mds maps for product attributesand market response: an...

Marketing Science � 1999 INFORMSVol. 18, No. 4, 1999, pp. 584–604

0732-2399/99/1804/0584/$05.001526-548X electronic ISSN

MDS Maps for Product Attributes andMarket Response: An Application to

Scanner Panel Data

Rick L. Andrews • Ajay K. ManraiDepartment of Business Administration, University of Delaware, Newark, Delaware 19716

[email protected] • [email protected]

AbstractThere is theoretical and empirical evidence that consumershave limited cognitive resources and thus cannot maintaindirect preferences for each choice alternative on the storeshelves. Instead, they likely form their overall preferences forchoice alternatives by evaluating the attributes describingeach item. Rather than mapping the locations of and pref-erences for all choice alternatives in a multidimensionalspace, as is the current practice in marketing research, it isinsightful to map the locations of and preferences for theattributes consumers use to evaluate the choice alternatives.The model proposed in this study unifies latent class pref-erence models (choice models or conjoint models) with latentclass multidimensional scaling (MDS) analysis. Dimensionalrestrictions are imposed on latent class preference modelssuch that the locations of attribute levels and market re-sponse parameters can be mapped in reduced-dimension

spaces. Interactions between attributes can be graphically ex-amined, which is not feasible with the traditional MDS ap-proach. Also, the effects of price reductions and promotionson the locations of attribute levels can be graphically exam-ined. An empirical application with scanner panel datashows the capabilities and limitations of the proposedmodel.In addition to the managerial insights provided by themodel, it is also much more parsimonious than existingmethods, and it forecasts holdout choices significantly better.In the empirical application, a model with two-dimensionalattribute maps has 50 fewer parameters than the best unre-stricted latent class choice model, yet the fit is comparable.The predictive performance of our model is shown to be su-perior to that of latent class MDS approaches and latent classconjoint approaches.(Brand Choice; Choice Models; Marketing Mix; Scaling Methods;Segmentation Research)

MDS MAPS FOR PRODUCT ATTRIBUTES AND MARKET RESPONSE:AN APPLICATION TO SCANNER PANEL DATA

Marketing Science/Vol. 18, No. 4, 1999 585

IntroductionLiterature on new product development is repletewithanalytical methods that summarize consumer judg-ment data to produce insights on optimal attributeconfigurations, brand positioning, and market seg-mentation (Urban and Hauser 1993). Recent researchon preference analysis has produced latent class con-joint analysis models (e.g., DeSarbo et al. 1992) andlatent class choice models (e.g., Kamakura and Russell1989) that estimate segment-specific preferences forproduct attributes and/or responses to marketing mixvariables. Recent research on latent class MDS has pro-duced models that map brand positions and prefer-ences of segments in a joint space (Bockenholt andBockenholt 1991, DeSarbo et al. 1991, DeSoete andHeiser 1993, DeSoete and Winsberg 1993, Chintagunta1994, Wedel and DeSarbo 1996).

This study develops a model that unifies latent classpreference models with latent class MDS. The modelimposes dimensional restrictions on a latent classchoice or conjoint model so that the preferences can bedepicted in reduced-dimension spaces. Unlike existingmodels, the proposed model produces a separate mapfor each product attribute, including a graphical ex-amination of interactions between attributes, as well asa map showing responses to marketing mix variables.As suggested by Wedel and DeSarbo (1996), the entirefamily of exponential distributions can form the basisof such models, allowing many possible types of re-sponse data (e.g., rating scale measures, choices, pick-any data, purchase frequencies, duration-type data,etc.). We provide an empirical application using scan-ner panel choice data for stock keeping units (SKUs,cf. Fader and Hardie 1996).

Most, if not all, research in marketing has focusedon mapping choice alternatives. From a managerialstandpoint, there are advantages to mapping productattributes rather than choice alternatives. First, from aconsumer behavior perspective, the attribute level is amore appropriate unit of analysis than the choice al-ternative. Though consumers choose from a large as-sortment of SKUs on the store shelves, Fader andHardie (1996) assert that consumers have limited cog-nitive resources and thus do not form preferences foreach individual SKU. With 50–150 SKUs on the shelvesin many product categories, consumers may not be

able to maintain well-defined preferences for each one.Instead, they probably form preferences for the attrib-utes describing each item (brand name, size, formula-tion, etc.) to derive their overall preference for an SKU,which is a much more manageable task. Fader andHardie cite theoretical justification for this assertionfrom both economics and psychology and provide em-pirical justification in their own study. Given the im-portant role of attributes in the formation of prefer-ences, it may be more meaningful to map theseattributes rather than the choice alternatives.

Second, the attribute level maps are easier to inter-pret than maps containing all choice alternatives. Ex-isting MDS approaches map the locations of all choicealternatives in a single space, possibly resulting in aspace that is very crowded and difficult to interpret.In a later application involving 57 SKUs of fabric soft-eners, a map produced by existing MDS methodswould show all 57 alternatives in a single space, alongwith segment-specific preference vectors. Our modelmaps each attribute in a separate space, resulting inmaps that are simpler to interpret. Instead of a singlemap containing all 57 SKUs, our method produces fivemaps (one for each attribute—brand name, size, prod-uct form, and formulation—and one for the marketingmix). No map contains more than 10 attribute levelsand the preference vectors. Graphical examination ofan interaction between two attributes does not requireadditional maps—it can be incorporated into one of themaps of main effects. Given that it is not at all uncom-mon to find product categories with many alternativeson the store shelves, the proposed method could beapplicable in many studies.

Third, the approach is very parsimonious. The ex-isting approach for mapping the locations of andpreferences for discrete product attributes, known assimultaneous reparameterization, requires that re-strictions be placed on the latent class MDS model.Frequently, these restrictions are rejected by the data(cf. Wedel and DeSarbo 1996), as we later demonstrateempirically. Our model instead imposes restrictions ona latent class preference model. In contrast, we havenot found a single instance in which the restrictionsrequired by our model are rejected by the data. Therestrictions we impose typically have little effect on the

ANDREWS AND MANRAIMDS Maps for Product Attributes and Market Response: An Application to Scanner Panel Data

586 Marketing Science/Vol. 18, No. 4, 1999

fit of the model, yet they often reduce parameter re-quirements significantly. In a later empirical applica-tion, we show that a model with two-dimensionalmaps for all attributes uses 50 fewer parameters thanthe best unrestricted latent class model, yet fits as welland forecasts better.

Finally, the proposed attribute-level maps provideunique and compelling managerial insights not obvi-ous from plots of part worths. The maps provide allthe preference information available from part worths,but they can provide managerial insights beyond thoseprovided by part worths. Attribute levels located inclose proximity have very similar preferences andprobably need to be differentiated from other levels.Interactions between attributes can be shown graphi-cally. Heterogeneous responses to marketing mix vari-ables such as price, aisle display, and store feature ad-vertising can be shown graphically. We are not awareof any MDS approach that maps marketing mix effects.Later, we extend the proposed model to show howprice reductions and promotional activities such asaisle displays and store feature advertisements affectthe mapped locations of brand names and otherattributes.

This paper first reviews recent developments in la-tent class preference analysis and MDS that are rele-vant to the current study. Following the review of lit-erature, the formulation of the proposed model ispresented. Finally, we present an application to scan-ner panel choice data for SKUs of fabric softeners.

Relevant LiteratureFigure 1 shows the positioning of the proposed modelin relation to existing models and serves as an outlinefor this review of relevant literature.

Conjoint analysis models that simultaneously seg-ment consumers and estimate segment-level conjointpart worths are now quite numerous (DeSarbo et al.1992, Green and Helsen 1989, Green and Krieger 1991,Hagerty 1985, Kamakura 1988, Kamakura et al. 1994,Ogawa 1987, Wedel and Kistemaker 1989, Wedel andSteenkamp 1989). Of particular interest in this study isthe latent class approach of DeSarbo et al. (1992). Intheir model, the segments and the utility function partworths are estimated simultaneously using a model in-volving mixtures of multivariate normal distributions.

The simulation study by Vriens et al. (1996) comparesnine metric conjoint segmentation methods and con-cludes that the latent class model and a fuzzy cluster-wise regression procedure (Wedel and Steenkamp1989) generally outperform other procedures with re-spect to coefficient and segment membership recovery.

Kamakura et al. (1994) develop a similar approachbased on logit in which the mixing weights are param-eterized as a function of consumer descriptor variables(see also Gupta and Chintagunta 1994 and Wedel andDeSarbo 1996). Fader and Hardie (1996) reparameter-ize the choice alternative-specific constants of a logitmodel into part worths in the context of scanner paneldata for SKUs of fabric softeners, bridging the gap be-tween logit analysis and conjoint analysis. Our modelbuilds on the contribution of Fader and Hardie by de-picting preferences in reduced-dimension joint spaces.Kamakura et al. (1994) and Fader and Hardie (1996)are examples of nonmetric or choice-based conjointanalysis.

In the MDS literature, latent class models have re-cently been developed to represent locations of choicealternatives and segment preferences in a joint space.DeSarbo et al. (1991) develop a vector model for nor-mally distributed ratings data, while DeSoete andHeiser (1993) formulate an ideal point model for nor-mally distributed data. Vector models are based on thenotion that more of a desirable dimension is better, butideal point models suggest that there is an ideal levelof any dimension. Bockenholt and Bockenholt (1991)estimate vector and ideal point models for binary (pickany or pick any/J) data. Chintagunta (1994) developsa vector model for multinomial choice data. DeSarboet al. (1994) and Wedel and DeSarbo (1996) review theliterature on latent class MDS models.

To improve the interpretability of the derived di-mensions of a joint space, researchers have used so-called property fitting methods (Bockenholt andBockenholt 1991, Carroll et al. 1989, Wedel andDeSarbo 1996). For example, the estimated MDS co-ordinates of choice alternatives can be regressed on adesign matrix describing the alternatives on a groupof attributes. The resulting weights for attributes canbe used to plot attribute vectors in the same joint spaceas the choice alternatives and preferences.

Others have proposed reparameterizing coordinates



Figure 1 Relation of Proposed Method to Existing Methods

of choice alternatives into attribute coordinates as partof an integrated MDS analysis (DeSarbo et al. 1982,DeSarbo and Rao 1986, DeSoete and Heiser 1993,DeSoete and Winsberg 1993, Wedel and DeSarbo1996), which is known as simultaneous reparameteri-zation. This procedure involves computing the coor-dinates of choice alternatives as the product of a designmatrix describing the alternatives on conjoint attrib-utes and a matrix of estimated attribute locations. Thus,simultaneous reparameterization imposes constraintson the conventional stochastic mixture MDS modeland requires the estimation of attribute coordinates in-stead of the coordinates of choice alternatives. The endresult is a single map that contains all attribute levelsand segment preferences (and perhaps choice alter-natives if desired). We note that such maps may be

crowded and difficult to interpret, as we demonstratein a later empirical application.

More importantly, the restrictions imposed by thesimultaneous reparameterization may not be consis-tent with the data. The study by Wedel and DeSarbo(1996) found that the restrictions were not consistentwith the data and relied on conventional property fit-ting methods instead. DeSoete and Heiser (1993) dem-onstrate the reparameterization with synthetic databut not data from an actual application. The simulta-neous reparameterization is rejected in our empiricalapplication as well. Though the studies by DeSarbo etal. (1982) and DeSoete and Winsberg (1993) did findthat simultaneous reparameterization restrictions areconsistent with the data, the technique is sometimes



not an acceptable way of depicting preference infor-mation in reduced-dimension spaces. Our proposedmodel is intended to remedy this problem.

In our model, dimensional restrictions imposed onlatent class preference models allow the preferences tobe shown graphically in reduced-dimension spaces, asin an MDS model. The result is a separate map for eachattribute, with each map containing the locations of theattribute levels as well as the segment-specific prefer-ences for those attribute level locations. As we dem-onstrate with simulated data in the next section, aninteraction between two attributes can be plotted in asingle map. Heterogeneous market responses to price,aisle display, and store feature advertising can also bemapped in a reduced-dimension space. As our empir-ical analysis demonstrates, these dimensional restric-tions can result in a significant reduction in parameterswithout a corresponding loss of fit. The forecastingperformance of our approach is shown to be superiorto that of all existing approaches as well.

An Integrated Latent ClassPreference-MDS ModelWe first present a latent class preference model basedon that of DeSarbo et al. (1992) and then impose di-mensional restrictions on the model so that the pref-erences can be graphically depicted in reduced-dimension spaces. Leti � 1, . . . , I, consumers;j � 1, . . . , J, choice alternatives;k � 1, . . . , K, derived segments;l � 1, . . . , L, variables (continuous or discrete) de-

scribing the alternatives;Yij � the response to choice alternative j by con-

sumer i;Yi � the J � 1 column vector of responses by con-

sumer i;Xjl � the value of the lth variable for the jth

alternative;Xj � the 1 � L row vector of variables for the jth

alternative;X � ((Xjl)), which is J � L;blk � the estimated response to the lth variable for

the kth segment;

bk � the L � 1 column vector of responses for thekth segment;

b � ((blk)), which is L � K;Rk � a J � J covariance matrix estimated for seg-

ment k; andR � (R1, R2, . . . , Rk).

The density function for the response vector Yi can bemodeled as a mixture of distributions

K

H(Y ; �, X, b, R) � � g (Y |X, b , R ), (1)i � k ik i k kk�1

where � � (�1, �2, . . . , �k) are the mixing weights, in-terpreted as segments sizes, such that 0 � �k � 1 and

� 1. When the preferences for alternatives Yi areR �k k

normally distributed ratings,

�1/2�J/2g Y |X, b , � (2p)ik i k � ��

k k

�1

exp[�0.5(Y � Xb )� (Y � Xb )], (2)i k � i kk

but when consumers make choices and Yi takes on val-ues of zero or one,

YijJ exp(X b )j kg (Y |X, b ) � . (3)ik i k � J

j�1exp(X b )� �� s k

s�1

DeSarbo et al. (1992) use Equation (2), while Fader andHardie (1996) and Kamakura et al. (1994) use Equation(3). Equation (1) is the likelihood function for an indi-vidual consumer. For a sample of I consumers, the loglikelihood function is

I k

ln L � ln � g . (4)� � k ik� �i�1 k�1

We discuss the estimation of parameters through max-imization of Equation (4) later in this section.

Restrictions to Reduce Dimensionality of LatentClass Preference ModelsThe model presented thus far estimates K sets ofsegment-specific preferences. The contribution of ourmodel is that it imposes dimensional restrictions on thelatent class preference model above so that preferences



may be graphically depicted in reduced-dimensionspaces.

We first describe maps for discrete attributes suchas brand name and later extend the maps to marketingmix attributes, which may be continuous (e.g., price)or discrete (e.g., store feature). Letf � 1, . . . , F, attributes;Lf � the number of levels for attribute f;fb k � the Lf � 1 vector of preferences for attribute f,

segment k;Mf � the dimensions of the space into which is tofb k

be mapped (to be determined by the data);Af � the Lf � Mf vector of household invariant lo-

cations of in the space; andfb kfwk � the Mf � 1 vector of importance weights

which segment k attaches to the Mf

dimensions.The estimated preference for an attribute f for seg-

ment k, , can be decomposed into the location of thefb kcorresponding attribute levels in multidimensionalspace,Af, and the segment-specific importanceweightsconsumers attach to those locations, (cf.fwkChintagunta 1994, Elrod 1988). Specifically, assumethat is a linear function of the levels’ locationswithinfb kan Mf-dimensional map such that

f f fb � Aw . (5)k k

For an attribute with Lf levels, only Lf � 1 part worthsare identified per segment, so one of the levels mustbe normalized to zero. Likewise, one attribute levelmust be normalized to the origin of themap, and henceone row ofAfwill contain only zeros (it does notmatterwhich row contains the zeros). This is analogous toarbitrarily setting one element of to zero in afb kdummy variable regression, which is required foridentification. Restrictions are also imposed on the im-portance weights , as explained below.fwk

Consider the simple example of an unrestricted five-segment latent class preference model that has onlyone attribute with four levels. Fifteen part worths (3for each segment) would be estimated for such amodel. The unrestricted model can be cast into theframework of model (5) above. The Af matrix for theunrestricted latent class model would be 4 � 5, reflect-ing a five-dimensional solution. (For an unrestrictedlatent class model, the number of dimensions is the

same as the number of segments; for our restricted la-tent class model, the number of dimensions is less thanthe number of segments). Since one row of Af containsall zeros for identification, there are 15 location param-eters in the matrix. It is important to note that the numberof parameters used for an attribute can never be greater thanthe number required for the unrestricted latent class pref-erence model. In this example, it would not be possibleto identify any importance weights since all 15 of theallowable parameters are used in the location matrix.Thus, the segments must have weight vectors �fw1

[1,0,0,0,0], � [0,1,0,0,0], � [0,0,1,0,0], �f f fw w w2 3 4

[0,0,0,1,0], and � [0,0,0,0,1]. This setup would pro-fw5

duce exactly the same solution as an unrestricted five-segment latent class model (DeSarbo et al. 1992).

Now if we impose a two-dimensional map restric-tion on the attribute, the location matrix Af is 4 � 2and contains 6 location parameters (one row again con-tains all zeroes). The dimensional restrictions are im-posed such that the two dimensions of the map rep-resent the first two segments’ preferences. Theremaining three segments’ preferences are linear com-binations of the first two segments’ preferences. Thesegments would have weight vectors � [1,0], �f fw w1 2

[0,1], � [w31, w32], � [w41, w42], and � [w51,f f fw w w3 4 5

w52]. The first dimension of the map represents thepreferences of segment 1, while the second dimensionof the map represents the preferences of segment 2.The other segments’ preferences are then linear com-binations of the first two segments’ preferences. Ratherthan estimate more part worths for the other three seg-ments (as in an unrestricted latent class preferencemodel), we simply estimate importance weights thatare applied to the first two segments’ part worths. Thetwo-dimensional model contains 12 parameters (6 lo-cation parameters and 6 weight parameters) ratherthan the 15 parameters required for the unrestrictedfive-segment latent class model, thus requiring 3 fewerparameters for that attribute.

If we impose a one-dimensional map restriction, theAf matrix is 4 � 1 and contains 3 location parameters.The first segment’s preferences are estimated directly,but the other four segments’ preferences are weightedfunctions of the first segment’s preferences. The onedimension of the map therefore represents the prefer-ences of segment 1. The segments would have weight



vectors � 1, � w2, � w3, � w4, and �f f f f fw w w w w1 2 3 4 5

w5, for a total of 7 parameters.Note that three- and four-dimensional maps for this

example would not constitute a reduction in parame-ters. The three-dimensional version would require 15parameters (9 locations and 6 preference weights),while the four-dimensional version would require 16(12 locations and 4 preference weights). It is not pos-sible to identify more than the 15 parameters requiredby the unrestricted latent class preferencemodel. Thus,for a four-level attribute with five preference segments,one would only consider two- and one-dimensionalmaps.

We have not encountered any data for which three-or higher-dimensional maps were needed; two-dimensional map restrictions are seldom, if ever, re-jected by the data. If two-dimensional maps are notrejected by the data, then one has no need to considerhigher-dimensional maps. One-dimensional maps im-pose quite severe restrictions and are often rejectedby the data even when two-dimensional restrictionsare not rejected. In addition, the managerial insightsprovided by one-dimensional maps are much lessinteresting.

For a map to be identified, the number of parametersrequired for any given attribute must be less than orequal to the number of parameters required for thatattribute in the unrestricted latent class model. For ex-ample, for a latent class model with five segments, anattribute with four levels would require 5 (4 � 1) �

15 parameters. The total number of parameters in Af

for all segments k must therefore be less than orfwkequal to 15. More generally, if Mf is the number of di-mensions for factor f, K is the number of segments, andLf is the number of levels, the identifiability restrictioncan be expressed as:

number of location parameters� number of preference weight parameters� number of parameters in unrestricted model

or

f f fM (L � 1) � M (K � M ) � K(L � 1). (6)f f

If a one-dimensional map is desired (Mf � 1), anynumber of segments (K � 1) can be used with any type

of attribute (Lf � 2). There are no unidentified combi-nations of K and Lf whenMf � 1. If a two-dimensionalmap is desired, there must be at least two segments (K� 2) regardless of the number of levels for the attri-bute. If there are only two attribute levels (Lf � 2), onlya two-segment map (K � 2) is identified; models withmore segments (K � 3) are not identified when Lf � 2.Thus, two-dimensional maps are identified for K � 2except for the combination of Lf � 2 and K � 2.

Otherwise, there do not appear to be any identifi-cation problems with the maps beyond those inherentin latent class models (see McLachlan and Basford1988, Titterington et al. 1985, Section 3.1). The DeSarboet al. (1992) article on which our model is based doesnot suggest that there are any special identificationproblems with latent class preference models. Giventhat our model imposes more restrictions on the latentclass preference model, we would not expect there tobe unidentified parameters. Titterington et al. (1985)give a formal definition of identifiability for finitemixtures, illustrate that nonidentifiability can occur,and derive both necessary and sufficient conditions foridentifiability. Of course, for a K-segment model, thereare K! permutations of the segment labels (e.g., whatwas called segment 1 on one run may emerge as seg-ment 2, 3, . . . , K on other runs). This is of no conse-quence and does not constitute lack of identifiability(McLachlan and Basford 1988).

In summary, if Mf-dimensional restrictions are im-posed on the latent class preference model with K seg-ments, we estimateMf unique sets of preferences, withthe remaining preferences (K � Mf sets of them) com-puted as linear combinations of the first Mf sets. Therestrictions pose no identification problems except inthe very limited circumstances noted above, yet theyoften save a significant number of parameters. Two-dimensional maps are sufficient in our experience; wehave not encountered any data for which higher-dimensional maps were needed. One-dimensionalmaps are rarely appropriate since they require strin-gent parameter restrictions; in addition, the manage-rial insights provided by one-dimensional maps aremuch less useful. Formal model selection criteria canbe used to supplement this heuristic for determiningthe appropriate number of dimensions, as we discussin a later section.



Representation of Preference Vectors and AttributeLevelsTo graphically depict themap for an attribute, one sim-ply plots the location values in the Af matrix. As withother mapping methods, the interpretation of map di-mensions is purely subjective. With a two-dimensionalmap, however, the horizontal dimension is alignedwith segment 1’s preferences, and the vertical dimen-sion represents segment 2’s preferences.

Given the locations, it is straightforward to plotsegment-specific preference vectors. Theweights in theweight vector give the direction of maximum pref-fwkerences.1 Assuming a two-dimensional map in whichattribute 1 is the horizontal dimension and attribute 2the vertical dimension, the preference vector for a seg-ment begins at the origin of the map and has a slopeequal to the ratio of the second element and the firstelement of .fwk

To determine a segment’s preferences for an attri-bute level, one drops a perpendicular line from the lo-cation of the attribute level in product space to thatsegment’s preference vector. The attribute level far-thest along the segment’s preference vector is the onemost preferred by that segment. One can extend thevector back through the origin if necessary to deter-mine the preferences for all attribute levels. It isstraightforward to show that all attribute levels on aline perpendicular to the vector have equivalent pref-erences (cf. Green et al. 1989). Multiplying the locationsof the attribute levels by the lengths of the preferencevectors produces the conventional part worths (cf.Equation 5).

Extension for Marketing Mix AttributesWe can also impose restrictions on the latent class co-efficients for marketing mix variables, whether theyare continuous or discrete. If we let be the Lc � 1cbkvector of marketing mix response coefficients of seg-ment k, can be decomposed into the location of thecbkattributes in Ac, an Mc-dimensional space, and thesegment-specific importance weights consumers at-tach to those locations, , as in Equation (5):cwk

1This assumes that higher values of the rating scale mean that re-spondents have higher preferences for the profile. If not, then theprofile evaluations need to be rescaled prior to analysis.

c c cb � A w . (7)k k

Unlike the maps for features, it is not necessary thatone of the coefficients (corresponding to one row ofAc)be normalized to the origin. Hence, if the map is two-dimensional, there are two more location parametersrequiring estimation compared to the case of a discreteattribute. The constraints on the weight vectors are thesame as for discrete attributes. The total number of pa-rameters used in Ac (which is McLc) and (which iscwkMc(K � Mc) cannot exceed the number of parametersrequired for the unrestricted latent class preferencemodel (which is KLc). Like the maps for discrete attrib-utes, we recommend two-dimensional maps for mar-keting mix coefficients.

One issue that arises is that the scaling of continuousvariables can affect their mapped locations. For in-stance, we can multiply the prices for all brands by 10(which is legitimate since only the relative prices mat-ter), and the coefficient that results from the estimationwill be 1/10 its previous value. The coordinates of theprice coefficient in space will be 1/10 their previousvalues as well. To make locations of continuous mar-keting mix attributes comparable in space, we recom-mend rescaling them prior to analysis so that they havethe same means and standard deviations (standardiz-ing is one option).

Extension for InteractionsThe proposed attribute-level maps can also provideuseful insights about interactions between attributesthat are not obvious from traditional mapping meth-ods. Consider an example in which there is a five-levelattribute and also a two-level attribute, resulting in 10choice alternatives. For example, the attribute with fivelevels may be brand name, and the attribute with twolevels may be size (say, small and large). An interactionbetween brand and size would suggest that brandpreferences differ depending onwhich sizewe are con-sidering (and vice versa).

We generated simulated ratings data from 200 con-sumers (2,000 observations) for this example, includ-ing significant interaction effects, such that each partworth was distributed normally. Figure 2A shows atraditional MDS solution to the problem, with 2 seg-ments2, in which all choice alternatives are mapped in

2We stopped at two segments for simplicity. It is possible that moresegments could have better explained the data.



Figure 2 Extension for Interactions

the same space. For example, B1S1 is the alternativethat has brand level 1 and size level 1. The locations ofthe alternatives in space reflect the brand name andsize main effects as well as the brand-size interaction.It is not straightforward to isolate the main effects or

the interaction with the traditional MDS approachsince attributes play no role in the analysis.

Our approach can be extended to take into accountinteractions between factors. In this example, a brand-size interaction means that there would be two brandmaps (one for each size). If the preference vectors arerestricted to be the same for the two maps, the brandmaps for the two size levels can be overlaid on thesame graph, as in Figure 2B. For example, B1S1 is thelocation of brand 1 when the size level is 1, and B1S2is the location of brand 1 when the size level is 2. Thisgraph clearly shows the main effect of brand name andthe interaction between brand and size. The larger theinteractions between brand and size, the farther apartthe two points for each brand level (e.g., B1S1 andB1S2) will be in the graph. In this example, the trueinteractions between size and brands 1, 2, 3, and 4 havenormal distributions with means 0.1, 0.2, 0.3, and 0.4,respectively.3 Notice that for each successive pair ofbrands, the two sizes become farther apart in Figure2B, as we would expect. That is, B1S1 and B1S2 aremuch closer together than B4S1 and B4S2 since the trueinteraction for the former pair is 0.1, and the true in-teraction for the latter pair is 0.4. If the interaction be-tween brand 1 and size had been zero, B1 would havebeen located in exactly the same location, except forrandom error, regardless of whether it was coupledwith S1 or S2. In this map, the interaction tells us thatbrand 4 benefits from being coupled with size 1 more thanbrand 1 does. This is not obvious from the MDS map inFigure 2A.

The brand name main effect is recovered in thegraph as well. Brands 1 through 5 have true partworths of 5, 3, 2, �2, and 0, which is roughly consistentwith the scales on both dimensions of the graph.Though the size main effect is not shown in the graph,the model correctly recovers it as well. The true sizemain effect is 3 for size 1 (with size 2 fixed at zero),and the recovered preferences for size 1 for segments1 and 2 were estimated as 2.90 and 3.13.

Though the fit of this model is exactly the same asthat of the traditional approach in Figure 2A, our ap-proach allows us to graphically isolate the nature of

3The brand-size interaction has (2 � 1)(5 � 1) � 4 components sincesize has two levels and brand has five levels.



the interaction without contaminating it with the sizemain effect. In applicationswithmore than two factors,the traditional MDS approach of mapping all profilesin the same space would reflect all the main effects andinteractions, which could be nearly impossible to dis-entangle. With our approach, one could selectively in-clude certain interactions if desired, without having allinteractions in the model. For example, if there arethree factors with three levels each, the MDS plot of all33 � 27 profiles would reflect all main effects, 2- and3-way interactions. With our approach, we could al-low, say, two of the factors to interact while not allow-ing any of the other possible interactions. If we did this,two maps would completely characterize the prefer-ences—the one map for the main effect of the nonin-teracting factor and one map that shows the interactionbetween the other two factors (similar to Figure 2B). Inthe absence of a priori expectations about which inter-actions to examine, a simple ANOVA could be used(at least with ratings data) to determine which inter-actions are statistically significant before specifying themapping model.

In this regard, the proposed attribute-level mapshave a significant advantage over the traditional MDSapproach of mapping all choice alternatives in thesame joint space. We can graphically examine as manyor as few interactions as we wish with our approach,but the locations of choice alternatives obtained froma traditional MDS are obtained without reference toattributes.

Estimation ProcedureOptimization procedures can be used to maximize (4)with respect to the unknown parameters in �, b, andR. Due to significant parameter reductions, the pro-posed restricted models take less time to estimate thanthe unrestricted latent class preference model.

There are two unknowns to be determined in esti-mating the proposed class of models—the dimension-ality of the maps and the number of preference seg-ments. In most applications, two-dimensional mapsare consistent with the data and provide useful man-agerial insights. However, a more formal strategy is asfollows.

1. Estimate unrestricted latent class preference mod-els with various numbers of segments (say, 3, 4, 5, and6) until the optimal number of segments is found.

2. Re-estimate these models with the two-dimensional restriction for all attributes. To determinewhether the two-dimensional restrictions are consis-tent with the data, compare the 3-segment unrestrictedmodel to the 3-segment model with two-dimensionalrestrictions using some criterion that takes into accountthe number of parameters required, such as AIC, BIC,or CAIC. Make similar comparisons of restricted vs.unrestricted models for the 4- and 5-segment models.We have never encountered data for which the two-dimensional restrictions are rejected.

3. If the two-dimensional restrictions are not re-jected, then the models could be estimated again withthe one-dimensional map restrictions. The one-dimensional map restrictions are much more severethan the two-dimensional restrictions and hence aremore likely to be rejected by the data. In any case,the managerial implications available from one-dimensional maps are more limited, and we do notrecommend using them.

4. From the sets of estimated models (unrestricted,2-dimensional, and possibly 1-dimensional), one cansimply pick the one with the lowest value of the infor-mation criterion.

It is possible that the optimal number of dimensionsvaries across attributes. For example, if brand nameand size are the two attributes of interest, it is possiblethat the brand name map should be two-dimensionaland the size map should be one-dimensional (or vice-versa). The dimensionality of one map has nothing todo with the dimensionality of another map—they canbe determined independently. However, if there are Ffeatures, then the researcher would have to estimate 2F

models to completely exhaust the combinations of one-and two-dimensional maps for that preference analy-sis. This is not practical for large data sets because ofexcess estimation time. It is our philosophy that over-specifying the dimensionality of the maps is less of aproblem than underspecifying the dimensionality. Inour empirical example, we specify two-dimensionalmaps for all attributes, thoughwe did estimate amodelwith all one-dimensional maps as well. Overspecifi-cation merely costs extra parameters, but fit and man-agerial implications may be sacrificed by underspeci-fying the number of dimensions. In any case, even if



models with all two-dimensional maps are overspeci-fied, they are still much more parsimonious and muchless overspecified than the commonly-used unre-stricted latent class preference model.

The problem of determining the number of dimen-sions and segments in latent class multidimensionalscaling models is still in need of research. We acknowl-edge that the violation of regularity conditions thatmakes likelihood ratio tests invalid for this type ofmodel technically affects such statistics as BIC as well(Titterington et al. 1985). As a result, some researchershave resorted to simulation methods such as boot-strapping for determining the number of dimensionsand number of segments (e.g., Bockenholt andBockenholt 1991, DeSoete and Heiser 1993, DeSoeteand Winsberg 1993). However, this is a computation-ally burdensome procedure when there are many pa-rameters and many observations. Thus, this importantproblem is not yet solved and is in need of furtherresearch.

Application to Choice Data: FabricSoftener SKUsFader and Hardie (1996) show how a set of discreteattributes can be used to describe every SKU on thestore shelves in a parsimonious manner. In their study,56 SKUs of fabric softeners sold by stores in the Phila-delphia market are described by brand name (10 lev-els), size (4 levels), product form (4 levels), and for-mulation (4 levels). Their data set consists of panelists’purchases of actual products in the marketplace, notevaluations of artificial stimuli used in a typical con-joint experiment. The natural and less obtrusive natureof routine grocery shopping makes the analysis ofscanner panel data an appealing complement to thestandard conjoint experiment (Fader and Hardie 1996).

We apply the proposed preference model to a sam-ple of the IRI fabric softener scanner panel used byFader and Hardie (the reader is referred to that studyfor details of the data set, including a detailed descrip-tion of the attributes of each SKU). Four hundred ran-domly chosen panelists comprise our sample. Twohundred of these (2,364 purchases) were randomlychosen for model estimation, with the remaining twohundred (2,527 purchases) used for model validation.

There are 594 panelists in the entire data set, over two-thirds of whom are used in our study.

Besides the dummy variables for the various attrib-utes, we also include regular price, price cut, aisle dis-play, and store feature variables in the utility functionsof the models since marketing mix variables are oftenimportant explanatory variables in choice models. Wemap the coefficients for regular price, price cut, aisledisplay, and store feature advertising in a commonspace, as described in Equation (7). Obviously, the scal-ing of continuous variables such as price can affect thesize of the coefficient and hence the coordinates ofprice in the map. In our data, regular price has an av-erage value of one and very low variance so that itsmaximum impact on utilities is comparable to that ofthe discrete variables, and so scaling is not an issue.

Latent structure probabilistic choice models are typ-ically parameterized to be consistent with the propo-sition that consumers follow a zero-order brandswitching pattern (Chintagunta 1994, Dillon et al. 1994,Kamakura and Russell 1989), so we do not include loy-alty variables in the specifications. In any case, Faderand Hardie (1996) find high correlations between partworths estimated with loyalty variables present andthose estimated without loyalty variables.

Since the SKUs are from a real-world market and nota carefully-designed conjoint experiment, the profilesare correlated, and hence it is not possible to includeinteractions in the model. The 57 SKUs represented inthe data are only a small fraction of the 640 (�10 � 4� 4 � 4) such profiles that would be used in a fullfactorial design.

The likelihood values and BIC statistics for the un-restricted latent class models and the restricted two-dimensional models and are shown in Table 1A. Theunrestricted latent class model has 18 dummy vari-ables for the four major attributes (brand name, size,product form, and formulation), in addition to fourmarketing mix variables (regular price, price cut, aisledisplay, and store feature). Thus, each additional seg-ment adds 22 parameters plus a weight parameter. Therestricted models have two-dimensional maps for allattributes and marketing mix effects. Forecasting sta-tistics for these and several benchmark models areshown in Table 1B.

As Table 1A shows, the two-dimensional restrictions



Table 1 Results for Fabric Softener Data

A. Fit Statistics for Unrestricted Latent Class Model and Proposed 2-Dimensional Model for Fabric Softener DataUnrestricted Latent Class Restricted Latent Class 2-Dimensional Restrictions

Model Log L P a BICb Log L P BIC

2 segment �8040 45 16429 — — —3 segment �7771 68 16070 �7779 56 159444 segment �7556 91 15819 �7564 67 156485 segment �7289 114 15464 �7327 78 152606 segment �7160 137 15384 �7184 89 150597 segment �7001 160 15244 �7089 100 149558 segment �6903 183 15228 �7015 111 148929 segment �6850 206 15300 �6970 122 14887

10 segment — — — �6895 133 1482211 segment — — — �6885 144 14889

B. Comparison to Benchmark ModelsEstimation Sample 200consumers, n � 2,364

Validation Sample 200 consumers,n � 2,527

Model P a Log L BICb Log L

I. Traditional logit with SKU-specific constants 60 �8237 16940 �8961II. Latent class extension of model I, with 2-dimensional

maps of SKU-specific constants, 7 segments (cf. Chintagunta 1994)156 �6959 15130 �7502

III. Model II with reparameterized stimulus coordinates (cf. DeSoeteand Winsberg 1993)

80 �7472 15566 �7683

IV. Unrestricted latent class with part worths instead of SKU-specific constants,8 segments

183 �6903 15228 �7413

V. Restricted 2-dimensional version of model IV, 10 segments 133 �6895 14822 �7304

aNumber of parameters.bBIC � �2 Log L � P ln(n), where n is the sample size.

are consistent with the data regardless of the numberof segments. For example, for the 3-segment models,the restricted model has 12 fewer parameters but onlygives up 8 likelihood points to the unrestricted latentclass model. BIC favors the restricted model. For the 8-segment models, the restricted model has 72 fewer pa-rameters but gives up about 112 likelihood points. BICagain favors the restricted model.

For the unrestricted latent class model, 8 segmentsare optimal according to BIC, while for the restricted2-dimensional models, 10 segments are optimal.

Table 1B compares the forecasting performance ofthe unrestricted latent class model (Model IV), themodel with 2-dimensional restrictions (Model V), and

several other benchmark models. The first model pre-sented in the table is the traditional logit model withSKU-specific constants. With 57 SKUs purchased bypanelists4, requiring 56 constants, the total parametercount is 60 with the four marketing mix variables in-cluded in the utility function.

Model II is a latent class version of the basic logitmodel I, but with restrictions on the dimensionality ofthe space occupied by the 57 SKUs. This model pro-duces a single 2-dimensional map containing all 57SKUs, similar in spirit to that by Chintagunta (1994).

4Fader and Hardie (1996) include only 56 SKUs in their estimationsample and forecast the market share of the 57th SKU.



Figure 3 7-Segment MDS Map with Simultaneous Reparameteriza-tion (DeSoete and Winsberg 1993)

The optimal 7-segment model requires 156 parametersand performs much better than the basic logit model Iaccording to all criteria.

We then re-estimate the 7-segment model II with thecoordinates of choice alternatives simultaneously re-parameterized as attribute coordinates, as described byDeSoete and Winsberg (1993). Since the map is two-dimensional, the matrix containing locations of choicealternatives in model II is 57 � 2. In model III, thedesign matrix that describes the 57 choice alternativeson the four attributes is 57 � 18 since 18 dummy vari-ables are required to code four attributes. The matrixof attribute level locations in the two-dimensional spaceis 18 � 2. Thirty-six (�18 � 2) location parametersare required for the reparameterized model III, com-pared to 112 (�56 � 2) for the unrestricted model II5,resulting in a savings of 112 � 36 � 76 location pa-rameters over model II. The simultaneous reparame-terization is definitively rejected by the data. The BICvalue for the reparameterized model is 15,566 com-pared to 15,130 for the unrestricted 7-segment modelII. Also, note that the validation likelihood (�7,683) isnot as good as that of the unrestricted 7-segmentmodelII (�7,502), which also suggests that the simultaneousreparameterization restrictions are inappropriate.

We show the map from the simultaneous repara-meterization in Figure 3 to further demonstrate the dif-ferences between our approach and the existing ap-proach. Unlike our proposed model, which mapsattributes in separate spaces, the existing approachproduces a joint space that is quite crowded anddifficult to interpret. The locations of the 18 dummyvariables used to code the attributes and the sevenpreference vectors are shown in the map. Muchinformation (sizes, formulations, etc.) is eliminatedfrom the map by necessity.

Model IV estimates part worths instead of SKU-specific constants (cf. Fader and Hardie 1996). The 8-segment model has 8 � 22 � 176 parameters plus an-other 7 parameters for segment weights, for a total of183. The forecasting performance is significantly betterthan that of models I–III.

The proposed model with 2-dimensional restrictions

5Recall that one row of the location matrix must be fixed at the originin model II; this is not necessary in model III.

(model V) imposes restrictions on the latent class pref-erence model (model IV) such that the preferences arerestricted to two-dimensional space. As mentionedearlier, the 10-segment restricted model achieves asgood a fit as the unrestricted 8-segment model, butwith 50 fewer parameters. Forecasting performance onthe validation sample is also impressive—the likeli-hood value of the restricted model is 109 likelihoodpoints better than that of the unrestricted latent classpreference model.6

We also estimated the proposed model with one-dimensional maps for all attributes, but the restrictionswere not consistent with the data. It is possible thatsome, but not all, of the maps could be reduced to onedimension. Obviously, we would need to estimate allpossible combinations of one- and two-dimensionalmaps for the four attributes (24 � 16 models) to fullyexplore whether the overall model could be improvedby allowing some maps to be one-dimensional and

6For all latent class models II, III, IV, and V, forecasting was accom-plished by first assigning consumers to their respective segmentsusing posterior probabilities (cf. Kamakura and Russell 1989).



some to be two-dimensional. Given that this is notpractical with large data sets such as this one, we arecomfortable risking overspecification of some maps.As mentioned earlier, overspecification does cost extraparameters, but underspecification may cause loss ofmodel fit as well as managerial insights. In any case,the proposed model cannot be nearly as overspecifiedas the unrestricted latent class preference model.

The Attribute Level MapsFigure 4 shows the attribute level maps for brandnames (A), sizes (B), product forms (C), formulations(D), and the marketing mix (E) produced by the opti-mal 10-segment model. The lengths of the preferencevectors are scaled such that they are proportional tothe sizes of the latent segments. For example, a vectorthat is twice as long as another means that the segmenthas twice as many consumers as the other segment.Attribute levels that are very close together in the space(e.g., Arm & Hammer, Bounce, and Downy in Figure4A) have more similar preferences than levels not closetogether (e.g., Generic and StaPuf). Brands locatedvery close to one another may suffer from lack of dif-ferential advantage.

Dropping perpendicular lines from each brandname location in Figure 4A to the vector of the largestsegment (segment 1), we see that segment 1’s prefer-ence ordering is Snuggle, Cling Free, Final Touch,7

Downy, Arm & Hammer, Bounce, Toss n’ Soft, StaPuf,Private Label, and the Generic brand. Segment 1 com-prises 23 percent of the sample. Segment 2 (13 percentof the sample) prefers StaPuf, Private Label, Toss n’Soft, Final Touch, Cling Free, Generic, Snuggle,Bounce, Arm & Hammer, and Downy. Despite thevery different preferences of segments 1 and 2, notethat Arm & Hammer, Bounce, and Downy (all brandnames in very close proximity on the map) are closeto each other in each segment’s preference ranking.

Segments 3 and 4 have somewhat similar prefer-ences. Segment 3 prefers Generic, StaPuf, and PrivateLabel, while segment 4 prefers StaPuf, Generic, andPrivate Label. These segments comprise 12% and 11%of the sample, respectively. Segments 5, 6, 7, 8, and 9,comprising 10%, 8%, 8%, 7%, and 4% of the sample,

7Final Touch is mostly obscured by the arrow of the preference vec-tor for segment 6.

respectively, prefer Final Touch, StaPuf, Final Touch,Downy, and Downy, respectively. The smallest seg-ment, segment 10, strongly prefers Generic to PrivateLabel and StaPuf. This segment, comprising over 3%of the sample, is most likely too small to be actionable.

The size map in Figure 4B shows that segment 1 pre-fers medium, small, large, and extra large.8 Segment 2prefers medium and is virtually indifferent betweensmall, large, and extra large. Similarly, segments 3, 6,7, and 9 prefer medium as well. Segments 4, 5, and 8prefer small. Segment 10, the smallest segment, ap-pears to be economy-minded, with strong preferencesfor the generic brand (Figure 4A) and the extra largesize (Figure 4B). Overall, there is less heterogeneity insize preferences than in brand name preferences sincemore of the preference vectors point in the samedirection.

As shown in the map for product forms in Figure4C, segment 1’s preference ordering is sheets, concen-trate, liquid, and refill, while segment 2 is almost in-different between sheets and refill, with liquid andconcentrate being less preferred. Segment 3 preferssheets, while segment 4 is nearly indifferent betweenconcentrate, liquid, and refill. Consistent with earlierresults, the economy segment (segment 10) prefers re-fill product forms.

The map for formulations in Figure 4D shows thatsegment 1 prefers regular and is then nearly indifferentbetween light and staingard. Segments 2, 3, 4, and 10have fairly similar preferences for formulations, witha slight preference for light over regular. Clearly, thereis less heterogeneity in preferences for formulationsthan for brand names and product forms.

Finally, the map for marketing mix variables in Fig-ure 4E shows that segments 1, 5, 8, 9, and 10 are notvery sensitive to marketing activity. Segments 2, 4, 6,and 7 are most sensitive to marketing activity, withsegments 3 and 5 having intermediate sensitivity. Dis-play and store feature ad activity add to utility for allsegments. For example, whenwe drop a perpendicularline from display to any of the preference vectors, wefind that the line intersects the preference vector be-yond the origin of the vector in the direction of the

8The segments are the same in each map—that is, the consumers insegment 1 in map A are the same as the consumers in segment 1 inmaps B, C, D, and E.



Figure 4 Fabric Softener Data Results



Table 2 Attribute Importances by Segment

Brand Size Form Formula R. Price P. Cut Display Store Ad

Segment 1 0.11 0.10 0.41 0.08 0.00 0.10 0.11 0.08Segment 2 0.26 0.05 0.05 0.30 0.05 0.01 0.17 0.11Segment 3 0.32 0.15 0.26 0.10 0.03 0.03 0.07 0.04Segment 4 0.11 0.14 0.15 0.07 0.09 0.06 0.23 0.15Segment 5 0.48 0.16 0.04 0.05 0.03 0.03 0.13 0.09Segment 6 0.38 0.04 0.12 0.19 0.04 0.04 0.12 0.07Segment 7 0.27 0.10 0.16 0.12 0.06 0.06 0.14 0.09Segment 8 0.23 0.18 0.22 0.09 0.02 0.06 0.12 0.08Segment 9 0.75 0.04 0.03 0.11 0.00 0.02 0.03 0.02Segment 10 0.46 0.09 0.23 0.10 0.00 0.04 0.05 0.04

Note: Since we are computing the maximum effects for each attribute, the largest changes found in the data set for regular price and price cut were used,which are 63% for regular price and 100% for price cut.

arrow. Regular price subtracts from utility for allsegments. Price cut adds to utility for some segments(e.g., segment 3) but subtracts from utility for others(e.g., segment 1). As we will later see, the price cuteffect is actually negligible for all segments.

Information can be integrated across maps to deter-mine which attribute influences each segment themost. Table 2 was computed by first finding the max-imum change in utility possible through each variable.For brand name, size, product form, and formulation,this was the difference between the largest and small-est part worth. For aisle display and store feature ad(both 0/1 variables), the maximum difference in utilityproduced by the variables is simply the value of theprojections onto the segment vectors.9 To compute themaximum change in utility possible through regularprice changes and price cuts, we use the maximumval-ues found in the data set, which are 63% and 100%,respectively. For each segment, we then normalize theutility changes across attributes within segment so thatthe importance weights sum to one. The largest seg-ment (1) is most concerned with product form, whilesegment 2 is most concerned with formulations andthen brand names. With the exception of segment 4,which is most sensitive to aisle display, the remaining

9Equation (5) calculates the projection onto a preference vector asthe sum (across dimensions) of the product of locations and impor-tance weights.

segments pay most attention to brand names. It is sur-prising that three of the four largest segments weightsome attribute other than brand name as most impor-tant. Given the amount of attention devoted in the lit-erature to the marketing environment relative to prod-uct attributes, it is also surprising that the productattributes usually have much more affect on utilitiesthan the marketing environment does.

Combining (i) the preferences for each segment, (ii)the relative attribute importances for each segment,and (iii) the relative sizes of each segment, interestingmanagerial insights can be obtained. For example, seg-ment 1, which comprises 23% of the sample, is mostinfluenced by the product form attribute (Table 2) andstrongly prefers sheets (Figure 4C). Segment 2, com-prising 13% of the sample, is most influenced by for-mulation and slightly prefers light over regular. Seg-ment 4, comprising 11% of the sample, is mostinfluenced by aisle display. Similar analyses can beperformed for the other segments as well.

As a basis for comparison, we have plotted the partworths from the unrestricted latent class solution inFigure 5. Eight latent segments were optimal for thatmodel according to BIC. It is our opinion that theattribute-level maps in Figure 4 convey informationmore readily than the plots of part worths in Figure 5.For example, it would take a considerable amount ofinspection to determine that two brands had similarpreferences across all segments. Our model conveys



Figure 5 Part Worths from Unrestricted Latent Class Solution



this information immediately by showing two suchbrands as being very close in the derived space.

Managerial Implications: The Effectsof Marketing Mix Variables onMapped Attribute LocationsIn this section we investigate some unique managerialimplications of the maps that cannot be gleaned read-ily from plots of part worths. Specifically, we suggestan extension of the model that allows an examinationof the effects of marketing activity such as price andpromotions on the mapped locations of attributes.

For the proposed maps appearing in Figure 4, thevector of 57 SKU utilities for segment k on purchaseoccasion t was computed as

t 1 1 2 2U � exp(Brand • A • w � Size • A • wk k k

3 3 4� Form • A • w � Formulation • Ak

4 5 5• w � X • A • w ),k t k

where the Af matrices contain coordinates of attributelevels in the spaces, the are the preference vectorsfwk(cf. Equation (5)), the Brand, Size, Form, and Formulationmatrices are design matrices which describe each of the57 SKUs on each of the attributes10, andXt is thematrixwhich describes each SKU on the price, price cut, aisledisplay, and feature variables at purchase occasion t.For example, Figure 4A maps the locations in A1 andthe preference vectors .1wk

We can respecify the above model so that the pref-erence vectors are common to selected attributes.fwkThe effect of this respecification is that the attributeshaving a common set of preference vectors share thesame reduced-dimension space. This allows us tostudy the impact of marketing mix variables on thelocations of attribute levels in the derived spaces. If weconstrain the above model so that the brand name andmarketing mix attributes share a common space, theutilities becomet 1 5 1.5 2U � exp[(Brand • A � X • A ) • w � Size • Ak t k

2 3 3 4 4• w � Form • A • w � Formulation • A • w ],k k k

10These matrices have dimensions 57 � 10, 57 � 4, 57 � 4, and 57� 4, respectively.

where is the preference vector for segment k1.5wkshared by the brand name and marketing mix attrib-utes. This constrained model fits very well (Logl �

�6,998, BIC � 14,905) compared to the unrestrictedlatent class models, though not quite as well as themodel with separate spaces for brand name and mar-keting mix variables (Model V in Table 1B).

Figure 6A plots the coordinates in A1 and A5 and thepreference vectors for k � 1, . . . , 10. First, note1.5wkthat the results from this model are quite consistentwith those of the model presented in Figure 4. Com-paring Figure 4A to Figure 6A, we see that segment 1(the largest segment) still prefers Snuggle, Cling Free,and Final Touch (though in slightly different order),and segment 2 still prefers StaPuf. Segment 2 is verysensitive to aisle display and store feature ads (consis-tent with Figure 4E). The segment is quite price sen-sitive as well, with the projection of price onto the seg-ment 2 vector having a value of �1.11. The Genericbrand name has a large negative effect for segment 2as well, with a projection on the segment 2 vector of�0.88. Such information can be used to determine howmuch consumers would have to be compensated withprice for switching from one brand name to another.For example, if a segment 2 consumer were to switchfrom Arm & Hammer (which has a projection value ofzero) to Generic, their utility would decrease by 0.88.However, if the regular price of the generic were lowerby 0.79 (�1.11 � �0.79 � 0.88), consumers would beindifferent between Arm & Hammer and Generic. Thismagnitude of price variation is not present in the data,so apparently it is not feasible to compensate segment2 consumers enough for them to use generics. Giventheir deal proneness, consumers in this segment ap-parently look for named brands that are on sale ratherthan resort to generics.

Using this model, we can graphically describe whathappens to the locations of attribute levels when thereare price reductions and promotional activity. Figure6B shows the brand locations at the average price level,assuming no aisle display or store feature advertise-ments. The locations plotted in the graph are

1 5¯A � XA ,

and the preference vectors are the same as in Figure6A. Also shown in the graph is the movement of the



Figure 6 Effects of Marketing Mix on Brand Name Locations

Bounce brand name when there is a 23.75% price cut(the average size of the price cut when there was aprice cut), a store feature advertisement, and an aisledisplay for that brand. As in Figure 6A, we see that

segment 2 is very sensitive to promotions. For this seg-ment, when Bounce was heavily promoted, the posi-tioning moved from among the least preferred brandnames to a location far preferred over the other brands.The promotion makes Bounce most preferred for seg-ment 1 consumers as well, but not for segment 3 con-sumers, who still prefer StaPuf, Generic, and PrivateLabel. In fact, the promotions improve the positioningof Bounce for every segment.

One could compute a map such as that in Figure 6Bfor each purchase occasion, depending on the pricesand promotional activity in the marketing environ-ment. Thus, this example shows that it is possible tomake the proposed maps dynamic to reflect the pur-chase environment. Similar maps could be developedfor the size, product form, and formulation attributesas well. For example, to study the effects of promotionson the positioning of sizes, we could plot

2 5¯A � XA

in a joint space with preference vectors constrained tobe the same for the size and marketing mix attributes.

ConclusionA fundamental premise of this paper is that consumersare much more likely to form preferences for attributesof products rather than for each individual product onstore shelves. Given limited cognitive resources, it ishighly unlikely that consumers form direct preferencesfor each choice alternative on the store shelves. Map-ping preferences for attributes is therefore much moreconsistent with consumer behavior than the traditionalMDS approach of mapping all choice alternatives. Inaddition to the behavioral basis formapping attributes,the model is much more parsimonious than an unre-stricted latent class preference model, it forecasts hold-out choices better, and it provides unique managerialinsights not readily obtainable from traditional MDSapproaches or plots of part worths.

The parsimony of the model is very important withlarge data sets such as the fabric softener data used inthis study. Compared to the best unrestricted latentclass preference model, our restricted model requires50 fewer parameters yet provides a comparable fit.Equally impressive is the fact that the model forecasts



holdout choices better by 109 likelihood points. In-deed, our model forecasted choices better than MDSapproaches that reparameterize choice alternatives asattributes as well.

In addition to parsimony and improved forecastingperformance, the proposed model provides someunique managerial insights not readily available fromMDS or conjoint approaches. First, the interpretationof the maps is very intuitive. With a glance, we candetermine which attribute levels have similar prefer-ences across segments. Attribute levels very close toeach other in the space have very similar preferences,and are likely to be prone to more substitution andswitching. A focal brand in close proximity to otherbrands needs some sort of differential advantage overthose brands.

Second, the model allows us to graphically examineinteractions between attributes. In an example withsimulated data, we show how an interaction between,say, brand names and sizes can be shown as separatebrand maps for each level of the size factor. We canthen see how the preferences for a brand name changewhen the brand name is coupled with the various sizelevels. In contrast, a map produced by a traditionalMDS approach reflects not only the brand and sizemain effects but also the brand-size interaction. Thereis no satisfactory approach for examining interactionswithin the traditional MDS approach—simultaneousreparameterization has limitations, as shown in thisstudy, and with two-stage property fitting procedures,there is no guarantee that all parameters are optimized.

Third, the effects of changes in the price and pro-motional environment on positioning and preferencescan be studied. In our empirical application, we res-pecified the maps so that marketing mix effects couldbe mapped in the same space as the attributes. Thismakes the maps dynamic since the locations couldchange for each purchase occasion depending on theprices and promotional activities being used on thatoccasion. For each segment, we can also study how anycombination of price changes, aisle display, or storefeature advertising affects the positioning of the focalbrand (or other attribute level). Since nearly all scannerpanel data contains some information on SKU attrib-utes such as brand name, size, formulation, flavor, etc.,

this approach could be used widely in applied choicemodeling.

The study by Fader and Hardie (1996) calls for acomprehensive research program on the topic of SKUchoice. We answer this call with an integrated MDS/preference analysis model that provides informationon the positioning of attribute levels and the segment-specific preferences for those positions. The resultsfrom the fabric softener category show that productattributes usually have more impact on choices thando marketing environment variables, which suggeststhe need for more research into the role of attributesin choice. Future research could investigate the possi-bility of developing ideal point models for attributes.Instead of the “more is better” interpretation associ-ated with vector models, ideal point models posit thatthere is an ideal level of each dimension of the map.Another possible next step for the model is to param-eterize the mixing weights directly as a function ofactionable segmentation variables (cf. Gupta andChintagunta 1994, Kamakura et al. 1994). Another pos-sibility is to incorporate profitability information intothe model (e.g., Green and Krieger 1991), leading to anintegrated model of optimal product design.

More research is also needed on the technical issuesinvolved in estimating such models. There is need fora thorough Monte Carlo analysis that examines algo-rithm performance, parameter recovery, model selec-tion, CPU time, etc., as a number of model, data, anderror factors are experimentally manipulated.11

ReferencesBockenholt, Ulf, Ingo Bockenholt. 1991. Constrained latent class

analysis: simultaneous classification and scaling of discretechoice data. Psychometrika 56 (December) 699–716.

Carroll, J. Douglas, Paul E. Green, Jinho Kim. 1989. Preference map-ping of conjoint-based profiles: an INDSCAL approach: J. Acad.Marketing Sci. 17 (Fall), 273–281.

Chintagunta, Pradeep K. 1994. Heterogeneous logit model implica-tions for brand positioning. J. Marketing Res. 31 (May), 304–311.

DeSarbo, Wayne S., Daniel J. Howard, Kamel Jedidi. 1991. MULTI-CLUS: a new method for simultaneously performing multidi-mensional scaling and cluster analysis. Psychometrika 56(March) 121–136.

———, J. Douglas Carroll, Donald R. Lehmann, and John O’

11The authors thank the editor, area editor, and two anonymous re-viewers for their contributions to this study and Pete Fader for pro-viding the data.



Shaughnessy. 1982. “Three-Way Multivariate Conjoint Analy-sis,” Marketing Science, 1 (Fall), 323–350.

———, Ajay K. Manrai, Lalita A. Manrai. 1994. Latent class multi-dimensional scaling: a review of recent developments in themarketing and psychometric literature. Richard P. Bagozzi, ed.Advanced Methods of Marketing Research. Blackwell Publishers,Cambridge, MA, 190–222.

———, Vithala Rao. 1986. A constrained unfolding methodology forproduct positioning. Marketing Sci. 5 (Winter) 1–19.

———, Michel Wedel, Marco Vriens, Venkatram Ramaswamy. 1992.Latent class metric conjoint analysis.Marketing Letters 3 (3) 273–288.

DeSoete, Geert, Willem J. Heiser. 1993. A latent class unfoldingmodel for analyzing single stimulus preference ratings. Psy-chometrika 58 (December) 545–565.

———, Suzanne Winsberg. 1993. A latent class vector model forpreference ratings. J. Classification 10 195–218.

Dillon, William R., Ulf Bockenholt, Melinda Smith De Borrero, HamBozdogan, Wayne DeSarbo, Sunil Gupta, Wagner Kamakura,Ajith Kumar, Venkatram Ramaswamy, Michael Zenor. 1994.Issues in the estimation and application of latent structuremod-els of choice. Marketing Letters 5 (4) 323–334.

Elrod, Terry. 1988. Choice map: inferring a product-market mapfrom panel data. Marketing Sci. 7 (Winter) 21–40.

Fader, Peter S., Bruce G. S. Hardie. 1996. Modeling consumer choiceamong SKUs. J. Marketing Res. 33 (November) 442–452.

Green, Paul E., Frank J. Carmone, Jr., Scott M. Smith. 1989. Multidi-mensional Scaling: Concepts and Applications. Allyn and Bacon,Boston, MA.

———, Kristiaan Helsen. 1989. Cross-validation assessment of alter-natives to individual-level conjoint analysis: a case study. J.Marketing Res. 26 (August) 346–350.

———, Abba M. Krieger. 1991. Segmenting markets with conjointanalysis. J. Marketing 55 (October) 20–31.

Gupta, Sachin, Pradeep K. Chintagunta. 1994. On using demo-graphic variables to determine segment membership in logitmixture models. J. Marketing Res. 31 (February) 128–136.

Hagerty, Michael R. 1985. Improving the predictive power of con-joint analysis: the use of attribute analysis and cluster analysis.J. Marketing Res. 22 (May) 168–184.

Kamakura, Wagner. 1988. A least squares procedure for benefit seg-mentation with conjoint experiments. J. Marketing Res. 25 (May)157–167.

———, Gary J. Russell. 1989. A probabilistic choicemodel formarketsegmentation and elasticity structure. J. Marketing Res. 26 (No-vember) 379–390.

———, Michel Wedel, Jagadish Agrawal. 1994. Concomitant vari-able latent class models for conjoint analysis. Internat. J. Res. inMarketing 11 451–464.

McLachlan, Geoffrey J., Kaye E. Basford. 1988. Mixture models: in-ference and applications to clustering. Marcel Dekker, NewYork.

Ogawa, Kohsuke. 1987. An approach to simultaneous estimationand segmentation in conjoint analysis.Marketing Sci. 6 (Winter)66–81.

Titterington, E. M., A. F. M. Smith, U. E. .Makov. 1985. StatisticalAnalysis of Finite Mixture Distributions. John Wiley & Sons, NewYork.

Urban, Glen L., John R. Hauser. 1993. Design and Marketing of NewProducts. Prentice-Hall, Englewood Cliffs, NJ.

Vriens, Marco, Michel Wedel, Tom Wilms. 1996. Metric conjoint seg-mentation methods: a Monte Carlo Comparison. J. MarketingRes. 33 (February) 73–85.

Wedel, Michel, Wayne S. DeSarbo. 1996. An exponential-familymul-tidimensional scaling mixture methodology. J. Bus. and Econom.Statist. 14 (October) 447–459.

———, Cor Kistemaker. 1989. Consumer benefit segmentation usingclusterwise linear regression. Internat. J. Res. in Marketing 6 45–59.

———, Jan-Benedict E. M. Steenkamp. 1989. A fuzzy clusterwise re-gression approach to benefit segmentation. Internat. J. Res. inMarketing 6 241–258.

This paper was received June 23, 1998, and has been with the authors 5 months for 4 revisions; processed by Wagner Kamakura.

mds maps for product attributesand market response: an...

Documents