Uniform vs. Non-Uniform DIF

Uniform DIF occurs when one group consistently outperforms the other across all ability levels — the ICCs are shifted but do not cross. Non-uniform DIF occurs when the direction of the group difference changes at some ability level, producing crossing ICCs. This distinction matters because some detection methods are sensitive only to uniform DIF.

Mantel-Haenszel Method

The Mantel-Haenszel (MH) procedure matches focal and reference groups on total score, then forms $2 \times 2$ contingency tables across score strata $s$:

\[\hat{\alpha}_{\text{MH}} = \frac{\sum_s A_s D_s / n_s}{\sum_s B_s C_s / n_s}\]

The common odds ratio $\hat{\alpha}{\text{MH}}$ is transformed to the ETS $\Delta$ scale: $\Delta{\text{MH}} = -2.35 \ln\hat{\alpha}_{\text{MH}}$. Items with $

\Delta_{\text{MH}}

\ge 1.5$ are flagged as moderate DIF (Category B) and $\ge 1.5$ with significance as large DIF (Category C).

Logistic Regression DIF

A logistic regression approach models item response as a function of matching variable $\theta$, group membership $G$, and their interaction:

\[\logit[P(U=1)] = \beta_0 + \beta_1\theta + \beta_2 G + \beta_3(\theta \times G)\]

$\beta_2 \ne 0$ indicates uniform DIF; $\beta_3 \ne 0$ indicates non-uniform DIF. The likelihood-ratio test comparing nested models provides a $\chi^2$ test with 1 or 2 degrees of freedom.

IRT-Based DIF: Lord’s Chi-Squared

Under IRT, Lord’s $\chi^2$ tests for simultaneous equality of item parameters across groups:

\[\chi^2 = (\hat{\mathbf{a}}_R - \hat{\mathbf{a}}_F)^\top \hat{\boldsymbol{\Sigma}}^{-1} (\hat{\mathbf{a}}_R - \hat{\mathbf{a}}_F)\]

where $\hat{\mathbf{a}}$ contains calibrated item parameters for reference (R) and focal (F) groups after scale linking.