A Non-parametric Bayesian Model for Detecting Differential Item Functioning: An Application to Political Representation in the US

Abstract: A common approach when studying the quality of representation involves comparing the latent preferences of voters and legislators, commonly obtained by fitting an item-response theory (IRT) model to a common set of stimuli. Despite being exposed to the same stimuli, voters and legislators may not share a common understanding of how these stimuli map onto their latent preferences, leading to differential item-functioning (DIF) and incomparability of estimates. We explore the presence of DIF and incomparability of latent preferences obtained through IRT models by re-analyzing an influential survey data set, where survey respondents expressed their preferences on roll call votes that U.S. legislators had previously voted on. To do so, we propose defining a Dirichlet Process prior over item-response functions in standard IRT models. In contrast to typical multi-step approaches to detecting DIF, our strategy allows researchers to fit a single model, automatically identifying incomparable sub-groups with different mappings from latent traits onto observed responses. We find that although there is a group of voters whose estimated positions can be safely compared to those of legislators, a sizeable share of surveyed voters understand stimuli in fundamentally different ways. Ignoring these issues can lead to incorrect conclusions about the quality of representation.