Testing for measurement invariance and latent mean differences across methods: interesting incremental information from multitrait-multimethod studies

Models of confirmatory factor analysis (CFA) are frequently applied to examine the convergent validity of scores obtained from multiple raters or methods in so-called multitrait-multimethod (MTMM) investigations. We show that interesting incremental information about method effects can be gained from including mean structures and tests of MI across methods in MTMM models. We present a modeling framework for testing MI in the first step of a CFA-MTMM analysis. We also discuss the relevance of MI in the context of four more complex CFA-MTMM models with method factors. We focus on three recently developed multiple-indicator CFA-MTMM models for structurally different methods [the correlated traits-correlated (methods - 1), latent difference, and latent means models; Geiser et al., 2014a; Pohl and Steyer, 2010; Pohl et al., 2008] and one model for interchangeable methods (Eid et al., 2008). We demonstrate that some of these models require or imply MI by definition for a proper interpretation of trait or method factors, whereas others do not, and explain why MI may or may not be required in each model. We show that in the model for interchangeable methods, testing for MI is critical for determining whether methods can truly be seen as interchangeable. We illustrate the theoretical issues in an empirical application to an MTMM study of attention deficit and hyperactivity disorder (ADHD) with mother, father, and teacher ratings as methods.