Testing Measurement Invariance of the Students' Affective Characteristics Model across Gender Sub-Groups

In the psychological and behavioral sciences, latent variables and traits models are popular subjects of research. The concept of "latent traits" or "latent characteristics" can be observed only by an indirect way, and generally they can be modeled only as very complicated structures. Because these models include many indirectly observed variables, their limitations are high.It is important that psychological structures meet invariance across sub-groups (like cultural or ethnical groups, gender or territorial sub-groups, etc.). If a structure is invariant, it provides evidence that there is no measurement bias among sub-groups as the systematic error (Little, 1997; Lord, 1980). Also, this is a confirmation in terms of reliability and validity of the results of the measurements (Meredith, 1993; Vandenberg & Lance, 2000). On the other hand, if invariance is not met, and if there is no validity and reliability problem, it indicates that there could be real differences among sub-groups in the limitations of the structure. So, it is obvious that a concept is needed to discuss these psychometric characteristics of the psychometric models. One of these concepts is defined as "measurement invariance" or "measurement equivalence. "Measurement invariance is defined as the equivalency of the latent structure across different groups or sub-groups. When some groups are compared to each other in a latent traits model, the parameters of this model, depending on group memberships, should be the same. Measurement invariance indicates that (a) psychological structures are generalizable to each sub-group; (b) these structures are not affected by subgroup differences, and (c) bias and errors of measurement are minimal. Also, in this context, invariance is defined as the psychometric characteristics of the measurement scale that includes configural invariance, metric invariance, scalar invariance, and residual invariance hierarchically. Configural invariance means that the latent structure model should be the same for each group or sub-group. Metric invariance, also evaluated as "weak invariance," means that item loadings should be the same across groups. This indicates that there is no item bias across groups as a systematic error. Scalar invariance, also evaluated as "strong invariance," means that the vectors of the item intercept should be equivalent across groups. This indicates that there are the same correlations across factors on the model. Residual invariance, also evaluated as "strict invariance," means that items or observed variables of the model should have the same measurement errors. If configural, metric, or scalar invariances are met, this can be evaluated as "partial invariance" (Byrne, 2006; Byrne, Shavelson, & Muthen, 1989; Kline, 2011; Little, 1997; Meredith, 1993). Confirmatory factor analysis (CFA) and item response theory (IRT) models are reported as two powerful methods for testing the comparability of psychological measurements (Joreskog & Sorbom, 1996; Kline, 2011; Lord, 1980; Reise, Widaman, & Pugh, 1993). Between these two techniques, the most common technique to test measurement invariance is "multi-group confirmatory factor analysis (MGCFA). " The baseline of this technique is associated with a covariance structure model developed by Joreskog (Byrne, 2006; Kaplan, 1995; Little, 1997). Joreskog (1971) defined the process of testing invariance as a hierarchical set of steps. These steps begin with the construction of a well-fitting multi-group model and this model is commonly known as "factorial or configural model. " In this initial model, all parameters remain free across groups and there is no restriction. Then, the parameters of this model (factor loading, error correlations, error variances, and factor variances) are restricted in a logically ordered way. After each restriction, the level of invariance across groups is evaluated. If violations are observed in the first step, that means there is no measurement invariance across groups. …