ALSCAL: A nonmetric multidimensional scaling program with several individual-differences options

Abstract

ALSCAL is a nonmetric (or metric) multidimensional scaling (MDS) program with a number of individualdifferences options unavailable in other nonmetric MDS programs. This is the only program which incorporates individual-differences MDS models (Carroll & Chang, 1970; McGee, 1968) with multidimensional unfolding (MDU) models (Coombs, 1964; Young, 1972) and nonindividual-differences MDS models (Guttman, 1968; Kruskal, 1964; Shepard, 1962; Torgerson, 1952). ALSCAL consolidates most of the important developments in multidimensional scaling into a single program. ALSCAL uses the Alternating LeastSquares approach to scaling proposed by Takane, Young, and de Leeuw (1977), as improved by Young, Takane, and Lewyckyj (in press). ALSCAL is capable of performing a wide range of analyses, including (but not limited to) analyses equivalent to those performed by SSA-l, SSAR-l, and MINISSA (Lingoes, 1973), MDSCAL (Kruskal, 1964), KYST (Kruskal, Young, & Seery, Note 1), POLYCON (Young, 1973), INDSCAL (Carroll & Chang, 1970), TORSCA (Young, 1968), EMD, CEMD, and DEMD (McGee, 1968), and ASYMSCAL (Baker, Young, & Takane, Note 2; Young, Note 3). ALSCAL is suitablefor any type of twoor threeway data (rectangular or square, symmetric or asymmetric, conditional or unconditional, replicated or unreplicated, with or without missing data) that may be measured at the nominal, ordinal, interval, or ratio level of measurement (or may be binary). ALSCAL permits the analysis of an unlimited number of points or subjects in as many as six dimensions. This is the fastest general-purpose MDS program currently available, being 2 to 15 times as fast as the programs mentioned above. Data and Models. ALSCAL accepts data that are either twoor three-way and that concern objects or events from one, two, or three distinct sets. Given below are the specific types of data appropriate to ALSCAL, examples of these data, and the types of analyses that can be performed for these data. Data concerning objects from one set must represent some relation between pairs of objects in the set. These are termed two-way one-mode data, referring to the fact that a single set of objects is being paired with itself. For this type of data the analysis performed by ALSCAL corresponds to the nonindividual-differences MDS proposals mady by Kruskal (1964), Shepard (1962), and Torgerson (1952). An example of the data appropriate to this type of analysis is judgments of the similarity of various automobiles to each other. These data are contained in a single square matrix, with rows and columns representing the objects and entries representing the degree of relation between the objects. The data may be symmetric or asymmetric and may have missing elements. ALSCAL represents these objects (automobiles) as points in a multidimensional euclidean space. ALSCAL can analyze two different types of data concerning objects from two sets. One of the types, termed two-way two-mode data, is data that represents the degree of relation between objects from two distinct sets. For this type of data the analysis performed by ALSCAL corresponds to the nonindividual-differences MDU proposal made by Coombs (1964). Examples of data for this type of analysis are judgments of preference for automobiles from several subjects (representing the degree of relation between automobiles and subjects) or ratings on several attribute rating scales of the degree of attribute present in an automobile (representing the degree of relation between the rating scales and the automobiles). Such data are contained in a single rectangular matrix, with rows representing one set of items (say rating scales) and columns representing the other set (automobiles). This matrix may have missing elements. ALSCAL represents both sets of objects (scales and automobiles or subjects and automobiles) as points in a single multidimensional euclidean space. The second type of two-mode data that ALSCAL can analyze is three-way two-mode data. These are data for which there are several observations of the relation between pairs of objects in one set. Analyses performed by ALSCAL on these data correspond to the individualdifferences MDS proposals advanced by Carroll and Chang (1970) and McGee (1968). An example of appropriate data is judgments of automobile similarity obtained from several subjects; the objects being judged (automobiles) are one of the modes, and the subjects are the other. These data are contained in several square matrices, there being one matrix for each subject, with each matrix having rows and columns for the objects being judged. Each matrix may be symmetric or asymmetric, and there may be any pattern of missing data. ALSCAL can analyze three-way two-mode data in any of three distinct ways. First, ALSCAL can scale only the objects, in which case the objects are represented as points in a multidimensional euclidean space and the subjects are treated as replications. This analysis parallels McGee's (1968) treatment of individual differences in response style. Second, ALSCAL can scale both the objects and the subjects, in which case the objects are represented as points in a multidimensional euclidean