This paper compares two approach of detecting measurement bias. One is multiple group factor analysis (MGFA), and the other is restricted factor analysis (RFA). MGFA can be used to detect both uniform and non-uniform measurement bias, however, on disadvantage of MGFA is that the grouping variable has to be categorical such that the sample can be partitioned into subsamples / groups. RFA does not have this limitation, but it is only suitable for detecting uniform measurement bias.
The authors proposed to implement latent moderated structural equations (LMS) into RFA method such that the combined method is enabled to detect non-uniform bias as well as uniform bias. A simulation study is carried out using Mplus to compare MGFA and RFA-LMS in terms of detecting uniform and non-uniform measurement bias. Conditions vary the size of bias, the type of bias (in intercepts or factor loadings or factor means), the sample size (100 to 500). In their simulation design, bias is uniform in intercepts, and non-uniform in factor loadings.
Type I error inflated under almost all conditions. MGFA leads to slightly greater power, and RFA-LMS controls the Type I error slightly better. Overall, two approaches perform equally well in testing measurement bias, either uniform or non-uniform. When small non-uniform bias exists, both approaches perform worse.
The idea of proposed RFA-LMS method is similar to Wood (2011)’s MIMIC-interaction method. Both methods introduced an interaction term in the model to account for the DIF. Generated responses are continuous. Additional to the advantage of addressing continuous grouping variable, RFA-LMS has another advantage over MGFA. It makes it possible to include multiple violator variables, which can be very important in practice. The simulation study is somewhat limited because under all conditions, the total number of items is six, and the number of DIF item is only one. It makes it impossible to examine different patterns of non-uniform bias. Actually, this paper does not give the detailed pattern of non-uniform bias. Moreover, they have run only 100 replications under each condition, which is obviously not enough.