The Finite Mixture Partial Least Squares Approach ...

The Finite Mixture Partial Least Squares Approach - Methodology and Application Christian M. Ringle, Sven Wende, and Alexander Will1 University of Hamburg - School of Business, Economics and Social Sciences c.ringle|s.wende|[email protected]

1 Introduction Structural equation modeling (SEM) and path modeling with latent variables (LVP) are applied in modern marketing research to measure complex cause-effect relationships [6, 19]. Covariance structure analysis (CSA) [12] and partial least squares analysis (PLS) [13] constitute the two corresponding, yet different, statistical techniques for estimating such models. A typical research issue in SEM and LVP is the measurement of customer satisfaction [7], which is closely related to the requirement of identifying distinctive customer segments [10]. In SEM, segmentation can be achieved based on the heterogeneity of scores for latent variables in the structural (inner) model. Jedidi et al. (1997) address this field of research and propose a procedure that blends finite mixture models and the expectation-maximization (EM) algorithm [14, 15, 21]. However, the original technique extends CSA and is inappropriate for PLS because of methodological assumptions [5]. Therefore, Hahn et al. (2002) propose the finite mixture partial least squares (FIMIX-PLS) approach that is strongly connected to the maximum likelihood (ML) theory [4]. According to their proposal, a finite mixture procedure is joined with an EM-algorithm specifically regarding the ordinary least squares (OLS)-based predictions of PLS. Hahn et al. (2002), the developers of FIMIX-PLS, present a formal description and discussion of this methodology, the aspects of model identification and an interpretation of results.

2 Methodology The first methodological step is to estimate path models by applying the basic PLS algorithm for LVP [13]. FIMIX-PLS is then employed using the predicted scores of latent variables and their modified presentation of relationships in the inner (structural) model:

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Christian M. Ringle, Sven Wende, and Alexander Will

Bηi + Γ ξi = ζi

(1)

Segment specific heterogeneity of path models is concentrated in the estimated relationships between latent variables. FIMIX-PLS captures this heterogeneity. The segment specific distributional function is defined as follows, assuming that ηi is distributed as a finite mixture of conditional multivariate normal densities fi|k (·): ηi ∼

K X

ρk fi|k (ηi |ξi , Bk , Γk , Ψk )

(2)

k=1

Substituting fi|k (ηi |ξi , Bk , Γk , Ψk ) results in the following equation: # " K X |Bk | − 12 (Bk ηi +Γk ξi )0 Ψk−1 (Bk ηi +Γk ξi )) √ p e ηi ∼ ρk M 2π |Ψk | k=1

(3)

It is sufficient to assume multivariate normal distribution of ηi because endogenous variables ηi are a function of exogenous variables ξi in the inner model. Equations (4) and (5) represent an EM-formulation of the likelihood function and the log-likelihood (lnL) as the corresponding objective function for maximization: YY z L= [ρk f (ηi |ξi , Bk , Γk , Ψk )] ik (4) i

lnL =

XX i

k

zik ln(f (ηi |ξi , Bk , Γk , Ψk )) +

XX i

k

zik ln(ρk )

(5)

k

The EM-algorithm is used to maximize the likelihood in this model in order to ensure convergence. The ’expectation’ of equation (5) is calculated in the E-step, where zik is 1 if subject i belongs to class k (or 0 otherwise). The segment size ρk , the parameters ξi , Bk , Γk and Ψk of the conditional probability function are given, and provisional estimates (expected values) for zik are computed as follows according to Bayes’ theorem: ρk fi|k (ηi |ξi , Bk , Γk , Ψk ) E(zik ) = Pik = PK k=1 ρk fi|k (ηi |ξi , Bk , Γk , Ψk )

(6)

Equation (5) is maximized in the M-step. Initially, new mixing proportions ρk are calculated by the average of adjusted expected values Pik that result from the previous E-step: PI ρk =

i=1

Pik

(7) I Thereafter, optimal parameters for Bk , Γk and Ψk are determined by independent OLS regressions (one for each relationship between latent variables

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in the structural model). ML estimators of coefficients and variances are assumed to be identical to OLS predictions. The following equations are applied to obtain the regression parameters for latent endogenous variables: (8)

Xmi = (Emi , Nmi )0

(9)



{ξ1 , ..., ξAm } , Am ≥ 1, am = 1, ..., Am ∧ ξam is regressor of m (10) ∅ else



{η1 , ..., ηBm } , Bm ≥ 1, bm = 1, ..., Bm ∧ ηbm is regressor of m (11) ∅ else

Emi =

Nmi =

Ymi = ηmi

The closed form OLS analytic formula for τmk and ωmk is expressed as follows: 0

τ = ((γam mk ), (βbm mk )) = P mk 0 −1 P 0 [ i Pik (Xmi Xmi )] [ i Pik (Xmi Ymi )] P ωmk = cell (m × m) of Ψk =

i

(12)

Pik (Ymi − Xmi τmk )(Ymi − Xmi τmk )0 (13) Iρk

The M-step determines new mixing proportions. Independent OLS regressions are used in the next E-step iteration to improve the outcomes for Pik . Based on an a priori specified convergence criterion, the EM-algorithm stops when the lnL hardly improves (see figure 1). However, the stop is more a measure of lack of progress than a measure of convergence, and there is evidence that the algorithm is often stopped too early [21]. When applying FIMIX-PLS, the EM-algorithm monotonically increases lnL and converges towards an optimum [8]. Experience shows that FIMIXPLS frequently stops in local optimum solutions. The cause of this is that the likelihood becomes multimodal so that the algorithm becomes sensitive to the starting values [21]. Moreover, the problem of convergence in local optima seems to increase in relevance whenever component densities are not well separated. The number of parameters estimated is large and the information embedded in each observation is limited. This causes a relatively weak update of the membership probabilities in the E-step. Some examples of simple strategies for escaping local optima include beginning the EM-algorithm from a wide range of (random) starting values or using various clustering procedures, such as k-means, to obtain an appropriate initial partition of data. If alternative initializations (starting values) of the algorithm result in different local optima, then the solution with the

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Christian M. Ringle, Sven Wende, and Alexander Will

// initial E-step set random starting values for Pik ; set lastlnL = V ; set 0 < S < 1 repeat do begin // theP M-step starts here I

Pik

i=1 ρk = ∀k I determine Bk , Γk , Ψk , ∀k calculate currentlnL ∆ = currentlnL − lastlnL

// the E-step starts here if ∆ ≥ S then begin ρ fi|k (ηi |ξi ,Bk ,Γk ,Ψk )

Pik = PKk

k=1

ρk fi|k (ηi |ξi ,Bk ,Γk ,Ψk )

∀i, k

lastlnL = currentlnL end end until ∆ < S

Fig. 1. The FIMIX-PLS algorithm

maximum value of likelihood is recommended as best solution. Concerns still remain whether this kind of an unsystematically selected solution reaches the global optimum. Simulations are required to determine appropriate strategies for identifying convergence towards local optimum FIMIX-PLS solutions. If the reliability of these methods is uncertain, then future research will need to alternatively test the applicability of metaheuristics [1] to avoid local optimum FIMIX-PLS results.

3 Segmentation and ex post analysis The number of segments is unknown and the identification of an appropriate number of K classes is not straightforward when applying FIMIX-PLS. A statistically satisfactory solution does not exist for several reasons [21], i.e., mixture models are not asymptotically distributed as chi-square and do not allow for the likelihood ratio statistic. For this reason, Hahn et al. (2002) propose repeated operation of FIMIX-PLS with consecutive numbers of latent classes K (e.g., 1 to 10) and to compare the class specific outcomes for criteria such as the lnL, the Akaike Information Criterion (AICK = −2lnL + cNK ; [2]), the AIC Controlled Criterion (AICCK = −2lnL + ln(I + 1)NK ) or the Bayesian Information Criterion (BICK = −2lnL + ln(INK ); [18]). The

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results of these heuristic measures provide evidence about an appropriate number of latent classes. Moreover, an entropy statistic [16], limited between 0 and 1, indicates the degree of separation in the individually estimated class probabilities. P P [ i k −Pik ln(Pik )] (14) ENK = 1 − Iln(K) The quality of separation of the derived classes will improve the higher ENK is. Values of EN that are above 0.5 will result in estimates for Pik that permit unambiguous segmentation. This criterion is especially relevant, e.g., for identifying different types of customers in the field of marketing. Furthermore, estimates of class specific relationships in the inner model need to provide statistically reasonable values after segmentation. Given these assumptions, FIMIX-PLS is only applicable for additional analytic purposes, if the segments are interpretable. For this reason, Hahn et al. (2002) suggest an ex post analysis of the estimated probabilities of membership using an approach proposed by Ramaswamy et al. (1993) to identify explanatory variables that allow further interpretation of the final segments. This kind of characterization of FIMIX-PLS results is essential for the methodology and its applicability. The findings of this new approach can be used to group data (e.g., into ’young customers’ and ’old customers’) and to calculate the LVP for each segment (set of data). The following examples, which use experimental data and empirical data, document this approach.

4 Example using experimental data SmartPLS 2.0 [17] is the first statistical software application for (graphical) path modeling with latent variables employing both the basic PLS algorithm [13] as well as FIMIX-PLS capabilities for the kind of segmentation proposed by Hahn et al. (2002). Applying this statistical software module to experimental data for a marketing-related path model demonstrates the potentials of the methodology for PLS-based research. Our hypothetical model is comprised of one endogenous latent variable Satisfaction and two exogenous latent variables Price and Quality in the structural model. • Price-oriented customers (segment 1) This segment is characterized by a strong relationship between Price and Satisfaction and a weak relationship between Quality and Satisfaction. • Quality-oriented customers (segment 2) This segment is characterized by a strong relationship between Quality and Satisfaction and a weak relationship between Price and Satisfaction. Each latent exogenous variable (Price and Quality) consists of seven reflective indicator-variables, and the latent endogenous variable (Satisfaction)

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Christian M. Ringle, Sven Wende, and Alexander Will

measured by two manifest variables in a reflective measurement model. According to the scheme in table 1, the underlying case values of the manifest variables are generated on a scale ranging from 1 to 7. For instance, the case values 1 to 20 of the manifest indicators Price1 to Price7, Satisfaction1 and Satisfaction2 are normally distributed random numbers, with µ = 6 and σ = 0.1. The case values 1 to 20 of the manifest indicators Quality1 to Quality7 are normally distributed random numbers, with µ = 4 and σ = 1. Case 1 − 20 21 − 40 41 − 60 61 − 80

Price µ = 6.0 / σ µ = 4.0 / σ µ = 2.0 / σ µ = 4.0 / σ

= 0.1 = 1.0 = 0.1 = 1.0

Quality µ = 4.0 / σ = 1.0 µ = 6.0 / σ = 0.1 µ = 4.0 / σ = 1.0 µ = 2.0 / σ = 0.1

Satisfaction µ = 6.0 / σ = 0.1 µ = 6.0 / σ = 0.1 µ = 2.0 / σ = 0.1 µ = 2.0 / σ = 0.1

Table 1. Data generation scheme

The standard PLS procedure is executed to measure the path model based on the experimental data for manifest indicator variables. FIMIX-PLS uses the results to process the latent variable scores in the inner model for two segments. The standard PLS inner model weights in table 2 show that both constructs Price and Quality have a relatively high effect on Satisfaction and result in a substantial R2 of 0.932. An overall goodness of fit (GoF) measure - that is limited between 0 and 1 - is proposed as the geometric mean of the average communality (outer model) and the average R2 (inner model) [20]: q √ GoF = communality · R2 = 0.865 · 0.932 = 0.898 (15) However, it is misleading to interpret these excellent estimates for a path model. The application of FIMIX-PLS permits additional analyses that lead to different conclusions.

Standard PLS FIMIX-PLS segment 1 FIMIX-PLS segment 2

Price → Satisfaction Quality → Satisfaction 0.7 0.7 0.7 0.0 0.0 0.7

Table 2. Inner model weights

According to the FIMIX-PLS results presented in table 2, both a priori created segments are identified in the experimental set of data, i.e., segment 1 shows a strong effect of Price on Satisfaction and a week relationship between Quality and Satisfaction, and segment 2 shows a strong effect of Quality on Satisfaction and a week relationship between Price and Satisfaction. As a consequence of the experimental design, segment specific regression variances

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are very low for the latent endogenous variable Satisfaction (0.001 for segment 1 and 0.002 for segment 2). This results in corresponding outcomes for R2 that nearly equal one. Cases 1 − 20 and cases 41 − 60 are perfectly assigned to segment 1, and cases 21 − 40 and cases 61 − 80 are perfectly assigned to segment 2 (Pik has either a value of 0 or 1 for the final segmentation). Consequently, EN has a value of 1.0 indicating perfect separation of these two a priori created (distinctive) segments.

5 Marketing example using empirical data FIMIX-PLS is often not as straightforward as demonstrated in the foregoing example using experimental data if researchers use empirical data and do not have a priori segmentation assumptions. Extensive use of FIMIX-PLS with empirical data in future research ought to furnish additional findings about the methodology and its applicability. For this reason, we apply the technique to a marketing related path model and to empirical data from Gruner + Jahr’s ’Brigitte Communication Analysis 2002’. Gruner + Jahr is one of the leading publishers of printed magazines in Germany. They have been conducting their Communication Analysis survey every other year since 1984. In the survey, over 5,000 women answer questions on brands in different product categories and questions regarding their personality. The women represent a cross section of the German female population. Answers to questions on the Benetton fashion brand name (on a four point scale from ’low’ to ’high’) are selected in order to use the survey as a marketing-related example of FIMIX-PLS-based customer segmentation. We assume that Benetton’s aggressive and provocative advertising in the 1990s resulted in a lingering customer heterogeneity that is more distinctive and easier to identify compared with other fashion brands in this survey (e.g., Esprit or S.Oliver). LVP is designed to demonstrate the applicability of FIMIX-PLS in the field of marketing. The scope of this article does not include a presentation of theoretically substantiated hypotheses and PLS-based empirical confirmation of them (see for example [9]). We therefore create a reduced path model for Benetton’s brand preference that consists of one latent endogenous ’BRAND PREFERENCE’ variable, and two latent exogenous variables, ’IMAGE’ and ’PERSON’, in the structural model. Figure 2 illustrates our inner model, the outer models of LVP that they reflect, and the manifest variables employed. PLS is applied for estimating the path model using the SmartPLS 2.0 software application. We follow the suggestions given by Chin (1998) for arriving at a brief evaluation of results. All relationships in the reflective measurement models have excellent factor loadings (the smallest loading has a value of 0.795). Moreover, the average variance extracted (AVE) and ρc are at relatively high levels (see section 7.2). The latent exogenous ’IMAGE’ variable (weight of 0.423) exhibits

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Fig. 2. The brand preference model

a strong relationship to the latent endogenous ’BRAND PREFERENCE’ variable. The influence of the latent exogenous ’PERSON’ variable is considerably weaker (weight of 0.174). Both relationships are statistically significant (see the results of the bootstrapping procedure in section 7.2). R2 of ’BRAND PREFERENCE’ equals 0.239 and is at a moderate level for PLS path models. This value is mainly determined by Benetton’s ’IMAGE’. The ’PERSON’ variable is not as relevant but exhibits a statistically significant influence. The average communality of the three reflective measurement models is relatively high, so that R2 of the latent endogenous ’BRAND PREFERENCE’ variable mainly causes (only) a moderate GoF outcome: q √ (16) GoF = communality · R2 = 0.748 · 0.239 = 0.423 The FIMIX-PLS module of SmartPLS 2.0 is applied for customer segmentation based on the estimated scores for latent variables. Table 3 shows heuristic FIMIX-PLS evaluation criteria for alternative numbers of classes K. According to these results, the choice of two latent classes seems to be appropriate for customer segmentation purposes, especially in terms of EN. Our EN result of 0.501 also reaches a proper level compared to the EN of 0.43 that Hahn et al. (2002) present for an FIMIX-PLS application. Their presentation is the only one found thus far in English language academic literature. Table 4 presents the FIMIX-PLS results for two latent classes. In a large segment (size of 0.809), the explained variance of the latent endogenous ’BRAND PREFERENCE’ variable is at a relatively weak level for PLS models (R2 = 0.108). This variance is explained by the latent exogenous ’IMAGE’ variable, with its weight of 0.343, and the latent exogenous ’PERSON’ vari-

The Finite Mixture Partial Least Squares Approach Number of latent classes Aggregate (K = 1) K=2 K=3 K=4 K=5

lnL −569, 461 −713.233 −942.215 −1053.389 −1117.976

AIC 1152.922 1448.466 1954.431 2192.793 2441.388

BIC 1181.593 1493.520 2097.784 2450.830 2846.874

AICC 1181.609 1493.545 2097.863 2450.972 2847.097

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EN 1.000 0.501 0.216 0.230 0.214

Table 3. Model selection

able, with its weight of 0.177. A smaller segment (size of 0.191) has a relatively high R2 for ’BRAND PREFERENCE’ (value of 0.930). The influence of the ’PERSON’ variable does not change much for this segment. However, the weight of the ’IMAGE’ variable is more than twice as high and has a value of 0.759. This result reveals that the preference for Benetton is explained to a high degree whenever the image of this brand is far more important than the individuals’ personality.

Segment size R2 (for ’BRAND PREFERENCE’) Path ’IMAGE’ to ’BRAND PREFERENCE’ Path ’PERSON’ to ’BRAND PREFERENCE’

K=1 0.809 0.108 0.343 0.177

K=2 0.191 0.930 0.759 0.170

Table 4. Disaggregate results (solution for two latent classes)

The next step in FIMIX-PLS involves the necessity to identify a certain variable that characterizes customer segments that have been formed. We implemented an ex post analysis of the estimated probabilities of membership in line with the suggestions of Hahn et al. (2002). The most significant explanatory variables among the several indicators examined are: ’I AM VERY INTERESTED IN THE LATEST FASHION TRENDS’, ’I GET INFORMATION ABOUT CURRENT FASHION FROM MAGAZINES FOR WOMEN’, ’BRAND NAMES ARE VERY IMPORTANT FOR SPORTS WEAR’ and ’I LIKE TO BUY FASHION DESIGNERS’ PERFUMES’ (t-statistics ranging from 1.462 to 2.177). These variables may be appropriate for explaining the segmentation of customers into two classes. Table 5 shows PLS results using the ’I LIKE TO BUY FASHION DESIGNERS’ PERFUMES’ variable for an a priori customer segmentation into two classes. Both outcomes for segment-specific LVP estimations satisfy the relevant criteria for model evaluation [3] (see section 7.3). Segment 1 represents customers that are not interested in fashion designers’ perfumes (size of 0.223). By contrast, segment 2 (size of 0.777) represents customers that are attracted to Benetton who would enjoy using this fashion brand’s products in other product categories, such as perfumes. From a marketing viewpoint,

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these customers are very important to fashion designers who are intending to create brand extensions.

GoF R2 (for ’BRAND PREFERENCE’) ’IMAGE’ → ’BRAND PREFERENCE’ ’PERSON’ → ’BRAND PREFERENCE’

Segment 1 Segment 2 0.387 0.470 0.204 0.323 0.394 0.562 0.164 0.104

Table 5. A priori segmentation based on ’I LIKE TO BUY FASHION DESIGNERS’ PERFUMES’

Besides their applicability to explaining the FIMIX-PLS segmentation for the structural model, the variables identified in the ex post analysis do not offer much potential for a meaningful a priori segmentation, except for the ’I LIKE TO BUY FASHION DESIGNERS’ PERFUMES’ variable. The other four variables with high t-statistics do not result in different measurements for segment specific path models when segmented a priori into two classes. We consider reasonable alternatives and test the ’CUSTOMERS’ AGE’ variable for an a priori segmentation of Benetton’s brand preference LVP. The ex post analysis of FIMIX-PLS results does not furnish evidence for the relevance of this variable (t-statistic = 0.690). Yet when creating a customer segment for females older than 28 years (segment 1; segment size: 0.793) and for female consumers between ages of 15 and 28 years (segment 2; segment size: 0.207), we achieve a result that is nearly identical to the a priori segmentation using ’I LIKE TO BUY FASHION DESIGNERS’ PERFUMES’ (see table 6; the evaluated model [3] reveals that the PLS model estimates are acceptable for each segment, see section 7.4). However, this variable and ’AGE’ are not highly correlated (correlation coefficient: −0.166).

GoF R2 (for ’BRAND PREFERENCE’) ’IMAGE’ → ’BRAND PREFERENCE’ ’PERSON’ → ’BRAND PREFERENCE’

Segment 1 Segment 2 0.355 0.521 0.172 0.356 0.364 0.559 0.158 0.110

Table 6. A priori segmentation based on ’CUSTOMERS’ AGE’

The findings that we present depict problems in identifying interpretable variables for an a priori segmentation of data. We therefore select several potential explanatory variables to create different kinds and numbers of customer segments before specifically estimating their LVP. These results reveal that variables, which are indicated as significant throughout the ex post analysis, do not necessarily provide for enhanced a priori segmentation. However, our

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tests do confirm the appropriateness of two classes for a priori segmentation as indicated by FIMIX-PLS. Using larger numbers of latent classes causes PLS model estimations with insignificant relationships within the structural model or relatively low values of R2 for the latent endogenous ’BRAND PREFERENCE’ variable. The example that uses empirical data demonstrates that FIMIX-PLS reliably identifies a suitable number of customer segments, if heterogeneity exists within the structural model. These kinds of findings, which result in segment specific LVP estimations, provide a platform for acquiring further differentiated PLS path modeling conclusions. However, an ex post analysis is essential in terms of methodology and offers various potentials for future research.

6 Summary FIMIX-PLS allows us to capture unobserved heterogeneity in the estimated scores for latent variables in path models by grouping data. This is advantageous to a priori segmentation because homogeneous segments are explicitly generated for the inner model relationships. The procedure is broadly applicable in business research. For example, marketing-related path modeling can exploit this approach for distinguishing certain groups of customers. In the first example involving experimental data, FIMIX-PLS reliably identifies and separates the two a priori created segments of price- and qualityoriented customers. The second example of a marketing-related path model for Benetton’s brand preference is based on empirical data, and it also demonstrates that FIMIX-PLS reliably identifies an appropriate number of customer segments if distinctive groups of customers exist that result in heterogeneity within the inner model. In this case, FIMIX-PLS enables us to identify: (1) a large segment of ’more mature’ female customers that shows similar results when compared to the original model estimation and (2) a smaller segment of ’young’ female customers that reveals a strong relationship between ’IMAGE’ and ’BRAND PREFERENCE’. We accordingly conclude that the methodology offers capabilities to extend and further differentiate PLS-based analyses of LVP. The most important fact within this context is that poor standard PLS results for the overall set of data, caused by the heterogeneity of estimates in the structural model, may result in good estimates of the inner relationships and substantial values for R2 of latent endogenous variables after segmentation. However, an accurate ex post analysis is essential for this kind of methodology to ensure a suitable characterization of the formed groups for further interpretation purposes and additional conclusions. Such FIMIX-PLS related analyses must be conducted by trial and error testing routines until future research presents reliable procedures. Extensive simulations with experimental data and broad use of empirical data are required to further illustrate the applicability of FIMIX-PLS.

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7 Appendix 7.1 Description of symbols Am am Bm bm γam mk βbm mk τmk ωmk c fi|k (·) I i J j K k M m Nk Pik R S V Xmi Ymi zik ζi ηi ξi B Γ ∆ Bk Γk Ψk ρ ρk

number of exogenous variables as regressors in regression m exogenous variable am with am = 1, ..., Am number of endogenous variables as regressors in regression m endogenous variable bm with bm = 1, ..., Bm regression coefficient of am in regression m for class k regression coefficient of bm in regression m for class k ((γam mk ), (βbm mk ))0 vector of the regression coefficients cell (m × m) of Ψk constant factor probability for case i given a class k and parameters (·) number of cases or observations case or observation i with i = 1, ..., I number of exogenous variables exogenous variable j with j = 1, ..., J number of classes class or segment k with k = 1, ..., K number of endogenous variables endogenous variable m with m = 1, ..., M number of free parameters defined as (K − 1) + KR + KM probability of membership of case i to class k number of predictor variables of all regressions in the inner model stop or convergence criterion large negative number case values of the regressors for regression m of individual i case values of the regressant for regression m of individual i zik = 1, if the case i belongs to class k; zik = 0 otherwise random vector of residuals in the inner model for case i vector of endogenous variables in the inner model for case i vector of exogenous variables in the inner model for case i M × M path coefficient matrix of the inner model M × J path coefficient matrix of the inner model difference of currentlnL and lastlnL M × M path coefficient matrix of the inner model for latent class k M × J path coefficient matrix of the inner model for latent class k M × M matrix for latent class k containing the regression variances (ρ1 , ..., ρK ), vector of the K mixing proportions of the finite mixture mixing proportion of latent class k

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7.2 PLS results for the ’BRAND PREFERENCE’ path model PLS results for the reflective measurement models I have a clear impression of this brand This brand can be trusted Is modern and up to date Represents a great style of living Fashion is a way to express who I am I often talk about fashion A brand name is very important to me I am interested in the latest trends Sympathy Brand usage

Image Person B.P. 0.860 0.899 0.795 0.832 0.801 0.894 0.850 0.831 0.944 0.930

AV E and ρc for the latent variables AV E ρc Image 0.710 0.910 Person 0.725 0.922 B.P. 0.881 0.937 PLS results for the structural model and the GoF R2 of the latent endogenous ’BRAND PREFERENCE’ variable = 0.239, GoF = 0.423. Bootstrapping, 500 samples, construct level changes. Image → B.P. Person → B.P. Original estimation 0.423 0.174 0.428 0.174 Mean of all bootstrapping sample Standard deviation 0.041 0.036 t-statistic 10.361 4.875

7.3 PLS results for segment 1 based upon ’I LIKE TO BUY...’ This segment with a size of 0.223 represents customers that are not interested in fashion designers’ perfumes.

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PLS results for the reflective measurement models I have a clear impression of this brand This brand can be trusted Is modern and up to date Represents a great style of living Fashion is a way to express who I am I often talk about fashion A brand name is very important to me I am interested in the latest trends Sympathy Brand usage

Image Person B.P. 0.851 0.894 0.783 0.821 0.761 0.878 0.851 0.823 0.949 0.930

AV E and ρc for the latent variables AV E ρc Image 0.703 0.904 Person 0.687 0.898 B.P. 0.882 0.938 PLS results for the structural model and the GoF R2 of the latent endogenous ’BRAND PREFERENCE’ variable = 0.204, GoF = 0.387. Bootstrapping, 500 samples, construct level changes. Image → B.P. Person → B.P. Original estimation 0.394 0.164 Mean of all bootstrapping sample 0.394 0.173 Standard deviation 0.041 0.037 t-statistic 9.562 4.492

7.4 PLS results for segment 2 based upon ’I LIKE TO BUY...’ This segment with a size of 0.777 represents customers that are attracted by the fashion brand and would enjoy using this brand for items of other product categories like perfumes.

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PLS results for the reflective measurement models I have a clear impression of this brand This brand can be trusted Is modern and up to date Represents a great style of living Fashion is a way to express who I am I often talk about fashion A brand name is very important to me I am interested in the latest trends Sympathy Brand usage

Image Person B.P. 0.873 0.906 0.832 0.870 0.788 0.645 0.747 0.718 0.906 0.934

AV E and ρc for the latent variables AV E ρc Image 0.758 0.926 Person 0.528 0.816 B.P. 0.847 0.917 PLS results for the structural model and the GoF R2 of the latent endogenous ’BRAND PREFERENCE’ variable = 0.323, GoF = 0.470. Bootstrapping, 500 samples, construct level changes. Image → B.P. Person → B.P. Original estimation 0.562 0.104 Mean of all bootstrapping sample 0.564 0.106 Standard deviation 0.034 0.063 t-statistic 15.020 1.662

7.5 PLS results for segment 1 based upon customers’ age This segment with a size of 0.793 represents women who are older than 28 years.

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PLS results for the reflective measurement models I have a clear impression of this brand This brand can be trusted Is modern and up to date Represents a great style of living Fashion is a way to express who I am I often talk about fashion A brand name is very impoertant to me I am interested in the latest trends Sympathy Brand usage

Image Person B.P. 0.860 0.879 0.777 0.807 0.769 0.894 0.867 0.831 0.936 0.925

AV E and ρc for the latent variables AV E ρc Image 0.692 0.900 Person 0.708 0.906 B.P. 0.866 0.928 PLS results for the structural model and the GoF R2 of the latent endogenous variable ’BRAND PREFERENCE’ = 0.172, GoF = 0.355. Bootstrapping, 500 samples, construct level changes. Image → B.P. Person → B.P. Original estimation 0.364 0.158 Mean of all bootstrapping sample 0.368 0.164 Standard deviation 0.041 0.034 t-statistic 8.796 4.639

7.6 PLS results for segment 2 based upon customers’ age This segment with a size of 0.209 represents women who are between 15 and 28 years.

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PLS results for the reflective measurement models Image Person B.P. I have a clear impression of this brand 0.850 This brand can be trusted 0.928 Is modern and up to date 0.888 Represents a great style of living 0.897 Fashion is a way to express who I am 0.849 I often talk about fashion 0.834 A brand name is very important to me 0.752 I am interested in the latest trends 0.818 Sympathy 0.958 0.940 Brand usage AV E and ρc for the latent variables AV E ρc Image 0.794 0.939 Person 0.663 0.887 B.P. 0.900 0.948 PLS results for the structural model and the GoF R2 of the latent endogenous ’BRAND PREFERENCE’ variable = 0.356, GoF = 0.521. Bootstrapping, 500 samples, construct level changes. Image → B.P. Person → B.P. Original estimation 0.559 0.110 Mean of all bootstrapping sample 0.558 0.116 Standard deviation 0.031 0.034 t-statistic 17.931 3.218

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