# A New Multidimensional Scaling Methodology For the Representation of Inter-Product Substitutability

ABSTRACT - Batsell and Polking (1985) have recently proposed a new class of market share models. The models demonstrate that it is possible to decompose the ratios of market shares for any pair of products in any choice set into a natural hierarchy of inter-product competitive effects. One level in this hierarchy is a measure of the degree to which a product, say k, cannibalizes the market share of product i relative to -product j. This paper proposes a new multidimensional scaling (MDS) methodology which uses the revealed substitutability between products to derive a representation of the competing products as points in a space of prescribed dimensionality. An interesting feature of the methodology is that it allows for the derivation of spaces which are either symmetric or asymmetric. In the symmetric space, each product is represented as a single set of coordinates. In the asymmetric space, each product is represented as 2 sets of coordinates: one set of coordinates for each product as a "drawer" of market share, and a second set of coordinates for each product as a "drawee" from which the "drawers" take share. The methodology and corresponding algorithm are discussed in detail and two applications are used to demonstrate the procedure.

##### Citation:

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Wayne S. DeSarbo and Richard R. Batsell (1988) ,"A New Multidimensional Scaling Methodology For the Representation of Inter-Product Substitutability", in NA - Advances in Consumer Research Volume 15, eds. Micheal J. Houston, Provo, UT : Association for Consumer Research, Pages: 518-527.
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But Coke's goal is to steal business from Diet Pepsi and not hurt Diet Coke in the process of rebuilding Tab's sales.

Wall Street Journal

Batsell and Polking (1985) have recently proposed a new class of market share models. The models demonstrate that it is possible to decompose the ratios of market shares for any pair of products in any choice set into a natural hierarchy of inter-product competitive effects. One level in this hierarchy is a measure of the degree to which a product, say k, cannibalizes the market share of product i relative to -product j. This paper proposes a new multidimensional scaling (MDS) methodology which uses the revealed substitutability between products to derive a representation of the competing products as points in a space of prescribed dimensionality. An interesting feature of the methodology is that it allows for the derivation of spaces which are either symmetric or asymmetric. In the symmetric space, each product is represented as a single set of coordinates. In the asymmetric space, each product is represented as 2 sets of coordinates: one set of coordinates for each product as a "drawer" of market share, and a second set of coordinates for each product as a "drawee" from which the "drawers" take share. The methodology and corresponding algorithm are discussed in detail and two applications are used to demonstrate the procedure.

INTRODUCTION

In a recent paper, Batsell and Polking (1985) have proposed a new class of market share models. This class of models is based on a theorem that demonstrates that the natural log of the ratio of the market shares for any two products in any choice set can be decomposed into a linear combination of a set of scalars, called alphas. These alphas are characterized in terms of a natural hierarchy and reflect the competitive effects products have on each other. Of particular interest here are the third order alphas, a k/ij, which reflect the degree to which product k pulls from product i's market share as opposed to product j's market share. In a sense, the third-order alphas scale the relative substitutability of product k for i versus j. Although these third-order alphas provide useful managerial insight into which products compete with which other products, there are a very large number of them. For N products there are (N/2 ) * (N-2) third order alphas. Thus, the competitive information contained in these alphas, although managerially useful, is not efficiently portrayed in the form of tables.

In this paper, we propose a new multidimensional scaling (MDS) methodology for representing the information contained in the third-order alphas. A metric unfolding MDS model is used to represent a set of competing products in a space of prescribed dimensionality. The relative distances between the products is inversely proportional to their revealed substitutability. An important feature of the model is that it allows for the portrayal of products in derived spaces which are either symmetric or asymmetric. That is, each product can be represented by either one set of coordinates as in traditional MDS, or by two sets of coordinates - one set as "drawers" of market share and another set as "drawees" from whom market share is stolen. A rationale for this feature is presented later in the body of the paper.

The remainder of this paper is divided into 3 sections. First, we briefly review the models proposed by Batsell and Polking (1985). Second, the MDS model sand its associated algorithm are discussed in detail. Finally, the two applications presented in Batsell and Polking (1985) are used as examples of the new MDS methodology.

REVIEW OF THE BATSELL-POLKING MODELS

Let T= [1, 2, ...., N] be a set of alternative products. For each subset ACT, let P_{"}(i) denote the probability that alternative i will be chosen when the set A is the set of available alternatives and where P_{"}(i) … 0 for all i e A and for all ACT.

For each non-empty subset ACT and for i, j e A, define

Batsell and Polking (1985) prove that there exist unique numbers, a I/ij, defined for ijeT,I1T with I1 {i,j}=f, such that for every A,

where, for convenience,

a I/ii=0

and

a I/ij=0 if I1{i,j}…f, for all I, i.

If A= {i,j} we have bij/ij=a f/ij. In the special case A = {i,j,k}, (5) becomes

Thus, a f/ij is the natural logarithm of the ratio of i's share to j's share, and a k/ij measures the effect of the natural logarithm of that measure of the effect of the competition between presence of alternative k on the ratio. In other words, a k/ij is the the presence of alternative k on alternatives i and j.

If A = {i,i,k,1}, (5) becomes

Thus, a kl/ij is the measure of the effect of the pair k and 1 on the competition between i and j with the effect of the individual products k and 1 already accounted for by a k/ij and a kl/ij. In effect, a kl/ij measures an interaction of the quadruple {i,j,k,1}. The authors call a kl/ij a fourth order effect.

In general, a I/ij measures the effect of the products in the set I on the competition between i and j with the simpler effects of all proper subsets JCI already accounted for in (5) by the factors a J/ij. Batsell and Polking (1985) show how OLS can be utilized to estimate the a I/ij=s for a specified nth order model.

If n= # (Ic{i,j}), they call a I/ij an nth order effect. The nth order model is defined as (5) with a I/ij = 0 if # (Ic{i,j}) > n. Of interest here are the third order alphas: a k/ij. For example, when product k is added to the set {ij}:

1) if a k/ij < 0, k pulls from i's share proportionally more than it pulls from j's share;

2) if a k/ij = 0, k pulls from both i's share and j's share in the same proportion; and,

3) if a k/ij > 0, k pulls from j's share proportionally more than it pulls from i's share.

PARAMETER ESTIMATES FROM THE 3RD ORDER MODEL: INDIVIDUAL-LEVEL, SNACK DATA

RESEARCH OBJECTIVES

In the two applications presented in Batsell and Polking (1985), third order models were found to fit the data significantly better than the corresponding second order models. Table 1 presents the individual parameter estimates from the third order model fitted using five stimuli (snack items). (In this study, an individual subject made repeated choices from all subsets of two or more of five snack items.) In examining the scaled third order effects, Batsell and Polking (1985) noted that Yodels pulls from Twinkies market share more that it did from that of either Milky Way (a 1/24 =-.451) or Three Musketeers (a 1/25 =-.554). Yodels pulls from Devil Dogs more than from either Milky Way (a 1/34 = -.345) or Three Musketeers (a 1/35 = -.449).

PARAMETER ESTIMATES FROM THE 3RD ORDER MODEL: GROUP-LEVEL, POLITICAL DATA

Table 2 presents the group parameter estimates from the third order model fitted using six politicians as stimuli. (In this study, the choices of forty graduate students from all subsets of two or more politicians were aggregated to form proportions of votes obtained by each candidate in each subset.) From this table note, for example, that Carter pulls dramatically more from Brown than Connally (a 3/24=-.545), and Connally pulls much more strongly from Reagan than from Kennedy (a 4/56=.652).

The research problem investigated here is to develop a more parsimonious manner to display the structure implied by these third order alphas. Clearly, the more choice alternatives or stimuli, the more problem one has in comprehending the managerial implications of these third order effects because of their increasing complexity. Just from progressing from N=5 in Table 1 to N=6 in Table 2, the number of third order alphas double (from 30 to 60)!

We present a new metric multidimensional scaling unfolding (Coombs, 1964) type model to provide a spatial representation of the structure contained in these third order alphas. Our analysis of these parameter estimates provides a joint space of stimuli as "drawers" and "drawees" of market share. That is, each snack food (politician) in Table 1 (Table 2) is represented by two points in the derived dimensional space. One point represents the brand as a drawer of market share where drawee brands located near it would be ones from whom this brand would take market share away. The other point for the brand represents the brand as a drawee of market share from whom drawer brands nearby would be taking away market share. This asymmetric notion of a joint space is discussed in the next section.

POSITIONING MAPS AND SYMMETRY VERSUS ASYMMETRY

In the development of product strategies, marketing management often relies on positioning maps. Whether derived from similarity, "brand-switching", or some other data, these maps are intended to reflect the degree to which products are competing with each other. The real value of the map stems from the implicit assumption that products close to each other are presumed to be pulling from each other's market shares. The very concept of positioning, for example, is based on the notion that a new product will draw share primarily from products close to the position for which the new product is targeted. But the measures on which the maps are based are not actual measures of relative pull, and the assumption of "distance is inversely proportional to relative pull" is seldom, if ever, validated.

Rather than derive the product map from measures of similarity or "brand-switching" data and assume it reflects the degree to which products pull from each other's market shares, we derive the map from revealed measures of the actual pull itself. Given that we are going to derive positioning maps from revealed substitutability, we next address the issue of symmetry versus asymmetry. Most positioning maps based on measures of perceived similarity have been symmetrical. Indeed, it would probably invite derision if a consumer were asked: "Is A more similar to B or is B more similar to A?" But the assumption of symmetry is easier to question in the context of relative pull, or substitutability.

For example, suppose we ran an experiment in which a subject made repeated choices from the choice sets listed in Table 3. We wish to illustrate a subtle point, concerning asymmetrical relationships between triples of products. First, compare choice sets 2 and 4. Notice that the ratio of DD's share to TM's share in set 2 is 2.03. But in set 4, this ratio increases to 3.41. Thus, when we add MW to the set {DD,TM} we discover that the ratio of DD's share to TM's share increases by 68%. Now compare sets 1 and 4. In set 1 the ratio of DD's share to MW's is 1.38. With the addition of TM, this ratio increases to 2.32, also an increase of 68%. Thus, the rate at which MW pulls from TM versus DD is the same as the rate at which TM pulls from MW versus DD. If we were to try to place these 3 products as points in a symmetrical map where distance is inversely proportional to pull, then MW and TM should be equidistant from DD. Assuming this were true, and therefore that DD was equidistant from TM and MW, the map would imply that DD should pull from MW versus TM in exactly the same proportion. But the data in Table 3 show this is clearly not the case. When DD is added to the set {MW,TM}, DD pulls more from TM than MW. The ratio of MW's share to TM's share goes from .72 to 1.47 because DD pulls more from TM than MW. To state the proposition more generally:

CHOICE DATA REVEALlNG ASYMMETRY

If two products, A and B, pull equally from a third product, this does not necessarily imply that the third product will pull equally from them.

The point here is that the pattern of substitutability revealed in Table 3 could not be accurately represented in a symmetrical product positioning map. Although this example, based on actual data, shows that an asymmetrical map may be necessary, it is only an example. The need for asymmetry map must be demonstrated empirically using a statistical test. Fortunately, the Batsell-Polking model described earlier allows us to do so. Recall that the third-order model involves second (a f/ij) and third (a k/ij) order alphas. A symmetric third order model is one in which:

Thus, it is possible to first fit the third-order model which does not assume symmetry and then fit a special case of the third-order model which is symmetric. Because the symmetrical third-order model can be expressed as a linear restriction on the parameters in the full third-order model, a significance test Using the F-statistic can be used to decide on the need for a symmetrical versus non-symmetrical representation.

Such a test was performed on the data in the original Batsell-Polking paper on the sets of data presented in Tables 1 and 2. As can be seen from Table 4, in both cases the asymmetric model fit significantly better than the symmetric model p < .01 and p < .01, respectively. Thus, in each example, there exists sufficient asymmetry so as to warrant an asymmetrical map.

This notion of asymmetry is quite appealing in modelling spatial competitive positioning via two sets of distinct points. It allows for brands to pull from other brands which do not necessarily pull from them. Lattin and McAllister (1983) discuss this issue of "competitive non-symmetry" with respect to their choice model. Tversky (1977) presents a set-theoretical approach to (brand) similarity in which competitive brands can be represented as collections of features, and similarity (or competitiveness) is described as a feature matching process which is not restricted to be symmetric.

STATISTICAL TESTS FOR SYMMETRIC VERSUS ASYMMETRIC MODELS ON THE SNACK AND POLITICAL DATA

AN UNFOLDING METHODOLOGY FOR NONSYMMETRIC INCOMPLETE PROXIMITIES

The model

As previously noted, we are interested in providing a spatial representation of brands based on the Batsell and Polking (1985) third order alpha effects where two sets of points are represented--drawers and drawees. Our metric (since the a k/ij are assumed to be interval scales) model assumes:

where:

d 2/ik = the squared distance between brand i and brand k in some euclidean space;

d 2/jk = the squared distance between brand j and brand k in the same euclidean space;

f[*] = a linear function.

In addition, we define:

a

_{it}= the t-th coordinate for "drawee" i;b

_{kt}= the t-th coordinate for "drawer" ki, j, k = 1 .... N objects;

t = 1 .... T dimensions.

Thus, a k/ij < 0 implies that "drawer" brand k pulls from i's share proportionately more than it pulls from j's share. Assuming f[*] is linear and positive, this would imply d 2/ik < d 2/jk which means that "drawer brand" k pulls more from "drawee brand" i than "drawee brand" j. Thus, similar to an ideal point unfolding model, the closer a "drawer brand" is to a "drawee brand", the proportionately more market share the "drawer brand" would pull from that "drawee brand". If a k/ij = 0, then d 2/ik = d 2/jk and the "drawer brand" k is equally distant from "drawee brands" i and j. Similarly, if a k/ij > 0, then d 2/ik > d 2/jk and the "drawer brand" k is closer to "drawee brand" j than to "drawee brand" i.

The general model in expression (6) is similar to Torgerson's (1958) method of triads where stimuli are presented in triples or triads to the respondent and he/she is asked to make similarity judgments. From these triadic comparisons, Torgerson (1958) obtains, over all subjects, the proportion of times any stimulus k is judged more similar to stimulus i than to j. He then models the true proportion above as a function of the corresponding difference in distance (d_{jk} - d_{ik}).

We assume that f[*] in (6) is linear, so that (6) becomes:

where c_{1} is a multiplicative constant [Note that c_{1} is not identifiable in the linear form of equation (7) since it can be embedded in both the a_{it}'s and b_{kt}'s and then be set to 1. However, c_{1} can be gainfully utilized to signal "anti-ideal point" solutions (c_{1} < 0) and is thus included in the model. In addition, its estimation in the two-stage algorithm employed proves to accelerate convergence.] (c_{1} > 0), and c_{2} is an additive constant. [c_{2} is not to be confused with the traditional definition of an additive constant defined as the smallest number added to the proximities so as to have a metric defined (i.e., satisfaction of the triangle inequality for all triples of points). Here, c_{2} is solely an empirical value to be estimated under assumptions of an interval scale in order for the predicted values to best fit the input proximities. Note that one can not easily apply the traditional solution to the additive constant problem to asymmetric proximities since one side of the corresponding triangle or triple of points will always be missing with such data. Furnas (1987) is currently pursuing a theoretically correct solution to this additive constant problem for asymmetric proximities using a four point condition.]

Substituting for d 2/ik and d 2/jk terms in (7) and expanding renders:

The estimation problem in this deterministic model is to estimate a_{it}, b_{kt}, c_{1} and c_{2}, for i, k = 1.... N, so as to minimize the sums of squares loss function below:

An alternating least-squares algorithm (Wold, 1966) is described in Appendix I which utilizes a conjugate gradient method to estimate a_{it} and b_{kt} in the first phase of the algorithm, and OLS to estimate c1 and c2 in the second phase of the algorithm. This alternating estimation procedure terminates when convergence is reached. Since the objective function in (9) is continuous and has a lower bound of z =0, and since each phase of the algorithm can be shown to conditionally reduce iC (holding the other set of parameters fixed), one can use a limiting sums argument (Courant, 1965) to show that the procedure will converge to-at least a locally optimum solution.

Depending upon assumptions made concerning the scale of a k/ij (either interval or ratio), one could use either a variance accounted-for measure (R2) or a sums-of squares accounted-for measure (see DeSarbo, et. al., 1982). A ratio scale assumption would entail setting c_{2} = 0, while an interval scale assumption would have c_{2} being freely estimated as an additive, empirical constant.

Note that the model degrees of freedom for expression (8) is equal to R - T(2N) + T(T+1)/2 - 1, where R equals the number of 3rd order alphas and T(T+1)/2 is added due to the centering and rotational indeterminacies inherent in MDS unfolding models [As in all two-way simple unfolding models, indeterminacies exists with respect to centering both sets of points and rotating both sets via a (nonsingular) orthogonal rotation. We can rewrite the squared distance d 2/ik as:

d 2/ik =(

a_{i}-b_{k})=(a_{i}-b_{k}),

where __a___{i} is a T X 1 vector of coordinates for "drawee brand" i and __b___{k} is a T X 1 vector of coordinates for "drawer brand" k. As such, one can pre-multiply each set of coordinates by a nonsingular matrix __M__ and still obtain the same d 2/ik above if __M__'__M__ = __I__, the T X T identity matrix. Our procedure has an option to center the resulting joint space with respect to the origin.].

MAJOR PROGRAM OPTIONS

Symmetry

The methodology also allows the user to estimate one set of coordinates for both "drawer" and "drawee" brands (a_{it}=b_{it}œi,t). The use of such a symmetry option is tantamount to assuming a symmetrical "drawer" and "drawee" competitive structure--that a brand pulls market share from brands that pull market share from it. One can view such a model being nested in the more general model expressed in equation (5) allowing for asymmetric effects.

Linear Restrictions

The methodology contains an option to allow the user to constrain the coordinates of "drawer brands" to be linear functions of some specified set of background or feature/attribute variables. As in CANDELINC (Carroll, Pruzansky, and Kruskal, 1979) and in GENFOLD2 (DeSarbo and Rao, 1984; 1986), one can impose the reparameterization that:

where:

X

_{kt}= the value of the t-th attribute/feature of drawer k,d

_{it}=s = the impact coefficient of the t-th attribute/feature on the t-th dimension;t = 1.... L attributes/features.

Such a reparameterization aids in the interpretation of the derived space, as well as providing a basis for "positioning" drawer brands by manipulating such attributes/features (as in GENFOLD2 (DeSarbo and Rao, 1984, 1986)). This option also aids in interpreting the derived dimensions by providing estimates of d_{it} which can be plotted in T dimensions to examine which attributes/features load highly on either positive or negative ends of the derived dimensions. The positioning by attributes/features can be done after the d_{it}=s have been estimated. Assuming a target location,b^{*}_{kt}, has been established for brand k, one can then solve for X^{*}_{kt} such that EQUATION is as close to b^{*}_{kt} as possible (see DeSarbo and Rao, 1984, 1986 for a further discussion of the use of such reparameterized models).

As in the previous discussion of the symmetric model, one can view the reparameterized model in expression (10) as nested within the general model as long as L < N. To date, such linear restrictions have only been implemented for "drawer" brands because of the more interesting marketing strategy questions that can be asked concerning their positioning and market share. Such an option could easily be extended to "drawee" brands also.

MONTE CARLO ANALYSES

Because of potential problems that could theoretically arise from local minimum solutions, some eight different Monte Carlo runs were performed using synthetic, error-free data. Here, synthetic data were generated which fit the model exactly. A 2 X 2 X 2 full factorial design was used to generate data according to different size N (5, 7), symmetric/asymmetric, and restrictions/no restrictions. In all eight runs, the global optimum solutions were obtained, producing R2's= 1.00. While this provides some evidence concerning the viability of the algorithm, more testing is required. The impact of adding error generated from various distributions should be investigated. A wider range of number of stimuli (N) should also be tested.

APPLICATIONS

Snack Foods

Recall that Table 1 presents the individual parameter estimates from the Batsell and PoLking (1985) third order model using N = 5 snack items that will be used as input to our new unfolding methodology. With only 30 third-order alphas, no more than 3 dimensions can be derived in an asymmetric space. (A 3-dimensional solution for an asymmetric representation would involve 30 observations and 25 independent parameters.) Table 5 presents the results for a 1- and 2-dimensional asymmetric representation. The 2-dimensional solution appears to better describe the data in that it accounted for 20% more variance than the l-dimensional solution and the R2 is .998. (Note that in the 2-dimensional asymmetric solution there are 30 observations and 18 independent parameters resulting in only 12 degrees of freedom.)

UNFOLDING RESULTS FOR ASYMMETRIC ANALYSIS OF SNACK FOOD DATA

Figure 1 presents the joint space snack food solution for "drawers" (white) and "drawees" (black). The solution shows that:

1) the 2 candy bars tend to pull share primarily from each other and not from the cake-like items;

2) Devil Dogs pulls the most from Yodels, but seems to exert about the same degree of pull on the candy bars as it does on the other cake-like items;

3) in contrast to Devil Dogs, Yodels and Twinkies pull primarily from the cake-like items as opposed to the candy bars;

4) Yodels and Twinkies pull more from Devil Dogs, than Devil Dogs pull from Yodels and Twinkies;

5) Yodels pull slightly more from Twinkies than Twinkies pull from Yodels.

Perhaps the unusual role played by Devil Dogs is caused by the fact that in contrast to the other 2 cake-like products which are yellow-cake, Devil Dogs is all chocolate - both cake and icing. This may help explain its unusual role since the candy bars are perceived as primarily chocolate candy bars.

To compare the asymmetric representation with a symmetric one, a symmetric (A = i model was estimated in t= 1, 2, and 3 dimensions. (Note that one can estimate more dimensions here because the explicit constraint that drawer = drawee coordinates improve the degrees of freedom for estimation.) Table 6 shows the results for each of the 3 solutions. Although the asymmetric model results are not much better than the symmetric model results for the l-dimensional solution, the difference between the goodness-of-fit statistics for T = 2 dimensions is striking.

ASYMMETRIC CANNIBALIZATION MAP FOR THE SANCK FOOD EXAMPLE

SYMMETRIC CANNIBALIZATION MAP FOR THE SNACK FOOD EXAMPLE

UNFOLDING ANALYSIS FOR SYMMETRIC ANALYSIS OF SNACK FOOD DATA

Figure 2 displays the 2-dimensional symmetric solution which fails to capture the asymmetries in competition as noted earlier.

Politicians

As noted, Table 2 presents the group parameter estimates from the Batsell and Polking (1985) third order model using N = 6 politicians. The unfolding model was run in 1 and 2 dimensions for the asymmetric solution (more dimensions can be estimated, but given the R2 = .990 for the T = 2 solution, the utility of estimating further dimensions is again questionable). Table 7 presents the goodness of fit statistics and convergence information for both runs. As shown, the 2-dimensional solution appears to better describe the structure in these third order alphas accounting for an additional 18% of the variance than the 1-dimensional solution. Figure 3 displays this 2-dimensional joint space where politicians in white are drawers and politicians in black are drawees. From this joint space plot, several interesting notions, as found in Batsell and Polking (1985). are apparent.

UNFOLDING RESULTS FOR ASYMMETRIC ANALYSIS OF POLITICIAN DATA

Reagan (drawer) pulls dramatically more from Connally (drawee) than the other candidates. Similarly, Connally (drawer) pulls dramatically more from Reagan (drawee) than the other candidates. Bush, Brown, and Carter, as drawers, pull away share from each other.

As in the previous example with snack foods, the symmetric model was also fitted to these data and compared to the asymmetric model. Table 8 displays the results for the 1-, 2-, and 3-dimensional (since there are more degrees of freedom) solutions. For the symmetric space, it is not as clear that two dimensions best describe the structure in the third order alphas. A comparison of the symmetric and asymmetric 2-dimensional solutions shows that the asymmetric solution does explain considerably more variance than the symmetric solution. The corresponding statistics displayed in Table 8 show that the additional third dimension adds little to the explanatory power of the symmetric model in this instance.

UNFOLDING RESULTS FOR SYMMETRIC ANALYSIS OF POLITICIAN DATA

The resulting 2-dimensional symmetric solution is displayed in Figure 4. The figure indicates that a 1-dimensional solution here might be the most parsimonious of the (inferior) symmetric solutions since most politicians (except Bush) lie close to a 450 line. Projecting coordinates onto this line, one obtains a Conservative-Liberal dimension where Connally, Bush, and Reagan are on the conservative end and Brown, Carter, and Kennedy are on the liberal end. But, the symmetric model does not adequately account for the asymmetric structure in these third order alphas.

ASYMMETRIC CANNIBALIZATION MAP FOR THE POLITICIAN EXAMPLE

SYMMETRIC CANNIBALIZATION MAP FOR THE POLITICAL EXAMPLE

DISCUSSION

We have proposed a new unfolding MDS methodology to investigate competitive positioning of various stimuli/brands in a joint space of "drawers" and "drawees" of market share. The process of estimating two sets of coordinates for a stimulus in a T dimensional Euclidean space is tantamount to specifying non-symmetric distance relationships between the stimuli (i.e., d_{ij} … d_{ji}). The resulting spatial representation greatly aids in easily interpreting the competitive structure that exists between various brands, as contrasted to investigating complex tables of third order alphas such as those presented in Table 2. This is especially the case when N becomes large. Such visual portrayal of competitive positioning sheds light into marketing strategy considerations concerning the respective brands of interest. For example, Figure 3 demonstrates that Reagan would have benefitted most if Connally withdrew from the race. It also demonstrates which subset of candidates are most competitive or threatening to some candidate of interest.

More work is required concerning additional Monte Carlo analyses for a variety of sizes of stimuli (N) and error levels introduced into the data. In addition, more experience with actual data sets is required. Perhaps the most interesting example would be one with linear restrictions where, say, there was a design matrix of background characteristics which were strongly related to the positioning of the brands in the coordinate space. Once impact coefficients fit were estimated, normative strategies could be tested by altering and examining new "repositionings" as in GENFOLD2 (DeSarbo and Rao, 1984, 1986).

AN ALTERNATING LEAST-SQUARES ALGORITHM FOR THE ASYMMETRIC UNFOLDING MODEL

REFERENCES

Batsell, R.R. and Polking, J.C. (1985), "A Generalized Model of Market Share," Marketing Science, 4, 177-198.

Carroll, J.D., Pruzansky, S., and Kruskal, J.B. (1980), "CANDELINC: A General Approach to Multidimensional Analysis of N-Way Arrays with Linear Constraints on Parameters" Psychometrika, 45, 3-24.

Coombs, C.H. (1964), A Theory of Data, New York: Wiley.

Courant, R. (1965), Differential and Integral Calculus, 2nd ed., Vol. 1, New York: Wiley.

DeSarbo, W S. and Rao, V.R. (1984), "GENFOLD2: A Set of Models and Algorithms for the General Unfolding of Preference/Dominance Data", Journal of Classification, 1, 147-186.

DeSarbo, W.S. and Rao, V.R. (1986), "A Constrained Unfolding Model Product Positioning", Marketing Science, 5, 1-19.

Fletcher, R. and Reeves, C.M. (1964), "Function Minimization by Conjugate Gradients", Computer Journal, 7, 149-154.

Furnas, G.W. (1987), "The Additive Constant Problem for Asymmetric Proximities," Work in Progress, Bell Communications Research, Morristown, NJ.

Himmelblau, D.M. (1972), Applied Nonlinear Programming, New York: McGraw Hill, Inc.

Lattin and McAllister (1984), "Identifying Competitive Brand Relationships When Consumers Seek Variety," Technical Report, Marketing Science Institute.

Luce, D.R. (1959), Individual Choice Behavior: A Theoretical Analysis, New York: Wiley.

Luce, D.R. (1977), "The Choice Axiom after Twenty Years," Journal of Mathematical Psychology, 15, 215-233.

Natanishi, M. and Cooper, L.G. (1974), "Parameter Estimation for a Multiplicative Competitive Interaction Model - A Least Squares Approach", Journal of Marketing Research, 11, 303-311.

Torgerson, W.S. (1958), Theory and Methods of Scaling, New York: Wiley.

Tversky, A. (1977), "Features of Similarity", Psychological Review, 84, 327-352.

Wold, H. (1966), "Estimation of Principal Component and Related Models by Iterative Least Squares", in Multivariate Analysis, e.d. P.R. Krishnaiah, New York: Academic Press; 391-420.

----------------------------------------

##### Authors

Wayne S. DeSarbo, Southern Methodist University

Richard R. Batsell, Rice University

##### Volume

NA - Advances in Consumer Research Volume 15 | 1988

##### Share Proceeding

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