Multivariate techniques for sorting data: DiSTATIS and Discriminant DiSTATIS Kriegsman, M.A. 1* , Raman, R. 1 , Tillman, B. 2 , Dowling, W.J. 1 , & Abdi, H. 1 1 The University of Texas at Dallas, Richardson, TX, USA 2 Lyon Neuroscience Research Center, Lyon, France *Michael.Kriegsman@utdallas.edu Introduction Sorting:  In a sorting task: assessors (participants) sort stimuli by similarity  Free vs constrained: Assessors may or may not be told the number of stimulus categories  Verbal descriptions of stimuli can also be analyzed, and can enrich interpretation 5  Fields: consumer preference (food, beer, wine, textile, perfume) 6,7 sensory evaluation (smell, taste, sound, touch), cognition (music)  Applications: R&D, quality control, marketing 7  Advantages of sorting:  Requires minimal training. Amateurs and experts often give similar results 5,6 , though similarity between amateurs and experts may differ by stimulus type 7  Relatively easy, fast, and not fatiguing, even for many stimuli 5  Does not require a priori selection of attributes/categories  Disadvantages of sorting:  May be less accurate than ratings (a.k.a., profiling) 5,6  Analysis  Data: Square dissimilarity matrices  Analyzed often by MDS. MDS maps overall perceived similarity of stimuli  DiSTATIS reveals individuals’ perceived similarity of stimuli  Other potential applications of (Di)DiSTATIS: brain data (functional and/or structural connectivity) References: 1. Abdi, H. (2007). Metric multidimensional scaling: Analyzing distance matrices. In N. J. Salkind (Ed.), Encyclopedia of Measurement and Statistics (pp. 598–605). Thousand Oaks (CA): Sage. 2. Abdi, H., & Williams, L. J. (2010a). Principal component analysis. Wiley Interdisciplinary Reviews: Computational Statistics, 2(August), 433–459. 3. Abdi, H., & Williams, L. J. (2010b, June 23). Barycentric Discriminant Analysis. In Encyclopedia of research design. 4. Abdi, H., Williams, L. J., Valentin, D., & Bennani-Dosse, M. (2012). STATIS and DISTATIS: Optimum multitable principal component analysis and three way metric multidimensional scaling. Wiley Interdisciplinary Reviews: Computational Statistics, 4, 124–167. 5. Cartier, R., Rytz, A., Lecomte, A., Poblete, F., Krystlik, J., Belin, E., & Martin, N. (2006). Sorting procedure as an alternative to quantitative descriptive analysis to obtain a product sensory map. Food Quality and Preference, 17(7-8), 562–571. 6. Chollet, S., Lelièvre, M., Abdi, H., & Valentin, D. (2011). Sort and beer: Everything you wanted to know about the sorting task but did not dare to ask. Food Quality and Preference, 22(6), 507–520. 7. Chollet, S., Valentin, D., & Abdi, H. (2014). Free Sorting Task. In P. V. Tomasco & G. Ares (Eds.), Novel Techniques in Sensory Characterization and Consumer Profiling (pp. 207–227). Boca Raton: Taylor and Francis. (Di)DiSTATIS: PCA + dissimilarity + multitable (+ discriminant) An Example in Music Cognition Experimental question:  Do assessors perceive the stylistic differences between 3 composers, Bach, Beethoven, and Mozart? Experimental Designs:  Constrained sort into 3  Unbeknownst to assessors, 3 composers  Two Experiments: (1) MIDI sound clips, (2) recordings (by 4 pianists: Richter, Arrau, Pires, Barenboim) Results Summary / Discussion:  Better discrimination for MIDI > Recordings, DiDiSTATIS > DiSTATIS.  Across all analyses: Beethoven opposed Mozart  Recordings showed: Barenboim was Beethoven-like, Richter was Mozart-like  No main effect of experience, except on recordings, where ↑ experience = ↑ sorting variability  Only some songs within each category were consistently sorted  DiSTATIS vs DiDiSTATIS: DiDiSTATIS had ↑ %var on Component 1, but only a modest ↑ in R 2 MDS (Metric Multi-Dimensional Scaling) - Dissimilarity STATIS (*A Long French Name) - Multitable BADA (Barycentric Discriminant Analysis) - Discriminant DiSTATIS = MDS + STATIS PCA (Principal Component Analysis):  The core of these multivariate methods  Eigen-decomposition (square psd matrices)  Singular Value Decomposition (rectangular matrices) Eigen-decomposition Singular Value Decomposition DiDiSTATIS = MDS + STATIS + BADA *Structuration des Tableaux `a Trois Indices de la Statistique *Roughly translated: Structuring Three-Way Statistical Tables MIDI – Low experience MIDI – High experience Recordings – Low experience Recordings – High experience Summed distance matrices