Structural equation modeling (SEM) has, in recent decades, become one of the primary data analysis techniques used by researchers across the social sciences, including sociology, psychology, education, management, and public policy (Hair et al., 2019b). Its growing dominance in applied social research reflects its unique capacity to examine multiple, interrelated dependence relationships simultaneously while explicitly accounting for measurement error in latent, unobserved constructs such as attitudes, satisfaction, trust, or social capital (Byrne, 2016; Kline, 2015). Unlike simpler techniques such as multiple regression or exploratory factor analysis considered in isolation, SEM integrates measurement theory and structural (path) theory into a single, testable statistical model (Jöreskog, 1969), which is a major reason it has displaced simpler multivariate techniques as the default confirmatory and predictive analysis method in journals across the social sciences.

Since Jöreskog’s (1969) pioneering work on confirmatory maximum-likelihood factor analysis and Wold’s (1982) development of partial least squares path modeling, SEM has evolved into a family of related but methodologically distinct techniques. Two estimation traditions dominate contemporary applied research: covariance-based SEM (CB-SEM), most commonly implemented in software such as AMOS, LISREL, and Mplus, and variance-based, or partial least squares, SEM (PLS-SEM), most commonly implemented in SmartPLS or WarpPLS (Hair et al., 2019b; Sarstedt et al., 2017). Although both approaches share the same conceptual foundation of relating latent constructs to their indicators and to one another, they differ substantially in their statistical objectives, distributional assumptions, sample size requirements, and, critically for applied researchers, the criteria used to judge whether an estimated model is acceptable (Hair et al., 2019b).

A recurring difficulty for graduate students and novice researchers is that the evaluation criteria developed for CB-SEM are not appropriate for PLS-SEM, and vice versa, yet these criteria are frequently confused or misapplied in student theses and even in published research (Hair et al., 2019b; Henseler et al., 2015). The purpose of this article is therefore twofold: first, to provide a clear, citation-supported explanation of what SEM is and how CB-SEM and PLS-SEM differ; and second, to consolidate, in tabular form, the goodness-of-fit, normality, and validity/reliability criteria recommended for each approach, together with their accepted threshold values. In addition, the article outlines how each approach estimates and assesses mediation and moderation, how formative measurement models should be evaluated, and what SEM results can and cannot establish about causal effects. The article is written primarily as a teaching and reference resource for social science researchers and students who are approaching SEM for the first time, or who need a single, well-referenced source to guide model evaluation and reporting.

Structural equation modeling is a family of multivariate statistical techniques that combines elements of factor analysis and multiple regression analysis to estimate a network of relationships between measured (observed) variables and unmeasured (latent) constructs (Byrne, 2016; Hair et al., 2019a). A latent construct, such as job satisfaction, quality of life, or social trust, cannot be observed or measured directly; instead, it is inferred from a set of observed indicators (also called items or manifest variables) that are assumed to reflect, or in some cases form, the underlying construct (Kline, 2015).

A complete SEM specification generally consists of two interrelated sub-models (Hair et al., 2019a; Kline, 2015). The measurement model specifies the relationships between each latent construct and its observed indicators. It is conceptually equivalent to a confirmatory factor analysis (CFA) and allows the researcher to assess how well the chosen indicators represent their intended latent constructs before proceeding to test substantive hypotheses (Byrne, 2016). The structural model specifies the hypothesized directional or non-directional relationships among the latent constructs themselves, analogous to a system of simultaneous regression equations. Constructs that only receive paths from other constructs are termed endogenous, while constructs that only send paths to other constructs are termed exogenous (Kline, 2015).

Although the paths in a structural model are drawn as single-headed arrows and are commonly described as “effects,” their direction is an assumption that the researcher specifies on theoretical grounds; it is not something that the estimation procedure discovers or tests. A model that fits the data well, or that yields statistically significant path coefficients, shows only that the data are consistent with the hypothesized structure, and equivalent models with reversed or rearranged paths will often reproduce the data equally well (MacCallum et al., 1993). This limitation is particularly important in the social sciences, where most SEM applications rely on observational, and frequently cross-sectional, survey data. In such studies, path coefficients estimate associations conditional on the model’s assumptions; interpreting them as causal effects additionally requires that the cause precedes the effect, that relevant confounders have been measured and modeled, and that the hypothesized direction is correct. These conditions must be justified by the research design and by theory rather than by SEM itself (Antonakis et al., 2010; Bollen & Pearl, 2013; Kline, 2015). Unless the design supports stronger claims, for example through random assignment, longitudinal panel data, or valid instrumental variables, researchers should describe directional paths in associational terms (e.g., “is positively associated with” or “predicts”) rather than as evidence that one construct “causes” or “leads to” another.

Because SEM estimates the measurement and structural components simultaneously, it explicitly separates random measurement error from the true relationships among constructs, which is a key advantage over path analysis performed on composite or observed scores alone (Hair et al., 2019a). This property makes SEM particularly well suited to social science research, where constructs of interest are almost always latent and are measured with survey items subject to both random and systematic error (Kline, 2015).

Two broad estimation traditions have emerged for fitting SEM models to data. Covariance-based SEM (CB-SEM), rooted in the work of Jöreskog (1969) and implemented through maximum-likelihood estimation, seeks parameter estimates that minimize the discrepancy between the observed sample covariance matrix and the covariance matrix implied by the specified model (Byrne, 2016). Partial least squares SEM (PLS-SEM), rooted in Wold’s (1982) soft-modeling tradition and popularized by Chin (1998) and Hair et al. (2019b), instead uses an iterative, component-based (ordinary least squares) algorithm that maximizes the explained variance of endogenous constructs.

Although CB-SEM and PLS-SEM are both referred to as “structural equation modeling,” they rest on different statistical logics and are suited to different research objectives (Hair et al., 2019b). CB-SEM is fundamentally a covariance-fitting, theory-confirmation technique. Because it evaluates how closely the model-implied covariance matrix reproduces the observed covariance matrix, it produces a set of global goodness-of-fit statistics that indicate whether the overall model is consistent with the data (Hu & Bentler, 1999). CB-SEM assumes multivariate normality of the observed indicators, generally requires a comparatively larger sample size for stable maximum-likelihood estimation, and is best suited to well-established theoretical models in which the researcher wishes to confirm or disconfirm a specific covariance structure (Byrne, 2016; Hair et al., 2019a).

PLS-SEM, in contrast, is a prediction-oriented, variance-maximizing technique. Because it does not rely on a common-factor model or on covariance reproduction, PLS-SEM does not produce a single chi-square-based test of overall model fit; instead, model quality is assessed through a series of measurement and structural model criteria focused on reliability, validity, and predictive power (Hair et al., 2019b). PLS-SEM makes minimal distributional assumptions, performs well with smaller sample sizes and more complex models involving many constructs and indicators, and can accommodate both reflective and formative measurement models, whereas CB-SEM handles formative constructs only with considerable additional constraints (Chin, 1998; Hair et al., 2019b). The main differences between the two approaches are summarized inTable 1.

Table 1.Comparison of covariance-based SEM and partial least squares SEM.

Note. Adapted from guidance in Chin (1998), Hair et al. (2019b), Jöreskog (1969), and Wold (1982).

In practice, the choice between CB-SEM and PLS-SEM should be guided by the research objective rather than by convenience or software familiarity. Hair et al. (2019b) recommend CB-SEM when the research goal is theory confirmation and the model is grounded in well-developed theory, and PLS-SEM when the goal is theory development, prediction, or exploratory analysis, especially under conditions of small sample sizes, non-normal data, formative measurement, or highly complex models. Sarstedt et al. (2017) similarly emphasize that PLS-SEM should not be viewed merely as a “fallback” option for CB-SEM but as a distinct methodological choice suited to prediction-oriented research questions.

Because CB-SEM produces a model-implied covariance matrix that can be directly compared with the observed covariance matrix, its evaluation is organized around three broad categories of statistics: goodness-of-fit indices, which assess how well the overall model reproduces the observed data; normality diagnostics, which check whether the assumptions of maximum-likelihood estimation are met; and reliability/validity measures, which assess the quality of the measurement model (Byrne, 2016; Hair et al., 2019a; Kline, 2015).

Because PLS-SEM does not rely on covariance reproduction, it is evaluated using a systematic two-step process that first assesses the measurement (outer) model and then the structural (inner) model, rather than a single global fit statistic (Hair et al., 2019b). This section summarizes the criteria recommended for each step.

Many social science models go beyond direct effects to ask how an effect is transmitted (mediation) or under what conditions it holds (moderation). Both CB-SEM and PLS-SEM can estimate such models, but they differ in how indirect and conditional effects are specified, estimated, and assessed. Because mediation describes a causal process, the caution in Section 2 about the causal interpretation of directional paths applies with particular force; in cross-sectional data, estimates of indirect effects can be substantially biased relative to the longitudinal processes they are meant to represent (Maxwell & Cole, 2007).

For researchers beginning to work with SEM, four practical recommendations follow from the material reviewed above. First, the choice between CB-SEM and PLS-SEM should be made before data collection, based on the research objective (theory confirmation versus prediction/theory development), the anticipated sample size, the expected distribution of the data, and whether the measurement model includes formative indicators (Hair et al., 2019b; Sarstedt et al., 2017). Switching approaches after the fact simply to obtain more favorable results is a questionable research practice and should be avoided (Hair et al., 2019b).

Second, model evaluation should always proceed in the correct order: the measurement model (reliability and validity) must be assessed and confirmed as adequate before the structural model, that is, the hypothesized relationships, is interpreted, regardless of whether CB-SEM or PLS-SEM is used (Hair et al., 2019b). Reporting structural path coefficients from a measurement model with poor reliability or validity produces results that cannot be meaningfully interpreted.

Third, researchers should resist the temptation to rely on a single statistic to justify model adequacy. Methodologists consistently recommend reporting a combination of indices (for example, RMSEA together with CFI and RMR in CB-SEM, or AVE, HTMT, and Q² together in PLS-SEM) and interpreting recommended threshold values as general guidelines rather than rigid pass/fail rules (Hu & Bentler, 1999; Kline, 2015; Hair et al., 2019b). Sample size, model complexity, and the number of indicators per construct can all influence fit statistics, and thresholds developed under one set of simulation conditions do not automatically generalize to every applied context (Kline, 2015). Ultimately, sound SEM practice combines statistical criteria with careful theoretical justification of the specified model (Byrne, 2016).

Fourth, researchers should match the strength of their conclusions to the strength of their research design. As noted in Section 2, a well-fitting model and significant directional paths do not by themselves establish that one construct causes another, particularly in observational and cross-sectional studies (Antonakis et al., 2010; Bollen & Pearl, 2013). Hypotheses may be derived from theory and tested as directional relationships, but the discussion of results should acknowledge plausible equivalent models, reverse causation, and omitted confounders, and should reserve causal language for designs, such as experiments or longitudinal panel studies, that support it.

Researchers should also be aware that the choice of software often follows directly from the choice of estimation approach. CB-SEM is most commonly estimated using IBM SPSS Amos, LISREL, Mplus, or the lavaan package in R; AMOS in particular produces the full ten-block Model Fit Summary summarized in Table 2, from CMIN and RMR/GFI through Hoelter’s critical N, although SRMR, which Table 2 also mentions, must be requested separately through the Standardized RMR plugin (Byrne, 2016; Hooper et al., 2008). PLS-SEM is most commonly estimated using SmartPLS, WarpPLS, ADANCO, or the seminr and plspm packages in R; SmartPLS in particular produces the bootstrapped measurement-model criteria in Tables 5-6, the SRMR/d_ULS/d_G/Chi-square/NFI/RMS_theta model fit report in Table 7, and the structural-model criteria in Table 8 (Hair et al., 2019b; Henseler et al., 2014). Because the two families of software report different default statistics, researchers should confirm, before beginning analysis, that the criteria they intend to report are actually appropriate for the estimation method their chosen software implements.

When writing up SEM results for a thesis, dissertation, or journal submission, Hair et al. (2019b) recommend that authors report, at a minimum: (a) the measurement model results, including indicator loadings, internal consistency reliability, and convergent and discriminant validity statistics, with reference to the threshold values summarized in this article; (b) the structural model results, including path coefficients, their significance levels, and, for CB-SEM, the full set of global fit indices, or, for PLS-SEM, R², f², and Q² values, together with PLSpredict results where prediction is a study objective, and, for models with mediation or moderation, the bootstrapped indirect or interaction effects with their confidence intervals; and (c) the specific software, version, and estimation algorithm used, so that the analysis can be reproduced. Following a consistent, criterion-based reporting structure of this kind not only strengthens the methodological rigor of a researcher’s own research but also makes it considerably easier for examiners, reviewers, and readers to evaluate the credibility of the reported model (Hair et al., 2019b; Kline, 2015).

Structural equation modeling offers social science researchers a powerful framework for testing complex, theory-driven relationships among latent constructs while explicitly modeling measurement error. As this article has shown, the two dominant SEM traditions, covariance-based SEM and partial least squares SEM, rest on different statistical logics and are therefore evaluated using different, non-interchangeable sets of criteria. CB-SEM, as implemented in AMOS, relies on a wide battery of global goodness-of-fit indices, including CMIN/DF, GFI/AGFI, the baseline-comparison indices (NFI, RFI, IFI, TLI, CFI), parsimony-adjusted indices, RMSEA, information criteria, and Hoelter’s critical N, alongside normality diagnostics, to confirm that a theoretically specified covariance structure adequately reproduces the observed data. PLS-SEM, by contrast, is evaluated through a systematic assessment of measurement model reliability and validity, namely loadings, composite reliability, AVE, and HTMT for reflective constructs, and redundancy analysis, indicator collinearity, and indicator weights for formative constructs, complemented by SmartPLS’s own model fit report (SRMR, d_ULS, d_G, Chi-square, NFI, and RMS_theta) and structural model criteria emphasizing explanatory and predictive power, namely R², f², Q², and, for genuine out-of-sample prediction, PLSpredict. For social science researchers and students, understanding these distinctions and applying the correct, well-referenced threshold values summarized in this article, is essential not only for producing methodologically sound theses and publications, but also for critically evaluating SEM-based research encountered in the wider literature. Both approaches can accommodate mediation and moderation, provided that indirect effects are tested with bootstrap confidence intervals and interaction effects are specified with methods appropriate to the estimator. Finally, whichever approach is chosen, the directional paths of a structural equation model represent theoretical assumptions rather than demonstrated causal effects, and the conclusions drawn from them should be no stronger than the research design allows.