Peter H. Schonemann
An algebraic solution for a class of subjective metrics models
Psychometrika, 1972, 37, 441-451
It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carrol and Chang.
In addition to being very simple, this algebraic solution also permits testing of the constraints of these models explicitly. A numerical example is given.
At the time this paper was written, the so-called "INDSCAL" model had become very popular because it promised to dispense with the need to worry about the nagging identification problem that plagues euclidean MDS solutions in the multidimensional case. In factor analysis, this problem reduces to a "rotation" problem. Since in metric MDS it involves the wider class of similarity transformations (translations, rotations, and central dilations) it becomes more bothersome when the number of dimensions exceeds 2 or 3. Therefore, most early MDS papers as if by magic usually wound up with no more than 2 or 3 dimensions.
The INDSCAL model (Carroll and Chang, Psychometrika, 1970, 35, 283-320) is a minor variation on a theme Horan had introduced one year earlier into the literature (Psychometrika, 1969, 34, 139-165): After choice of an origin - usually taken at the centroid - N pxp matrices Ck of scalar products derived from presumed euclidean distances between p stimuli can be computed by solving the euclidean embedding problem (cf, e.g., Schonemann, 1970, Appendix 1).
In Horan's model, these scalar product matrices are postulated to have the form
Ck = A DkA', k = 1, N
where k is the subject index, and the N Dk are positive diagonal weight matrices indicating the salience the k'th subject attaches to each of the m dimensions. Thus, the basic idea was to account for the Np2 data in terms of a pxm common coordinate matrix A, and N subject-specific positive diagonal weight matrices of order p.
The same model had also been discussed by U. Schulz (Zu einem Decompositionsmodell der multidimensionalen Skalierung mit individueller Gewichtung der Dimensionen, Psychologische Beitrage, 1975, 17, 167-187). This paper was based on a talk he had given in 1972. Carroll and Chang (1970) added as a new wrinkle N subject-specific constants c(k). This change enabled them to norm the resulting scalar product matrices, but it results in a minor loss of identifiablity of the N D(k).
Carroll and Chang (1970) had offered a program that solved for the common space matrix A and the N subject-specific diagonal weight matrices D(k) by iteration. It quickly became very popular because it dispensed with the worry about the orientation of the resulting m dimensions and it always seemed to work. What went unnoticed - or unmentioned at least - was that this strong identifiability property comes at a price: If the underlying strong model doest not fit the data, the presumed orientation of the common space dimensions becomes be illusory.
The paper presents an exact algebraic solution of Horan's model that dissects the model into two basic assumptions, each of which can be tested separately:
(a) a common space assumption and
(b) a diagonality assumption.
As might be expected, the latter - which is needed to ensure the desired rotational determinacy - is much harder to meet in practice than the former.
In Schonemann, James, and Carter (1979): Statistical inference in multidimensional scaling: A method for fitting and testing Horan's model (in Lingoes, ed., Geometric Representations of Relational Data, Ann:Arbor: Mathesis, 791-826), a program is presented that is based on this algebraic solution, together with norms for testing the two constraints statistically.