||
A_RA | 0.43 | 0.12 | 0.23 | 0.27 | 0.32 | ||||||||
A_SG | 0.46 | 0.18 | 0.24 | 0.30 | 0.30 | ||||||||
M107101 | NonA | 0.43 | 0.69 | 0.73 | 0.65 | ||||||||
A_ET | 0.21 | 0.38 | 0.43 | 0.38 | |||||||||
A_RA | 0.12 | 0.39 | 0.42 | 0.35 | |||||||||
A_SG | 0.18 | 0.43 | 0.47 | 0.39 | |||||||||
M106601 | NonA | 0.77 | 0.73 | ||||||||||
A_ET | 0.47 | 0.43 | |||||||||||
A_RA | 0.43 | 0.42 | |||||||||||
A_SG | 0.46 | 0.47 | |||||||||||
M106401 | NonA | 0.42 | |||||||||||
A_ET | 0.22 | ||||||||||||
A_RA | 0.23 | ||||||||||||
A_SG | 0.24 | ||||||||||||
M106801 | NonA | 0.77 | 0.42 | 0.73 | |||||||||
A_ET | 0.47 | 0.22 | 0.43 | ||||||||||
A_RA | 0.43 | 0.23 | 0.42 | ||||||||||
A_SG | 0.46 | 0.24 | 0.47 | ||||||||||
M144501 | NonA | 0.77 | 0.74 | 0.73 | |||||||||
A_ET | 0.47 | 0.51 | 0.43 | ||||||||||
A_RA | 0.43 | 0.47 | 0.42 | ||||||||||
A_SG | 0.46 | 0.53 | 0.47 | ||||||||||
M107601 | NonA | 0.47 | 0.73 | ||||||||||
A_ET | 0.26 | 0.43 | |||||||||||
A_RA | 0.23 | 0.42 | |||||||||||
A_SG | 0.25 | 0.47 | |||||||||||
M072901 | NonA | 0.74 | 0.45 | ||||||||||
A_ET | 0.51 | 0.34 | |||||||||||
A_RA | 0.47 | 0.32 | |||||||||||
A_SG | 0.53 | 0.30 | |||||||||||
M107201 | NonA | 0.43 | 0.45 | 0.62 | |||||||||
A_ET | 0.21 | 0.34 | 0.43 | ||||||||||
A_RA | 0.12 | 0.32 | 0.41 | ||||||||||
A_SG | 0.18 | 0.30 | 0.46 | ||||||||||
M144401 | NonA | 0.77 | 0.43 | 0.69 | 0.73 | 0.65 | |||||||
A_ET | 0.47 | 0.21 | 0.38 | 0.43 | 0.38 | ||||||||
A_RA | 0.43 | 0.12 | 0.39 | 0.42 | 0.35 | ||||||||
A_SG | 0.46 | 0.18 | 0.43 | 0.47 | 0.39 | ||||||||
M105801 | NonA | 0.42 | |||||||||||
A_ET | 0.22 | ||||||||||||
A_RA | 0.23 | ||||||||||||
A_SG | 0.24 | ||||||||||||
M106701 | NonA | 0.43 | 0.42 | 0.65 | |||||||||
A_ET | 0.21 | 0.22 | 0.38 | ||||||||||
A_RA | 0.12 | 0.23 | 0.35 | ||||||||||
A_SG | 0.18 | 0.24 | 0.39 | ||||||||||
M145001 | NonA | 0.77 | 0.43 | 0.69 | 0.54 | ||||||||
A_ET | 0.47 | 0.21 | 0.38 | 0.31 | |||||||||
A_RA | 0.43 | 0.12 | 0.39 | 0.27 | |||||||||
A_SG | 0.46 | 0.18 | 0.43 | 0.30 | |||||||||
M013131 | NonA | 0.65 | |||||||||||
A_ET | 0.43 | ||||||||||||
A_RA | 0.37 | ||||||||||||
A_SG | 0.45 | ||||||||||||
M145101 | NonA | 0.47 | 0.43 | 0.42 | 0.73 | 0.65 | |||||||
A_ET | 0.26 | 0.21 | 0.22 | 0.43 | 0.38 | ||||||||
A_RA | 0.23 | 0.12 | 0.23 | 0.42 | 0.35 | ||||||||
A_SG | 0.25 | 0.18 | 0.24 | 0.47 | 0.39 | ||||||||
M075801 | NonA | 0.77 | 0.42 | 0.73 | |||||||||
A_ET | 0.47 | 0.22 | 0.43 | ||||||||||
A_RA | 0.43 | 0.23 | 0.42 | ||||||||||
A_SG | 0.46 | 0.24 | 0.47 | ||||||||||
M106301 | NonA | 0.47 | |||||||||||
A_ET | 0.26 | ||||||||||||
A_RA | 0.23 | ||||||||||||
A_SG | 0.25 | ||||||||||||
M144001 | NonA | 0.77 | 0.45 | ||||||||||
A_ET | 0.47 | 0.34 | |||||||||||
A_RA | 0.43 | 0.32 | |||||||||||
A_SG | 0.46 | 0.30 | |||||||||||
M144201 | NonA | 0.47 | 0.73 | 0.54 | |||||||||
A_ET | 0.26 | 0.43 | 0.31 | ||||||||||
A_RA | 0.23 | 0.42 | 0.27 | ||||||||||
A_SG | 0.25 | 0.47 | 0.30 | ||||||||||
M013531 | NonA | 0.47 | |||||||||||
A_ET | 0.26 | ||||||||||||
A_RA | 0.23 | ||||||||||||
A_SG | 0.25 | ||||||||||||
M105601 | NonA | 0.77 | |||||||||||
A_ET | 0.47 | ||||||||||||
A_RA | 0.43 | ||||||||||||
A_SG | 0.46 | ||||||||||||
M105901 | NonA | 0.47 | |||||||||||
A_ET | 0.26 | ||||||||||||
A_RA | 0.23 | ||||||||||||
A_SG | 0.25 | ||||||||||||
M144301 | NonA | 0.47 | 0.43 | 0.69 | 0.73 | 0.65 | |||||||
A_ET | 0.26 | 0.21 | 0.38 | 0.43 | 0.38 | ||||||||
A_RA | 0.23 | 0.12 | 0.39 | 0.42 | 0.35 | ||||||||
A_SG | 0.25 | 0.18 | 0.43 | 0.47 | 0.39 |
Item ID | Group | P7 | P9 | P10 | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S10 | S11 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
M143601 | NonA | ||||||||||||
A_ET | |||||||||||||
A_RA | |||||||||||||
A_SG | |||||||||||||
M091701 | NonA | ||||||||||||
A_ET | |||||||||||||
A_RA | |||||||||||||
A_SG | |||||||||||||
M144901 | NonA | 0.66 | 0.58 | 0.73 | 0.80 | ||||||||
A_ET | 0.42 | 0.29 | 0.51 | 0.54 | |||||||||
A_RA | 0.41 | 0.27 | 0.50 | 0.54 | |||||||||
A_SG | 0.43 | 0.32 | 0.51 | 0.54 | |||||||||
M107101 | NonA | ||||||||||||
A_ET | |||||||||||||
A_RA | |||||||||||||
A_SG | |||||||||||||
M106601 | NonA | 0.58 | |||||||||||
A_ET | 0.29 | ||||||||||||
A_RA | 0.27 | ||||||||||||
A_SG | 0.32 | ||||||||||||
M106401 | NonA | 0.64 | 0.58 | ||||||||||
A_ET | 0.38 | 0.29 | |||||||||||
A_RA | 0.36 | 0.27 | |||||||||||
A_SG | 0.38 | 0.32 | |||||||||||
M106801 | NonA | 0.78 | |||||||||||
A_ET | 0.52 | ||||||||||||
A_RA | 0.50 | ||||||||||||
A_SG | 0.56 | ||||||||||||
M144501 | NonA | 0.53 | 0.77 | ||||||||||
A_ET | 0.43 | 0.59 | |||||||||||
A_RA | 0.44 | 0.59 | |||||||||||
A_SG | 0.38 | 0.59 | |||||||||||
M107601 | NonA | 0.67 | |||||||||||
A_ET | 0.39 | ||||||||||||
A_RA | 0.37 | ||||||||||||
A_SG | 0.44 | ||||||||||||
M072901 | NonA | 0.58 | 0.58 | 0.78 | 0.77 | ||||||||
A_ET | 0.29 | 0.33 | 0.52 | 0.52 | |||||||||
A_RA | 0.27 | 0.29 | 0.50 | 0.51 | |||||||||
A_SG | 0.32 | 0.32 | 0.56 | 0.56 | |||||||||
M107201 | NonA | 0.77 | |||||||||||
A_ET | 0.52 | ||||||||||||
A_RA | 0.51 | ||||||||||||
A_SG | 0.56 | ||||||||||||
M144401 | NonA | 0.53 | 0.64 | 0.77 | |||||||||
A_ET | 0.43 | 0.38 | 0.52 | ||||||||||
A_RA | 0.44 | 0.36 | 0.51 | ||||||||||
A_SG | 0.38 | 0.38 | 0.56 | ||||||||||
M105801 | NonA | 0.58 | |||||||||||
A_ET | 0.29 | ||||||||||||
A_RA | 0.27 | ||||||||||||
A_SG | 0.32 | ||||||||||||
M106701 | NonA | 0.58 | |||||||||||
A_ET | 0.29 | ||||||||||||
A_RA | 0.27 | ||||||||||||
A_SG | 0.32 | ||||||||||||
M145001 | NonA | 0.58 | |||||||||||
A_ET | 0.29 | ||||||||||||
A_RA | 0.27 | ||||||||||||
A_SG | 0.32 | ||||||||||||
M013131 | NonA | 0.67 | |||||||||||
A_ET | 0.39 | ||||||||||||
A_RA | 0.37 | ||||||||||||
A_SG | 0.44 | ||||||||||||
M145101 | NonA | 0.58 | 0.80 | ||||||||||
A_ET | 0.29 | 0.54 | |||||||||||
A_RA | 0.27 | 0.54 | |||||||||||
A_SG | 0.32 | 0.54 | |||||||||||
M075801 | NonA | 0.78 | 0.80 | 0.77 | |||||||||
A_ET | 0.52 | 0.54 | 0.52 | ||||||||||
A_RA | 0.50 | 0.54 | 0.51 | ||||||||||
A_SG | 0.56 | 0.54 | 0.56 | ||||||||||
M106301 | NonA | 0.73 | 0.64 | 0.58 | 0.67 | 0.80 | 0.77 | ||||||
A_ET | 0.49 | 0.38 | 0.29 | 0.39 | 0.54 | 0.52 | |||||||
A_RA | 0.46 | 0.36 | 0.27 | 0.37 | 0.54 | 0.51 | |||||||
A_SG | 0.49 | 0.38 | 0.32 | 0.44 | 0.54 | 0.56 | |||||||
M144001 | NonA | 0.53 | 0.64 | 0.58 | 0.78 | ||||||||
A_ET | 0.43 | 0.38 | 0.29 | 0.52 | |||||||||
A_RA | 0.44 | 0.36 | 0.27 | 0.50 | |||||||||
A_SG | 0.38 | 0.38 | 0.32 | 0.56 | |||||||||
M144201 | NonA | 0.64 | 0.80 | ||||||||||
A_ET | 0.38 | 0.54 | |||||||||||
A_RA | 0.36 | 0.54 | |||||||||||
A_SG | 0.38 | 0.54 | |||||||||||
M013531 | NonA | 0.64 | 0.58 | ||||||||||
A_ET | 0.38 | 0.29 | |||||||||||
A_RA | 0.36 | 0.27 | |||||||||||
A_SG | 0.38 | 0.32 | |||||||||||
M105601 | NonA | 0.58 | 0.78 | ||||||||||
A_ET | 0.33 | 0.52 | |||||||||||
A_RA | 0.29 | 0.50 | |||||||||||
A_SG | 0.32 | 0.56 | |||||||||||
M105901 | NonA | 0.58 | 0.67 | ||||||||||
A_ET | 0.29 | 0.39 | |||||||||||
A_RA | 0.27 | 0.37 | |||||||||||
A_SG | 0.32 | 0.44 | |||||||||||
M144301 | NonA | 0.66 | 0.53 | 0.64 | 0.77 | ||||||||
A_ET | 0.42 | 0.43 | 0.38 | 0.52 | |||||||||
A_RA | 0.41 | 0.44 | 0.36 | 0.51 | |||||||||
A_SG | 0.43 | 0.38 | 0.38 | 0.56 |
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Svetina, D., Dai, S. & Wang, X. Use of cognitive diagnostic model to study differential item functioning in accommodations. Behaviormetrika 44, 313–349 (2017). https://doi.org/10.1007/s41237-017-0021-0
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DOI: https://doi.org/10.1007/s41237-017-0021-0