Multidimensional Scaling of Binary Dissimilarities: Direct and Derived Approaches

Abstract

Given a matrix of dissimilarities, it has been debated whether researchers should perform multidimensional scaling on this original matrix or on a new one derived by comparing rows in the original matrix. Careful comparison studies (Drasgow & Jones, 1979; Van der Kloot & Van Herk, 1991) in the context of sorting data indicated that most of the initial enthusiasm for the derivative approach was unfounded. The current work, a Monte Carlo study of structured binary data derived from known two-dimensional configurations using ALSCAL, complements and extends the previous studies. We discuss a weakness in the squared difference (δ) row-comparison rule used previously and propose an alternative row-comparison measure based on the Jaccard coefficient. Scaling the binary data directly gave better performance, as gauged by Procrustes statistics, than did scaling A data across a range of noise levels. The quality of solutions obtained by scaling Jaccard data was always better or essentially equal to that from scaling δ data, and in certain parameter regions improved upon that of direct scaling. Another alternative approach, applying the δ rule after first row-centering the binary data, was found to be generally ineffective. These findings are pertinent to the analysis not just of stimulus sorting data but of coarse dissimilarities generally, for example from direct pairwise judgment tasks and in fields outside statistical psychology.

Designing a frame: rhetorical strategies of architects

“…For our SOQ data 32.5 per cent of our frequency matrix cells were zero. Thus, we assessed the similarities in standardized frequencies by using the Jaccard coefficient algorithm designed to overcome inflation of similarities coefficient due to inclusion of ''0'' matches (Knoke, 2003;Simmen, 1996). In essence, architect firms used only a few words from the cultural register frequently such as design, building, and project.…”

Comparison of Methods for Collecting and Modeling Dissimilarity Data: Applications to Complex Sound Stimuli

“…However, the price of the increased efficiency is seldom considered: for example, free sorts are known to be less accurate than dissimilarity ratings (Subkoviak & Roecks, 1976). Further, other differences between free sorting and dissimilarity ratings are simply unknown: no study has compared their reliability, and redundancy studies (Bertino & Lawless, 1993;Bonebright, 1996;Cartier et al, 2006;Ward, 1977) are often carried out by focusing on MDS models, rather than on raw data (for an exception, see Harbke, 2003), despite the known inaccuracies of the MDS analysis of the binary freesorting dissimilarities (Goodhill, Simmen, & Willshaw, 1995;Kendall, 1975;Simmen, 1996) and the vulnerability of the fit of these models to variations in the distributional properties of the input data (Pruzansky, Tversky, & Carroll, 1982).…”

An evaluation of the use of multidimensional scaling for understanding brain connectivity

“…As the new matrix generally has more distinct data values than the original one, this approach might appear useful for overcoming the problem of ties in poorly differentiated data-sets. However, Monte Carlo studies show that nmds solutions obtained using higher-order dissimi larities either do not give improved recovery of structure (Drasgow & Jones 1979), or -for binary datado so only for a particular row-comparison measure for data with low PN (Simmen 1994), a regime in which the absolute levels of recovery are in any case rather poor. However, there is a rather different type of derivation rule relevant only to binary data.…”

“…The reason that in neither case does the RSQ approach 1 despite the success of the reconstruction is that there is an inherent upper bound on RSQ for scaling binary data using the tied approach (as first pointed out by Simmen (1994)). The precise value of this RSQ "ceiling" depends on the particular configuration obtained.…”

“…Results for P N = 50% are shown in figure 7. Performance decreases as the added error increases, with greater rapidity for small N than large N. For further discussion see (Simmen, 1994).…”