Covariance-based structural equation modeling (CB-SEM): a SmartPLS 4 software tutorial

Hair, Joseph F.; Babin, Barry J.; Ringle, Christian M.; Sarstedt, Marko; Becker, JanMichael Article — Published Version Covariance-based structural equation modeling (CB-SEM): a SmartPLS 4 software tutorial Journal of Marketing Analytics Provided in Cooperation with: Springer Nature Suggested Citation: Hair, Joseph F.; Babin, Barry J.; Ringle, Christian M.; Sarstedt, Marko; Becker, JanMichael (2025) : Covariance-based structural equation modeling (CB-SEM): a SmartPLS 4 software tutorial, Journal of Marketing Analytics, ISSN 2050-3326, Palgrave Macmillan, London, Vol. 13, Iss. 3, pp. 709-724, https://doi.org/10.1057/s41270-025-00414-6 This Version is available at: https://hdl.handle.net/10419/330731 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Journal of Marketing Analytics (2025) 13:709–724 https://doi.org/10.1057/s41270-025-00414-6 SOFTWARE REVIEW Covariance‑based structural equation modeling (CB‑SEM): aSmartPLS 4 software tutorial JosephF.Hair1 · BarryJ.Babin2 · ChristianM.Ringle3,4 · MarkoSarstedt5,6 · Jan‑MichaelBecker7 Revised: 7 May 2025 / Accepted: 8 May 2025 / Published online: 7 June 2025 © The Author(s) 2025 Abstract Covariance-based structural equation modeling (CB-SEM) enables researchers to estimate models with hypothesized causeeffect relationships between latent variables (i.e., constructs), each of which is operationalized by several items (i.e., indicators). To conduct CB-SEM analyses, researchers can rely on a range of software applications. However, many of these applications require researchers to engage in sometimes complicated and error-prone programming tasks. While IBM SPSS AMOS provides a graphical user interface(GUI), it does not fully meet the expectations of contemporary software. In order to address these challenges, the statistical SmartPLS 4 software has recently introduced a new CB-SEM module, which improves the user experience through a modern and intuitive graphical interface and comprehensive result reports. This tutorial describes the key CB-SEM analysis steps (i.e., model setup, estimation, and results evaluation) using the SmartPLS software. Keywords CB-SEM· CFA· Confirmatory factor analysis· Covariance-based structural equation modeling· SEM· SmartPLS Introduction Structural equation modeling (SEM) is a general multivariate analysis framework that allows researchers to empirically test theoretically established models with relationships between constructs, typically operationalized by multiple indicators (Sarstedt etal. 2016). To estimate structural equation models, researchers can draw on various estimators, which differ in terms of how they statistically approximate constructs and model relationships (Cho etal. 2022). Arguably the most prominent and widely developed estimator is covariance-based SEM (CB-SEM) via maximum likelihood estimation (MLE; Jöreskog 1978, 1993). Numerous articles provide introductions to CB-SEM, document its widespread impact on the management and marketing literature, and offer guidance on best practices (e.g., Bagozzi and Yi 1988; Rigdon 1998; Iacobucci 2010; Diamantopoulos and Riefler 2011; Hair etal. 2017; Baumgartner and Weijters 2020; Zyphur etal. 2023). Additionally, several textbooks offer a comprehensive understanding of how to use CB-SEM in research and practice (e.g., Diamantopoulos and Siguaw 2000; Raykov and Marcoulides 2006; Byrne 2016; Whittaker and Schumacker 2022; Kline 2023). While the origins of SEM go back to the early twentieth century, it was only with the advent of late twentieth century computing power and the development of multiple commercial software applications, such as EQS, LISREL, and Mplus, that SEM procedures became prominent among academic researchers. These software applications relied predominantly on manual specification of the model in some * Christian M. Ringle [email protected] 1 Cleverdon Chair ofBusiness, Mitchell College ofBusiness, University ofSouth Alabama, Mobile, AL, USA 2 Chair oftheDepartment ofMarketing, Analytics, andProfessional Sales (MAPS), Ole Miss Business School, The University ofMississippi, Oxford, MS, USA 3 Institute ofManagement andDecision Sciences, Hamburg University ofTechnology, Hamburg, Germany 4 College ofBusiness, Law andGovernance, James Cook University, Townsville, Australia 5 LMU Munich School ofManagement, Ludwig-Maximilians-University Munich, Munich, Germany 6 Faculty ofEconomics andBusiness Administration, Babes-Bolyai University, Cluj-Napoca, Romania 7 Department ofMarketing, BI Norwegian Business School, Oslo, Norway 710 J.F.Hair et al. form of coded syntax. Today, the lavaan package of the R statistical software (R Core Team 2022) offers a free and open-source alternative that is widely applied by researchers to perform CB-SEM analyses, but still requiring users to specify the model using syntax. An approach to avoid coding the model in a syntax emerged with the introduction of AMOS: Analysis of Moment Structures (Arbuckle 1989). A few years after AMOS’s introduction as a standalone option that relied predominantly on a graphical user interface(GUI), IBM acquired the SPSS software and it became a widely used SEM application. Academics and practitioners appreciated the ease of using the AMOS point-and-click interface to build their models and to visualize the results (Hair etal. 2014). IBM SPSS AMOS was developed more than 30years ago, however, and has changed very little in terms of the GUI. The software’s user interface and overall usability therefore do not fully align with current standards. As an alternative, the statistical SmartPLS software (https:// www. smar t pls. com; Ringle etal. 2024), which has been developed for SEM analyses using partial least squares (Wold 1982; Lohmöller 1989; Chin 1998), now includes a CBSEM module with a modern GUI—among other methods such as multiple regression analysis, logistic regression, necessary condition analysis, path analysis, and generalized structured component analysis (GSCA; Hwang and Takane 2004). Prior reviews of the SmartPLS software emphasize the software’s intuitive design and ease of use (e.g., Sarstedt and Cheah 2019). Several textbooks (e.g., Hair etal. 2022, 2024; Chua 2024), articles (Matthews etal. 2016; Sarstedt etal. 2024b; Merkle 2025), and software reviews (e.g., Memon etal. 2021; Cheah etal. 2024; Sarstedt etal. 2024b) have highlighted the potentials of the SmartPLS software and provided guidance on its use for a wide range of statistical analyses. However, the application of the SmartPLS software in a CB-SEM context has not yet been documented. In light of the above, this tutorial article demonstrates how to conduct a CB-SEM analysis using the SmartPLS 4 software. To do so, we draw on the case study used in Hair etal. (2019; i.e., Chaps. 9 to 12), which ranks among the most widely used textbook on multivariate data analysis in the social sciences (e.g., Black and Babin 2019). Our illustrations aim at helping researchers to reliably run CB-SEM analyses, thereby facilitating the use of the full spectrum of SEM estimators in order to safeguard the results’ robustness when using methods with different assumptions (e.g., Sarstedt etal. 2024a). In the following, we first describe the case study and principles of CB-SEM estimation, followed by a step-by-step description of model setup, estimation, and results evaluation using SmartPLS4. This software tutorial article concludes with additional observations and SmartPLS software extensions that can be expected in the near future. Case study andmodel estimation Our illustrations draw on the employee retention model (Fig.1) and the data (N = 400) used in Hair etal. (2019). The model has two mainelements (Anderson and Gerbing 1988): Structural and measurement models. The structural model defines the dependence (singleheaded arrows) and correlation (double-headed arrows) among constructs of interest (larger circles in Fig.1). Researchers typicallydistinguish between endogenous and exogenous constructs. Endogenous constructs are dependent in that they are being explained by other constructs in the model; unexplained variance is captured by error terms, represented by the small circles in Fig.1. Exogenous constructs only explain other constructs in the model and are thus independent. The employee retention model’s objective is to understand and explain the effects that organizational commitment (OC) and job satisfaction (JS) have on employees’ staying intentions (SI). In addition, the model considers the work environment perceptions (EP) of employees and their attitudes toward their co-workers (AC) as antecedents of OC and JS (Fig.1). The directed paths demonstrate the hypothesized relationships in the model and the double-headed arrows depict the correlations between exogenous (i.e., independent) constructs. Hair etal. (2019) allow the covariance between AC and EP to be freely estimated (i.e., unconstrained), as indicated by the double-headed arrow between these constructs, since the two constructs are both related to the environment in which they work. Leaving this arrow out would constrain the covariance between the two constructs to zero, which may result in substantial differences in the model fit and could also influence (change) other parameter estimates for the relationships between the constructs. The measurement models specify how indicators or items (rectangles in Fig.1) represent the constructs of interest. More specifically, the indicators are seen as manifestations or reflections whose variance the underlying construct explains. Analogous to the structural model, the small circles in Fig.1 represent the indicators’ error terms, capturing their unexplained variance.1 A construct is usually operationalized by several indicators aimed at ensuring the reliability and validity of the constructs meets established guidelines. Measurement theory specifies which items are associated with a particular construct and the estimates either confirm or reject the measurement theory. Research articles, especially those focusing on scale development (e.g., Relling etal. 2016; Becker etal. 2024) and scale handbooks (e.g., for marketing; Bearden etal. 2011; Bruner 2021) provide researchers with information on how to operationalize 1 Note that the measurement models can also consider potential correlations between error terms. 711Covariance-based structural equation modeling (CB-SEM): aSmartPLS 4 software tutorial constructs. Each of the constructs in the employee retention model is operationalized withbetween four and five indicators. Table10.2 in, Chapter10 Hair etal. (2019) shows the scales used for the indicators and the survey questions. To estimate a model such as in Fig.1, CB-SEM combines aspects of both confirmatory factor analysis (CFA) and multiple regression analysis. By constructing a series of equations depicting both the direct relationships between constructs and their indirect effects (i.e., through another construct in the structural model), SEM can simultaneously analyze multiple dependent relationships, enabling researchers to establish complex theoretical models that assume a network of interdependencies among constructs. More specifically, CB-SEM simultaneously estimates parameters for all equations involved in a model by trying to reproduce the observed covariance matrix and, at the same time, meet the requirements of the theoretically imposed constraints. In a model with no theoretically imposed constraints, the procedure would perfectly reproduce the observed covariance matrix. Such a model would be referred to as saturated because it has no degrees of freedom and is generally not of theoretical interest. MLE is the most widely applied method for calculating CB-SEM results. While other options are available, including generalized least squares, weighted least squares, and a variety of distribution-free estimators (e.g., Boomsma and Hoogland 2001), MLE provides a relatively robust estimation approach (Iacobucci 2009; Hair etal. 2017). The goal of MLE is to determine the parameters of the specified model (given some constraints) to obtain the best model fit. The model fit represents how well the data (more specifically, the observed covariance matrix S) matches the a-priori theoretical structure (more specifically, the model-implied covariance matrix Σ). The results can guide researchers in testing whether the hypothesized theoretical model with its estimated relationships reflects the observed data structure. In addition, researchers apply several additional criteria to evaluate and ensure the quality of the results obtained for the estimated model. Case study illustration using theSmartPLS 4 software Hair etal. (2019, Chapter10) suggest thata comprehensive CB-SEM analysis comprises of six stages: • Stage 1—Defining individual constructs. • Stage 2—Developing the overall measurement model. • Stage 3—Designing a study to produce empirical results. • Stage 4—Assessing the measurement model (CFA). • Stage 5—Specifying the structural model. • Stage 6—Assessing the structural model (CB-SEM). Stages 1 through 4 focus on the construct operationalization on the grounds of measurement theory and their Fig. 1 Employee retention model (Hair etal. 2019, Chap. 11) 712 J.F.Hair et al. validation using CFAs as highlighted by Anderson and Gerbing (1988); see also Baumgartner and Weijters (2020) and Hair etal. (2019, Chapter10). Stages 5 and 6 focus on the test of latent construct structure; that is, the directional relationships between the constructs as implied by structural theory. In practically all cases, this analysis is more constrained than the first because the typical assumption when configuring a CFA is that all constructs are related to all other constructs. Thus, the structural theory assessment in Stages 5 and 6 is formed by specifying directional relationships between relevant theoretical constructs, and by adding constraints to the CFA to indicate where relationships between factors should not exist.2 After creating Hair etal.’s (2019, Chap. 11) model using SmartPLS, and estimating it by means of CB-SEM, we specifically focus on Stages 4 and 6, which deal with the assessment procedures for the measurement and structural models. The measurement model assessment essentially applies the criteria used in a CFA, which Hair etal. (2019), for example, explain in their Chapter10. In Stage 6, assessing the structural model, researchers are primarily interested in the model fit. If the measurement model demonstrates adequate fit and other indicators of construct reliability and validity, then researchers proceed with the interpretation of the structural model. Model anddata The SmartPLS 4 software has the employee retention model included as a sample project file. The project includes the model, as described above, and the original data from Hair etal. (2019). The data are synthetic, representing employee responses as collected by an established marketing research company. Specifically, the data comprise N = 400 responses from HBAT Industries (HBAT), an international paper product manufacturer, and meet all the assumptions of CBSEM, including normally distributed data and to some extent homoskedasticity. To import the project, open the Workspace view and go to Sample projects. Navigate to the CB-SEM/CFA sample projects and tick the boxes next to Covariance-based SEM (CB-SEM) and Confirmatory factor analysis (CFA) (Fig.2). Next, the Example—Confirmatory factor analysis (CFA) and Example—Covariance-based SEM (CB-SEM) appear in the SmartPLS Workspace. The sample projects include several models and datasets. For illustrative purposes, we delete all the models except the Hair etal. MDA textbook model in the Example—Confirmatory factor analysis (CFA) and Example—Covariance-based SEM (CB-SEM) projects. When we right-click on this selection, a dialog with several options opens. Select the Delete resource option and confirm the deletion in the subsequent dialog box. Similarly, we delete all datasets except the Hair etal. (HBAT) [400]. Each Example—Confirmatory factor analysis (CFA) and Example—Covariance-based SEM (CB-SEM) project in the Workspace now only contain one model and one dataset, as shown in Fig.3. Double-clicking on Hair etal. (HBAT) [400] opens the Data View, which shows the indicators and their descriptive statistics, such as the minimum and maximum values, the mean values, skewness, kurtosis, etc. In addition, there is an option to examine the correlation matrix. Next, to start with a CFA, double-click on the Hair etal. MDA textbook model in the Example—Confirmatory factor analysis (CFA) project in order to open the Modeling View, which displays the CFA model as shown in Fig.4. The model shows the constructs and their indicators. One loading estimate per measurement model is constrained to 1 (i.e., the first indicator per construct in this example). Such a procedure allows us to identify the scale of the construct. You can set and change constraints by double-clicking on a relationship, which opens a dialog for this purpose. Constraints can also be set on constructs (or error terms) to constrain the parameter, which would be an alternative to identify the scale of a construct. The model also displays the indicators’ error terms. If reasonable, you can also add covariances/correlations between the error terms by using the Correlation option in the menu bar. They are represented Fig. 2 Importing sample projects into the SmartPLS software Fig. 3 Projects in the SmartPLS software 2 This would be the case for recursive models. In the rare instance of a nonrecursive model, the CFA may be more constrained. 713Covariance-based structural equation modeling (CB-SEM): aSmartPLS 4 software tutorial by double-headed arrows between the constructs.3 It is also possible to constrain these correlations.4 Next, we follow the steps in the SmartPLS software to obtain this model’s CFA results. CFA model estimation Before the model estimation, you need to safeguard unidimensionalty (Hair etal. 2019, Chap. 10). Unidimensional measures mean that a set of measured variables (indicators) can be explained by only one underlying construct. Unidimensionality becomes critically important when more than two constructs are involved. In such a situation, each indicator is hypothesized to relate to only a single construct. All cross-loadings and error variance and covariance are hypothesized to be zero when unidimensional constructs exist. Moreover, you need to ensure the model is identified as explained by Hair etal. (2019, Chap. 10). Model identification ensures that enough information exists to compute a solution for a set of structural equations using CB-SEM. In contrast, an identification problem (an unidentified model) leads to an inability of the proposed model to generate estimates as finding a solution is mathematically impossible. The three possible conditions of identification are overidentified, just-identified, and underidentified. The model used in this example is identified as demonstrated in detail by Hair etal. (2019, Chap. 10). The SmartPLS software uses a simple test to evaluate if the model is underidentified and issues a warning. To estimate the model by using the maximum likelihood CB-SEM algorithm, click on the Calculate button in the menu bar and select Basic CB-SEM algorithm. The SmartPLS software opens a dialog box (Fig.5), which enables the user to specify several algorithm settings, such as the maximum number of iterations and the stop criterion (Table1). We recommend researchers should keep the default settings. Next, in the Basic CB-SEM algorithm start dialog box, ensure that the box next to Open report has been ticked and click on the Start calculation button. The SmartPLS software will then estimate the CFA model, and the Results View will open. The Results View initially shows the model and selected parameter estimates on the right-hand side(Fig.6), while different result report elements appear on the left-hand side. Fig. 4 CFA of the employee retention model constructs (Hair etal. 2019, Chap. 10) 3 If you want, you can also display the correlation between the constructs in a curved manner. To do this, left-click on a connection and select it. Then a point appears in the middle of the connection, which you can click on with the left mouse button and move to change the straight lines into curved ones. 4 By default, if no correlation is drawn between error terms or exogenous constructs the correlation is constrained to zero. Drawing a correlation allows freely estimating the correlation until it is constrained to a specific value by the user by double-clicking on the correlation and setting a fixed value. Correlations (as well as all other parameters) can also be constrained to be equal by using the same string (e.g., “a”) for all correlations that should be equal. 714 J.F.Hair et al. The box labeled Graphical output on the lower left side enables you to choose different types of parameter estimates to be shown in the displayed model. For example, when selecting Path coefficients (standardized) under Structural model, and Weights/loadings (standardized) under Measurement model, the standardized coefficients’ results are shown on the model’s graphical output. Moving to the results report, we can, for example, select Factor loadings—Matrix (standardized) under Final results. When clicking on this menu item, the Results View appears on the right-hand side, showing the Factor loadings—Matrix (standardized) results. In this new output, factor loadings above 0.70 appear in green, while loadings below 0.70 are red—as we will discuss in the measurement model assessment stage. Finally, we can save the results report in the menu bar of the SmartPLS software (i.e., the results report appears under the project in the Workspace View), or in Excel format, and in HTML files for use outside of the software. Measurement model assessment Following the systematic procedure for measuring model assessment as outlined by Hair etal. (2019, Chaps. 9 and 10), we carry out the following analyses to evaluate the model fit as well as the reliability and validity of the constructs: • Overall fit. • Reliability and factor loadings. Fig. 5 Basic CB-SEM algorithm start dialog box Table 1 CB-SEM algorithms settings in SmartPLS Source: SmartPLS webpage, https:// www. smart pls. com/ docum entat ion/ algor ithmsandtechn iques/ cbsem/, Accessed May 2025 Setting Explanation Maximum iterations The maximum number of iterations the optimizer will perform. This parameter should be high enough to ensure a good model solution. The default value is 1,000, but could be higher in more complex models Starting value strategy Apply configured starting values. By checking this option, the user can specify its own starting values for the free model parameters. If this option is not selected, the software will use the default starting values Default strategy. This strategy mimics Lavaan’s default starting values. It uses Fabin-style estimates for its loadings, 0.0 for path coefficients and covariances, a 0.5*indicator variance for its error variances, and 0.05 for factor error variances One zero strategy. This strategy applies more simple starting values, 1.0 for loadings and variances, and 0.0 for path coefficients and covariances Stop criterion (gradient) The optimizer stops when one of the two stop criteria is fulfilled, and convergence to the optimum is assumed. In this case, the optimizer terminates when ||g||< stop criterion * max(1, ||x||), where ||.|| denotes the Euclidean (L2) norm. The default value is 10^-6 Stop criterion (function value) The optimizer stops when one of the two stop criteria is fulfilled and convergence to the optimum is assumed. In this case, the optimizer terminates when the decrease in the objective function (maximum likelihood value) is smaller than the recommended minimum. The condition is met if (f′–f)/f < stop criterion, where f′ is the objective value of the previous iteration and f is the objective value of the current iteration. The default value is 10^−9 Special assumptions Imply latent variable correlations. Select this option if you want to freely estimate the correlations between all the exogenous constructs. Usually, if no correlation arrow is drawn in the model, the correlation between the exogenous constructs is constrained to zero. With this option, the correlations are also estimated freely when no arrow is drawn Imply causal indicator correlations per construct. Select this option if you want to estimate the correlations between all the causal indicators of a construct. Usually, if no correlation arrow is drawn in the model, the correlation between the causal indicators is constrained to zero. With this option, the correlations are also estimated freely when no arrow is drawn Imply a variance of 1.0 for causal indicators. If we choose this option, all the variances of causal indicators are constrained to 1.0. This also overwrites use-specified values. This option should help mimic the default Lavaan results 715Covariance-based structural equation modeling (CB-SEM): aSmartPLS 4 software tutorial • Validity (i.e., convergent validity, nomological validity, discriminant validity).5 The above metrics are interpreted to confirm the measurement models. When these metrics meet established guidelines, as recommended by Hair etal. (2019), the measurement models can be confirmed. The assessment of the model’s fit builds on a comparison of the observed indicator covariance matrix (S) and the model-implied covariance matrix (Σ) by means of a χ2 test. The smaller the difference between the two covariance matrices, the better the model fit. In “close-fitting” models, the χ2 would not differ significantly from 0. However, statistical power and model complexity serve to make that possibility rare in large models with large samples. Both conceptually and practically, the model fit is the most critical result for testing a theoretical model in CB-SEM. Research has proposed a series of different metrics that quantify the degree of model fit. Hair etal. (2019, Chap. 9) describe these metrics in greater detail and offer suggestions for cutoff values (i.e., their Table9.4), depending on (1) the sample size (N), and (2) the number of observed variables in the model (m). These metrics include: • The χ2 value and the associated degrees of freedom (df). • Absolute fit indices, such as the goodness-of-fit index (GFI), the root mean square error of approximation (RMSEA), or the standardized root mean residual (SRMR). • Incremental fit indices, such as the comparative fit index (CFI) or the Tucker-Lewis index (TLI). • Badness-of-fit indices (e.g., RMSEA, SRMR). • The adjusted theoretical fit index (ATFI) offers a useful scrutiny of the theoretical structural and measurement models’ relative fit. No single “magic” fit index value separates poor-fitting and well-fitting models. Further, applying a single set of cut-off rules to all measurement models, and for that matter, to all models of any type, is not reasonable. The quality of the fit depends strongly on the model characteristics, including the sample size and the model complexity. Therefore, multiple fit indices should be used to assess a model’s goodness-of-fit. Researchers may also apply flexible cutoff values which consider the specific data (sample size) and model parameters (Niemand and Mai 2018; McNeish and Wolf 2023). To obtain the model fit results, return to the SmartPLS Workspace and double-click on the Hair etal. MDA Fig. 6 Graphical output(displaying factor correlations and standardized loadings) 5 Note: In addition, you need to ensure face validity, which is the extent to which the content of the items is consistent with the construct definition. Face validity is based solely on the researcher’s judgment and was established based on the content of the corresponding items (Hair etal. 2019, Chap. 10). 716 J.F.Hair et al. textbook model in the Example—Confirmatory factor analysis (CFA) project. The Modeling View opens, which enables you to click on the Calculate button. Select the Basic CB-SEM algorithm option for obtaining results of the CFA model, Start calculation, and Open results. In the results report, under Quality criteria, the SmartPLS software displays the Model fit outcomes as shown in Fig.7. Note that the null model is hypothesized to be the simplest model that can be theoretically justified. It serves as the baseline or comparison standard used in incremental fit indices (i.e., that’s why other indices show n/a as result for the null model in Fig.7). We focus on the estimated model’s results, which represent the outcomes of the theoretically established and specified model shown in Fig.6. For the overall fit assessment (i.e., to test the discrepancy between the sample and the model-implied covariance matrices), researchers often revert to the χ2 value, which represents a badness of fit measure in the sense that generally, a higher value is associated with relatively worse fit (depending on df). For a close fit, the statistical null hypothesis that the observed and model-implied covariance matrices do not differ would be supported. In that case, the χ2 value would not be statistically significant and, thus, not indicate that the difference between observed and modelimplied covariance matrix is different from zero (e.g., Kline 2023; Chap. 10). However, the χ2 test typically displays statistical significance (i.e., indicating a poor-fitting model)— as in our example (Fig.7). Because of this inherent limitation of the χ2-test, researchers typically report other fit statistics, most of which are mathematical variations of the χ2 value, null model χ2 value, df and sample size. The normed χ2, which is the χ2 value relative to the degree of freedom (df), give an alternative picture: Researchers consider that ChiSqr/df value of 3 (in some cases, even up to 5) or less represents a good model fit (Dash and Paul 2021). In our example, the ChiSqr/df value is 1.327 (Fig.7), which support the model fit. However, reporting the normed χ2 value without the actual χ2 value and df is inappropriate because while it is easy to know the normed value from actual (and df), the reverse is not true. For additional model fit assessment criteria, Hair etal. (2019, Chap. 9) provide critical cut-off values. For the results shown in Fig.7, we find that the root mean square error of approximation (RMSEA) has a value of 0.029, with a 90 percent confidence interval of 0.018 to 0.038. Although 0 is not in the confidence interval, the RMSEA value is relatively low. The comparative fit index (CFI) and Tucker-Lewis index (TLI) outcomes of 0.986 and 0.984 are above 0.94. The standardized root mean square residual (SRMR) has a value of 0.035, which is below 0.08. Based on these outcomes, we conclude that the model has relatively good fit. Next, we focus on the indicator reliability metric. This metric is evaluated by examining the standardized indicator loadings. Hair etal. (2019, Chaps. 9 and 10) provide detailed explanations and rules of thumb. To view the standardized loadings results in the SmartPLS software, click on Final results → Outer loadings → Matrix (standardized). The results in Fig.8 show most of the indicators’ standardized loadings are above the assumed minimum value of 0.70. Four indicators have loadings between 0.50 and 0.70, which is below the recommended guideline. While these standardized loadings are lower than desired, they do contribute in a meaningful way and are therefore acceptable in principle. Thus, in line with Hair etal. (2019, Chap. 10), we retain these indicators in the model to support content validity for these constructs.6 High indicator reliability usually results in high internal consistency reliability for the construct, which is assessed based on the coefficient alpha (i.e., Cronbach’s α). This criterion remains widely used, even though researchers acknowledge that this metric probably underestimates reliability. High values in these metrics indicate that the items consistently reflect the same underlying construct. Researchers usually expect that Cronbach’s α to be above 0.7 (Hair Fig. 7 Model fit results of the confirmatory factor analysis (CFA) 6 Note that loadings with a standardized value of 0.50 or higher are usually statistically significant. Consequently, we do not examine the individual items’ statistical significance at this stage of the measurement model assessment. However, interested researchers could check whether the loadings are significant as a component of the significance testing of the overall structural model relationships. This analysis is available in List (unstandardized) and shows all indicator loadings are significant. 723Covariance-based structural equation modeling (CB-SEM): aSmartPLS 4 software tutorial Marketing and Business Research, ed. B. J. Babin and M. Sarstedt, 121–130. Cham: Springer. Boomsma, A., and J. J. Hoogland. 2001. The Robustness of LISREL Modeling Revisited. In Structural Equation Modeling: Present and Future, ed. R. Cudeck, S. du Toit, and D. Sörbom, 139–168. Chapel Hill: Scientific Software International. Bruner, G. C. 2021. Marketing Scales Handbook: Multi-Item Measures for Consumer Insight Research, vol. 11. Fort Worth: GCBII Productions. Byrne, B. M. 2016. Structural Equation Modeling with AMOS: Basic Concepts, Applications, and Programming (Multivariate Applications), 3rd ed. Milton Park: Routledge. Cheah, J.-H., F. Magno, and F. Cassia. 2024. Reviewing the SmartPLS 4 Software: The Latest Features and Enhancements. Journal of Marketing Analytics 12:97–107. Chin, W. W. 1998. The Partial Least Squares Approach to Structural Equation Modeling. In Modern Methods for Business Research, ed. G. A. Marcoulides, 295–358. Mahwah: Lawrence Erlbaum. Cho, G., M. Sarstedt, and H. Hwang. 2022. A Comparative Evaluation of Factorand Component-based Structural Equation Modelling Approaches Under (In)Correct Construct Representations. British Journal of Mathematical and Statistical Psychology 75 (2): 220–251. Chua, Y. P. 2024. A Step-by-Step Guide to SMARTPLS 4: Data Analysis Using PLS-SEM, CB-SEM, Process and Regression. Researchtree Education. Dash, G., and J. Paul. 2021. CB-SEM vs PLS-SEM Methods for Research in Social Sciences and Technology Forecasting. Technological Forecasting and Social Change 173 : 121092. Davison, A. C., and D. V. Hinkley. 1997. Bootstrap Methods and Their Application. Cambridge: Cambridge University Press. Diamantopoulos, A., and P. Riefler. 2011. Using Formative Measures in International Marketing Models: A Cautionary Tale Using Consumer Animosity as an Example. Advances in International Marketing 10 (22): 11–30. Diamantopoulos, A., and J. A. Siguaw. 2000. Introducing LISREL. Thousand Oaks: Sage. Efron, B., and R. J. Tibshirani. 1993. An Introduction to the Bootstrap. Boca Raton: Chapman & Hall. Franke, G. R., and M. Sarstedt. 2019. Heuristics versus Statistics in Discriminant Validity Testing: A Comparison of Four Procedures. Internet Research 29 (3): 430–447. Hair, J. F., M. Gabriel, and V. Patel. 2014. AMOS Covariance-Based Structural Equation Modeling (CB-SEM): Guidelines on Its Application as a Marketing Research Tool. Brazilian Journal of Marketing -BJM 13:44–55. Hair, J. F., B. J. Babin, and N. Krey. 2017. Covariance-Based Structural Equation Modeling in the Journal of Advertising: Review and Recommendations. Journal of Advertising 46 (1): 163–177. Hair, J. F., W. C. Black, B. J. Babin, and R. E. Anderson. 2019. Multivariate Data Analysis, 8th ed. Boston: Cengage Learning. Hair, J. F., G. T. M. Hult, C. M. Ringle, and M. Sarstedt. 2022. A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM), 3rd ed. Thousand Oaks: Sage. Hair, J. F., M. Sarstedt, C. M. Ringle, and S. P. Gudergan. 2024. Advanced Issues in Partial Least Squares Structural Equation Modeling (PLS-SEM), 2nd ed. Thousand Oaks: Sage. Hayes, A. F. 2022. Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach, 3rd ed. New York: Guilford Press. Henseler, J., C. M. Ringle, and M. Sarstedt. 2015. A New Criterion for Assessing Discriminant Validity in Variance-based Structural Equation Modeling. Journal of the Academy of Marketing Science 43 (1): 115–135. Hwang, H., and Y. Takane. 2004. Generalized Structured Component Analysis. Psychometrika 69 (1): 81–99. Hwang, H., and Y. Takane. 2014. Generalized Structured Component Analysis: A Component-Based Approach to Structural Equation Modeling. Boca Raton: Chapman & Hall. Iacobucci, D. 2009. Everything You Always Wanted to Know About SEM (Structural Equation Modeling) but Were Afraid to Ask. Journal of Consumer Psychology 19 (4): 673–680. Iacobucci, D. 2010. Structural Equations Modeling: Fit Indices, Sample Size, and Advanced Topics. Journal of Consumer Psychology 20 (1): 90–98. Jöreskog, K. G. 1978. Structural Analysis of Covariance and Correlation Matrices. Psychometrika 43 (4): 443–477. Jöreskog, K. G. 1993. Testing Structural Equation Models. In Testing Structural Equation Models, ed. K. A. Bollen and J. S. Long, 294–316. Thousand Oaks: Sage. Kline, R. B. 2023. Principles and Practice of Structural Equation Modeling, 5th ed. New York: Guilford Press. Lohmöller, J.-B. 1989. Latent Variable Path Modeling with Partial Least Squares. Cham: Springer. MacCallum, R. C., M. Roznowski, and L. B. Necowitz. 1992. Model Modifications in Covariance Structure Analysis: The Problem of Capitalization on Chance. Psychological Bulletin 111 (3): 490–504. Matthews, L., M. Sarstedt, J. F. Hair, and C. M. Ringle. 2016. Identifying and Treating Unobserved Heterogeneity with FIMIXPLS: Part II—A Case Study. European Business Review 28 (2): 208–224. McNeish, D., and M. G. Wolf. 2023. Dynamic Fit Index Cutoffs for Confirmatory Factor Analysis Models. Psychological Methods 28:61–88. Memon, M. A., T. Ramayah, J.-H. Cheah, H. Ting, F. Chuah, and T. H. Cham. 2021. PLS-SEM Statistical Program: A Review. Journal of Applied Structural Equation Modeling 5 (1): i–xiii. Merkle, A. C. 2025. Item Level Correction: Detecting, Removing, and Reporting Common Methods Variance. Journal of Marketing Analytics. https:// doi. org/ 10. 1057/ s4127002500399-2. Niemand, T., and R. Mai. 2018. Flexible Cutoff Values for Fit Indices in the Evaluation of Structural Equation Models. Journal of the Academy of Marketing Science 46 (6): 1148–1172. Preacher, K. J., and A. F. Hayes. 2008. Asymptotic and Resampling Strategies for Assessing and Comparing Indirect Effects in Multiple Mediator Models. Behavior Research Methods 40 (3): 879–891. R Core Team. 2022. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https:// www.Rproje ct. org. Raykov, T., and G. A. Marcoulides. 2006. A First Course in Structural Equation Modeling, 2nd ed. Mahwah: Lawrence Erlbaum. Relling, M., O. Schnittka, C. M. Ringle, H. Sattler, and M. Johnen. 2016. Community Members’ Perception of Brand Community Character: Construction and Validation of a New Scale. Journal of Interactive Marketing 36:107–120. Rigdon, E. E. 1998. Structural Equation Modeling. In Modern Methods for Business Research, ed. G. A. Marcoulides, 251–294. Mahwah: Lawrence Erlbaum. Ringle, C. M., M. Sarstedt, N. Sinkovics, and R. R. Sinkovics. 2023. A Perspective on Using Partial Least Squares Structural Equation Modelling in Data Articles. Data in Brief 48 : 109074. Ringle, C. M., S. Wende, and J. M. Becker. 2024. SmartPLS 4. SmartPLS. https:// www. smart pls. com/ Sarstedt, M., and J. H. Cheah. 2019. Partial Least Squares Structural Equation Modeling Using SmartPLS: A Software Review. Journal of Marketing Analytics 7 (3): 196–202. Sarstedt, M., J. F. Hair, C. M. Ringle, K. O. Thiele, and S. P. Gudergan. 2016. Estimation Issues with PLS and CBSEM: Where the Bias Lies! Journal of Business Research 69 (10): 3998–4010. 724 J.F.Hair et al. Sarstedt, M., J. F. Hair, C. Nitzl, C. M. Ringle, and M. C. Howard. 2020. Beyond a Tandem Analysis of SEM and PROCESS: Use of PLS-SEM for Mediation Analyses! International Journal of Market Research 62 (3): 288–299. Sarstedt, M., S. J. Adler, C. M. Ringle, G. Cho, A. Diamantopoulos, H. Hwang, and B. D. Liengaard. 2024a. Same Model, Same Data, But Different Outcomes: Evaluating the Impact of Method Choices in Structural Equation Modeling. Journal of Product Innovation Management 41 (6): 1100–1117. Sarstedt, M., N. F. Richter, S. Hauff, and C. M. Ringle. 2024b. Combined Importance–performance Map Analysis (cIPMA) in Partial Least Squares Structural Equation Modeling (PLS–SEM): A SmartPLS 4 Tutorial. Journal of Marketing Analytics. 12:746. Sharma, P. N., M. Sarstedt, C. M. Ringle, J.-H. Cheah, A. Herfurth, and J. F. Hair. 2024. A Framework for Enhancing the Replicability of Behavioral MIS Research Using Prediction Oriented Techniques. International Journal of Information Management 78 : 102805. Voorhees, C. M., M. K. Brady, R. Calantone, and E. Ramirez. 2016. Discriminant Validity Testing in Marketing: An Analysis, Causes for Concern, and Proposed Remedies. Journal of the Academy of Marketing Science 44 (1): 119–134. Whittaker, T. A., and R. E. Schumacker. 2022. A Beginner’s Guide to Structural Equation Modeling, 5th ed. Milton Park: Routledge. Wold, H. 1982. Soft Modeling: The Basic Design and Some Extensions. In Systems Under Indirect Observations: Part II, ed. K. G. Jöreskog and H. Wold, 1–54. Amsterdam: North-Holland. Zyphur, M. J., C. V. Bonner, and L. Tay. 2023. Structural Equation Modeling in Organizational Research: The State of Our Science and Some Proposals for Its Future. Annual Review of Organizational Psychology and Organizational Behavior 10:495–517. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Joseph F. Hair is the Director of the PhD program and the Cleverdon Chair of Business at the University of South Alabama. He has authored over 100 books, including MKTG (Marketing), Cengage Learning, 14th edition 2024; Multivariate Data Analysis, Cengage Limited, U.K., 8th edition 2019 (cited 156,000+ times and is in the top five all time social sciences research methods textbooks); Essentials of Business Research Methods, Routledge, 5th edition 2024; Essentials of Marketing Research, McGraw-Hill, 6th edition 2024; A Primer on Partial Least Squares Structural Equation Modeling, Sage, 3rd edition 2022, and Advanced Issues in Partial Least Squares Structural Equation Modeling, Sage, 2nd edition 2024. He also has published numerous articles in scholarly journals such as the Journal of Marketing Research, Journal of Academy of Marketing Science, Organizational Research Methods, European Journal of Marketing, Journal of Advertising Research, Journal of Business Research, Journal of Long-Range Planning, Industrial Marketing Management, Journal of Retailing, and others. His work has been cited more than 436,000 times in academic literature and since 2018 he has been included in the Clarivate Analytics' Highly Researchers list. More information: http:// www. south alaba ma. edu/ colle ges/ mcob/ marke ting/ hair. html. Barry J. Babin Ph.D. (Louisiana State University, 1991) is Phil B. Hardin Professor of Marketing and Chair of the Department of Marketing, Analytics, and Professional Sales (MAPS) at the Ole Miss Business School. Barry also serves as the Executive Director of the Academy of Marketing Science® (www. amsweb. org). He has authored over 200 professional publications with research appearing in the International Journal of Wine Business Research, Journal of the Academy of Marketing Science, Journal of Marketing, Journal of Retailing, Journal of Business Research, Journal of Consumer Research, International Journal of Research in Marketing, the Journal of Wine Research, and others. His research emphasis areas include wine marketing, marketing analytics, meta-analysis, sales management, and designing effective value-delivering retail customer experiences. Barry is past president of the Academy of Marketing Science® (AMS) and a previous recipient of the AMS Harold W. Berkman Distinguished Service Award. He co-authored several leading books including CB: A Consumer Value Framework, Multivariate Data Analysis, and Essentials of Marketing Research. He is internationally known as an expert in marketing research and is a guest speaker at universities and conferences across the world. He has directed over 20 Dissertations/Theses and has served in total on more than 75 doctoral dissertation or HDR committees. Christian M. Ringle is a Chaired Professor and the Director of the Institute of Management and Decision Sciences at the Hamburg University of Technology (Germany), andan Adjunct Professor at the James Cook University (Australia). His research, which has been cited more than 300,000 times (Google Scholar), focuses on management and marketing topics, method development, business analytics, artificial intelligence,machine learning, and the application of business research methods to decision making. His contributions have been published in journals such as Industrial Marketing Management, International Journal of Research in Marketing, Information Systems Research, Journal of Business Research, Journal of Service Research, Journal of the Academy of Marketing Science, Long Range Planning, and MIS Quarterly. Since 2018, Christian has been included in the Clarivate Analytics’ Highly Researchers list. He regularly teaches doctoral seminars on business analytics and multivariate statistics. Christian is a co-founder and co-developer of the statistical software SmartPLS (https:// www. smart pls. com). More information: https:// www. tuhh. de/ mds/ team/ profdr-cmringle. Marko Sarstedt is a full professor of marketing at the Ludwig-Maximilians-University Munich (Germany) and an adjunct research professor at Babes-Bolyai-University Cluj-Napoca (Romania). His main research interest is the advancement of research methods to further the understanding of consumer behavior. His research has been published in Nature Human Behaviour, Journal of Marketing Research, Journal of the Academy of Marketing Science, Multivariate Behavioral Research, Organizational Research Methods, MIS Quarterly, and Psychometrika, among others. His research ranks among the most frequently cited in the social sciences with more than 200,000 citations according to Google Scholar. Marko has won numerous best paper and citation awards, including five Emerald Citations of Excellence awards and two AMS William R. Darden Awards. Marko has been repeatedly named member of Clarivate Analytics’ Highly Cited Researchers List, which includes the “world’s most impactful scientific researchers.” In March 2022, he was awarded an honorary doctorate from BabesBolyai-University Cluj-Napoca for his research achievements and contributions to international exchange. More information: https:// linktr. ee/ marko sarst edt Jan‑Michael Becker is an associate professor in the Department of Marketing at the BI Norwegian Business School (Norway). His research interests and expertise focus on the digital transformation of marketing strategy and consumer behavior as well as marketing analytics, behavioral research methods, causal inference, machine learning, and computational statistics. His research has been published in several premier academic journals, includingthe Journal of the Academy of Marketing Science, International Journal of Research in Marketing, Information Systems Research, MIS Quarterly, Psychometrika, Nature Human Behavior, Multivariate Behavioral Research, Journal of Business Research, and Marketing Letters. He is a co-developer and cofounder of the statistical software SmartPLS (https://www.smartpls. com). More information: https:// www. bi. edu/ aboutbi/ emplo yees/ depar tmentofmarke ting/ janmicha elbecker/.