A generalization/specialization of the PARAFAC model is developed that improves its properties when applied to multi-way problems involving linearly dependent factors. This model is called PARALIND (PARAllel profiles with LINear Dependences). Linear dependences can arise when the empirical sources of variation being modeled by factors are causally or logically linked during data generation, or circumstantially linked during data collection. For example, this can occur in a chemical context when end products are related to the precursor or in a psychological context when a single stimulus generates two incompatible feelings at once. For such cases, the most theoretically appropriate PARAFAC model has loading vectors that are linearly dependent in at least one mode, and when collinear, are nonunique in the others. However, standard PARAFAC analysis of fallible data will have neither of these features. Instead, latent linear dependences become high surface correlations and any latent nonuniqueness is replaced by a meaningless surface-level 'unique orientation' that optimally fits the particular random noise in that sample. To avoid these problems, any set of components that in theory should be rank deficient are re-expressed in PARALIND as a product of two matrices, one that explicitly represents their dependency relationships and another, with fewer columns, that captures their patterns of variation. To demonstrate the approach, we apply it first to fluorescence spectroscopy (excitation-emission matrices, EEM) data in which concentration values for two analytes covary exactly, and then to flow injection analysis (FIA) data in which subsets of columns are logically constrained to sum to a constant, but differently in each of two modes. In the PARAFAC solutions of the EEM data, all factors are 'unique' but this is only meaningful for two of the factors that are also unique at the latent level. In contrast, the PARALIND solutions directly display the extent and nature of partial nonuniqueness present at the latent level by exhibiting a corresponding partial uniqueness in their recovered loadings. For the FIA data, PARALIND constraints restore latent uniqueness to the concentration estimates. Comparison of the solutions shows that PARALIND more accurately recovers latent structure, presumably because it uses fewer parameters and hence fits less error.
Parallel proportional profiles, intrinsic axes, DEDICOM, PARAFAC2, Cattell, trilinear models, quadrilinear models, factor rotation problem, multidimensional scaling, principal components, oblique confactor,
Over the last decade, numerous methods for the multidimensional scaling (MDS) of perceptions and preferences have been applied by researchers in marketing. However, one notable gap in MDS methodology has been the lack of suitable models for analyzing inherently asymmetric data relationships. Recently, Harshman (Harshman, R. A. 1978. Models for analysis of asymmetrical relationships among objects or stimuli. Paper presented at the First Joint Meeting of the Psychometric Society and the Society for Mathematical Psychology. McMaster University, Hamilton, Ontario, August; Harshman, R. A. 1982a. DEDICOM: A family of models generalizing factor analysis and multidimensional scaling for decomposition of asymmetric relationships. Unpublished manuscript, University of Western Ontario.) has proposed a new family of models—called DEDICOM (DEcomposition into Directional COMponents)—for analyzing data matrices that are intrinsically asymmetric. In this article, the single-domain DEDICOM model is described and applied to two illustrative cases in marketing research. The examples demonstrate that DEDICOM solutions will sometimes make more substantive sense and provide significantly better fits to asymmetric data than solutions obtained by factor analysis or MDS. DEDICOM also provides a novel type of information—a description of asymmetric relations among dimensions or clusters. Such information will often have useful marketing implications.multidimensional scaling, factor analyses
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.