DIF in previous studies is commonly viewed as a secondary dimension (called as nuisance dimensions versus to the primary dimension) can result in DIF between manifest groups. The current study combined the second dimension with latent DIF (DIF between latent classes) and developed so call as a mixed dimensional approach.
It was argued that the proposed mixed dimensional approach differs from the traditional multidimensional model for DIF (MMD) and dimensionality approach such as MIMIC. The MMD approach assumes that there is a secondary dimension in both of reference group and the focal group, whereas the MIMIC approach does not include a secondary dimension in the class. For the mixed dimensional approach, it implies a secondary dimension, but only in the DIF latent class.
(Why not a secondary dimension in both of DIF and non-DIF latent class? The authors did not provide much argument or judgment, but only from the results of applications. )
Three main questions are address:
(1) Whether a secondary dimension helps with the explanation of latent DIF?
(2) Whether there is discrimination DIF related to the primary dimension?
(3) Does the secondary dimension need to extend to the non-DIF latent class?
Four types of models (as below) were developed and were compared in terms of goodness of fit (log likelihood, AIC, BIC) so as to answer the three questions above.
The four types of models are:
(1) One-dimensional mixture models. This model does not introduce the secondary dimension, and serves as the reference model. (model 0)
(2) One-dimensional difficulty mixture model. The difficulty of DIF items may different in the DIF latent class (model 1a) or both of the difficulty and discrimination of the items are different in the DIF latent class (model 1b).
(3) A secondary (nuisance) dimension in the DIF latent class. But the same primary dimension applies to both latent groups (model 2a), or the discrimination of the DIF items differ for the primary dimension (model 2b).
(4) Model 2a and model 2b can be extended by a secondary dimension in both of the DIF latent class and non-DIF latent class (model 3a, model 3b).
Hence, comparing model 2a with model 1a or model 2b with model 1b can answer Question 1. If model 2a and model 2b fit better than model 1a and model 1b, it implies that a secondary dimension helps with explanation of DIF. Similarly, comparing model 1b with model 1a or comparing model 2b with model 2a can answer Question 2. And comparing model 3a with model 2a or comparing model 3b with model 2b can answer Question 3.
Two applications of the models (Speededness and arithmetic operations) were used. It was found that a secondary dimension does help with the explanation of DIF. And the secondary dimension do not need to extend to the non-DIF latent class.
Questions:
(1) How to define an item property to be a secondary dimension is not sufficiently developed which limit the application of the model.
(2) Though it was found that the mixed dimensional approach have a better fit from the two applications, the arguments and adjustment are hard to propose.
(3) The reliability of the mixed dimensional approach needs to be confirm in the future study.