Segmentation for path models and unobserved heterogeneity: The finite mixture partial least squares approach

MPRA Munich Personal RePEc Archive Segentation for path odels and unobserved heterogeneity: The finite ixture partial least squares approach Christian M. Ringle University of Haburg, Departent of Business and Econoics 006 Online at http://pra.ub.uni-uenchen.de/10734/ MPRA Paper No. 10734, posted 6. October 008 03:40 UTC

Christian M. Ringle Segentation for Path Models and Unobserved Heterogeneity: The Finite Mixture Partial Least Squares Approach Research Papers on Mareting and Retailing University of Haburg No. 035 Noveber 006 Tor zur Welt der Wissenschaft

Departent Wirtschaftswissenschaften Institut für Handel und Mareting Arbeitsbereich Mareting und Branding Geschäftsführender Diretor Prof. Dr. Henri Sattler Nr. 035 Christian M. Ringle* Segentation for Path Models and Unobserved Heterogeneity: The Finite Mixture Partial Least Squares Approach Noveber 006 ISSN 1618-8985 *) Dr. Christian M. Ringle holds a postdoc position at the Institute of Industrial Manageent, School of Business, Econoics and Social Sciences, University of Haburg, Von-Melle-Par 5, 0146 Haburg, Gerany; eail: cringle@econ.uni-haburg.de

Segentation for Path Models and Unobserved Heterogeneity: The Finite Mixture Partial Least Squares Approach Abstract Partial least squares-based path odeling with latent variables is a ethodology that allows to estiate coplex cause-effect relationships using epirical data. The assuption that the data is collected fro a single hoogeneous population is often unrealistic. Identification of different groups of consuers in connection with estiates in the inner path odel constitutes a critical issue for applying the path odeling ethodology to for effective areting strategies. Sequential clustering strategies often fail to provide useful results for segent-specific partial least squares analyses. For that reason, the purpose of this paper is fourfold. First, it presents a finite ixture path odeling ethodology for separating data based on the heterogeneity of estiates in the inner path odel, as it is ipleented in a software application for statistical coputation. This new approach perits reliable identification of distinctive custoer segents with their characteristic estiates for relationships of latent variables in the structural odel. Second, it presents an application of the approach to two nuerical exaples, using experiental and epirical data, as a eans of verifying the ethodology's usefulness for ultigroup path analyses in areting research. Third, it analyses the advantages of finite ixture partial least squares to a sequential clustering strategy. Fourth, the initial application and critical review of the new segentation technique for partial least squares path odeling allows us to unveil and discuss soe of the technique's probleatic aspects and to address significant areas of future research. Key words: segentation, latent variable odels, ixture odels easureent, custoer satisfaction, brand preference Universität Haburg Tor zur Welt der Wissenschaft Von-Melle-Par 5 0146 Haburg www.henrisattler.de

1. Introduction Structural equation odeling (SEM) and path odeling with latent variables (LVP) are used to easure coplex cause-effect relationships (Fornell/Larcer 1981; Chin 1998a; Steenap/ Baugartner 000). Such odels are often applied in areting to perfor research on brand equity (Yoo et al. 000), consuer behavior (Sargeant et al. 006) or custoer satisfaction (Anderson/Sullivan 1993; Chun/Davies 006). Covariance-based structural equation odeling (CBSEM, Jöresog 1978) and partial least squares analysis (PLS, Lohöller 1989) constitute the two corresponding, yet distinctive (Schneeweiß 1991), statistical techniques for assessing causeeffect-relationship-odels with latent variables. Copared to CBSEM, Wold's (198) basic PLS design or basic ethod of soft odeling is rather a different than alternative ethodology for estiating these odels (Fornell/Boostein 198). Soft odeling refers to the ability of PLS to be ore flexible in handling various odeling probles in situations where it is difficult or ipossible to eet the hard assuptions of ore traditional ultivariate statistics. Within this context, "soft" is only attributed to distributional assuptions and not to the concepts, the odels or the estiation techniques (Lohöller 1989). However, CBSEM is custoarily used in areting to estiate relationships in cause-effect odels via latent variables and epirical data. Apparently, there has been little concern about the frequent inability of areting data to eet ethodological requireents or about the coon occurrence of iproper solutions (Fornell/ Boostein 198; Baugartner/Hoburg 1996). Representing a well-substantiated alternative to CBSEM, PLS is relatively unnown and rarely used in areting research, which fails to appreciate its iportance for estiating LVP in a variety of contexts, ranging fro theoretical and applied research in areting, anageent and other social sciences disciplines. Recognizable PLS-based LVP analyses in business research are presented by, for exaple, Fornell et al. (1985); Fornell et al. (1990); Fornell et al. (1996); Gray/Meister (004); Venatesh/Agarwal (006). Nevertheless, the statistical instruents needed to copleent the PLS ethod for business research are not well developed. This paper addresses a ey extension of PLS for segenting data on the heterogeneity in inner path odel estiates. We focus on custoer satisfaction to identify and treat heterogeneity aong consuers by segentation as a eans of presenting the benefits of the ethod for PLS path odeling in areting research. Custoer satisfaction has becoe a fundaental and well docuented construct in areting that is critical to deand and to any business's success given its iportance and established relation to custoer retention and corporate profitability (Anderson et al. 1994; Mittal et al. 005; Morgan et al. 005). Although it is often acnowledged that truly hoogeneous segents of consuers do not exist, recent studies report that unobserved custoer heterogeneity does exists within a given product or service class (Wu/Desarbo 005). This is critical for foring groups of consuers that are hoogeneous in ters of the benefits they see or their response to areting progras (e.g. product offering, price discounts). Segentation is therefore a ey eleent for areters in developing and iproving their targeted areting strategies. Since its foral introduction in the 1950s, aret segentation reains one of the priary areting ideas for product developent, areting strategy and understanding custoers. However, the true distribution heterogeneity is never nown a priori and, thus, there are cases where it is hard to find hoogeneous custoer segents. The developent of analytic ethods for segenting arets has lagged behind their need in business applications (Cohen/Raasway 1998). In SEM, for exaple, sequential clustering techniques, such as K-eans or tree - 1 -

clustering, usually do not provide outcoes that further distinguish segent-specific estiates because they cannot account for heterogeneity of latent variables and their relationships within the structural odel. We believe that latent segentation odels will assue an iperative role in enhancing CBSEM and PLS in the next wave of analytic procedures. Latent class odeling (Kaaura/Russell 1989) is a useful classification tools for uncovering groups of persons with siilar preferences and sensitivities. In CBSEM, Jedidi et al. (1997) pioneered this field of research and proposed the finite ixture SEM approach, i.e., a procedure that blends finite ixture odels and the expectation-axiization (EM) algorith (McLachlan/ Basford 1988; Wedel/Kaaura 000; McLachlan/Krishnan 004). Although the original technique extends CBSEM, and is ipleented in software pacages for statistical coputations, e.g. Mplus (Muthén/Muthén 1998), it is inappropriate for PLS because of dissiilar ethodological assuptions (Fornell/Boostein 198). For this reason, Hahn et al. (00) introduced the finite ixture partial least squares (FIMIX-PLS) ethod that cobines a finite ixture procedure with an EM-algorith specifically coping with the ordinary least squares (OLS)-based predictions of PLS. Figure 1: Analytical steps of FIMIX-PLS Building on the guiding articles by Jedidi et al. (1997) and Hahn et al. (00), this paper presents FIMIX-PLS as it is ipleented for the first tie in a statistical software application and, thereby, is ade broadly applicable for research in social sciences as priary approach for segenting data based on PLS path odeling results. This research is iportant to expand the ethodological toolbox in PLS. We systeatically apply FIMIX-PLS, as depicted in Figure 1, on the first nuerical exaple presented in literature with experiental data and on the second application with epirical data. These inds of segentation results allow to further differentiate standard PLS path odeling estiates and, hence, to deonstrate the potentials of FIMIX-PLS for uncovering distinctive groups of custoers for the relationships within the inner PLS path --

odel. Furtherore, these analyses reveal iportant ethodological iplications that have not been addressed yet. The following sections of this paper, which introduce the ethodology, evaluate results and cover ex post analysis of FIMIX-PLS results, are dedicated to the different analytical steps presented in Figure 1. We particularly focus on step two to show that FIMIX-PLS accurately identifies a priori created segents for the siulated data presented in section 4. In section 5, we eploy a areting related LVP exaple and epirical data to apply all four steps of FIMIX- PLS for identifying a ey custoer segent for shaping targeted areting strategies. Finally, we address the issue of whether the new segentation technique is advantageous for PLS path odeling copared with a sequential data analysis strategy (section 6). Before drawing our conclusions (section 8), we distinguish, in section 7, sets of probles and discuss the need for future research that eerges fro our nuerical exaples and initial review of the FIMIX-PLS ethodology.. Methodology SartPLS.0 (Ringle et al. 005) was the first statistical software application for (graphical) path odeling with latent variables eploying both the basic PLS algorith (Wold 198, 1985; Lohöller 1989) as well as FIMIX-PLS capabilities for the ind of segentation proposed by Hahn et al. (00). This section concisely presents the new approach for segentation as it is ipleented into the statistical software application. In the first step of FIMIX-PLS (Figure 1), a path odel is estiated by using the PLS algorith for LVP and (epirical) data for anifest variables in the outer easureent odels. The resulting scores for latent variables in the inner path odel are then eployed to run the FIMIX-PLS algorith in a second ethodological step (Figure 1). The equation below expresses a odified presentation of the relationships (Table 11 in the appendix provides a description of all of the sybols used in the equations presented in this paper): Β ηi + Γξi = ζi (1) Segent-specific heterogeneity of path odels is concentrated in the estiated relationships between latent variables. FIMIX-PLS captures this heterogeneity and calculates the probability of each observation so that it fits into each of the predeterined K nubers of classes. The segent-specific distributional function is defined as follows, assuing that η i is distributed as a finite ixture of conditional ultivariate noral densities f i ( ) : K i ~ i i i =1 η ρ f ( η ξ,b, Γ, Ψ ) () Substituting fi ( η i ξi, B, Γ, Ψ ) results in the following equation: K 1 η +Γ ξ Ψ B (B ) 1(B η +Γ ξ )) i i i i ηi ~ ρ e (3) M =1 π Ψ Equation (4) represents an EM-forulation of the log-lielihood ( lnl ) as the objective function for axiization: lnl = z iln(f ( ηi ξi, B, Γ, Ψ )) + z iln( ρ ) (4) i i - 3 -

An EM-forulation of the FIMIX-PLS algorith (figure Figure ) is used for statistical coputations to axiize the lielihood and to ensure convergence in this odel. The expectation of Equation (4) is calculated in the E-step, where z i is 1 if subject i belongs to class (or 0 otherwise). The segent size ρ, the paraeters ξ i, B, Γ and Ψ of the conditional probability function are stated (as results of the M-step), and provisional estiates (expected values), E (zi ) = P i, for z i are coputed according to Bayes' (1763/1958) theore (E-step in Figure ). ------------------------------------------------------------------------------------- // initial E-step set rando starting values for P i ; set last lnl = V ; set 0 < S < 1 repeat do begin // the M-step starts here I P i=1 i ρ = I deterine B, Γ, Ψ, calculate current lnl = current lnl last lnl // the E-step starts here if S then begin ρfi ( ηi ξi, B, Γ, Ψ ) Pi = i, K ρfi ( ηi ξi, B, Γ, Ψ ) =1 last lnl = current lnl end end until < S ------------------------------------------------------------------------------------- Figure : The FIMIX-PLS algorith Equation (4) is axiized in the M-step (Figure ). This part of the FIMIX-PLS algorith accounts for the ost iportant changes to fit the finite ixture approach to PLS copared with the original finite ixture structural equation odeling technique (Jedidi et al. 1997). Initially, new ixing proportions ρ are calculated by the average of adjusted expected values P i that result fro the previous E-step. Thereafter, optial paraeters for B, Γ and Ψ are deterined by independent OLS regressions (one for each relationship between latent variables in the structural odel). ML estiators of coefficients and variances are assued to be identical to OLS predictions. The following equations are applied to obtain the regression paraeters for latent endogenous variables: Y = η and X = (E, N ) (5) i i i i i {ξ 1,...,ξ A }, A 1,a = 1,...,A ξ is regressor of E = a i (6) else - 4 -

{η 1,..., ηb }, B 1,b = 1,..., B η is regressor of N = i else (7) The closed for OLS analytic forula for τ and ω is expressed as follows: 1 τ = ( γ ),( β ) = P (X X ) P (X Y ) a i i i i i i b i i (8) Pi (Yi Xiτ )(Yi Xiτ ) i ω = cell ( ) of Ψ = (9) Iρ As a result, the M-step deterines new ixing proportions ρ, and the independent OLS regressions are used in the next E-step iteration to iprove the outcoes for P i. The EM-algorith stops whenever lnl hardly iproves, and an a priori specified convergence criterion is reached. This and other probleatical issues lined to FIMIX-PLS are pointed out in the following sections and suarized in section 7. 3. Identifying an appropriate nuber of segents and ex post analysis The ost iportant FIMIX-PLS coputational results are the probability P i, the ixing proportions ρ, class-specific estiates B and Γ for the inner relationships of the path odel and Ψ for the regression variances. In particular, with regard to the finite ixture's probabilities P i of observations to fit into the predeterined nuber classes, it ust be decided if FIMIX-PLS allows to detect and treat heterogeneity aong consuers in the inner PLS path odel estiates by (unobservable) discrete oderating factors. This is explored in the next analytical step with analyses for different nubers of K classes (step 3 in Figure 1). The nuber of segents is usually unnown and the process of identifying an appropriate nuber of classes is not clear-cut when applying FIMIX-PLS. A statistically satisfactory solution for this analytical procedure does not exist for several reasons (Wedel/Kaaura 000). One reason is that the ixture odels are not asyptotically distributed as chi-squares and disallow the lielihood ratio statistic. Consequently, the FIMIX-PLS procedure ust be repeatedly perfored with consecutive nubers of latent classes K (e.g. to 10 ). Another reason is that the algorith (Figure ) converges for any given nuber of K classes. This eans that statistically noninterpretable outcoes for the class-specific estiates B and Γ of the inner path odel relationships and for Ψ of the regression variances for latent endogenous variables are coputed when the nuber of classes is increased. Segent size is a useful indicator for stopping the analysis of additional nubers of latent classes for the sae of avoiding unreasonable FIMIX- PLS results is the segent size (section 5). At a certain point, an additional class has only a sall segent size, which explains a arginal portion of heterogeneity in the overall set of data. The eerging statistically coprehensible FIMIX-PLS estiates for different K nubers of classes are then copared for criteria such as the lnl K, the Aaie Inforation Criterion ( AIC K ), the AIC Controlled ( AICC K ) or the Bayesian Inforation Criterion ( BIC K ). These heuristic easures perit an evaluation of FIMIX-PLS coputations and the quality of their - 5 -

segentation. The ain goal of this analysis is to capture the heterogeneity of the inner PLS path odel grouping data in accordance with the FIMIX-PLS results. Within this context, the nored entropy statistic (Raasway et al. 1993) is a critical criterion for analyzing class specific FIMIX-PLS results. This criterion indicates the degree of separation for all observations and their estiated ebership probabilities P i on a case-by-case basis, and it subsequently reveals the ost appropriate nuber of latent classes for segentation: Piln(Pi ) i ENK = 1 (10) Iln(K) The EN is liited between 0 and 1, and the quality of separation of derived classes coensurate with the increase in EN K. Application of FIMIX-PLS furnishes evidence that values of EN above 0.5 result in estiates for P i that perit unabiguous segentation. The exaple with epirical data in section 5 deonstrates this ind of segentation. In this case, ost observations are associated with high probabilities of ebership in a certain class. Hence, the entropy criterion is especially relevant for assessing whether a FIMIX-PLS solution is interpretable or not. The segents are fuzzy in situations where a certain nuber of classes is identified as ost appropriate, based on the heuristic evaluation, but the EN K is considerably below 0.5. This eans that only parts of the subjects belong to a certain class. Fuzzy class eberships prevent eaningful a priori segentation for specific PLS estiations, a coprehensible interpretation of results and sound establishent of anagerial iplications. Under such circustances, and in cases where the differences between the evaluation criteria for FIMIX-PLS results of different nubers of classes only slightly differ, the highest probability per observation and its distribution regarding the entire set of data needs to be analyzed (section 5). The ore that observations exhibit high ebership probabilities, e.g. higher than 0.8, the better they uniquely belong to a specific class and can be well separated. An explanatory variable ust be uncovered in the ex post analysis (step 4 in Figure 1) in situations where FIMIX-PLS results indicate that heterogeneity in the overall set of data can be reduced by segentation using the best fitting nuber of K classes. Correspondingly, data is segented and used as new input for segent-specific LVP coputations with PLS. This process produces differentiated PLS odeling results and facilitates ultigroup PLS analyses (Chin/ Dibbern 006). An explanatory variable ust include both siilar grouping of data, as indicated by the FIMIX-PLS results, and interpretability of the distinctive clusters. This ind of analysis is essential for exploiting FIMIX-PLS findings for PLS path odeling, and it is the ost challenging analytical step to accoplish. For this reason, an ex post analysis of the estiated FIMIX- PLS probabilities of ebership eploys an approach proposed by Raasway et al. (1993). While this systeatical search uncovers explanatory variables that fit well with the FIMIX-PLS results in ters of data grouping, a logical search alternatively focuses for the ost part on the interpretation of results. In this case, certain variables with high relevance for explaining the expected differences in segent-specific PLS path odel coputations are exained for their ability to for groups of observations that atch FIMIX-PLS results. Both approaches ay lead to different nonetheless reasonable results as deonstrated by a full FIMIX-PLS analysis in the exaple using epirical data (section 5). - 6 -

4. Nuerical exaple using experiental data Suppose that a aret researcher has forulated a LVP on theoretically well developed causeeffect relationships. The researcher suspects, however, that an unobserved oderating factor accounts for heterogeneity or that the data belongs to a finite nuber of segents. In this instance, the theoretical assuptions can be used to identify a priori oderating factors that account for consuer heterogeneity in PLS path odels. But this strategy is not feasible in any areting applications (Jedidi et al. 1997) and gives rise to analytical techniques lie FIMIX- PLS. To deonstrate the potentials of FIMIX-PLS to detect and treat unobserved heterogeneity in the inner path odel relationships, this ethodology is applied on a nuerical exaple using experiental data for the anifest indicator variables in the outer easureent odel. Although the process of naing variables in this test set is not relevant, we use a areting related exaple to deonstrate that FIMIX-PLS reliably identifies and separates distinctive groups of custoers in the overall set of siulated data. In ters of heterogeneity in the structural odel, it ight be desirable to identify and describe price sensitive consuers (Ki et al. 1999) and consuers who have the strongest preference for another particular product attribute (Allenby et al. 1998), e.g. quality. Thus, the path odel for our exaple with experiental data consists of one endogenous latent variable Satisfaction, two exogenous latent variables Price and Quality in the inner odel (Dillon et al. 1997; Desarbo et al. 001). The used experiental set of data consist of the following equally sized segents: Price-oriented custoers (segent 1): This segent is characterized by a strong relationship between Price and Satisfaction and a wea relationship between Quality and Satisfaction. Quality-oriented custoers (segent ): This segent is characterized by a strong relationship between Quality and Satisfaction and a wea relationship between Price and Satisfaction. Each latent exogenous variable (Price and Quality) is operationalized with seven reflective indicator variables. The latent endogenous variable (Satisfaction) is easured by two anifest variables in a reflective easureent odel. According to forat presented in Table 1, the underlying case values of the anifest variables are generated on a scale ranging fro 1 to 7. For instance, the case values 1 to 0 of the anifest indicators Price1 to Price7, Satisfaction1 and Satisfaction are norally distributed rando nubers, with µ = 6 and σ = 0.1. The case values 1 to 0 of the anifest indicators Quality1 to Quality7 are norally distributed rando nubers, with µ = 4 and σ = 1. Case Price Quality Satisfaction 1 0 µ = 6.0 / σ = 0.1 µ = 4.0 / σ = 1.0 µ = 6.0 / σ = 0.1 1 40 µ = 4.0 / σ = 1.0 µ = 6.0 / σ = 0.1 µ = 6.0 / σ = 0.1 41 60 µ =.0 / σ = 0.1 µ = 4.0 / σ = 1.0 µ =.0 / σ = 0.1 61 80 µ = 4.0 / σ = 1.0 µ =.0 / σ = 0.1 µ =.0 / σ = 0.1 Table 1: Data generation schee We use the SartPLS.0 (Ringle et al. 005) software application for the PLS path odel estiation and the FIMIX-PLS analysis. The standard PLS procedure is executed with the overall set of siulated data for anifest variables as input to easure the LVP (step 1 in Figure 1). All - 7 -

estiates for path coefficients are at high levels. We follow the suggestion for a PLS odel evaluation by Chin (1998b), and all iniu requireents for the outer and inner easureent odel are et (Table 1, appendix). For exaple, the outer loadings are all above 0.7 while the inner odel weights of Price and Quality on Satisfaction both have a value of 0.7 and therefore attain highly significant levels. This causes a substantial R of 0.93 for the latent endogenous variable Satisfaction. This ind of analysis deterines the quality of PLS path odel estiations by assessing outer and inner easureent odels and, thus, partial odel structures for certain non-paraetric evaluation criteria that ust satisfy iniu requireents (Wold 1980; Chin 1998b). Methodological iplications of PLS path odeling (Wold 198, 1985; Lohöller 1989), especially its distribution-free character, do not perit the application of paraetric global goodness of fit easures that are used for CBSEM (Jöresog 1993). As a substitute, Tenenhaus et al. (005) propose the geoetric ean of the average counality (outer odel) and the average R (inner odel) that is liited between values of 0 and 1 as overall goodness of fit (GoF) easure for PLS: GoF = counality R = 0.865 0.93 = 0.898 (11) An interpretation of these PLS path odeling results - without testing for different segents in relation to heterogeneity in the estiates for the inner odel - causes insufficient and isleading conclusions. In our exaple, Satisfaction could be coprehended as being explained by the Price and by the Quality construct for all custoers, accopanied by recoendations that are adjusted accordingly for the areting strategy (Dillon et al. 1997). By contrast, application of FIMIX-PLS uncovers heterogeneity and perits further differentiated conclusions. The consequences are deonstrated by applying FIMIX-PLS to K = classes and the latent variable scores that eerge fro the standard PLS procedure (step in Figure 1). Since the custoer segents for the underlying siulated data is nown, testing different nubers of classes and coparing the results for FIMIX-PLS evaluation criteria is not required for this exaple. An analysis for an unnown nuber of K classes is presented in section 5. The FIMIX-PLS results in Table show that both groups of consuers in the experiental set of data are identified. Segent 1 has a strong effect of Price on Satisfaction and indicates a wee relationship between Quality and Satisfaction. Segent has a strong effect of Quality on Satisfaction and indicates a wee relationship between Price and Satisfaction. Cases 1 0 and cases 41 60 are perfectly assigned to segent 1, and cases 1 40 and cases 61 80 are perfectly assigned to segent ( P i has either a value of 0 or 1 for the final segentation). Consequently, EN has a value of 1.0 indicating perfect separation of the two groups of observations. Price Satisfaction Quality Satisfaction Standard PLS 0.7 0.7 FIMIX-PLS segent 1 0.7 0.0 FIMIX-PLS segent 0.0 0.7 Table : Inner odel weights In the above nuerical exaple with siulated data, we do not perfor step 3 of the full FIMIX- PLS analysis. Rather we test the FIMIX-PLS results for segent-specific PLS analyses according - 8 -

to step 4 in Figure 1. Therefore, the full set of experiental data is divided into two sets of data regarding the probability of ebership P i of case i to class. This data is separately used as input atrices for anifest variables to estiate the path odel for each group of consuers with PLS. This final analytical step essentially achieves the FIMIX-PLS results for segent-specific relationships in the structural odel. While the lower relationship in the inner path odel for each group of price or quality oriented consuers reains at a value of 0.0, the stronger relationship rises above 0.7 causing excellent levels in the values for R of Satisfaction. As illus- trated by this nuerical exaple with siulated data, FIMIX-PLS is capable to identify and treat heterogeneity of inner path odel estiates by segentation. Subsequent group-specific PLS analyses are iportant for further differentiated path odel estiates for heterogeneous groups of consuers in the overall set of observations to guide additional interpretations and specific recoendations for the targeted use of the areting-ix instruents. 5. Mareting exaple using epirical data When researchers wor with epirical data and do not have a priori segentation assuptions to capture unobserved heterogeneity in the inner PLS path odel relationships, FIMIX-PLS is often not as clear-cut as deonstrated in the foregoing exaple that is based on siulated data. Until now, research efforts to apply FIMIX-PLS and to assess its usefulness for expanding the ethodological toolbox in areting was restricted by the unavailability of a statistical software application for this ind of analysis. Since such functionalities are provided as presented in section, extensive use of FIMIX-PLS with epirical data in future research ought to furnish additional findings about the ethodology and its applicability. For this reason, we ae use of that technique for a areting related path odel and epirical data fro Gruner+Jahr's 'Brigitte Counication Analysis 00'. Gruner+Jahr is one of the leading publishers of printed agazines in Gerany. They have been conducting their Counication Analysis Survey every other year since 1984. In the survey, over 5,000 woen answer nuerous questions on brands in different product categories and questions regarding their personality. The woen represent a cross section of the Geran feale population. We choose answers to questions on the Benetton fashion brand nae (on a four-point scale fro 'low' to 'high') in order to use the survey as a areting-related exaple of FIMIX- PLS-based custoer segentation. We assue that Benetton's aggressive and provocative advertising in the 1990s resulted in a lingering custoer heterogeneity that is ore distinctive and easier to identify copared with other fashion brands in the Counication Analysis Survey (e.g. Esprit or S.Oliver). The scope of this paper does not include a presentation of a theoretically hypothesized LVP in the field of areting and its PLS-based estiation by epirical eans (Bagozzi 1994). Consequently, we do not provide an extensive presentation of the survey data or a discussion if one ought use CBSEM or PLS to estiate odel paraeters (Bagozzi/Yi 1994) or if the easureent odels of the latent variables should be operationalized as forative or reflective (Diaantopoulos/Winlhofer 001; Rossiter 00; Jarvis et al. 003; MacKenzie et al. 005). Our goal is to deonstrate the applicability of FIMIX-PLS to epirical data and to illustrate a reduced cause-effect-relationship-odel on branding (Yoo et al. 000) that principally guides all inds of areting-based LVP analyses that eploy the new segentation technique. The PLS - 9 -

path odel for Benetton's brand preference consists of one latent endogenous Brand Preference variable, and two latent exogenous variables, i.e., Iage and Person, in the inner odel. All latent variables are operationalized via a reflective easureent odel and the anifest variables fro Gruner+Jahr's 'Brigitte Counication Analysis 00'. Figure 3 illustrates the path odel that eploys the latent and the particular anifest variables. The basic PLS algorith Loh89 is applied to estiate LVP using the SartPLS.0 software application (step 1 in Figure 1). I have a clear ipression of this brand The brand can be trusted 0.899 0.860 Iage Is odern and up to date Represents a great style of living 0.795 0.83 0.43 Brand preference 0.944 Sypathy Fashion is a way to express who I a 0.801 0.39 0.930 Brand usage I often tal about fashion 0.894 0.177 A brand nae is very iportant to e I a interested in the latest fashion brands 0.850 0.831 Person Figure 3: The brand preference odel As in section 4, we follow the suggestions of Chin (1998b) for evaluating PLS estiates. An overview of the evaluation criteria for the results of PLS path odeling is provided in table Table 13 in the appendix. All relationships in the reflective easureent odels have factor loadings at sufficiently high levels (the sallest loading has a value of 0.795 ). Moreover, the average variance extracted (AVE) and ρ c exhibit satisfactory results. Both relationships in the inner path odel are at statistically significant levels for explaining the latent endogenous variable (Table 13 in the appendix provides results of the bootstrapping procedure). The latent exogenous Iage variable (weight of 0.43) exhibits a strong relationship to the latent endogenous Brand Preference variable while the influence of the latent exogenous Person variable is considerably weaer (weight of 0.174 ). Thus, the R of Brand Preference has a value of 0.39, which is a oderate level for PLS path odels. The average counality of the three reflective easureent odels is relatively high, so that R of the latent endogenous Brand Preference variable ainly causes a GoF outcoe that only is at a oderate level: GoF = counality R = 0.748 0.39 = 0.43 (1) In the next analytical step, the FIMIX-PLS odule of SartPLS.0 is applied to custoer segentation based on the estiated scores for latent variables (step in Figure 1). In this exaple, - 10 -

FIMIX-PLS results are coputed for two classes. Thereafter, the nuber of K classes is successively increased. A coparison of the class-specific FIMIX-PLS coputations for heuristic evaluation criteria (section 3), as presented in Table 3, reveals that the choice of two groups is appropriate for custoer segentation purposes. All relevant evaluation criteria considerably decrease in the ensuing nubers of classes. Nuber of segents lnl AIC BIC AICC EN K = 713.33 1448.466 1493.50 1493.545 0.501 K = 3 94.15 1954.431 097.784 097.863 0.16 K = 4 1053.389 19.793 450.830 450.97 0.30 K = 5 1117.976 441.388 846.874 847.097 0.14 K = 6 1140.018 36.037 40.41 40.93 0.70 Table 3: Model selection The choice of two groups for a priori segentation of the data is priarily supported by the EN of 0.501 for K = classes. It is at a relatively high level copared to the EN of 0.43 arrived at in the only other proficient FIMIX-PLS segentation presented thus far in literature by Hahn et al. (00). As illustrated in Table 4, two thirds of all our observations are well assigned to one of the two classes with a probability P i of ore than 0.7. These probabilities considerably decline for higher nubers of K classes, which indicates an increased fuzziness of segentation that is also depicted by the lower EN. Consequently, a result of EN of 0.5 or higher for a certain nuber of FIMIX-PLS classes perits unabiguous segentation of data. P i K = K = 3 K = 4 K = 5 K = 6 0.900 1.000 0.491 0.800 0.900 0.063 0.00 0.700 0.800 0.083 0.011 0.600 0.700 0.18 0.50 0.00 0.306 0.500 0.600 0.34 0.30 0.011 0.1 0.339 0.400 0.500 0.387 0.761 0.7 0.3 0.300 0.400 0.043 0.5 0.3 0.369 0.00 0.300 0.100 0.00 0.000 0.100 Su 1.000 1.000 1.000 1.000 1.000 Table 4: Overview of observations' highest probability of assignent to a certain class Table 5 indicates additional inds of assuptions regarding the nuber of classes and the ind of data heterogeneity in the inner PLS path odel. (a) As the nuber of classes increases, the large sized class is ainly reduced to create additional segents, while the size of the saller class reains relatively stable (about 0.19 for K = to K = 4 classes and about 0.145 for K = 5 and K = 6 classes). (b) Considering the decline of the outcoes for additional nubers of classes based on the EN criterion, it can be concluded that the overall set of observations for this particular analysis of the Benetton brand preference consists of a large, fuzzy group of feale consuers and a sall hoogeneous group of feale consuers. (c) The fuzziness of the larger segent cannot be further reduced by FIMIX-PLS. In the process of selecting additional classes, FIMIX- - 11 -

PLS can still identify the saller group of consuers with coparable probabilities of ebership, but is soehow abivalent when processing the large group with heterogeneous observations. The additional classes are ainly created by splitting the larger segent. As a consequence the probability of ebership P i declines resulting in the drop of EN. This indicates ethodological coplexity in the process of assigning the observations in this set of data to additional classes. FIMIX-PLS coputation forces observations to fit into a given nuber of K classes. As a result, FIMIX-PLS generates outcoes that are statistically probleatical for the segentspecific estiates B and for Γ, i.e., for the inner relationships of the path odel, and for Ψ, i.e., for the regression variances of latent endogenous variables. In this exaple, such results exhibiting inner path odel relationships and/or regression variances above one are obtained for K = 7 classes. Consequently, the analysis of additional nubers of classes can stop at this juncture in accordance with developent of segent sizes in Table 5. Nuber of K classes Class 1 Class Class 3 Class 4 Class 5 Class 6 Su 0.809 0.191 1.000 3 0.55 0.88 0.188 1.000 4 0.390 0.191 0. 0.197 1.000 5 0.487 0.56 0.149 0.09 0.017 1.000 6 0.441 0.33 0.143 0.115 0.038 0.031 1.000 Table 5: Segent sizes for different nubers of classes Table 6 presents the FIMIX-PLS results for two latent classes. In a large segent (size of 0.809 ), the explained variance of the latent endogenous Brand Preference variable is at a relatively wea level for PLS path odels ( R = 0.108 ). The variance is explained by the latent exogenous Iage variable, with its weight of 0.343, and the latent exogenous Person variable, with its weight of 0.177. A saller segent (size of 0.191 ) has a relatively high R for Brand Preference (value of 0.930 ). The influence of the Person variable does not change uch for this segent. However, the weight of the Iage variable is ore 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 iage of this brand is far ore iportant than the individuals' personality. K = 1 K = Segent size 0.809 0.191 R (of Brand Preference) 0.108 0.930 Iage Brand Preference 0.343 0.759 Person Brand Preference 0.177 0.170 Table 6: Disaggregate results (solution for two latent classes) The next step of FIMIX-PLS involves identification of a certain variable to characterize the two uncovered custoer segents. For this reason, we conducted an ex post analysis for finite ixture odels according to the approach proposed by Raasway et al. (1993). We reviewed several potential explanatory variables. Our review analysis identifies: I a very interested in the latest fashion trends, I get inforation about current fashion fro agazines for woen Brand naes are very iportant for sports wear and I lie to buy fashion designers' perfues as significant descriptors for segentation (t-statistics ranging fro 1.46 to.177 ). These variables ay be appropriate for Benetton's brand preference PLS path odel as a eans of explaining the - 1 -

separation of feale consuers in the underlying survey into two groups to account for heterogeneity in the inner odel relationships. Table 7 shows PLS results using the I lie to buy fashion designers' perfues variable for an a priori custoer segentation into two classes. Both outcoes for segent-specific LVP estiations (Table 13 in the appendix) satisfy the relevant criteria for odel evaluation (Chin 1998b). Segent 1 represents custoers that are not interested in fashion designers' perfues (size of 0.3). By contrast, segent (size of 0.777 ) is characterized by feale consuers that are attracted to Benetton and who would enjoy using Benetton products in other product categories, such as perfues. Fro a areting viewpoint, these custoers are very iportant to fashion designers who want to plan brand extensions. Segent 1 Segent GoF 0.387 0.470 R (of Brand Preference) 0.04 0.33 Iage Brand Preference 0.394 0.56 Person Brand Preference 0.164 0.104 Table 7: A priori segentation based on "I lie to buy fashion designers' perfues" Besides their applicability to explaining the FIMIX-PLS segentation for the structural odel, the variables identified in the ex post analysis do not offer uch potential for eaningful a priori segentation, except for the I lie to buy fashion designers' perfues variable. The other three variables with high t-statistics do not result in different easureents for segent-specific path odels when segented a priori into two classes. We therefore consider reasonable alternatives and test the Custoers' age variable for an a priori segentation of Benetton's brand preference LVP. The ex post analysis of FIMIX-PLS results using the ethod of Raasway et al. (1993) fails to furnish evidence for the relevance of this variable (t-statistic = 0.690 ). This fact reveals potential probles of this technique for PLS path odeling. Yet, when creating a custoer segent for feales over age 8 years (segent 1; segent size: 0.793 ) and for younger woen (segent ; segent size: 0.07 ), we do achieve a result (Table 8) that is nearly identical to the a priori segentation using I lie to buy fashion designers' perfues. Evaluation of the PLS path odeling estiates (Chin 1998b) for these two a priori segented sets of data related to Custoers' age confirs satisfactory results (Table 13 in the appendix). In this case, another iportant areting result is achieved by separating out Benetton's target segent of custoers. Segent 1 Segent GoF 0.355 0.51 R (for Brand Preference) 0.17 0.356 Iage Brand Preference 0.364 0.559 Person Brand Preference 0.158 0.110 Table 8: A priori segentation based on "Custoers' age" Overall PLS path odel estiates ust provide satisfactory results Jedidi et al. (1997) whenever FIMIX-PLS is applied for additional analytic purposes. This rule applies to our nuerical exaple with epirical data. FIMIX-PLS can be eployed for guiding essential differentiations of overall PLS path odel estiations by detecting and treating unobserved heterogeneity in the inner path odel. According to the results of analyzing K = classes, a saller and a larger group of distinctive feale consuers is identified for Benetton's brand preference. Two inds of analytical results can be obtained fro this FIMIX-PLS analysis: - 13 -

Identification of two hoogeneous segents that both have iproved results copared to the overall PLS path odel estiates. This is the ore coon ind of result that one would expect fro the exaple with siulated data in section 4. Distinction of one segent with iproved results and another segent with estiates at coparable levels in ters of overall PLS odel estiates. This is the ore unusual ind of outcoe that fits the exaple with epirical data presented in this section. The larger segent tends to have a lower R for the latent endogenous variable Brand Preference copared to overall odel estiates. Most iportant, the larger segent is fuzzy and cannot be segented into further nubers of classes. This group of custoers is not the subject of suppleentary findings. Thus, this group of custoers is not the subject of suppleentary findings and the analysis ust focus on the saller segent. According to the sall changes of its segent size when additional nubers of K classes are analyzed as well as the results for EN, R and probabilities P i, this group of consuers is relative hoogeneous and well separable. The sall segent exhibits a substantial relationship between Iage and Brand Preference, and it is highly relevant fro a areting perspective. Regarding this group of feale consuers, Brand Preference of Benetton is priarily explained by aspects that areting activities can potentially control to create a specific Iage. Characteristics of the individual Person that are ore difficult to influence are not an issue for Benetton's brand preference in this segent of costuers. Furtherore, two inds of explanatory variables are uncovered. The saller group of consuers is accordingly characterized by feales who would also lie to buy Benetton's perfue or by younger feale consuers. The process of foring these segents a priori and subsequently perforing a specific PLS path analysis on the produces further differentiated results. In the exaple of Benetton's brand preference odel, the PLS outcoes for the saller group of custoers are particularly significant for originating areting strategies in ters of potential brand extensions or this brand's target group of custoers. 6. Coparison of FIMIX-PLS to sequential data analysis The subject arises, when applying FIMIX-PLS, of whether siilar results could be achieved with traditional clustering ethods for anifest variable scores. K-eans Wedel/Kaaura (000) is one of the best clustering techniques for aret segentation. Parallel to reviewing analyses by Jedidi et al. (1997) or Hofstede et al. (1999), we copare the effectiveness of FIMIX-PLS to a sequential data analysis strategy by perforing a K-eans cluster analysis of the observed variables followed by ultigroup PLS path odeling. We ensure coparability with prior FIMIX-PLS findings by using K-eans to split both anifest experiental data and epirical data into two clusters. We liewise cluster the latent variables scores of both previous LVP exaples into two segents with K-eans since FIMIX-PLS uses inner path odel estiates. The clustering results are then used for an a priori segentation of data and for coputing their specific PLS path odel (Table 9 and Table 10). - 14 -

A priori K-eans segentation (for two classes) anifest variables A priori K-eans segentation (for two classes) latent variables PLS path odel estiations for segent 1 PLS path odel estiations for segent Segent size: 0.5 Segent size: 0.5 R : 0.931 R : 0.933 GoF: 0.860 GoF: 0.861 Price Satisfaction: 0.5 Price Satisfaction: 0.468 Quality Satisfaction: 0.474 Quality Satisfaction: 0.59 Segent size: 0.5 Segent size: 0.5 R : 0.151 R : 0.158 GoF: 0.30 GoF: 0.33 Price Satisfaction: 0.447 Price Satisfaction: 0.439 Quality Satisfaction: 0.464 Quality Satisfaction: 0.413 Table 9: K-eans segentation of experiental data The above analysis of anifest variables for the exaple with experiental and epirical data produces estiated weights in the structural odel where one relationship is higher than the other. Values for R and GoF reain at levels coparable to PLS results for the full set of data. In contrast, K-eans clustering of latent variable scores and segent-specific LVP coputation with PLS provides very siilar estiates for the two relationships in the inner path odel. Although these weights only slightly differ for segent one and two of the experiental set of data, they are considerably lower for segent two copared to segent one in the exaple with epirical data. Eploying K-eans clustering results of latent variable scores ight cause segent-specific inner path odel weights at considerable levels. Nevertheless, R and GoF are extreely low in the exaples. These PLS estiates are subsequently not useful for additional segent-specific LVP interpretations and conclusions. A priori K-eans segentation (for two classes) anifest variables A priori K-eans segentation (for two classes) latent variables PLS path odel estiations for segent 1 PLS path odel estiations for segent Segent size: 0.476 Segent size: 0.53 R : 0.41 R : 0.17 GoF: 0.407 GoF: 0.385 Price Satisfaction: 0.480 Price Satisfaction: 0.371 Quality Satisfaction: 0.131 Quality Satisfaction: 0.195 Segent size: 0.39 Segent size: 0.671 R : 0.005 R : 0.016 GoF: 0.165 GoF: 0.10 Price Satisfaction: 0.193 Price Satisfaction: 0.090 Quality Satisfaction: 0.156 Quality Satisfaction: 0.078 Table 10: K-eans segentation of epirical data Using traditional segentation techniques such as K-eans should be based on data for anifest variables. This will provide two equally sized groups for the epirical exaple, with each group having siilar relationships in the inner path odel copared to the original LVP estiation (section 5). In this instance, the weight is slightly higher for one path and soewhat lower for the other path in the structural odel of segent one, while the opposite finding hold for segent two. This process will identify a group of custoers with higher differences (segent one) and a - 15 -

group with lower (segent two) differences between the two weights in the structural odel. However, these groups of custoers are not as distinctive as they are in FIMIX-PLS segentation. It is also iportant to note, regarding the nuerical exaple with experiental data (section 4), that K-eans clustering does not identify the two groups of price- and quality-oriented custoers and their distinctive segent-specific path coefficients within the structural odel. The results of these inds of coparisons show that FIMIX-PLS suppleents traditional segentation techniques as a useful ethodology for further differentiating PLS-based LVP. Both techniques have uch in coon, i.e., the extraction of several hoogeneous groups for achieving a ore diverse and heterogeneous data set (Cohen/Raasway 1998). FIMIX-PLS nevertheless consistently outperfors the sequential data analysis strategy (Jedidi et al. 1997) for two ain reasons: In contrast to deterinistic classification with K-eans, FIMIX-PLS uses a probabilistic classification ethod that accounts for different segent sizes (Hofstede et al. 1999). FIMIX-PLS differs fro the data-driven traditional approach in ters of its siultaneous odel-based segentation and prediction. One regression equation for each segent captures the predictor-outcoe relationship at the sae tie that the uncovered segents are captured, and this process accounts for heterogeneity in the inner path odel. Cluster analysis, by contrast, is a descriptive ethodology with no independent-dependent, predictor-outcoe relationship (Cohen/Raasway 1998). K-eans perfors well in classifying observations but is poor in paraeter recovery if data is separated well into a finite nuber of classes. K-Means perfors poor in both classification and paraeter recovery, if data is not well separated. In contrast, finite ixture perfors well in all cases and provides useful diagnostic inforation (Jedidi et al. 1997). Distinctive groups of observations in the overall set of data cause heterogeneity in the estiates for the structural odel. FIMIX-PLS, in coparison, provides superior results copared to sequential data analysis techniques whenever the researcher is interested in iproved inner path odel estiates for ore hoogeneous groups of observations. 7. Methodological probles and fields of future research The FIMIX-PLS approach is theoretically lined to finite ixture odels. It shares ethodological iplications and probles. Incorporating this approach into statistical software and applying it, also reveals new probles that were not addressed prior to its initial introduction to social science research. Most iportant probles and fields of future research are connected to local optiu solutions, inappropriate FIMIX-PLS estiates and the identification of an explanatory variable in the ex post analysis. Finite ixture odels are subject to local optiu probles (Jedidi et al. 1997). The EMalgorith always converges and onotonically increases lnl towards an optiu. The stop is ore a easure of lac of progress than a easure of convergence, and there is evidence that the algorith is often stopped too early (Wedel/Kaaura 000). Experience shows that FIMIX-PLS frequently stops in local optiu solutions. This is caused by ultiodality of the lielihood causing the algorith's sensitivity to starting values. Moreover, the proble of convergence in - 16 -