A Penalty Approach to Differential Item Functioning in Rasch Models

Abstract

A new diagnostic tool for the identification of differential item functioning (DIF) is proposed. Classical approaches to DIF allow to consider only few subpopulations like ethnic groups when investigating if the solution of items depends on the membership to a subpopulation. We propose an explicit model for differential item functioning that includes a set of variables, containing metric as well as categorical components, as potential candidates for inducing DIF. The ability to include a set of covariates entails that the model contains a large number of parameters. Regularized estimators, in particular penalized maximum likelihood estimators, are used to solve the estimation problem and to identify the items that induce DIF. It is shown that the method is able to detect items with DIF. Simulations and two applications demonstrate the applicability of the method.

Appendix

Proposition

Let for the parameters of the general model with predictor $\eta _{pi}=\theta_{p} -\beta_{i}-\boldsymbol {x}^{T}_{p} \boldsymbol {\gamma }_{i}$ be constrained by β _I =0, $\boldsymbol {\gamma }_{I}^{T}=(0,\dots,0)$ and let the matrix X with rows $(1,\boldsymbol {x}^{T}_{1}),\dots,(1,\boldsymbol {x}^{T}_{P})$ have full rank. Then parameters are identifiable.

Proof

Let two sets of parameters that fulfill the constraints be given such that

$$\eta_{pi}=\theta_p -\beta_i- \boldsymbol {x}^T_p \boldsymbol {\gamma }_i=\tilde{\theta}_p -\tilde{\beta}_i-\boldsymbol {x}^T_p \tilde{\boldsymbol {\gamma }}_i $$

for all persons and items. From considering item I and person p, one obtains by using $\beta_{I}=\tilde{\beta}_{I}=0$, $\boldsymbol {\gamma }_{I}^{T}=\tilde{\boldsymbol {\gamma }}_{I}^{T}=(0,\dots,0)$ $\theta_{p}=\tilde{\theta}_{p}$. Therefore, one has $\beta_{i}+\boldsymbol {x}^{T}_{p} \boldsymbol {\gamma }_{i}=\tilde{\beta}_{i}+\boldsymbol {x}^{T}_{p} \tilde{\boldsymbol {\gamma }}_{i}$ for all p, i, which for item i can be written in matrix form as

$$\boldsymbol {X}(\beta_i, \boldsymbol {\gamma }_i)^T = \boldsymbol {X}(\tilde{\beta}_i, \tilde{\boldsymbol {\gamma }}_i)^T. $$

One can multiply on both sides of the equation with X ^T, and, since X has full rank, with the inverse (X ^T X)⁻¹, obtaining $(\beta_{i}, \boldsymbol {\gamma }_{i})^{T}=(\tilde{\beta}_{i}, \tilde{\boldsymbol {\gamma }}_{i})^{T}$. Alternatively, one can use the single value decomposition of X. □