Nonlinear structural equation modeling made easy
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Current stable version: 6.0 (released 2017).
Previous stable version: 5.0 (released 2015).
Original version: 1.0 (released 2009).
- Powerful PLS-based structural equation modeling (SEM) software.
- Very easy to use, with a step-by-step user interface guide.
- Implements classic (composite-based) as well as factor-based PLS algorithms.
- Identifies nonlinear relationships, and estimates path coefficients accordingly.
- Also models linear relationships, using classic and factor-based PLS algorithms.
- Models reflective and formative variables, as well as moderating effects.
- Calculates P values, model fit and quality indices, and full collinearity coefficients.
- Calculates effect sizes and Q-squared predictive validity coefficients.
- Calculates indirect effects for paths with 2, 3 etc. segments; as well as total effects.
- Calculates several causality assessment coefficients.
- Provides a number of graphs, including zoomed 2D graphs, and 3D graphs.
WarpPLS is available through self-installing .exe files. These trial versions, valid for 3 months, are full implementations of the software, not demo versions. They are being used on various platforms, the most stable of which seem to be Windows (2000, XP, 7, 8 and 10). Non-Windows users (e.g., Mac OS X and Linux users) are advised to create a Windows partition on their computers using virtualization software, of which one of the most popular is VMWare, and install WarpPLS on that Windows partition.
Please read the instructions below carefully, particularly those related to the MATLAB Compiler Runtime. Additional installation details and discussions, as well uninstallation instructions, are available from the User Manual, linked below. In most cases, previous versions of WarpPLS and of the MATLAB Compiler Runtime may be retained on a user’s computer. Different versions of WarpPLS and of the MATLAB Compiler Runtime generally do not interfere with one other.
Current stable version: 6.0
- New users and WarpPLS 1.0 users. These instructions are for users who do not have WarpPLS installed on their computers, or who have only WarpPLS 1.0 installed on their computers. Download self-installing .exe file (about 170 MB) from: Download.com, Google Drive, Dropbox.com, or Southern USA. This file is large because you must install the MATLAB Compiler Runtime 7.14, which is included in this installation file.
- WarpPLS 2.0-5.0 users. These instructions are for users who have WarpPLS 2.0-5.0 (any version within this range) installed on their computers. Download self-installing .exe file (about 330 KB) from: Google Drive, Dropbox.com, or Southern USA. This file is small because you do not have to reinstall the MATLAB Compiler Runtime 7.14, which is not included in this installation file.
Previous stable version: 5.0
- New users and WarpPLS 1.0 users. These instructions are for users who do not have WarpPLS installed on their computers, or who have only WarpPLS 1.0 installed on their computers. Download self-installing .exe file (about 170 MB) from: Google Drive, Dropbox.com, or Southern USA. This file is large because you must install the MATLAB Compiler Runtime 7.14, which is included in this installation file.
- WarpPLS 2.0-4.0 users. These instructions are for users who have WarpPLS 2.0-4.0 (any version within this range) installed on their computers. Download self-installing .exe file (about 250 KB) from: Google Drive, Dropbox.com, or Southern USA. This file is small because you do not have to reinstall the MATLAB Compiler Runtime 7.14, which is not included in this installation file.
MATLAB Compiler Runtime 7.14
Once you download and run one of the large files above (about 170 MB), two main components are installed. The first is the free-distribution MATLAB Compiler Runtime 7.14, which contains code that enables WarpPLS to run on multiple platforms. The second component is WarpPLS.
The MATLAB Compiler Runtime 7.14 is used in versions 2.0-5.0 of WarpPLS. If you already have one of these versions installed on your computer, you should not try to reinstall the Runtime. The same Runtime cannot be installed twice on the same computer.
Register .prj extension
During the installation process you may be able to check or uncheck the option of associating the extension .prj with WarpPLS project files. This option, if checked, allows you to set, using the "Open with" option of Windows Explorer, your most recent version of WarpPLS as the default program to be launched when you double-click on .prj files.
WarpPLS' approach to have indirect effects calculated saves one from
the painful steps and time if one is to analyze mediating effects in
SPSS. (Chun M. Tang; Senior Lecturer; UCSI University, Malaysia)
Kock, N. (2018). WarpPLS User Manual: Version 6.0. Laredo, TX: ScriptWarp Systems.
Kock, N. (2015). WarpPLS User Manual: Version 5.0. Laredo, TX: ScriptWarp Systems.
Videos with overviews of steps 1 to 5
- SEM Analysis with WarpPLS (all steps) - Open or Create Project File to Save Work with WarpPLS (Step 1) - Read Raw Data Used in SEM Analysis with WarpPLS (Step 2) - Pre-process Data for SEM Analysis with WarpPLS (Step 3) - Define Variables and Links in SEM Model with WarpPLS (Step 4) - Perform SEM Analysis and View Results with WarpPLS (Step 5) -
Videos discussing specific features
- View Skewness and Kurtosis in WarpPLS - View Moderating Effects via 3D and 2D Graphs in WarpPLS - Create and Use Second Order Latent Variables in WarpPLS - Conduct a Moderating Effects Analysis in WarpPLS - View Indirect and Total Effects in WarpPLS - View and Change Settings in WarpPLS - Conduct a Multi-group Analysis with Range Restriction in WarpPLS - Chart Non-standardized Data with WarpPLS - View Nonlinear Relationships in WarpPLS - Warped Paths Become Significant in WarpPLS -
Videos discussing problems and respective solutions
- Conduct a Factor-Based PLS-SEM Analysis with WarpPLS - Use Consistent PLS Factor-Based Algorithms in WarpPLS - View and Change Missing Data Imputation Settings in WarpPLS - Isolate Mediating Effects in WarpPLS - Identify and Deal with Outliers in WarpPLS - Solve Indicator Problems in WarpPLS - Solve Collinearity Problems in WarpPLS - Use Data Labels to Better Understand Small Samples in WarpPLS -
Videos discussing advanced issues
- Explore Statistical Power and Minimum Sample Size in WarpPLS - Explore T Ratios and Confidence Intervals in WarpPLS - Explore Conditional Probabilistic Queries in WarpPLS - Explore Full Latent Growth in WarpPLS - Explore Multi-Group Analyses in WarpPLS - Explore Measurement Invariance in WarpPLS - Explore Analytic Composites in WarpPLS - Test and Control for Endogeneity in WarpPLS - Estimate Reciprocal Relationships in WarpPLS - Explore Numeric-to-Categorical Conversion in WarpPLS - Explore Categorical-to-Numeric Conversion in WarpPLS - Explore Dijkstra's Consistent PLS Outputs in WarpPLS - Explore Indicator Correlation Matrix Fit Indices in WarpPLS - Explore True Composite and Factor Reliabilities in WarpPLS -
Videos discussing domain-specific data analytics applications
- Analyze School District Data with WarpPLS - Analyze School District Data Nonlinearly with WarpPLS - Analyze School District Data Considering a Total Effect with WarpPLS - Analyze Government Corruption Data Considering Moderating Effects with WarpPLS -
- Annual full-day workshop on PLS-SEM, using WarpPLS, immediately before the annual PLS Applications Symposium. Click here and then on "Workshop on PLS-SEM".
- The annual PLS Applications Symposium is a great opportunity to collaboratively learn with audience and presenters about WarpPLS and PLS-SEM in general.
- For customized onsite face-to-face training sessions, as well as customized online training, please contact us for more details.
WarpPLS has been used to study a number of topics in a variety of disciplines and fields, including: accounting, anthropology, clinical psychology, ecology, economics, education, global environmental change, epidemiology, evolutionary psychology, exercise physiology, information systems, international business, finance, management, marketing, medicine, nursing, organizational psychology, and sociology. Click on the link below for a Google Scholar list of links to academic publications using or discussing WarpPLS, some of which are available in full text.
The full text publications below are publicly available on the web, and are also made available here with the goal of timely and wide dissemination of scholarly work. Individuals who decide to use the publications below as a basis for their research, scholarly activities, and/or educational efforts are cautioned against using the publications in ways that abuse and/or violate current laws in connection with the "fair use" of copyrighted material. For example, it is generally prohibited for an individual or organization to obtain financial advantage from the distribution of copyrighted material, if the individual or organization is not the copyright holder.
Methodological discussions based on WarpPLS
The publications below address various methodological issues, often through hands-on discussions of statistical tests that can be performed with WarpPLS. In addition to the publications below, see the User Manual, linked above; the full reference is provided.
Kock, N., & Hadaya, P. (2018). Minimum sample size estimation in PLS-SEM: The inverse square root and gamma-exponential methods. Information Systems Journal, 28(1), 227–261.
Kock, N., Avison, D., & Malaurent, J. (2017). Positivist information systems action research: Methodological issues. Journal of Management Information Systems, 34(3), 754-767.
Kock, N. (2017). Which is the best way to measure job performance: Self-perceptions or official supervisor evaluations? International Journal of e-Collaboration, 13(2), 1-9.
Kock, N. (2016). Advantages of nonlinear over segmentation analyses in path models. International Journal of e-Collaboration, 12(4), 1-6.
Kock, N. (2016). Hypothesis testing with confidence intervals and P values in PLS-SEM. International Journal of e-Collaboration, 12(3), 1-6.
Kock, N. (2015). Common method bias in PLS-SEM: A full collinearity assessment approach. International Journal of e-Collaboration, 11(4), 1-10.
Kock, N. (2015). One-tailed or two-tailed P values in PLS-SEM? International Journal of e-Collaboration, 11(2), 1-7.
Kock, N. (2014). Using data labels to discover moderating effects in PLS-based structural equation modeling. International Journal of e-Collaboration, 10(4), 1-14.
Kock, N. (2014). Advanced mediating effects tests, multi-group analyses, and measurement model assessments in PLS-based SEM. International Journal of e-Collaboration, 10(3), 1-13.
Kock, N. (2013). Using WarpPLS in e-collaboration studies: What if I have only one group and one condition? International Journal of e-Collaboration, 9(3), 1-12.
Kock, N., & Lynn, G.S. (2012). Lateral collinearity and misleading results in variance-based SEM: An illustration and recommendations. Journal of the Association for Information Systems, 13(7), 546-580.
Kock, N., & Verville, J. (2012). Exploring free questionnaire data with anchor variables: An illustration based on a study of IT in healthcare. International Journal of Healthcare Information Systems and Informatics, 7(1), 46-63.
Kock, N. (2011). Using WarpPLS in e-collaboration studies: Mediating effects, control and second order variables, and algorithm choices. International Journal of e-Collaboration, 7(3), 1-13.
Kock, N. (2011). Using WarpPLS in e-collaboration studies: Descriptive statistics, settings, and key analysis results. International Journal of e-Collaboration, 7(2), 1-18.
Kock, N. (2010). Using WarpPLS in e-collaboration studies: An overview of five main analysis steps. International Journal of e-Collaboration, 6(4), 1-11.
Philosophical and mathematical foundations of WarpPLS
WarpPLS builds on the methods of path analysis and partial least squares, developed by Sewall Wright and Herman Wold respectively. The publications below address issues related to the philosophical and mathematical foundations of WarpPLS; including issues related to evolutionary biology, numerical calculus, and mathematical statistics.
Kock, N. (forthcoming). From composites to factors: Bridging the gap between PLS and covariance‐based structural equation modeling. Information Systems Journal.
Kock, N. (2018). Single missing data imputation in PLS-based structural equation modeling. Journal of Modern Applied Statistical Methods, 17(1), 1-23.
Kock, N. (2018). Should bootstrapping be used in PLS-SEM: Toward stable p-value calculation methods. Journal of Applied Structural Equation Modeling, 2(1), 1-12.
Kock, N., & Sexton, S. (2017). Variation sharing: A novel numeric solution to the path bias underestimation problem of PLS-based SEM. International Journal of Strategic Decision Sciences, 8(4), 46-68.
Kock, N. (2017). Structural equation modeling with factors and composites: A comparison of four methods. International Journal of e-Collaboration, 13(1), 1-9.
Kock, N., & Moqbel, M. (2016). Statistical power with respect to true sample and true population paths: A PLS-based SEM illustration. International Journal of Data Analysis Techniques and Strategies, 8(4), 316-331.
Kock, N., & Gaskins, L. (2016). Simpson’s paradox, moderation, and the emergence of quadratic relationships in path models: An information systems illustration. International Journal of Applied Nonlinear Science, 2(3), 200-234.
Kock, N. (2016). Non-normality propagation among latent variables and indicators in PLS-SEM simulations. Journal of Modern Applied Statistical Methods, 15(1), 299-315.
Kock, N., & Moqbel, M. (2016). A six-stage framework for evolutionary IS research using path models: Conceptual development and a social networking illustration. Journal of Systems and Information Technology, 18(1), 64-88.
Kock, N., & Mayfield, M. (2015). PLS-based SEM algorithms: The good neighbor assumption, collinearity, and nonlinearity. Information Management and Business Review, 7(2), 113-130.
Kock, N. (2015). A note on how to conduct a factor-based PLS-SEM analysis. International Journal of e-Collaboration, 11(3), 1-9.
Kock, N. (2015). How likely is Simpson’s paradox in path models? International Journal of e-Collaboration, 11(1), 1-7.
Kock, N. (2011). A mathematical analysis of the evolution of human mate choice traits: Implications for evolutionary psychologists. Journal of Evolutionary Psychology, 9(3), 219-247.
Wold, H. (1980). Model construction and evaluation when theoretical knowledge is scarce. In J. Kmenta and J. B. Ramsey (Eds.), Evaluation of econometric models (pp. 47-74). Waltham, MA: Academic Press.
Wright, S. (1934). The method of path coefficients. The Annals of Mathematical Statistics, 5(3), 161-215.
Empirical studies employing WarpPLS
WarpPLS has been used in a wide variety of areas. The publications below are a sample that highlights the multidisciplinary reach of the software. Several of these publications are in highly selective refereed journals targeting broad disciplines and specialized fields.
Berglund, E., Lytsy, P., & Westerling, R. (2012). Adherence to and beliefs in lipid-lowering medical treatments: A structural equation modeling approach including the necessity-concern framework. Patient Education and Counseling, 91(1), 105-112.
Biong, H., & Ulvnes, A.M. (2011). If the supplier's human capital walks away, where would the customer go? Journal of Business-to-Business Marketing, 18(3), 223-252.
Brewer, T.D., Cinner, J.E., Fisher, R., Green, A., & Wilson, S.K. (2012). Market access, population density, and socioeconomic development explain diversity and functional group biomass of coral reef fish assemblages. Global Environmental Change, 22(2), 399-406.
Gebauer, J., Kline, D., & He, L. (2011). Password security risk versus effort: An exploratory study on user-perceived risk and the intention to use online applications. Journal of Information Systems Applied Research, 4(2), 52-62.
Gountas, S., & Gountas, J. (2016). How the ‘warped’ relationships between nurses' emotions, attitudes, social support and perceived organizational conditions impact customer orientation. Journal of Advanced Nursing, 72(2), 283-293.
Guo, K.H., Yuan, Y., Archer, N.P., & Connelly, C.E. (2011). Understanding nonmalicious security violations in the workplace: A composite behavior model. Journal of Management Information Systems, 28(2), 203-236.
Iconaru, C. (2013). The moderating role of perceived self-efficacy in the context of online buying adoption. Broad Research in Accounting, Negotiation, and Distribution, 4(1), 20-29.
Ketkar, S., Kock, N., Parente, R., & Verville, J. (2012). The impact of individualism on buyer-supplier relationship norms, trust and market performance: An analysis of data from Brazil and the U.S.A. International Business Review, 21(5), 782–793.
Kim, M.J., Park, C.G., Kim, M., Lee, H., Ahn, Y.-H., Kim, E., Yun, S.-N., & Lee, K.-J. (2012). Quality of nursing doctoral education in Korea: Towards policy development. Journal of Advanced Nursing, 68(7), 1494-1503.
Kock, N. (2015). Wheat flour versus rice consumption and vascular diseases: Evidence from the China Study II data. Cliodynamics, 6(2), 130–146.
Kock, N., & Chatelain-Jardón, R. (2011). Four guiding principles for research on evolved information processing traits and technology-mediated task performance. Journal of the Association for Information Systems, 12(10), 684-713.
Kock, N., & Gaskins, L. (2014). The mediating role of voice and accountability in the relationship between Internet diffusion and government corruption in Latin America and Sub-Saharan Africa. Information Technology for Development, 20(1), 23-43.
Kock, N., Moqbel, M., Jung, Y., & Syn, T. (2018), Do older programmers perform as well as young ones? Exploring the intermediate effects of stress and programming experience. Cognition, Technology & Work, 20(3), 489-504.
Kock, N., Verville, J., Danesh, A., & DeLuca, D. (2009). Communication flow orientation in business process modeling and its effect on redesign success: Results from a field study. Decision Support Systems, 46(2), 562-575.
Molina, C.M., Moreno, R.R., & Moreno, M.R. (2011). El papel moderador de la cultura en la generación de satisfacción y lealtad. Investigaciones Europeas de Dirección y Economía de la Empresa, 17(1), 57-73.
Moqbel, M. and Kock, N. (2018). Unveiling the dark side of social networking sites: Personal and work-related consequences of social networking site addiction. Information & Management, 55(1), 109-119.
Mulkeen, J., Abdou, H.A., & Parke, J. (2017). A three stage analysis of motivational and behavioural factors in UK internet gambling. Personality and Individual Differences, 107(1), 114-125.
Orzan, G., Serban, C., Iconaru, C., & Macovei, O.I. (2013). Modeling the impact of online social marketing campaigns on consumers’ environmentally friendly behavior. Research Journal of Recent Sciences, 2(3), 14-21.
Rasoolimanesh, S.M., Jaafar, M., Kock, N. and Ahmad, A. G. (2017). The effects of community factors on residents’ perceptions toward World Heritage Site inscription and sustainable tourism development. Journal of Sustainable Tourism, 25(2), 198-216.
Schmiedel, T., vom Brocke, J., & Recker, J. (2014). Development and validation of an instrument to measure organizational cultures’ support of business process management. Information & Management, 51(1), 43-56.
Schmitz, K. W., Teng, J. T., & Webb, K. J. (2016). Capturing the complexity of malleable IT use: Adaptive structuration theory for individuals. Management Information Systems Quarterly, 40(3), 663-686.
Ulvnes, A. M., & Solberg, H. A. (2016). Can major sport events attract tourists? A study of media information and explicit memory. Scandinavian Journal of Hospitality and Tourism, 16(2), 143-157.
Doctoral dissertations employing WarpPLS
WarpPLS is widely used by doctoral students in universities around the world. Many of these doctoral students are housed in highly ranked departments. The publications below are a sample of doctoral dissertations employing WarpPLS, which are often praised due to the extensive set of outputs provided by the software.
Al-Alawi, A.N.S. (2017). Holistic approach to the factors affecting individual investor’s decision making in the GCC markets: Evidence from Oman and Saudi Arabia. Plymouth, England: Plymouth University.
Alhayyan, K.N. (2012). Economic culture and trading behaviors in information markets. Tampa, FL: University of South Florida.
Bakay, A. (2012). Trust in peers, supervisor, and top management: A two-country study. Laredo, TX: Texas A&M International University.
Dohan, M. (2017). The importance of healthcare informatics competencies (HICs) for service innovation in paramedicine: A mixed-methods investigation. Hamilton, Canada: McMaster University.
Garza, V. (2011). Online learning in accounting education: A study of compensatory adaptation. Laredo, TX: Texas A&M International University.
Gaskins, L. (2013). The effect of information and communications technology (ICT) diffusion on corruption and transparency (a global study). Laredo, TX: Texas A&M International University.
Khanlarian, C.J. (2010). A longitudinal study of web-based homework. Greensboro, NC: University of North Carolina at Greensboro.
Lemos, A.Q. (2016). Effectuation e causation: Um estudo sobre o processo decisório empreendedor em redes de micros e pequenos supermercados. São Paulo, Brazil: Fundação Getúlio Vargas. (In Portuguese.)
Mohamadali, N.A. (2012). Exploring new factors and the question of ‘which’ in user acceptance studies of healthcare software. Nottingham, England: University of Nottingham.
Moqbel, M. (2012). The effect of the use of social networking sites in the workplace on job performance. Laredo, TX: Texas A&M International University.
Morrow, D.L. (2018). An exploration of the reciprocal relationship between job crafting techniques and job demands-resources job crafting. Dallas, TX: University of Dallas.
Ogara, S.O. (2011). Design for social presence and exploring its mediating effect in mobile data communication services. Denton, TX: University of North Texas.
Owen, K.D. (2016). Motivation and demotivation of hackers in the selection of a hacking task – a contextual approach. Hamilton, Canada: McMaster University.
Peroba, T.L.C. (2013). Modelo de avaliação de capital intelectual para os cursos de mestrado profissional em administração: Uma contribuição para a gestão das instituições de ensino superior. Rio de Janeiro, Brazil: Fundação Getúlio Vargas. (In Portuguese.)
Reijsen, J.V. (2014). Knowledge perspectives on advancing dynamic capability. Utrecht, The Netherlands: Utrecht University.
Roni, M.S.M.M. (2015). An analysis of insider dysfunctional behavours in an accounting information system environment. Perth, Australia: Edith Cowan University.
Taskin, N. (2011). Flexibility and strategic alignment of enterprise resource planning systems with business strategies: An empirical study. Vancouver, Canada: University of British Columbia.
Resources (with links to various files)
Right-click on the links and choose the option "Save link as ..." to save the files to a folder on you computer.
Excel files with sample datasets
- E-collaboration technologies study dataset. Spreadsheet (.xls file) containing a sample dataset from a study of teams that used e-collaboration technologies to different degrees. See this video, which uses a similar dataset. See also this publication, which includes a discussion of the notion of lateral collinearity. The data contains one instance of formative measurement. While it is based on a real study, the data has been modified (e.g., through the addition of error) for pedagogical reasons.
- SAT scores study dataset. Spreadsheet (.xls file) containing a sample dataset from a data mining analysis focused on SAT scores in school districts of a state in the USA. See this video, which uses a slightly modified dataset. The data is based on a real analysis, but has been modified (e.g., through the addition of error) for pedagogical reasons.
- E-collaboration moderation simulated dataset. Spreadsheet (.xlsx file) containing a sample dataset with data and data labels illustrating a moderating effect. See this video, which uses a slightly modified dataset. The data has been created based on a Monte Carlo simulation, and is discussed in this publication.
- Latitude and cancer rates dataset. Spreadsheet (.xlsx file) containing a sample dataset that can be used to illustrate the use of data labels to better understand small samples. See this video, which uses a slightly modified dataset. The data is based on a real study, but has been modified (e.g., through the addition of error) for pedagogical reasons.
- Job perceptions and performance dataset. Spreadsheet (.xlsx file) containing a sample dataset that can be used to illustrate a reciprocal relationship and endogeneity assessment. See this video, which also illustrates these issues based on a different dataset. The data has been created through a Monte Carlo simulation, and is based on past theory and empirical research.
Powerpoint files with model templates
- Sample model with hypotheses (.ppt file). This is a sample .ppt file with a model and hypotheses. Users can modify this file and copy and paste the model into their papers.
- Sample model with results (.ppt file). This is a sample .ppt file with model results. Specific symbols for path coefficients and R-squared values are used. Users can modify this file and copy and paste the model into their papers.
Excel files with formulas
- Mediating effects using the Sobel's standard error method. Spreadsheet (.xls file) with formulas for assessment of mediating effects using Sobel's standard error method, which is discussed in this publication. Mediating effects can be also assessed using the automatic indirect effects calculation feature of WarpPLS. This feature allows for the full assessment of mediating effects of varying levels of complexity.
- Multi-group analysis using the pooled and Satterthwaite standard error methods. Spreadsheet (.xls file) with formulas for conducting a multi-group analysis using the pooled and Satterthwaite standard error methods. The basis for comparison are coefficients generated by WarpPLS, including path coefficients and their standard errors. This spreadsheet employs an approach discussed in this publication. Multi-group analyses can also be conducted automatically using the "Explore multi-group analyses" feature of WarpPLS; related features are: "Explore measurement invariance" and "Explore full latent growth".
FAQ (questions and links to answers)
What are the main features that make WarpPLS different from other SEM software, such as PLS-Graph, SmartPLS, Amos and LISREL?
Where can I find a quick, 5-minute, overview of WarpPLS?
What is the MATLAB Compiler Runtime and how can it cause installation problems?
Can I run WarpPLS on a Mac?
How can I change the resampling method using in WarpPLS?
How can I view nonlinear relationships in WarpPLS?
Do warped paths always increase, and related P values decrease, in WarpPLS?
How can I solve problems with indicators that load poorly on their latent variables, and that have high cross-loadings, in WarpPLS?
Why are pattern cross-loadings so low in WarpPLS?
How can I solve latent variable collinearity problems in WarpPLS?
How are the model fit indices calculated by WarpPLS?
How is the warping done in WarpPLS?
What are the advantages of using WarpPLS for multiple regression analysis?