Multivariate Models for Threedimensional Data
James W. Grice · Applied Multivariate Research · 2009
As early as the publication of Raymond Cattell's (1966) 'basic data relation matrix', or 'data box' researchers have been interested in data structures that extend beyond two dimensions.The development of statistical and mathematical models as well as accessible computer software for the analysis of threedimensional data structures, particularly, has accelerated in recent years.A threedimensional data structure might, for example, be comprised of 500 persons who rated themselves on 50 items from a personality questionnaire on 10 different occasions.The resulting 500 × 50 × 10 data cube could be analyzed to explore the multivariate relationships between the three facets: persons, items, occasions.One general strategy would entail applying traditional factor analyses to different pairwise combinations of the facets by collapsing over the third facet.Gorsuch (1983, pp.322-327) describes a number of variations on this approach.Another strategy, originating with Tucker (1964), would entail factoring all three facets simultaneously, yielding a 'core matrix' that describes their interrelations.In this special issue of Applied Multivariate Research Kroonenberg, Harshman, and Murakami offer an excellent review of Tucker's original model and also compare it to Harshman's (Harshman & Lundy, 1984) popular Parafac model for analyzing three-dimensional data matrices.They furthermore provide clear guidance regarding practical decisions that must be made when employing either model, and they use a genuine parenting styles data set to exemplify the issues.While the models may appear complex at first glance, Kroonenberg and his colleagues show how the analyses parsimoniously uncover common styles of parenting while simultaneously revealing individual variability between families.Leenen and Ceulemans also compare and contrast two different models for analyzing thee-dimensional data matrices.Their paper, however, is centered around the HICLAS method introduced by De Boeck and Rosenberg (1988) for modeling the hierarchical relations among binary variables.HICLAS essentially weds a form of binary factor analysis with set theory, and in this issue Leenen and Ceulemans compare and contrast two different models: INDCLAS and Tucker-3 HICLAS.Like Kroonenber's paper, their contribution reveals Tucker's legacy to analyzing three-dimensional data matrices.Leenen and Ceulemans analysis of two genuine data sets suggest that the Tucker-3 HICLAS model may yield more parsimonious results than the INDCLAS model.In an interesting example that is near and dear to my own heart, given my interests in person-centered statistics (see Grice, 2007), the authors also show how their technique can be applied to a single case.The intra-personal perceptions (or 'object relations') of a young