Multiplicative Models and MTMM Matrices
Robert A. Cudeck · Journal of Educational Statistics · 1988
The rationale for multiplicative models in the context of MTMM covariance matrices, as developed by Swain and Browne, is described and illustrated with several sets of empirical data. Comparisons with restricted factor anal-ysis models are made, and certain criticisms of this approach are noted. An extension of the multiplicative model to three-facet data that includes occa-sions, methods, and traits is also presented. A research design of great practical importance in educational settings employs a multitrait-multimethod (MTMM) matrix (Campbell & Fiske, 1959) of correlations among test scores or other behavioral measures to investigate convergent and discriminant validity. The rationale for the MTMM design rests upon the compellingly simple argument that two or more measures that assess a given trait should correlate substantially among themselves, while measures that putatively assess distinct traits should produce correlations of much smaller magnitude. Studies that use the MTMM design are pivotal in applied research because measurement development and validation is fundamental to a great many research activ-ities. While the study of relationships among measurement methods and be-havioral traits is the most common MTMM design, many other extensions of the basic formulation are possible (Fiske, 1982). Two interesting varia-tions, for example, are a multitrait-multioccasion design to study trait stability over time (e.g., Schneider & Dachler, 1978), or a multitrait-multirater design to examine similarities and differences of judges ' trait ratings (e.g., Einhorn, 1974). Other possibilities could be described. In evaluating the extent to which a particular set of traits and methods exhibits trait validity, Campbell and Fiske (1959) discussed four criteria Thanks to M. W. Browne for a critical reading of this paper, and to Auke Tellegen, Susan Henly, and Kelli Klebe for several useful discussions about this material.