# Two-Way Spatial Methods For Modeling Individual Differences in Preference

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J. Douglas Carroll and Wayne S. DeSarbo (1985) ,"Two-Way Spatial Methods For Modeling Individual Differences in Preference", in NA - Advances in Consumer Research Volume 12, eds. Elizabeth C. Hirschman and Moris B. Holbrook, Provo, UT : Association for Consumer Research, Pages: 571-576.

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http://acrwebsite.org/volumes/6452/volumes/v12/NA-12

Our paper surveys the literature on two-way multidimensional scaling models for representing individual differences in preferential choice or other "dominance" data (to use Coombs' generic term for data in which observations relate to the dominance of one object or entity over another). This paper represents an update of the Carroll (1972, 1980) papers. The models range from the simple (linear) vector model, in which preferences for a given subject arc assumed to relate to projections of stimulus points onto a vector representing that subject, through the (quadratic) unfolding model and generalizations of that model. In the unfolding model. a subject's preference is assumed to relate in a simple fashion (c.g., linearly or monotonically) to distance from an ideal stimulus point for that subject. This "ideal point" represents that subject's most preferred stimulus and may or may not correspond to an actual stimulus point. Generalizations allow each subject to have differential weights for dimensions (as well as a different ideal point) or a possible idiosyncratic orientation of the coordinate system (to which differential weights are applied).

INTRODUCTION

We consider various two-way multidimensional (or multi-attribute) models for preferential choice data (although these models could be applied to any type of what Coombs (1964) has called "dominance data"). With one or two exceptions, these models apply to situations in which preference (or dominance) data arc available for a number of subjects (or other data sources) on a number of different stimuli (c.g., products), while the multidimensionality emerges from systematic individual differences among subjects or other data sources. Clearly, the multidimensional structure could be extracted from such data by, for example, calculating some form of "profile similarity" or "profile dissimilarity" measure among stimuli (over subjects or other data sources) and then applying standard methods of multidimensional (proximity) scaling to these derived dyadic proximities data C or applying, say, non-hierarchical or hierarchical clustering methods to obtain discrete multiattribute representations. We focus rather on explicitly formulated spatial models aimed at accounting for such systematic individual differences, and methods of analysis appropriate to such models. In general we shall be interested in deriving parameters representing subjects or other data sources as well as parameters defining stimuli as points in multidimensional space. While recognizing that dominance data other than preferential choice data could be involved, and that data sources other than subjects may be utilized, we henceforth will assume that preference judgments by each of a number. of subjects for each of a number of objects define the basic data.

INDIVIDUAL DIFFERENCES IN PREFERENCE

A number of different models have been proposed to account for individual differences in preference. Foremost among these arc the vector model, first proposed by Tucker (1960), and the simple unfolding model, proposed in the unidimensional form by Coombs ( 1964) and generalized to the multidimensional case by Bennett and Hays (1960). We shall consider a hierarchy of models (Carroll 1972, 1980) beginning with these two and continuing to further generalizations of the unfolding model in differential weights or saliences arc allowed for different individuals, or an even more general model in which the possibility of differential rotation of the system of coordinate axes as well as differential weighting is allowed.

MODELS FOR MULTIDIMENSIONAL PREFERENCE ANALYSIS

We now consider these models, which Carroll ( 1972, 1980) calls the linear-quadratic hierarchy of models in order of their complexity.

The Vector Model

The vector model, illustrated in Figure 1, assumes (in common with practically all the models to be discussed here) a set of stimulus points (such as A-E) embedded in a multidimensional space.

VECTOR MODEL ILLUSTRATED. SOURCE: CARROLL (1972)

In this model. different subjects are represented by distinct vectors, or directed line segments. Two of these vectors arc shown in Figure 1. The preference order for a given subject is assumed to be given by the projection of stimuli onto the vector representing that subject. As can be seen in the figure, quite different preference orders can be accommodated in such a model.

One way of interpreting vectors in this model is in terms of the relative importance of the dimensions to the preference judgment. The cosines of the angles the vector forms with the coordinate axes directly measure these relative importances. In the case of the vector model. these importances act like coefficients in a linear combination of dimensions. Its algebraic structure is defined below in equation (1):

where:

P_{ij}=the observed preference scale value of j^{th} stimulus for i^{th} subject

X_{jt}=the value of j^{th} stimulus on t^{th} dimension

b_{it}=the importance of dimension t for subject i (it is proportional to the direction cosine of the angle subject i's vector makes with the dimension t coordinate axis);

t=1....T dimensions;

i=1....I subjects;

j=1....J stimuli.

In matrix form:

II = BA' , (2)

where II = ||r_{ij}|| is the matrix of transformed preference values, p_{ij}, transformed by the function F_{i}, which takes observed preference scale values (p) into underlying "true" values (p). F_{i} will generally be assumed to be known in the case of "metric" analyses, and monotonic in the case of "nonmetric" analyses. B is the I x T matrix ||b_{it}|| of importances and X is the J x T matrix ||x_{jt}|| of coordinates of the stimulus space. (In the above, equations "=" is taken to mean "equals except for unspecified error terms," or that a least squares solution for unknown model parameters is sought.) Also, except where explicitly specified to the contrary, we assume complete preference scale values are obtained for all subjects, as specified by the matrix P. Carroll's (1972, 1980) PREFMAP and PREFMAP2 allows for the metric or non-metric estimation of this model as does his MDPREF (Carroll 1964) procedure.

An intuitively unattractive property of the vector model is that it assumes preference to change monotonically with all dimensions. That is to say, it assumes that if a certain amount of a given thing is good, even more must be better (ad infinitum). We know that this is not realistic for most quantities or attributes in the real world.

The Simple Unfolding Model

Coombs (1964) first introduced the unidimensional unfolding model which is illustrated in Figure 2.

UNIDIMENSIONAL UNFOLDING MODEL ILLUSTRATED. SOURCE: CARROLL (1972)

Here the stimuli can be described in one dimension. as represented by their positions along the abscissa in the figure. Different subjects correspond to different "ideal points" which represent their respective optimal values on that stimulus continuum. In the unidimensional unfolding model, the farther a stimulus is from a subject's ideal point, the less the subject will like that stimulus. In Figure 2, three hypothetical subjects are represented by ideal points, and the preference order generated by these different ideal points is indicated. It is clear that, even in one dimension, quite different orders can be generated by this model. Coombs called this the "unfolding" model because the preference order for a given individual can be generated by folding the stimulus scale at that individual's ideal point. To recover the stimulus scale from the preference data, then, it is necessary simultaneously to unfold all these preference scales for individuals (what Coombs calls the I scales) to find the common, or joint, stimulus scale (the J scale). The three curves drawn above the abscissa in Figure ' show hypothetical preference functions that go with the three individuals represented here. Of major importance is that they are. to use Coombs' term, single peaked (have a single maximum, which occurs of course at the ideal point) and symmetric. The particular shape is not important, so long as these two conditions are met, nor do they all have to have the same shape.

Bennett and Hays (1960) generalized this model to the multidimensional case. The two-dimensional case of this "multidimensional unfolding" model is illustrated in Figure 3.

MULTIDIMENSIONAL UNFOLDING MODEL ILLUSTRATED. SOURCE: CARROLL (1972)

Here the stimuli and subjects are both represented as points in the same multidimensional space. The points for individuals represent ideal stimuli, or optimal sets of stimulus values, for those individuals. In the Bennett and Hays model. the farther a given stimulus point is from an individual's ideal point, the less that individual likes that stimulus. This notion of relative distance implies a metric on the space. Bennett and Hays assumed this to be Euclidean, an assumption we shall share. The assumption of the Euclidean metric means that the "isopreferenee contours" (or loci of equal preference) are, in two dimensions, a family of concentric circles centered at the individual's ideal point. In three dimensions, they are concentric spheres, and hyperspheres in higher dimensions. Several authors have proposed algorithms for estimating stimulus scale values and ideal point coordinates from preference judgments assumed to follow the unfolding model (Lingoes 1972, 1973; Bennett and Hays 1960; Roskam 1973; Young and Torgerson 1967; Kruskal et. al. 1977; Kruskal 1964a,b; Schonemann 1970; Carroll 1972, 1980; Takane, Young, and deLeeuw 1977; Heiser 1981; Spence 1980; Greenaere and Browne 1982; DeSarbo and Rao 1983, 1984) . This approach of estimating both ideal points and stimulus coordinates is known as internal analysis (Carroll 1972), as opposed to external analysis methods which estimate only ideal points given fixed stimulus coordinates (obtained from perhaps an MDS analysis of similarities).

While the unfolding model and the vector model seem. superficially, to be quite different. it is in fact the case that the vector model is a special case of the unfolding model. One can see this by conceptually moving the ideal point for an individual farther and farther out along a fixed line from the origin, while holding the stimuli constant. As one does this, the rank order of distances from the ideal will approach that of projections of stimuli onto a vector whose direction is the same as that of the line along which the ideal point is moved. One can see it geometrically by noting that, as the ideal point is moved farther and far her out, the family of circular is preference contours looks (in the region occupied by the stimuli) more and more like a family of parallel straight lines perpendicular to the line joining the centroid to the ideal point, and asymptotically would be precisely such a family of straight lines. But the isopreference contours for the vector model comprise just such a family of parallel straight lines perpendicular to the vector. This geometric argument is illustrated in Figure 4.

The basic algebraic structure of the simple unfolding (or ideal point) model is contained in equations (3) and (4) below:

y_{it}=the t^{th} coordinate of the "ideal point" for the i^{th} subject;

x_{jt} is as before (as are p_{ij}, F_{i}, p_{ij} and T).

Note that the simple (linear) vector model is indeed linear in its variables (the x's, or stimulus coordinates) while the unfolding model is quadratic, (since it adds to the linear terms one term that involves the sum of squares of stimulus coordinates C i.e., a quadratic term). Schonemann (1970) presents an analytic internal solution for a strong case (treating the data as unconditional) of this unfolding model. Ross and Cliff (1964) provide methods for obtaining a stimulus configuration through a singular value decomposition of the double centered P=((P_{ij})) matrix under suitable assumptions. Schonemann and Wang (1972) combine this metric unfolding model with the Bradley-Terry-Luce choice model (Luce 1959), to produce a stochastic unfolding approach that is applicable to paired comparisons data. Zinnes and Griggs (1974) present a probabilistic multidimensional analogue of this model. Davidson (1972, 1973) presents a geometrical analysis for this type of unfolding model. Carroll's (1972, 1980) PREFMAP and PREFMAP2 allows for the estimation of this model using either an external or internal analysis. DeSarbo and Rao's (1984) GENFOLD2 allows for options for internal or external metric or nonmetric, and/or constrained or unconstrained analyses.

While the simple unfolding model assumes that a given difference (on a dimension) makes as much difference to one subject as to another, as well as assuming that all individuals relate to the same set of dimensions within the space, we now consider two generalizations of this model that allow one or both of these assumptions to be dropped.

The Weighted Unfolding Model

In this first generalization of the unfolding model (Carroll 1972, 1980) we continue, as in the simple unfolding model. to assume different ideal points for different individuals, but also allow distinct individuals to weight the dimensions differently. That is, in place of the usual Euclidean distance formula of the form

(where y_{it}, is the tth coordinate of individual i's ideal point, x_{jt} is the t^{th} coordinate of the j^{th} stimulus point, and d_{ij} is the distance between ideal point i and stimulus point j in a space of T dimensions), we substitute the formula

where the weighting factor w_{it} can be thought of as the "salience" or "importance" of the t^{th} dimension for subject i. For the moment we will assume the w_{it}'s are all positive or zero, but we will later consider the case where they may be negative. The generalized version of the unfolding model following expression (6) above assumes that one man's meat may be another man's irrelevancy; a dimension that is very important to one individual may be of no significance to another. It should be stressed that one need not assume these weights to reflect any differential perceptual discrimination of the dimensions. While a man may clearly perceive the difference between women who are blondes and those who are redheads, this perceptual dimension may be irrelevant to his choice of or preference for dating (or mating) partners. To another, of no greater perceptual acumen, this dimension may make all the difference in the world in making such judgments. Of course. the weighted distance formula of equation (6) does not distinguish between differential weights due to differences in perception, and those due to different "saliences" for preference of equally well discriminated dimensions,

This weighted unfolding model is illustrated in figure 5.

The only difference is that now the isopreference contours are ellipses. ellipsoids, or hyperellipsoids instead of circles, spheres, or hyperspheres. The larger the weight, the smaller the corresponding axis of the isopreference ellipse or ellipsoid. reflecting the fact that it takes a smaller change to make the same amount of difference (to be precise the ratios of the axes of the ellipsoids are reciprocally related to the square roots of the ratios of the weights that appear in equation (6)). In Figure 5, Subject I weights the two dimensions equally, Subject II weights dimension one more than two, while III weights dimension two more than one. The preference orders for the three hypothetical subjects are also shown. These are generated by applying the generalized Euclidean distance defined in equation (6), which can be characterized roughly as the number of rings of concentric ellipsoids out from the center (or ideal point).

Srinivasan and Shocker (1973) present a nonmetric external unfolding analysis with this model using linear programming methods including non-negativity constraints for the dimension weights. The same constraints are provided in a metric procedure using quadratic programming described by Davison (1976). Spence (1980) presents an interesting generalization of this model allowing for linear constraints on X = |x_{jt}|| and/or Y = ||y_{it}||. DeSarbo and Rao's (1984) GENFOLD2 allows for all of these options in the estimation of the model.

The General Unfolding Model (Differential Rotations and Weights)

In this first generalization we have assumed that, while individuals may differ both in ideal point and weighting of dimensions, the same basic set of dimensions are involved in the judgments of all individuals. The second generalization allows us to relax this assumption. While we shall assume all individuals to share in common a single perceptual space, we allow distinct individuals additional freedom in choosing a set of "reference axes" within that space (Carroll 1972, 1980). Thus each individual is allowed to rotate the reference frame of the perceptual spaces and then to weight differentially the dimensions defined by this rotated reference frame (in addition to being permitted an idiosyncratic ideal point).

Here, the formula for Euclidean distance can be written as:

d_{ij} = [(y_{i} - x_{j}) W^{i} (y_{i} - x_{j})']^{1/2} (7)

where

y_{i}=the I x T vector of deal point coordinates for subject i,

x_{j}=the I x T vector of coordinates for stimulus j,

W^{i}=the i-th subject's TxT symmetric transformation matrix.

One point that should be mate with respect to this model is that the rotation alone does not make the model different from the simple unfolding model (since orthogonal rotations leave Euclidean distances unchanged). It is only the rotation in combination with differential weighting of dimensions that makes this a genuinely new model. Bechtel (1976), and DeSarbo and Rao (1984) also present a related general powered distance model for unfolding. Ramsay (1982), and deLeeuw and Heiser (1979) also consider such metrics for dissimilarity data.

Figure 6 illustrates isopreference contours that could arise from this most general unfolding motel.

ILLUSTRATION OF THE GENERALIZED UNFOLDING MODEL. SOURCE: CARROLL (1972)

For simplicity, only a single ideal point is shown, and all isopreference ellipses have the same eccentricities, but neither of these conditions would necessarily hold in general.

The Possibility of Negative Weights

While the weights (w's) have heretofore been spoken of as though they were assumed to be positive, in the preference domain it is possible to argue that we may sometimes want to allow negative weights. A negative weight, in this context, means that there is a minimally (rather than maximally) preferred value for that dimension, as is certainly true for some dimensions. It has been argued (Carroll 1972) that such a minimally preferred value characterizes many subjects in the case of the temperature dimension for tea (i.e., many subjects like hot and cold but not lukewarm tea). In the unidimensional case a negative weight simply means that the preference functions are inverted, or that the folding of the J scale produces the negative of the I scale. In the multidimensional case, however, negative weights may lead to more interesting consequences. Of course, if all dimensions have negative weights it is just as though the preference scales were inverted in direction (so that the "ideal point" is transformed into an '*anti-deal"). If, however, some dimensions have positive and others have negative weights, the situation is more complex. Instead of either an "ideal point" or an "anti-deal point" we now have a saddle point, that is a point that is optimal with respect to some dimensions, and "pessimal" with respect to others.

This is shown more graphically in Figures 7, 8, and 9. Figure 7 shows a prototypical preference function for the usual case of both weights positive in the two dimensional case.

(Horizontal slices are taken through this response surface, and the intersections projected down onto the two dimensional plane to generate the elliptical isopreference contours). Figure 8 shows a section of a typical saddle-shaped preference function for the case in which one dimension (dimension one) is negatively weighted while the other (dimension two) has a positive weight.

Figure 9 contrasts the isopreference contours appropriate for the two cases.

In the case of the positive-negative pattern of weights, the isopreference contours comprise a family of hyperbolas. The two straight lines which are asymptotes of the hyperbolas define the locus of points equal in preference to the saddle point, which is the point at which the two lines intersect.

While there may well be a justification (both theoretical and empirical) for such negative weights in at least some cases, it is very often the case that they occur because of chance fluctuations in parameter estimates due to noise in the data. In such cases it may be desirable to constrain the weights to be non-negative. One way to do this is to use quadratic programming methods (see Davison 1976) . Another way involves the reformulation of the problem as a linear programming problem proposed by Srinivasan and Shocker (1973). DeSarbo and Rao (1984) allow for options to constrain these weights to be positive or to be estimated freely.

The Hierarchial Structure of Preference Models

It has already been argued that the vector model is a special case of the simple unfolding model. It is clear that this simple unfolding model in turn is a special case of the weighted unfolding model (since the weights can, as a special case, be all equal to one), while the weighted model is a special case of the general unfolding model (since the rotation matrix may be an identity). By transitivity, each simpler model is a special case of all the more general onesChence the hierarchy of models alluded to earlier.

DISCUSSION

We have presented a review of four general classes of spatial multidimensional scaling models used in the analysis of preference and/or dominance data: the vector model, the simple unfolding model, the weighted unfolding model, and the general unfolding model. The assumptions and implications of each type of model have been illustrated and discussed. as well as the nested features of these four models. The use of such models in marketing has had pronounced effects in the areas of market segmentation, product positioning, and consumer behavior.

There are other areas in multidimensional scaling that have not been discussed in this paper. Topics such as three-way analyses, discrete network models, reparameterization models, problems of solution degeneracy, different types of analyses, preprocessing effects. property fitting and interpretations of solutions, etc., are all relevant areas that should be addressed in future endeavors. In conclusion, we feel that marketing has only begun to benefit from the applications of such psychometric procedures, and that the future looks quite bright concerning the integration of these techniques into general marketing practice.

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