A MODEL-BASED APPROACH FOR VISUALIZING ... - Springer Link

PSYCHOMETRIKA — VOL . 73, NO . 1, M ARCH 2008 DOI : 10.1007/ S 11336-007-9015-2

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A MODEL-BASED APPROACH FOR VISUALIZING THE DIMENSIONAL STRUCTURE OF ORDERED SUCCESSIVE CATEGORIES PREFERENCE DATA

WAYNE S. D E S ARBO PENNSYLVANIA STATE UNIVERSITY

J OONWOOK PARK SOUTHERN METHODIST UNIVERSITY

C RYSTAL J. S COTT UNIVERSITY OF MICHIGAN-DEARBORN A cyclical conditional maximum likelihood estimation procedure is developed for the multidimensional unfolding of two- or three-way dominance data (e.g., preference, choice, consideration) measured on ordered successive category rating scales. The technical description of the proposed model and estimation procedure are discussed, as well as the rather unique joint spaces derived. We then conduct a modest Monte Carlo simulation to demonstrate the parameter recovery of the proposed methodology, as well as investigate the performance of various information heuristics for dimension selection. A consumer psychology application is provided where the spatial results of the proposed model are compared to solutions derived from various traditional multidimensional unfolding procedures. This application deals with consumers intending to buy new luxury sport-utility vehicles (SUVs). Finally, directions for future research are discussed. Key words: ordered successive categories, maximum likelihood, consumer psychology, multidimensional unfolding.

1. Introduction In many social science applications employing multidimensional scaling (MDS) for the analysis of two- or three-way dominance data (e.g., preference, choice, profile data), subjects’ judgments are often measured on ratings scales with a relatively small number of response categories collected over different contexts. For example, in a consumer psychology investigation of brand preference in a designated product/service class, a respondent may be asked to evaluate each brand presented in terms of his or her consideration of buying in the near future (intentions) at different times or consumptive situations. Here, descriptive names for such ordered response categories might be: very likely to consider, somewhat likely to consider, somewhat unlikely to consider, and very unlikely to consider. Such ordered “successive categories” (Torgerson, 1952) have traditionally been analyzed either as metric scales or as nonmetric ordered scales depending upon whether the analyst believes that the implicit differences between the categories are equal (metric) or unequal (nonmetric). MDS procedures that can provide joint space representations (row and column entities) for general two-way dominance data abound in terms of either unfolding representations (see Lingoes, 1972, 1973; Roskam, 1973; Young & Torgerson, 1967; The author names are presented alphabetically as all coauthors contributed equally to this manuscript. The authors wish to thank the editor, the associate editor, and three anonymous referees for their excellent constructive comments which resulted in the considerable improvement of this manuscript. Requests for reprints should be sent to Wayne S. DeSarbo, Marketing Department, Smeal College of Business, Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected]

© 2007 The Psychometric Society

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Kruskal, 1964a, 1964b; Kruskal & Carroll, 1969; Schönemann, 1970; Carroll, 1972; Greenacre & Browne, 1986; Heiser, 1981; Takane, Young, & DeLeeuw, 1977; DeSarbo & Rao, 1984, 1986; DeSarbo, Young, & Rangaswamy, 1997; DeSarbo & Carroll, 1985; DeSarbo & Hoffman, 1986; Gifi, 1990; Borg & Groenen, 2005), vector or scalar product representations (see Tucker, 1960; Slater, 1960; Carroll, 1980; Gifi, 1990; Borg & Groenen, 2005), or correspondence analysis or optimal scaling-type approaches (see Benzécri, 1973, 1992; Nishisato, 1980; Greenacre, 1984; Gifi, 1990; Cox & Cox, 2001). Ordered successive category measurement has been the topic of much research in the classical psychometric literature (see Messick, 1956; Torgerson, 1958; Stevens, 1966, 1971; Luce & Edwards, 1958; Galanter & Messick, 1961; Coombs, Dawes, & Tversky, 1970; Cliff, 1973; Adams & Messick, 1958; Maurin, 1983), although the vast majority of such research has been in the area of modeling two-way proximity judgments. Schönemann & Tucker (1967) developed a maximum likelihood estimation (MLE) procedure for the scaling of dissimilarities measured on such ordered successive category scales in obtaining estimates for stimulus scale values, boundary scale values, and stimulus dispersions associated with the traditional successive category scale model. Zinnes and Wolff (1977) generalized this approach to same–different judgments. Takane (1981) was first to explicitly take into account the content of measurement (ordered successive category scales pertaining to similarity) and not only scale the data multidimensionally, but also represent them by a distance model. Takane and Carroll (1981) later generalized this MLE approach to the multidimensional scaling of directional rankings of similarities. The objective of this manuscript is to generalize the Takane (1981) and Takane and Carroll (1981) MDS approaches (that deal with the more traditional MDS analysis of similarities) to the multidimensional unfolding (MDU) of such two- or three-way ordered successive category scales for measuring preference or other forms of dominance data. We provide a rather unique unfolding-like approach where we model the preference/utility structure surrounding each stimulus via an adjusted multivariate normal distribution. A cyclical conditional maximum likelihood procedure is devised for joint space estimation (both subjects and stimuli) in S dimensions (Takane, 1981 and Takane & Carroll, 1981) only represent stimuli). We then present the results of a modest-sized Monte Carlo experiment which examines parameter recovery with the proposed procedure and the quality of the various information heuristics utilized for dimensionality selection. We present a consumer psychology application involving consumers’ consideration likelihood for new luxury SUV brands. We later compare the results to those obtained from a number of existent MDU procedures, and compare predictions for a holdout set of brands not utilized in the model calibration. Finally, directions for future research are provided.

2. The Proposed Model Our objective is to provide a new joint space MDU procedure for the analysis of two- or three-way ordered successive category dominance judgments that jointly displays both subjects and stimuli such that the distribution masses of stimuli preference/utility structures are represented spatially via multivariate normal distributions. In this section we describe the technical aspects of the proposed model. For i = 1, . . . , I subjects who make judgments toward j = 1, . . . , J stimuli in r = 1, . . . , R situations (e.g., time periods, occasions, experimental conditions, etc.), let c = 1, . . . , C denote the ordered successive preference/choice/consideration categories. Assume that there are C − 1 common and ordered cutoff points {tc : tc−1 ≤ tc , c = 1, . . . , C − 1}, where t0 = −∞ and tC = +∞. Let Qij r denote the observed category of stimulus j for subject i in situation r such that Qij r = {1, 2, . . . , C}; s = 1, . . . , S dimensions; xj s = the sth coordinate (centroid) for stimulus j ; yis = the sth coordinate for subject i; αr = a situation specific additive constant; and λj = a multiplicative constant for stimulus j .

WAYNE S. DESARBO, JOONWOOK PARK, AND CRYSTAL J. SCOTT

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Now, we define a latent, unobservable utility measure as Uij r = Mij + αr + eij r ,

(1)

with error eij r ∼ N (0, δr2 ). Without excessive loss of generality, we assume that δr = 1. As per the consumer psychology applications to follow, we describe the model in terms of C response categories, and we interchange subjects with consumers and stimuli with brands. We assume the consideration rating of consumer i for brand j in situation r has latent cutoff points such that −∞ < t1 < t2 < · · · < tC−1 < ∞, and the probability of consumer i’s rating of C consideration categories to brand j can be expressed as follows: Pij r1 = P (Qij r = 1) = P (Uij r < t1 ) = P (αr + Mij + eij r < t1 ) = P (eij r < t1 − αr − Mij ) = Φ(t1 − αr − Mij ),

(2)

and Pij rc = P (Qij r = c) = P (tc−1 ≤ Uij r < tc ) = Φ(tc − αr − Mij ) − Φ(tc−1 − αr − Mij ).

(3)

Finally, 

Pij rC = P (Qij r = C) = P (Uij r > tC−1 ) = 1 − Φ(tC−1 − αr − Mij ),

(4)

since c Pij rc = 1. Here, Φ represents the cumulative standard normal distribution as this is akin to a standard ordered probit formulation thus far (see Greene, 2003). Now, Mij in the latent utility in expression (1) is modeled as   −T −T 1 Mij = λj (2π) 2 | j | 2 exp − (yi − xj )  −1 (y − x ) . (5) i j j 2 We assume that  j is symmetric and positive semidefinite; hence the term (yi − xj )  −1 j (yi −xj ) represents the weighted squared distance from yi to xj . Note that Mij is jointly determined by the coordinates of stimulus j (xj ),  j , λj , and subject i’s ideal point (yi ). As such, assuming λj = 1 for all j , Mij represents the value of that multivariate normal probability function for subject i toward stimulus j . Note that we accommodate the distinct effect of different stimuli on preferences via λj and  −1 j . If we assume  j to be an identity matrix for all j , then the contour of the iso-preference regions is the same for all stimuli. Further, the covariance elements of  j determines the major axis of Mij vis-à-vis its eigenvalues. One can thus argue that the magnitudes of  −1 j (i.e., norm of eigenvalues) and λj determine the overall attractiveness of stimulus j to subjects. Finally, we introduce an additive constant αr to the unobservable utility function as Mij is restricted to be positive. Here, αr represents a situation-specific additive constant to accommodate potential shifts in preferences or utility over situations to reflect the effects of learning, adaptation, preference change, fatigue, etc. Representing preferences via  −1 j only, however, is somewhat restricted since the area or mass around coordinates of stimulus j (xj ) is constant (as a probability distribution, its integral sums to one). That is, one would expect to have a mass of Mij equal to 1 without a multiplicative constant λj . This may result in a mass that either has a high peak and narrow base, or a low peak and wide base. To cope with this restriction, we incorporate a stimulus-specific, positive multiplicative constant λj into the deterministic part of utility Mij so that a mass can have both a high peak and a wide base.

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F IGURE 1. Illustration of the proposed MDU model.

To more intuitively explain the workings of the proposed model, we construct Figure 1 which illustrates a hypothetical representation of Mij in two dimensions. Since the application to follow deals with the analysis of consumer preferences for various brands of luxury sports utility vehicles (SUVs), let us adopt this consumer psychology scenario for better explaining the model. Here, for ease of exposition, let’s assume that four brands (stimuli) exist in this product category. Note that dots represent consumers’ (subjects’) ideal points, and the stimulus mounds represent the distribution of preference for brand j . In Figure 1, Brand D has the highest peak but relatively narrow tails, while some other brands have lower peaks yet wider tails. In other words, Brand D appears to be a very attractive brand to few subjects. However, for the set of consumers who find Brand D appealing, such consumers have very strong preference for Brand D, as indicated by the associated high peak. As an example, Volvo automobiles tend to be very strongly preferred by smaller market segments whose major concern is safety. Compare this to the estimated distribution for Brand B where much wider tails are seen with a lower distribution peak. Here, Brand B has wider appeal to a much larger customer base but with lower preference intensity (e.g., the more common Chevrolet brand may be an example). Brand A tends to be more unique given its distance from the other three brands which tend to be located closer together (for example, Hummer). From a consumer psychology perspective, the structure of such a spatial model can be linked somewhat to the extensive literature on branding and brand equity. Several definitions of brand equity are offered in the consumer literature, as well as several ways to measure the construct. From a cognitive psychology perspective, Keller (1993) defines customer-based brand equity as the differential effect that brand knowledge has on consumer response to the marketing of that brand. He suggests that brand equity arises from two major sources: awareness and associations.

WAYNE S. DESARBO, JOONWOOK PARK, AND CRYSTAL J. SCOTT

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Srinivasan, Park, and Chang (2005) take a financial view, defining brand equity as the incremental contribution (dollars) per year obtained by the brand in comparison to the base product or service without brand-building initiatives. They propose three sources of brand equity: brand awareness, attribute perception biases, and nonattribute preference. More generally, Farquhar (1989) defines brand equity as the added value to the firm or consumer with which a brand endows a product. Other definitions of brand equity share the view that the value of the brand to the firm is its effect on consumers through brand associations (Srinivasan, 1979; Aaker, 1991, 1996; Kamakura & Russell, 1993; Keller, 1993). A common method of measuring brand equity and brand associations involves the overall evaluation of the brand, and like the proposed model in this paper, uses random utility theory. Swait, Erdem, Louvier, and Dubelaar (1993) operationalized brand equity as the monetary equivalent of the total utility consumers attach to a brand. Park and Srinivasan (1994) suggest that brand associations contribute to brand equity effects on utility through attributebased components of equity and nonattribute-based components of equity. They define attributebased equity as the difference between subjectively perceived and objectively measured attributes. The nonattribute component captures brand associations of product imagery and brand personality (e.g., the rugged and masculine image conveyed by the Marlboro Man). In terms of Keller’s (1993) framework, brand equity arises when a consumer is aware of a brand and forms perceptions toward the brand based on these brand associations. When brand associations are strong, favorable, and unique they are said to drive brand preference/consideration (Keller, 2003). The spatial structure of the proposed model relates somewhat to the elements of brand equity through the above-mentioned elements of brand preference/consideration. First, intensity of preference for a brand (see Figure 1, Brand D) is reflected in the height of the distribution mass surrounding the brand of interest where a higher peak corresponds with stronger preference. The width or coverage of the mound is indicative of the brand’s appeal. Second, a favorable brand association is displayed through the examination of the joint space with respect to the derived dimensions, as well as through the use of subjectively perceived or objectively measured attributes that allow us to interpret both favorable and nonfavorable brand associations. This latter aspect may be observed through the post-hoc property fitting of explicit attributes as is common with all MDS procedures. Finally, given our spatial model, the uniqueness of a brand is readily observed given its distance to other brands in the derived joint space. Thus, the resulting spatial representation of the proposed model allows the elements of brand equity to be visually examined. Thus, we wish to construct a joint space map containing each consumer and brand, given the ordered preference rating concerning consideration set inclusion. Note that although Pij rc is modeled at the individual level, the ordered cutoff point tc is estimated at the aggregate level. As can be seen, we chose to use a model-based parametric approach akin to the familiar ordered probit model employing maximum likelihood given the ordered nature of the measurement scale dealt with. With Pij rc expressed in terms of parameters, xj , yi , t1 , t2 , . . . , tC−1 , αr ,  j , and λj , we can write the complete likelihood function as L=

 i

j

r

d

Pijijrcrc ,

(6)

c

where dij rc = 1 if Qij r = c

and

dij rc = 0

if Qij r = c.

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Thus the corresponding log likelihood is  Ln(L) = dij rc ln(Pij rc ) i

j

r

i

j

r

c

     dij r1 ln Φ(t1 − αr − Mij ) + dij r2 ln Φ(t2 − αr − Mij ) =  −Φ(t1 − αr − Mij ) + · · · + dij rC−1 ln Φ(tC−1 − αr − Mij ) − Φ(tC−2 − αr − Mij ) 

(7) +dij rC ln 1 − Φ(tC − αr − Mij ) . The estimation of the parameters entails maximizing the log likelihood function (or, equivalently, minimizing the negative of the log likelihood function). A number of issues arise with respect to equations (1–7) above. First, there are a rather large number of parameters to estimate given the need for a joint space map of individual level ideal points, brand positions, multiplicative constants, and brand specific covariance matrices. Here, we use an iterative conjugate gradient-based cyclical conditional maximum likelihood estimation procedure employing rational starts (see Jedidi & DeSarbo, 1991). Second, in order to ensure the threshold parameters have the order such that t1 < t2 < · · · < tC−1 , we need to impose a constraint on the threshold parameters t1 , t2 , . . . , tC−1 . Thus, in the case of four response categories, for example, we reparametrize t2 and t3 as t2 = t1 + ε12 and t3 = t1 + ε12 + ε22 , respectively. As is common in ordinal probit analysis, we restrict t1 to be zero since additive constants (αr ) are estimated. Third, we impose symmetry and positive semidefiniteness constraints on  j as stated in earlier assumptions. Here, we reparametrize  j as Vj Vj where Vj is a lower triangular matrix to ensure  j is symmetric and positive semidefinite. Finally, we reparametrize λj as λ˜ 2 to ensure its nonnegativity j

(λ˜ 2j ≥ 0). Note, however, that if λ˜ 2j = 0, the whole mass for brand j would become degenerate (i.e., a point). The total number of unique model parameters to be estimated is N P = (I · S) + (J · S) + (S(S + 1)/2 · J ) + J + R + C − 2. Since we impose symmetry and positive semidefiniteness in  j , we only need to estimate the diagonal and off-diagonal elements of  j ((S(S + 1)/2) · J ). Note, NP needs to be adjusted for the scale, rotation, and origin indeterminacies commonly associated with such a model to obtain the number of free parameters. As such, the total number of free parameters reflecting these adjustments becomes     S(S + 1) S(S + 1) NP = (I · S) + (J · S) + ·J +J +R+C −2− S + . (8) 2 2 As noted previously, the estimation of the parameters xj , yi , t1 , t2 , . . . , tc−1 ,  j , α, and λj entails maximizing the log likelihood function (or minimizing the negative log likelihood function) presented in (7). Among the many methods available for such a nonlinear optimization problem, we use an iterative conjugate gradient-based cyclical conditional maximum likelihood with rational starts. The conjugate gradient algorithm has been shown to be a good choice, especially when there are large numbers of parameters to estimate, as it only requires the estimation of its objective function and gradients (see Polak & Ribiére, 1969; Powell, 1977; Fletcher, 1987). Furthermore, unlike the Newton–Raphson or the quasi-Newton method, this conjugate gradient method does not need to update the Hessian matrix but only needs to update the gradient, which is often preferred when estimating a large number of parameters (Nocedal & Wright, 1999). However, given the large number of parameters involved and the highly nonlinear nature of the objective function, globally optimum solutions are not guaranteed; thus, the procedure should therefore be rerun a number of times with different random starts (Gilbert & Nocedal, 1992).

WAYNE S. DESARBO, JOONWOOK PARK, AND CRYSTAL J. SCOTT

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During the iterative procedure for the conjugate gradient algorithm, a suitable step size is needed along a direction of descent to minimize the negative of the log likelihood function. We use cubic interpolation to calculate the step size since it meets the Strong–Wolfe condition, which in turn, guarantees the direction to be a descent direction (Nocedal & Wright, 1999). Next, we briefly describe each stage of the estimation cycle. 2.1. Rational Starting Values Either the user or program can provide the starting values or generate them from the proposed procedure (see Jedidi & DeSarbo, 1991). In addition to generating random starting values, there is also an option to calculate rational starting values which use the structure in the input data to hopefully generate “better” starting values. Below, we describe the procedure to generate rational starting values directly from the program. For starting values for xj and yi , we first average Qij r over situations, compute a column standardized Q∗ij , and calculate the correlation matrix M = Q∗ Q∗ . Second, we take an eigenvalue decomposition of M in S dimensions (i.e., ˆ as the column standardized U UVU = M). Thus, U is a J × J orthogonal matrix. We define X and use it as a starting configuration. Then, we modify a PREFMAP2 regressionlike approach to generate the starting values for y as described below (Carroll, 1980): (a) let Wi be a transposed Q∗i for consumer i   (b) we set up the linear regression Wij = τ0 + Ss=1 τis xˆj s + βi Ss=1 xˆj2s (c) we calculate the starting value of yis as yˆis = −0.5(τis /βi ) We then compute equally distributed percentiles of Qij for the starting values of tc , respectively. Also, we use an identity matrix for the initial starting value for each  −1 j . 2.2. Estimate xj , yi , t1 , t2 , . . . , tC−1 ,  j , α, and λj This stage of the algorithm consists of several conditional estimation phases. That is, in each phase we estimate a set of parameters while setting the remaining parameters fixed at their current values. For example, we estimate xj while setting the remaining parameters, yi , t1 = 0, t2 , . . . , tC−1 ,  j , α, and λj fixed at their current values. This iterative conjugate gradient-based cyclical conditional maximum likelihood is cycled across all sets of parameters until convergence is reached. Options exist for either internal or external estimation with respect to any set of parameters, restricted analyses (e.g.,  j = I, α = α), constrained analyses (e.g., λj > 0), etc. In each phase, estimates of parameters are sought to minimize the negative log likelihood function in (7). We utilize a conjugate gradient procedure for estimating each set of parameters in turn (i.e., xj , yi , Vj , α, threshold values, and λj ). 2.3. Model Selection As in Takane (1981) and Takane and Carroll (1981), dimension selection is accomplished by inspection of various information heuristics such as the AIC, BIC, CAIC, and MAIC heuristics (see Akaike, 1974; Bozdogan, 1987), in addition to a measure of the prediction accuracy of the ordered response categories. Using the derived parameter estimates, we calculate predicted conˆ ij r and compare it with observed consideration Qij r . As the observed consideration sideration Q is measured on an ordinal scale, we utilize the following measure: R N |Qij r − Qˆ ij r | . (9) F = 1 − r=1 i=1 RN (C − 1) For example, if the predicted consideration is 1 while the actual consideration is 4 (e.g., C = 4), the F measure would be 0 if we consider only one observation.

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PSYCHOMETRIKA TABLE 1. Independent factors for Monte Carlo analysis.

Factors

Description

I

Number of subjects

II

Number of brands

III

Degree of error

IV

Number of dimensions

V

Number of situations

Levels 100 200 5 10 0.01 × Var(Mij ) 0.05 × Var(Mij ) 2 3 1 3

TABLE 2.

Averages and standard deviations of three measures in the Monte Carlo analysis.∗

Factors Number of subjects Number of brands Degree of error Number of dimensions Number of situations

Levels

Recovery of joint space

Overall fit

Number of iterations

100 200 5 10 0.01 × Var(Mij ) 0.05 × Var(Mij ) 2 3 1 3

0.899 (0.05) 0.894 (0.04) 0.893 (0.05) 0.900 (0.05) 0.896 (0.05) 0.898 (0.05) 0.897 (0.06) 0.896 (0.03) 0.903 (0.05) 0.890 (0.05)

0.971 (0.02) 0.971 (0.01) 0.968 (0.02) 0.974 (0.01) 0.970 (0.01) 0.972 (0.02) 0.969 (0.01) 0.973 (0.02) 0.973 (0.01) 0.969 (0.02)

40.77 (18.16) 71.70 (35.54) 43.83 (27.61) 68.64 (31.69) 57.16 (35.02) 55.31 (29.17) 43.63 (35.80) 68.84 (21.78) 54.22 (34.19) 58.25 (30.03)

∗ Value in parentheses indicates the standard deviation.

3. Monte Carlo Simulations A Monte Carlo simulation was performed to examine the performance of the proposed model with a number of factors hypothesized to impact the performance of the model. These independent factors—number of subjects, number of brands, degree of error, number of dimensions, and number of situations—are presented in Table 1 with the various levels tested. As in the application to be discussed, we set C = 4 to mirror those response scales in each data set. These five factors are used to generate a full factorial (25 ) design. Given four replications per cell of the design, we estimated the proposed model in 128 experimental trials. For each experimental trial, the model parameters xj , yi , and  j were randomly generated from a multivariate normal distribution. λj and αr were generated using uniform distributions; the αr were set to be negative to reflect the preference shift. Note that we reparametrized  j = Vj Vj in which Vj is a lower triangular matrix to ensure the variance–covariance matrix,  j , to be symmetric and positive-semidefinite. Next, we calculated Mij with these generated parameter sets, and add random errors from the standard Normal distribution. Finally, tc was set to be 25, 50, and 75 percentiles of Mij . This was performed for each of the 128 trials. Using these generated parameters now as those to be recovered by the procedure, we then ran the proposed procedure five times per cell with different starting values and selected the best fitting solution per experimental trial.

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WAYNE S. DESARBO, JOONWOOK PARK, AND CRYSTAL J. SCOTT

TABLE 3. Type III sum of squares of various measures for the Monte Carlo analysis.

Corrected model Intercept Subjects Brand Degree of error Dimensions Situations Subjects × Brands Subjects × Degree of error Brands × Degree of error Subjects × Brands × Degree of error Subjects × Dimensions Brands × Dimensions Subjects × Brands × Dimensions Degree of error × Dimensions Subjects × Degree of error × Dimensions Brands × Degree of error × Dimensions Subjects × Brands × Degree of error × Dimensions Subjects × Situations Brands × Situations Subjects × Brands × Situations Degree of error × Situations Subjects × Degree of error × Situations Brands × Degree of error × Situations Subjects × Brands × Degree of error × Situations Dimensions × Situations Subjects × Dimensions × Situations Brands × Dimensions × Situations Subjects × Brands × Dimensions × Situations Degree of error × Dimensions × Situations Subjects × Degree of error × Dimensions × Situations Brands × Degree of error × Dimensions × Situations Subjects × Brands × Degree of error × Dimensions × Situations Error Total Corrected total R squared Adjusted R squared ∗∗ (p < 0.05), ∗ (p < 0.1).

Recovery of joint space

Overall fit

Number of iterations

MAIC

0.049 102.889∗∗ 0.001 0.002 0.000 0.000 0.005 0.001 0.000 0.000 0.000 0.010∗∗ 0.000 0.000 0.000 0.000 0.001 0.003

0.01∗∗ 120.636∗∗ 0.000 0.001∗∗ 0.000 0.001∗ 0.001∗ 0.000 0.000 0.000 0.000 0.000 0.002∗∗ 0.000 0.000 0.000 0.000 0.000

105909.969∗∗ 404775.031∗∗ 30628.125∗∗ 19701.125∗∗ 108.781 20351.531∗∗ 520.031 9765.031∗∗ 153.125 78.125 132.031 351.125 4232∗∗ 3938.281∗∗ 5.281 312.500 861.125∗∗ 0.781

2.680 0.070 0.195∗ 0.195∗ 0.195∗ 0.195∗ 0.195∗ 0.070 0.008 0.070 0.008 0.070 0.383∗∗ 0.008 0.070 0.008 0.008 0.070

0.000 0.000 0.000 0.000 0.001 0.003

0.001∗ 0.000 0.001∗∗ 0.000 0.000 0.000

1740.500∗∗ 780.125∗ 850.781∗ 399.031 98.000 595.125

0.008 0.070 0.008 0.195∗ 0.195∗ 0.070

0.001

0.000

3.781

0.070

0.003 0.006∗∗ 0.006 0.000 0.000 0.001

0.000 0.001 0.000 0.000 0.000 0.000

399.031 2112.500∗ 2312.000∗∗ 1023.781∗ 157.531 153.125

0.070 0.008 0.008 0.070 0.070 0.070

0.000

0.000

2701.125∗∗

0.008

0.004

0.000

1444.531∗∗

0.008

0.220 103.150 0.260 0.185 –0.078

0.020 120.660 0.030 0.360 0.150

25061.000 535746.000 130970.969 0.810 0.750

6.250 9.000 8.930 0.300 0.074

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PSYCHOMETRIKA TABLE 4. Comparison of original and estimated value for the Monte Carlo analysis.

Brand (x) Ideal point (y) Variance (Σ) Threshold (t) Additive constant (α) Multiplicative constant (λ)

Original value

Estimated value

RMSE

–0.01 (0.83) 0.00 (1.07) 0.34 (0.43) 1.30 (0.12) –0.60 (0.17) 9.90 (2.98)

–0.02 (0.67) 0.00 (0.87) 0.31 (0.38) 1.73 (0.13) –0.98 (0.31) 10.26 (3.92)

0.24 0.24 0.11 0.14 0.24 1.01

∗ Value in parentheses indicates the standard deviation.

We then compared actual vs. recovered parameters for the stimuli coordinates, ideal points, and joint space.1 Table 2 presents the descriptive statistics for each of these three dependent measures. As shown in this table, the average recovery measures for the various parameters and data are consistent and above 0.90 with relatively small standard deviations. We also performed ANOVAs for each of these various dependent measures which are shown in Table 3. Outside of a few significant interaction terms, the recovery of the joint space appears somewhat robust across the cells of the design. As one would expect, overall fit was better for trials involving more data and smaller parameter sets (the brand, situation, and dimension factors were significant for the F recovery variable). In addition, additional computation time was required for larger data sets and larger numbers of parameters. We also measured the recovery of parameters of interest, i.e., brand (xj ), ideal point (yi ), threshold (tc ), additive constant αr , multiplicative constant λj , and variance–covariance ( j ) using RMSE (Root Mean Squared Error). The average RMSE, and means and standard deviations of original and estimated parameters, are reported in Table 4. Here we see consistent parameter recovery across the various parameters in the model. Note that the larger RMSE for the multiplicative constants was due to the larger scale and variances of these generated parameters compared to all the other parameters. Another analysis was conducted regarding these same 128 trials to examine the performance of various dimension selection heuristics in selecting the appropriate number of dimensions with these known synthetic data structures. This analysis uses the same data generated in the previous Monte Carlo analysis. Here, we estimate the model for each experimental trial in one to four dimensions and compare the four dimension selection heuristics AIC, MAIC, BIC, and CAIC in accurately denoting the appropriate dimensionality. For coding, we utilized a (−1) when the heuristic inappropriately selected the solution less than S dimensions, (+1) when the heuristic inappropriately selected the solution with more than S dimensions, and (0) when the heuristic correctly selected the solution with S dimensions. Table 5 presents the frequency of these codings across all 128 trials for the four information heuristics. As seen clearly in Table 5, the MAIC outperforms all other heuristics in correctly selecting the appropriate dimensionality in 93% of the trials. Consistent with the literature on such information heuristics, the more conservative BIC and CAIC statistics tend to lean more towards under-parametrized models. On the basis of this Monte Carlo analysis, we will use the MAIC heuristic to select the dimensionality of the application presented. Recall the last column in Table 3 presented the corresponding ANOVA results for only the MAIC recoded (−1, 0, +1) dependent variable across the 128 trials. There, 1 We use Procrustes rotations to maximize congruence between centered actual and recovered joint spaces, and compute average product moment correlations between target and transformed recovered across dimensions as suggested in Borg and Groenen (2005), as well as computational requirements (number of major iterations), and data recovery or overall fit (F in (9)).

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TABLE 5. Information heuristics descriptive frequencies for the Monte Carlo analysis.

Information heuristics AIC

MAIC

BIC

CAIC

Dimensions

Frequency

Percent

−1 0 1 −1 0 1 −1 0 1 −1 0 1

3 117 8 6 119 3 63 65 0 63 65 0

2.3% 91.4% 6.3% 4.7% 93.0% 2.3% 49.2% 50.8% 0.0% 49.2% 50.8% 0.0%

only one term in the ANOVA is significant at p < 0.01 (brands x dimensions) indicating somewhat robust performance of the MAIC as a dimensionality selection heuristic for this particular spatial model.

4. Application: Consumer Study of Preference/Consideration Structures for Luxur Sport Utility Vehicles 4.1. Study Background A large US automotive consumer research supplier administered a tracking study to gauge automotive marketing awareness and shopping behavior among typical consumers in the market place. The surveys used in these tracking studies were conducted among new vehicle intenders and were collected from an automotive consumer panel of more than 600,000 nationally representative households. This “Brand Image Study” had been conducted among new vehicle intenders semiannually, June and December, for the past 20 years. An “intender” is a consumer that will be “in-market” or has plans to purchase a new vehicle within the next 6–12 months. Each survey respondent rates each brand/model corresponding to the particular product segment in which he or she intends to purchase. The ending completed sample results in approximately 200 to 300 respondents per segment per wave. Information collected includes familiarity with each make/model, advertising recall, overall opinion, purchase consideration, image attribute ratings, and awareness of model “redesign.” The Image Study is administered across 16 car segments and 10 light truck and sport utility vehicle (SUV) segments. The luxury sports utility segment was selected for use in this study. The data was collected in December 2002 with 210 consumer intenders rating 17 different luxury utility vehicles. Using a 4-point ordered consideration scale (4—“Definitely Would Consider,” 3—“Probably Would Consider,” 2—“Probably Would Not Consider,” and 1—“Definitely Would Not Consider”), the respondents rated how likely they would be to consider purchasing each brand in this product segment given they would soon be in-market. Table 6 lists the brands classified by the industry to be in this luxury sports utility product segment together with their 2002 market shares. In addition to rating their likeliness to consider the vehicle, the respondents used a 5-point interval scale to subjectively rate each vehicle on some 24 image attributes, which were later augmented with information from the Consumer’s Guide (2002) for various objective features and engineering specifications. Table 7 lists both sets of subjective and objective SUV attributes.

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PSYCHOMETRIKA TABLE 6. Luxury sport utility vehicle market share.

Brand Lexus RX-Series Acura MDX BMW X5 Mercedes Benz M-Class Cadillac Escalade Lincoln Navigator Hummer H2 Land Rover Discovery Infiniti QX4 Lexus LX-Series Land Rover Range Rover Toyota Land Cruiser Volvo XC90 Lexus GX470 Linoln Aviator Infiniti FX45 Porsche Cayenne

2002 sales

Market share (%)

72,963 52,995 42,742 39,680 36,114 30,613 18,861 17,417 16,938 9,231 8,549 6,752 4,379 2,190 1,856 n/a n/a

20.20% 14.67% 11.83% 10.98% 10.00% 8.47% 5.22% 4.82% 4.69% 2.56% 2.37% 1.87% 1.21% 0.61% 0.51% n/a n/a

361,280

100%

TABLE 7. Subjective and objective attributes.

Subjective SUV attributes

Objective SUV attributes

Gas mileage Value for money Workmanship Ride/handling Hard to load/unload Luxurious Good safety Good ride handling/off road Built rugged and tough Excellent towing and capacity Reasonably priced Sporty Good looking Good vehicle for family use Fun to drive Good interior passenger room Hard to enter/exit Dependable Excellent acceleration Excellent cargo space Lasts a long time Prestigious Technically advanced High trade-in value

Wheelbase (in.) Overall length (in.) Overall width (in.) Overall height (in.) Curb weight (lbs.) Cargo volume (cu ft.) Seating capacity Front head room (in.) Max. front leg room (in.) Rear head room (in.) Min rear leg room (in.) Payload (lbs.) Towing (lbs.) Platform Average MSRP ($)

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Market trends reveal that SUV sales had continued to increase at that time as new vehicle buyers abandon traditional sedans and minivans. In 1997, midsize cars represented 27.1% of the total market and SUVs were 12.3%. In the first four months of 2002, SUV sales were up to 20.2% and midsize cars had fallen to 17.1% (J.D. Power & Associates, 2002). The luxury sport utility segment experienced more growth between 2001 and 2002 than any other subsegment in the SUV arena. In 2002, light trucks (includes SUVs) outsold cars for the first time claiming a full 51% of the market. This growth was driven by SUVs, with the luxury utility segment experiencing a sales increase of 10.46% from 2001 to 2002, compared to a 0.36% increase for all light trucks and SUVs during the same time period; segments experiencing decreasing sales were fullsize SUVs, vans, and pick-up trucks (J.D. Power & Associates, 2002). In 2003 however, SUV sales slumped for the first time in 13 years, from a record 2,974,466 in 2002 to 2,864,616 in 2003 (2004 Ward’s Automotive Yearbook). Note, when gasoline prices spiked to more than $3 a gallon after Hurricane Katrina, SUV sales plummeted, especially those of large/luxury SUVs. In 2005, the Cadillac Escalade and Lincoln Navigator both had a decrease in sales of approximately 20% (Forbes.com 4/11/06). We reduced the 16 vehicles included in the Brand Image Study down to 10 calibration brands (used for estimation) based on 2002 market share information by using the top 10 brands. The resulting data set for this analysis includes the following brands: Lexus RXSeries (20.20%), Acura MDX (14.67%), BMW X5 (11.83%), Mercedes Benz M-Class (10.98%), Cadillac Escalade (10.00%), Lincoln Navigator (8.47%), Hummer H2 (5.22%), Land Rover Discovery (4.82%), Lexus LX-Series (2.56%), and Land Rover Range Rover (2.37%). These ten brands account for 90% of the 2002 luxury sport utility sales. Other luxury utility vehicles included in the data set but not included among the top 10 market share brands were the Toyota Land Cruiser (1.87%), Volvo XC90 (1.21%), Lexus GX470 (0.61%), Lincoln Aviator (0.51%), Infinity FX45, and Porsche Cayenne.2 The 2002 market share sales data is congruent with these consideration ratings in this Image Study as the top five vehicles in sales are also the top five considered vehicles among new vehicle intenders in the luxury utility market according to the Image Study data set. 4.2. The Proposed Model Solution Table 8 presents the various information heuristics for the obtained solutions by dimension. As shown, the two-dimensional solution appears most parsimonious as indicated from all presented heuristics. Figure 2 presents the estimated two-dimensional joint space plot for the proposed model. Here, as is commonplace in all MDS procedures, we property fit (post-hoc) the complete set of subjective and objective attributes via simple regression without intercepts so that these attributes would emanate from the origin. We normalized the length of the vectors for the sake of convenience in terms of ease of interpretation. Dimension 1 (horizontal axis) appears to relate to Ruggedness where brands on the right-hand side tend to be the tougher, off-road vehicles. Dimension 2 (vertical axis) can be characterized as a Size dimension where brands located TABLE 8. Goodness-of-fit heuristics for the proposed MDS model.

Number of dimensions

Number of parameters

Log likelihood

MAIC

Overall fit

1 2 3 4

241 478 724 979

–2300 –1741 –1580 –1393

5324 4916 5332 5723

78% 87% 90% 93%

2 Market shares for Infinity FX45 and Porsche Cayenne were not available at the time data was collected.

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F IGURE 2. Two-dimensional joint space and property fitting vector plot for the proposed MDU model.

at the upper portion of the figure tend to be larger and heavier. This dimension is also related to a Domestic–Import dimension since all SUVs in the upper quadrants are US made as opposed to the fact that all SUVs in the lower quadrants are Imports. Figure 2 also shows that imported SUVs compete with each other more closely than domestic SUVs. For instance, the Lexus Series, Acura MDX, BMW X5, and Mercedes Benz M-Class are located somewhat close together in the third quadrant, while domestic SUVs such as the Lincoln Navigator, Cadillac Escalade, and Hummer H2 seem to have more unique positions in these consumers’ eyes. Figure 3 portrays the brand density plot (Mij ) for each of the calibration brands indicating the utility structure estimated by brand. Some brands are particularly noteworthy to discuss here. The Cadillac Escalade appears peaked with small tails. From inspection of the frequency distributions, one sees a bimodal distribution across the four response categories with peaks at “1” and “4.” In fact, the Cadillac Escalade possesses the highest percentage of “4” responses across all ten calibration brands. Thus, consumers either tend to like the vehicle very much or don’t like it at all, accounting for this particular form of the utility distribution seen here. This is in direct contrast, for example, for what is displayed for the BMW X5 where a flat distribution is portrayed with larger tails. A quick inspection of the actual frequency distribution of the ordered response categories for the BMW X5 shows a near uniform distribution with nearly equal fre-

WAYNE S. DESARBO, JOONWOOK PARK, AND CRYSTAL J. SCOTT

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F IGURE 3. Brand density representations two-dimensional joint space and property vector plot for the proposed MDU model.

quencies across the four response categories. Finally, the Lincoln Navigator appears to possess the largest, volumewise, of all the brand utility Mij structures. 4.3. Traditional MDS Unfolding Results We chose to compare the results of four existing unfolding procedures for this data set. These existing unfolding procedures are: (a) ALSCAL (Takane, Young, & DeLeeuw, 1977) in SPSS; (b) PREFSCAL (Busing, Groenen, & Heiser, 2005) in SPSS; (c) GENFOLD3 (a metric only version of DeSarbo & Rao, 1984, 1986); and (d) the nonmetric procedure of Kim, Rangaswamy, and DeSarbo (1999, 2000) which we label KRD99. All analyses were run treating the data as row conditional using all software default values for all other options. In each case, we first utilized scree plots of the respective goodness-of-fit statistics to determine the dimensionality since no formal tests of dimensionality are provided by these deterministic approaches. Second, after centering the joint spaces of these solutions, we utilized Procrustes rotations to maximize congruence with the centered joint space of the proposed procedure displayed in Figure 2. As can be seen in Figure 4, these four MDU techniques show somewhat different results. The separation between ideal points and brand locations seems most pronounced in the ALSCAL solution. The interpretations of the resulting dimensions also appear different across these solutions. All solutions except KRD99 show that Cadillac Escalade/ESV and Lincoln Navigator are closely competing with each other. While imported SUVs compete closely with other imported SUVs in GENFOLD3 and KRD99, German SUVs (i.e., BMW X5 and Mercedes Benz M-Class) are more

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F IGURE 4. Two-dimensional joint space plots for various MDU models.

closely competing with Discovery and/or Range Rover in ALSCAL and PREFSCAL—perhaps reflecting a European vs. Japanese submarket. Table 9 presents the prediction accuracy rates for the ten calibration brands as well as the six validation brands for the various MDU models. As shown in the upper part of Table 9, the overall fit value of the proposed model is 87% followed by PREFSCAL (82%), ALSCAL (79%), GENFOLD3 (74%), and KRD99 (74%) for the calibration brands. Surprisingly, both ALSCAL and PREFSCAL show somewhat better fits than GENFOLD3 and KRD99 despite the fact that their solutions look less informative given the separation of brand and the ideal points than other two MDU techniques. The lower part of Table 9 shows the prediction accuracy for the six different luxury SUV brands that were not utilized in the initial analysis. First, based upon stepwise multiple regression property fitting with the objective and subjective attributes portrayed in Table 7, predicted locations were derived for these six validation brands in the derived two dimensions for all five

17

WAYNE S. DESARBO, JOONWOOK PARK, AND CRYSTAL J. SCOTT TABLE 9. Prediction rates by brand for the various MDU models.

Calibration brands Land Rover Discovery Range Rover BMW X5 Mercedes Benz M-Class Lincoln Navigator Lexus LX470 Lexus RX300 Acura MDX Hummer H2 Escalade/ESV Overall fit

Proposed model

ALSCAL

PREFSCALE

GENFOLD3

KRD99

82% 85% 87% 83% 89% 87% 90% 89% 88% 91%

79% 82% 76% 74% 81% 76% 79% 80% 80% 81%

81% 83% 79% 79% 85% 81% 83% 82% 80% 83%

75% 76% 69% 69% 78% 70% 75% 77% 76% 76%

73% 76% 72% 73% 72% 75% 75% 75% 79% 70%

87%

79%

82%

74%

74%

Proposed model

ALSCAL

PREFSCALE

GENFOLD3

KRD99

Toyota Land Cruiser Porsche Cayenne Infiniti FX45 Lincoln Aviator Volvo XC90 Lexus GX470

65% 73% 75% 68% 68% 75%

66% 63% 66% 68% 72% 76%

59% 74% 67% 60% 55% 62%

65% 72% 68% 66% 67% 64%

66% 77% 67% 68% 66% 67%

Overall fit

75%

71%

68%

69%

69%

Holdout brands

procedures. Based on these predicted locations, we formulated predictions for these six brands, and then compared the predictions to the actual response data (not used in the initial analysis). It should be noted that we performed an ordinal probit analysis for the existing MDU techniques to obtain such out-of-sample predictions in ordered integer format. As seen in the lower part of Table 9, the proposed model shows 75% in overall fit followed by ALSCAL (71%), GENFOLD3 (69%), KRD99 (69%), and PREFSCAL (68%) for these six validation brands. In sum, the proposed model dominates all competing procedures in terms of fit (note, expression (7) is not optimized by either of these five procedures). 5. Discussion We have introduced a new MDS procedure for the analysis of ordered successive preference categories based on a novel graphical representation involving multivariate normal distributions. The technical description of the underlying model and the MLE-based estimation procedure have been presented. A modest-sized Monte Carlo analysis is reported in terms of recovering synthetic data structures and identifying which information heuristics function better for this experiment. An application in consumer psychology has been provided concerning consumer buying consideration for luxury SUVs (a two-way analysis). Comparison with ALSCAL, GENFOLD3, PREFSCAL, and KRD99 unfolding procedures are provided for the application in contrast with the results obtained by the proposed methodology. In all cases involving goodness-of-fit with respect to both calibration and validation brands, the proposed procedure performed better than its traditional rivals. An inspection of the calibration brands’ multivariate normal surfaces rendered additional insight into the preference distributions surrounding each brand/stimulus (brand equity).

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Obviously, further work needs to be conducted in this arena. More complete Monte Carlo analyses with synthetic data of known structure should be conducted to examine parameter recovery and local optima (given the problems with stochastic unfolding models as stated in Johnson & Junker, 2003) as a number of data, error, and model specification forms are varied. One of the problems concerning analyses with two-way data without replications with the proposed procedure relates to the issue of incidental parameters whose order vary according to the size of the input data (here, rows and columns). Such difficulties may lead to inconsistent parameter estimates and questionable use of the information heuristics for dimension selection. Generalizing Takane’s (1997) marginal maximum likelihood method for pick-any data to such ordered successive categories would be a productive avenue for further methodological research here, especially with the use of two-way data. Further comparisons with more recent joint space approaches that allow for predictive validation would also be worthwhile directions for future research. The exploration of the use of permutation tests and bootstrapping to examine the asymptotic properties of the derived estimators would prove beneficial (see Good, 2005). Like most MDU procedures, the proposed model can also suffer from the occurrence of degenerate solutions where there may be less informative joint space configurations derived with, for example, large separations between ideal points and stimulus locations (see Heiser, 1981; DeSarbo & Rao, 1984, 1986; Busing et al. 2005). Future work should proceed so as to minimize the occurrence of such solutions in line with either the use of constraints, additional information, penalized terms, etc. Finally, additional applications across a number of social science areas should be executed with this approach. References Aaker, D. (1991). Managing brand equity. New York: The Free Press. Aaker, D. (1996). Building strong brands. New York: The Free Press. Adams, E., & Messick, S. (1958). An axiomatic formulation and generalization of successive intervals scaling. Psychometrika, 23, 355–368. Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723. Benzécri, J.P. (1973). L’analyse des données: Tome II. Analyse de correspondances. Paris: Dunod. Benzécri, J.P. (1992). Correspondence analysis handbook. New York: Dekker. Borg, I., & Groenen, P. (2005). Modern multidimensional scaling: Theory and application (2nd edn.). New York: Springer. Bozdogan, H. (1987). Model selection and Akaike’s Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370. Busing, F.M.T.A., Groenen, P.J.F., & Heiser, W. (2005). Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70, 71–98. Carroll, J.D. (1972). Individual differences and multidimensional scaling. In R.N. Shepard, A.K. Romney, & S. Nerlove (Eds.), Multidimensional scaling: Theory and applications in the behavior sciences: Vol. I. Theory. New York: Seminar Press. Carroll, J.D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance) data. In E.D. Lantermann, H. Feger (Eds.), Similarity and choice. Vienna: Hans Huber. Cliff, N. (1973). Scaling. Annual Review of Psychology, 24, 473–506. Consumer quide 2002 automobile book (2002). Lincolnwood, IL: Publications International Ltd. Coombs, C.H., Dawes, R.M., & Tversky, A. (1970). Mathematical psychology. Englewood Cliffs: Prentice-Hall. Cox, T.F., & Cox, M.A. (2001). Multidimensional scaling (2nd edn.). London: Chapman & Hall. DeSarbo, W.S., & Carroll, J.D. (1985). Three-way metric unfolding via weighted least-squares. Psychometrika, 50, 275– 300. DeSarbo, W.S., & Hoffman, D. (1986). Simple and weighted unfolding MDS threshold models for the spatial analysis of binary data. Applied Psychological Measurement, 10, 247–264. DeSarbo, W.S., & Rao, V.R. (1984). GENFOLD2: A set of models and algorithms for the general unfolding analysis of preference/dominance data. Journal of Classification, 1, 147–186. DeSarbo, W.S., & Rao, V.R. (1986). A constrained unfolding methodology for product positioning. Marketing Science, 5, 1–19. DeSarbo, W.S., Young, M.R., & Rangaswamy, A. (1997). A parametric multidimensional unfolding procedure for incomplete nonmetric preference/choice set data in marketing research. Journal of Marketing Research, 34, 499–516. Farquhar, P.H. (1989). Managing brand equity. Marketing Research, 1, 24–33. Fletcher, R. (1987). Practical methods of optimization. New York: Wiley.

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