The focus of this study is to consider random differential item functioning (DIF) for polytomous items from a multilevel (3 level) logistic regression perspective. Often, in educational studies, three levels with nested variables are common (e.g., items scores for students nested in schools). A statistical model for detecting random DIF for polytomously scored items will be presented.
The random-effect DIF model will incorporate a multilevel (3 levels) approach. In order to parameterize this model for polytomous outcomes, a hierarchical generalized linear model (HGLM) will be utilized. This approach will be modified to include an item response theory (IRT) model for ordinal response data. In this model, DIF may be present between any levels of the categorical response. This can be referred to as “inner-response DIF” or IDIF. In order to allow the DIF effect to randomly vary, the DIF parameters are given a random component in the level-3 model. This approach allows for the DIF effect to not be consistent across the level-3 groupings.
In this modeling framework, the number of random effects can rapidly increase because of multiple threshold parameters which can all be random. More traditional maximum likelihood estimation procedures may not be feasible computationally due to the high-dimensional integrals in the likelihood function. Since a Bayesian approach does not deal with numerical integrations, estimation will be feasible even when the model contains many random effects. Thus, this study incorporates a Bayesian approach in parameter estimation. A Bayesian approach would consider these unknown parameters as random variables with appropriate prior distributions. The estimation of parameters for the three-level random DIF model would be based on the joint posterior distribution. A Bayesian estimation procedure will be derived, and tested using Monte Carlo simulation methods. Finally, the model will be used to analyze an actual data set involving polytomous data, with a discussion of the results.
