Decoding the Meaning of Factorial Invariance and Updating the Practice of Multi-group Confirmatory Factor Analysis: A Demonstration With TIMSS Data

TLDR

Measurement invariance has been developed in highly technical language, limiting its accessibility to social and behavioral researchers. The paper aims to decode measurement invariance within factor analysis using familiar regression and linear modeling concepts. The authors explain MI in factor analysis, advocate modeling mean and covariance structures (MACS), and update testing practices using chi‑squared likelihood ratio tests and fit‑index changes per Cheung and Rensvold. They illustrate the approach with 21 cross‑country MI comparisons of 1999 TIMSS mathematics scores.

Abstract

Measurement invariance (MI) has been developed in a very technical language and manner that is generally not widely accessible to social and behavioral researchers and applied measurement specialists. Primarily relying on the widely known concepts in regression and linear statistical modeling, this paper decoded the concept of MI in the context of factor analysis. The paper began by describing what is MI (and lack of MI) and how the concept can be realized in the context of factor analysis. Next, we explained the need for modeling the mean and covariance structure (MACS), instead of the traditionally applied covariance structure, in detecting factorial invariance. Along the way, we addressed the related matter of statistically testing for MI using the Chi-squared likelihood ratio test and fit indices in multi-group MACS confirmatory factor analysis. Bringing to bear current developments by Cheung and Rensvold (2002) and others, we provided an update on the practice of using change in fit statistics to test for MI. Throughout the paper we concretized our discussion, without lack of generality to other constructs and research settings, with an example of 21 crosscountry MI comparisons of the 1999 TIMSS mathematics scores.