Abstract
Assessing the factorial invariance of two-way rating designs such as ratings of concepts on several scales by different groups can be carried out with three-way models such as the Parafac and Tucker models. By their definitions these models are double-metric factorially invariant. The differences between these models lie in their handling of the links between the concept and scale spaces. These links may consist of unrestricted linking (Tucker2 model), invariant component covariances but variable variances per group and per component (Parafac model), zero covariances and variances different per group but not per component (Replicated Tucker3 model) and strict invariance (Component analysis on the average matrix). This hierarchy of invariant models, and the procedures by which to evaluate the models against each other, is illustrated in some detail with an international data set from attachment theory.
Highlights
Two-way rating designs may consist of, for instance, ratings of concepts on several rating scales
THREE-WAY ANALYSIS OF VARIANCE To acquire an initial perspective on the differences between samples, we carried out a three-way analysis of variance of the Strange Situation data
INVESTIGATING TYPE OF INVARIANCE VIA MODEL FIT Because the procedure outlined for assessing factorial invariance for two-way rating designs is an exploratory one, deciding on the degree of invariance is a substantive and subjective matter, based on numerical information
Summary
Two-way rating designs may consist of, for instance, ratings of concepts on several rating scales. In this paper we tackle the problem of the invariance of the factorial structure of data arising from such designs when the data have been collected from several groups. In particular we will show that three-mode component models are ideally suited to assess factorial invariance for such designs. We will specify a hierarchy of models with increasing restrictions on the parameters resulting in more and more invariant factorial structures across groups. Because in this paper we are dealing with component models we will use the term “components” rather than “factors,” unlessfactors are explicitly indicated. To stay within the standard terminology we will use the term factorial invariance, rather than subspace invariance or component invariance. A detailed treatment of the differences between factor analysis and component analysis for two-way data can for instance be found in Widaman (2007)
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