Finite Mixture and Genetic Algorithm Segmentation in Partial Least Squares Path Modeling: Identification of Multiple Segments in Complex Path Models

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1 Finite Mixture and Genetic Algorithm Segmentation in Partial Least Squares Path Modeling: Identification of Multiple Segments in Complex Path Models Christian M. Ringle, Marko Sarstedt, and Rainer Schlittgen Abstract When applying structural equation modeling methods, such as partial least squares (PLS) path modeling, in empirical studies, the assumption that the data have been collected from a single homogeneous population is often unrealistic. Unobserved heterogeneity in the PLS estimates on the aggregate data level may result in misleading interpretations. Finite mixture partial least squares (FIMIX- PLS) and PLS genetic algorithm segmentation (PLS-GAS) allow the classification of data in variance-based structural equation modeling. This research presents an initial application and comparison of these two methods in a computational experiment in respect of a path model which includes multiple endogenous latent variables. The results of this analysis revealparticularadvantagesand disadvantages of the approaches. This study further substantiates the effectiveness of FIMIX-PLS and PLS-GAS and provides researchers and practitioners with additional information they need to proficiently evaluate their PLS path modeling results by applying a systematic means of analysis. If significant heterogeneity were to be uncovered by the procedures, the analysis may result in group-specific path modeling outcomes, thus allowing further differentiated and more precise conclusions to be formed. Keywords Finite mixture Genetic algorithm Heterogeneity PLS path modeling Segmentation. 1 Introduction Applications of structural equation models (SEMs) are usually based on the assumption that the analyzed data stem from a single population (i.e., a unique global model represents all the observations well). However, in many real-world applications, Christian M. Ringle (B) University of Hamburg, Institute of Industrial Management, Von-Melle-Park 5, Hamburg, Germany, cringle@econ.uni-hamburg.de and University of Technology Sydney (UTS), Centre for Management and Organisation Studies (CMOS), 1-59 Quay Street, Haymarket, NSW 2001, Australia, christian.ringle@uts.edu.au A. Fink et al., (eds.), Advances in Data Analysis, Data Handling and Business Intelligence, Studies in Classification, Data Analysis, and Knowledge Organization, DOI / , c Springer-Verlag Berlin Heidelberg

2 168 C.M. Ringle et al. this assumption of homogeneity is often unrealistic, as individuals are likely to be heterogeneous in their perceptions and evaluations of latent constructs. Traditionally, heterogeneity in SEMs is taken into account by assuming that observations can be assigned to segments a-priori, on the basis of, for example, geographic variables or stated preferences. However, postulating homogenous segments based on such prior knowledge usually provides unsatisfactory outcomes as observable characteristics often gloss over heterogeneity (Wedel & Kamakura, 2000). Owing to the increasing dissemination of PLS (Wold, 1982; Lohmöller, 1989), research interest has only recently been devoted to the question of clustering in PLS path modeling. Not only have newly developed multi-group comparison procedures for comparing segments derived from a-priori information or clustering methods been proposed lately, but also different approaches to detect latent classes, which generalize, for example, finite mixture, fuzzy regression, or genetic algorithm approaches to PLS. Sarstedt (2008) has reviewed the latent class approaches to segmentation in PLS path modeling, such as the decision-tree-based PATH- MOX (Sánchez & Aluja, 2007) method, the distance-measure-based PLS-TPM and REBUS-PLS approaches (Esposito Vinzi, Ringle, Squillacciotti, & Trinchera, 2007) PLS-GAS (Ringle & Schlittgen, 2007) and fuzzy regression techniques (Palumbo, Romano, & Esposito Vinzi, 2008), as well as the FIMIX-PLS (Hahn, Johnson, Herrmann, & Huber, 2002; Ringle, 2006; Ringle, Sarstedt, & Mooi, 2009) approach, which builds on a finite mixture of latent classes. According to Sarstedt (2008), FIMIX-PLS can currently be regarded as the primary approach to capture heterogeneity in PLS path modeling. The method has been integrated in the statistical software application SmartPLS 2.0 (Ringle, Wende, & Will, 2005) and, hence, is most commonly in PLS path modeling applications (Sarstedt, Schwaiger, & Ringle, 2009). However, FIMIX-PLS also reveals some disadvantages (Esposito Vinzi, Ringle, Squillacciotti, & Trinchera, 2007). Firstly, the approach relies on distributional assumptions to form latent classes, which is contrary to the methodological character of PLS. Moreover, the methodology only accounts for the heterogeneity in the inner path model estimates and does not provide a final clustering of data, but rather determines the probabilities of segment membership in respect of each observation. Other distance measure-based PLS typological segmentation techniques have been designed to overcome this and other limitations. However, while PLS-TPM and REBUS-PLS are limited to reflective measurment models, PLS-GAS has been specifically designed as a universal segmentation approach for PLS path modeling. PLS-GAS is a genetic search/hill-climbing hybrid that first uses a nondeterministic genetic algorithm to find a good partition by exploring the search space. The genetic algorithm does not process all possible assignments of N observations of a given number of K segments and, thus, has to cope with the potential problem of complexity in data partitioning (Maulik & Bandyopadhyay, 2000). Since there is no guarantee that the genetic algorithm will provide the global optimum solution, the best solution subsequently becomes the starting partition for a deterministic hill-climbing approach to locally improve (if possible) the initial outcome and to determine the final partition.

3 FIMIX-PLS and PLS-GAS 169 As PLS-GAS depicts the most recent advances in distance-based segmentation procedures for PLS path modeling, knowledge about the method s performance is very limited. In addition, the literature does not provide a comparative analysis of this methodology. Consequently, the goal of this research is to provide an assessment of PLS-GAS and the most commonly used approach FIMIX-PLS by conducting a computational experiment. A comparison of the outcomes exposes the strengths and weaknesses of both approaches. 2 Computational Experiment Computational experiments on causal models with latent variables require data to be generated for indicator variables that meet, after model estimation, the pre-specified parameters for relationships in the inner model, as well as in the outer models. Only few Monte Carlo simulation studies and computational experiments that require artificial data for pre-specified parameters have been presented for PLS thus far (Chin, Marcolin, & Newsted, 2003; Ringle & Schlittgen, 2007; Ringle, Wende, & Will, 2009; Ringle, Sarstedt, et al., 2009). All of these publications generate data for latent variable scores in accordance with the pre-specified relationships in the inner model, thus subsequently obtaining data for the manifest variables with respect to the measurement models parameter pre-specifications. Consequently, these studies confine their PLS analyses to reflective measurement models. However, the literature does not present suitable approaches that permit artificial data to be generated for pre-specified SEM parameters when formative measurement models are involved. Hence, this FIMIX-PLS computational experiment on artificial data also only draws on reflective measurement models. In contrast to previous experimental data analyses of FIMIX-PLS in the literature, this research employs a PLS path model with greater complexity in that it uses more than one endogenous latent variable and a higher number of a-priori defined groups. Figure 1 presents the sample path model of the simulation study with three exogenous and two endogenous latent variables. Each latent variable has four manifest variables in its reflective measurement model operationalization with pre-specified outer loadings above 0:9. Measurement invariance does not represent a critical issue in this computational experiment. Table 1 provides the a-priori determined inner weights 11, 21, 22 and 32 between the three exogenous latent variables ( 1, 2 and 3 ) and the two endogenous latent ( 1 and 2 )aswellasthe inner weight ˇ12 between the between 1 and 2 in the PLS path model. Each of the four data groups consists of 200 artificial observations. A computer program by Ringle and Schlittgen (2007) in the matrix-oriented programming language GAUSS 9.0 (Aptech, 2008), which employs a fast acceptance-rejection algorithm, allows us to obtain data for the manifest variables that meet pre-specified PLS path model parameters and distributional characteristics very precisely. This approach is consistent with the descriptions, functionalities, and examples in the PRELIS 2 software package and manual (Jöreskog & Sörbom, 1999), which was used, for example, by Chin et al. (2003) in an earlier PLS simulation study.

4 170 C.M. Ringle et al. Fig. 1 PLS path modeling results on the aggregate data level Table 1 Artificial data generation scheme k D 1 k D 2 k D 3 k D :9 0:9 0:1 0:1 21 0:1 0:1 0:9 0:9 22 0:3 0:1 0:8 0:9 32 0:8 0:9 0:3 0:1 ˇ12 0:1 0:4 0:1 0:4 The basic PLS algorithm is executed with the complete set of artificially generated data (800 cases) to estimate the path model in Fig. 1. We use the SmartPLS 2.0 (Ringle et al., 2005) software application for the PLS and FIMIX-PLS computations and follow the suggestions by Chin (1998) for a concise assessment of model estimations. The methodological implications of PLS (Wold, 1982; Lohmöller, 1989), especially with respect to its distribution-free character, do not permit the application of parametric global goodness of fit measures that are utilized in covariancebased SEMs. The results of partial model structures must satisfy the minimum requirements of certain non-parametric evaluation criteria (Chin, 1998) such as, for example, construct reliability c (>0.6), indicator reliability (>0.5), average variance extracted (>0.5) and bootstrap significance of formative measurement model and inner model relationships (Henseler, Ringle, & Sinkovics, 2009). The evaluation reveals the reliability and validity of the PLS results outer and inner path model structures. For example, all outer loadings are well above the required minimum value of 0:7 and, thus, the average variance extracted (AVE) and the composite reliability c clearly exceed the critical minimum value of each evaluation criterion. Additional tests provide support for the measures discriminant

5 FIMIX-PLS and PLS-GAS 171 validity. Despite of these good results, the inner path model exhibits only three out of five significant (p < 0:05) relationships ( 11, 21 and ˇ12 ) and two paths ( 22 and 32 ) that are not significantly different from zero. Moreover, only the R 2 value of the latent endogenous variable 1 is at a substantial level. In contrast, the latent exogenous variables explain almost no variance in respect of the latent endogenous variable 2, which subsequently exhibits a R 2 value at a level which is close to zero. At this stage, an interpretation of PLS results that do not account for unobserved heterogeneity only concentrates on the significant relationships between the latent exogenous variables 1 and 2 and the latent endogenous latent variable 1, which entail a substantial level of R 2. However, the outcomes on the aggregate data level may be ambiguous in that they involve ineffective conclusions if unobserved heterogeneity significantly affects the PLS path model estimates. In the next step, we apply FIMIX-PLS on the experimental data set. Initially, FIMIX-PLS results are computed in respect of two segments. Thereafter, the number of segments is successively increased and information criteria such as the consistent Akaike s information criterion (CAIC) or the Bayesian information criterion (BIC) are computed. All heuristics reveal the best fitting outcome (i.e., the minimum value) for K D 4 classes, which is in accordance with the prespecified data characteristics. Consequently, FIMIX-PLS probabilities of segment membership in respect of four classes are used to classify the individuals into segments. Each observation is therefore assigned to one of the four groups according to its highest probability of segment membership. The four data sets are separately (group-wise) used as input matrices for the manifest variables to estimate the PLS path model for each group of individuals. In practical applications, the next step would be to identify an observable explanatory variable which best reflects the grouping of data, as indicated by FIMIX-PLS results (Ringle, 2006; Ringle, Sarstedt, et al., 2009). The FIMIX-PLS results therefore suggest an a- priori segmentation of data into four explainable groups of observations that permit multi-group analyses and group-specific interpretations of results. Subsequently, we apply the PLS-GAS procedure on the data. As the approach does not give any indication regarding the number of segments to retain from the data, we base our model selection decision on FIMIX-PLS results and run PLS-GAS for K D 4 segments. 3 Results The final FIMIX-PLS and PLS-GAS segmentation results for this computational experiment (Table 2) are comparable with the segment-specific path model estimation results of the artificially a-priori generated data sets (Table 1). Each of the four group-specific path model estimations fulfills the reliability and validity requirements for the reflective measurement of latent variables. The group-wise R 2 outcomes increase substantially. For example, the R 2 value of the endogenous variable 2 increases from 0:074 on the aggregate data level to a value between 0:664

6 172 C.M. Ringle et al. Table 2 Segment-specific PLS path modeling results k D 1 k D 2 Orig. FIMIX-PLS PLS-GAS Orig. FIMIX-PLS PLS-GAS 11 0:920 0:909 0:944 0:919 0:931 0: :076 0:067 0:025 0:080 0:095 0: :303 0:338 0:320 0:132 0:183 0: :747 0:767 0:865 0:847 0:889 0:813 ˇ12 0:143 0:115 0:029 0:400 0:438 0:454 SSE 0:003 0:03 0:006 0:006 R 2 1 0:86 0:829 0:897 0:861 0:896 0:867 R 2 2 0:627 0:664 0:825 0:866 0:917 0:856 Size 0:250 0:292 0:188 0:250 0:228 0:299 k D 3 k D 4 Orig. FIMIX-PLS PLS-GAS Orig. FIMIX-PLS PLS-GAS 11 0:146 0:126 0:111 0:145 0:142 0: :916 0:949 0:951 0:916 0:915 0: :797 0:878 0:981 0:565 0:561 0: :301 0:201 0:250 0:092 0:090 0:073 ˇ12 0:033 0:066 0:165 0:422 0:431 0:376 SSE 0:019 0: :007 R 2 1 0:875 0:927 0:923 0:875 0:873 0:834 R 2 2 0:639 0:707 0:798 0:94 0:948 0:887 Size 0:250 0:227 0:191 0:250 0:253 0:322 * Significant at p < 0:10; ** significant at p < 0:05 and 0:948 in the final group-specific path model estimations. Instead of employing bootstrapping or permutation-based testing routines for PLS multi-group analysis, this paper draws on Henseler s (2007) non-parametric procedure for testing the difference in group-specific PLS path model estimates. This method has been specifically designed for multi-group PLS analysis and, thus, has certain advantages compared to alternatively proposed approaches (Henseler et al., 2009). Consequently, when comparing the PLS results of each segment, the inner path coefficients differ significantly (p < 0:01). As illustrated by this numerical example with simulated data, PLS path modeling results on the aggregate data level may be misleading when unobserved heterogeneity affects the inner path model estimates. FIMIX-PLS and PLS-GAS are capable of uncovering unobserved heterogeneity in the inner relationships and of dealing with it by means of segmentation. The group-specific outcomes change drastically in comparison to aggregate data-level results (Fig. 1; Table 2). However, the group-specific inner model FIMIX-PLS and PLS-GAS path coefficients fulfill the a-priori expectations very well. Table 2 shows that the segment-specific inner model PLS results of the artificially generated data sets as well as of the FIMIX-PLS and PLS-GAS segmentation results are very well matched in that they arrive at

7 FIMIX-PLS and PLS-GAS 173 comparable levels. The minimum difference between these group-specific results is j0:001j, while the maximum is j0:184j. FIMIX-PLS and PLS-GAS exhibit different performances when forming data groups. While FIMIX-PLS uncovers a slightly larger segment with a relatively low average R 2 values of the endogenous latent variables.k D 1/, it forms another segment with a relatively high average R 2 outcome.k D 4/. The FIMIX-PLS segments sizes meet the expectations from the artificial data generation procedure. These kinds of observation are in accordance with the findings by Esposito Vinzi et al. (2007). In contrast, PLS-GAS generates two larger and two smaller segments. While the FIMIX-PLS R 2 outcomes exhibit a similar pattern and heterogeneity than the originally generated sets of data, PLS-GAS shows the ability to provide relatively high segment-specific R 2 results for both endogenous latent variables in all four segments, which are more homogenous in comparison with FIMIX-PLS. Thereby, PLS-GAS forms groups of data with the highest group-specific average R 2 results for the four segments. Finally, we calculate the sum of squared errors (SSE) regarding the pre-specified and estimated path coefficients, as well as the overall root mean squared error (RMSE). Across all segments, the SSE values of FIMIX-PLS range below the respective PLG-GAS results. Consequently, with a value of 0.083, the overall RMSE of the FIMIX-PLS results is almost half of what is achieved by PLS-GAS.RMSE D 0:158/. In conclusion, FIMIX-PLS performs extremely well in respect of artificially generated data sets with low RMSE and relatively high average R 2 values for the latent endogenous variables (e.g., k D 4). PLS-GAS, on the other hand, demonstrates the capability to form very homogenous groups of data with even higher inner model R 2 values. Nevertheless, higher levels of R 2 values are obtained at the cost of parameter recovery accuracy. Maximizing the overall predictivity through segmentation fulfills the goal of PLS path modeling applications and outcomes at similar levels, which is advantageous for interpretation. However, exploiting the beneficial features of PLS-GAS in this respect requires a considerably greater computational effort. In this example, this effort entails a time factor of about 100 and above in comparison with FIMIX-PLS computations. 4 Summary and Conclusion Applications of PLS path modeling usually focus on analyzing the results of the overall data set and do not address issues related to heterogeneity. However, assuming that the data have been derived from a single population is often unrealistic; furthermore, analyses on an aggregate level can seriously distort calculation outcomes. Research interest has only recently been devoted to clustering in PLS and several novel approaches have been proposed to response-based segmentation. Whereas FIMIX-PLS has been identified as the primary and most often used approach to date (Sarstedt, 2008), PLS-GAS has been developed to overcome the former procedure s limitations. The key difference between the two

8 174 C.M. Ringle et al. approaches is that PLS-GAS provides a discrete assignment of data to certain classes, while FIMIX-PLS allows the classification of data via probabilities of membership. Furthermore, PLS-GAS overcomes shortcomings of FIMIX-PLS, for example by accounting for heterogeneity in the outer models. The initial assessment and comparison of the two approaches reveal that both detect and deal with unobserved heterogeneity very well. Whereas FIMIX-PLS achieves a higher model fit in terms of parameter recovery, PLS-GAS achieves higher and more homogenous R 2 values for all endogenous latent variables across the generated segments. However, this beneficial result of PLS-GAS, which matches the predictionoriented character of the PLS method, comes at the cost of a considerable computational demand. Thus, the application of PLS-GAS in complex models set-ups with a greater number of latent and manifest variables might prove problematic from an application-oriented point of view. Nevertheless, this weakness is compensated by the fact that PLS-GAS allows the integration of formative measurement models. Considering that formative measurement models have received increasing research interest over the last years (e.g., Gudergan, Ringle, Wende, & Will, 2008), this point especially is of great practical interest. However, the literature does not yet offer a means with which to accurately generate data with pre-specified parameters for such models. More research is necessary in this field to evaluate the performance of PLS-GAS in models with formative measures. In addition, researchers should analyze more complex models that incorporate a greater number of variables, segments and segment sizes. This would allow for a more detailed evaluation of the approaches performance. Finally, studies on the PLS-GAS parameter settings represent critical avenues of future research. Applying FIMIX-PLS and PLS-GAS to empirical examples with typical heterogeneous data is required to illustrate the applicability and problematic aspects of the procedures from a practical point of view. FIMIX-PLS effectively provides segmentation outcomes and global evaluation criteria. Comparing the FIMIX-PLS evaluation criteria for alternative numbers of segments allows to uncover unobserved heterogeneity and to determine an appropriate number of classes (Ringle, 2006; Ringle, Sarstedt, et al., 2009). This aspect is an important benefit of FIMIX- PLS, which builds on probabilities of membership for each observation. In contrast, the PLS-GAS algorithm is computationally demanding and does not provide global evaluation criteria. But PLS-GAS has the advantage of providing clearcut groups of data via PLS typological segmentation. Future studies may aim at combining the advantageous features of both methodologies. An initial analysis should build on FIMIX-PLS for determining an appropriate number of classes. Then, PLS-GAS uses the FIMIX-PLS results as starting partition for obtaining the final segmentation outcome. This kind of approach offers very promising capabilities to effectively address the problem of segmentation in PLS path modeling applications.

9 FIMIX-PLS and PLS-GAS 175 References Aptech. (2008). GAUSS for Windows Version 9.0. Black Diamond: Aptech Systems. Chin, W.W. (1998). The partial least squares approach to structural equation modeling. In G.A. Marcoulides (Ed.), Modern methods for business research (pp ). Mahwah: Lawrence Erlbaum. Chin, W. W., Marcolin, B. L., & Newsted, P. N. (2003). A partial least squares approach for measuring interaction effects: Results from a Monte Carlo simulation study and an electronic mail emotion/adoption study. Information Systems Research, 14(2), Esposito Vinzi, V., Ringle, C. M., Squillacciotti,S., &Trinchera, L. (2007). Capturing and treating unobserved heterogeneity by response based segmentation in PLS path modeling: A comparison of alternative methods by computational experiments (ESSEC Research Center Working Paper 07019). Paris: ESSEC Business School. Gudergan, S. F., Ringle, C. M., Wende, S., & Will, A. (2008). Confirmatory tetrad analysis in PLS path modeling. Journal of Business Research, 61(12), Hahn, C., Johnson, M. D., Herrmann, A., & Huber, F. (2002). Capturing customer heterogeneity using a finite mixture PLS approach. Schmalenbach Business Review, 54(3), Henseler, J. (2007). On the convergence of the partial least squares path modeling algorithm. In H. Martens, T. Næs, & M. Martens (Eds.), PLS 07 International Symposium on PLS and Related Methods Causalities Explored by Indirect Observation (pp ). Matforsk, Ås. Henseler, J., Ringle, C. M., & Sinkovics, R. R. (2009). The use of partial least squares path modeling in international marketing. Advances in International Marketing (Vol. 20, pp ). Bingley, UK: Emerald. Jöreskog, K. G., & Sörbom, D. (1999). PRELIS 2: User s reference guide. Lincolnwood: Scientific Software International. Lohmöller, J.-B. (1989). Latent variable path modeling with partial least squares. Heidelberg: Physica. Maulik, U., & Bandyopadhyay, S. (2000). Genetic algorithm-based clustering technique. Pattern Recognition, 33(9), Palumbo, F., Romano, R., & Esposito Vinzi, V. (2008). Fuzzy PLS path modeling: A new tool for handling sensory data. In C. Preisach, H. Burkhardt, L. Schmidt-Thieme, & R. Decker (Eds.), Data Analysis, Machine Learning and Applications. Proceedings of the 31st Annual Conference of the Gesellschaft fur Klassifikation e.v., Albert-Ludwigs-Universitat Freiburg, March 7 9, 2007 (pp ). Heidelberg: Springer. Ringle, C. M. (2006). Segmentation for path models and unobserved heterogeneity: The finite mixture partial least squares approach (Research Papers on Marketing and Retailing 35). University of Hamburg. Ringle, C. M., Sarstedt, M., & Mooi, E. A. (2009). Response-based segmentation using finite mixture partial least squares. Theoretical foundations and an application to American Customer Satisfaction Index data. Annals of Information Systems (Vol. 8). Berlin: Springer. Ringle, C.M., & Schlittgen, R. (2007). A genetic algorithm segmentation approach for uncovering and separating groups of data in PLS path modeling. In H. Martens, T. Næs, & M. Martens (Eds.), PLS 07 International Symposium on PLS and Related Methods Causalities Explored by Indirect Observation (pp ). Matforsk, Ås. Ringle, C. M., Wende, S., & Will, A. (2005). SmartPLS 2.0 M3 (beta). Hamburg. Retrieved from Ringle, C. M., Wende, S., & Will, A. (2009). The finite mixture partial least squares approach: Methodology and application. In V. Esposito Vinzi, W. W. Chin, J. Henseler, & H. Wang (Eds.), Handbook of partial least squares: Concepts, methods, and applications, Computational Statistics (Vol. II). Heidelberg: Springer. Sánchez, G., & Aluja, T. (2007). A simulation study of PATHMOX (PLS path modeling segmentation tree) sensitivity. In H. Martens, T. Næs, & M. Martens (Eds.), PLS 07 International Symposium on PLS and Related Methods Causalities Explored by Indirect Observation (pp ). Matforsk, Ås.

10 176 C.M. Ringle et al. Sarstedt, M. (2008). A review of recent approaches for capturing heterogeneity in partial least squares path modelling. Journal of Modelling in Management, 3(2), Sarstedt, M., Schwaiger, M., & Ringle, C. M. (2009). Do we fully understand the critical success factors of customer satisfaction with industrial good? - Extending Festge and Schwaiger s model to account for unobserved heterogeneity. Journal of Business Market Management, 3. Wedel, M., & Kamakura, W. (2000). Market segmentation: Conceptual and methodological foundations (2nd ed.). Dordrecht: Kluwer. Wold, H.O. (1982). Soft modeling: The basic design and some extensions. InK. G.Jöreskog, & H. O. Wold (Eds.), Systems under indirect observations, Part II (pp. 1 54). Amsterdam: North- Holland.