Structural Equation Modelling (SEM) Explained: CFA, Path Models, Fit Indices, and Choosing Between AMOS and SmartPLS

TL;DR — Quick Answer

Structural Equation Modelling (SEM) is a statistical technique that tests an entire theoretical model at once — measuring latent constructs from observed items and estimating the relationships between those constructs simultaneously. It combines two parts: the measurement model (Confirmatory Factor Analysis, which checks that your survey items actually measure the constructs you claim) and the structural model (the path relationships between constructs, which tests your hypotheses). Unlike regression, SEM accounts for measurement error and can test multiple dependent variables in one model. You assess it using fit indices — CMIN/df, CFI, TLI, RMSEA, SRMR — alongside reliability and validity measures (CR, AVE). Use covariance-based SEM (AMOS, Lisrel, R’s lavaan) when you are confirming established theory with a large sample; use PLS-SEM (SmartPLS) when your model is exploratory, your sample is smaller, or your data departs from normality.

MK — Dr. Madhuri Kanojiya, Founder & Director · Empire Research Press

Most postgraduate researchers meet statistics as a sequence of separate tests. You run a t-test here, a correlation there, a regression for the main hypothesis. Each test answers one narrow question, and the thesis becomes a patchwork of findings that the researcher must stitch into a story afterwards.

Structural Equation Modelling works the other way around. You begin with the story — the theoretical model, the diagram of constructs and arrows you built during your conceptual framework stage — and you test the whole thing in one estimation. That is why SEM has become the dominant analytical method in management, marketing, education, psychology, and organisational research, and why so many doctoral candidates encounter it for the first time at precisely the moment they can least afford to be confused by it.

This guide explains what SEM actually does, how the measurement model and structural model fit together, what the fit indices mean and what thresholds to report, and how to choose between the two major software approaches.

What SEM Is — and What Makes It Different

Structural Equation Modelling is best understood as three ideas working together.

First, it models latent variables. Constructs like job satisfaction, perceived usefulness, organisational commitment, or brand trust cannot be measured directly. You measure them indirectly, through several survey items that each capture part of the construct. SEM treats the construct as a latent variable — unobserved but inferred — and the survey items as observed indicators of it. This is a fundamentally different assumption from regression, which treats a summed scale score as if it were measured perfectly.

Second, it accounts for measurement error. No survey item measures its construct flawlessly. Regression ignores this and quietly attributes all imperfection to the residual, which biases the estimated relationships — usually downward. SEM models the error term for each indicator explicitly, so the estimated relationships between constructs are corrected for the unreliability of measurement. This is the single most important technical advantage of SEM, and the reason findings from an SEM analysis are often stronger than the equivalent regression on the same data.

Third, it estimates a whole system at once. Regression handles one dependent variable at a time. SEM allows a variable to be an outcome in one relationship and a predictor in another — which is exactly what mediation looks like — and estimates all these paths simultaneously rather than in a sequence of separate models.

The Two Halves of Every SEM Analysis

Every properly conducted SEM study proceeds in two stages, and reporting them in the wrong order is one of the most common errors in submitted theses.

Stage One: The Measurement Model (Confirmatory Factor Analysis)

Before you can test whether Construct A influences Construct B, you must first establish that your items genuinely measure Construct A and Construct B as distinct things. That is what Confirmatory Factor Analysis does.

CFA takes your proposed factor structure — you specify in advance which items load onto which construct — and tests how well that structure fits the observed data. It is confirmatory precisely because the structure is theory-driven and specified beforehand. This distinguishes it from Exploratory Factor Analysis, where you let the data suggest how many factors exist and which items belong where.

In practice, the sequence is: use EFA when you are developing a new scale or working with items whose structure is uncertain; use CFA when you are validating an established instrument or confirming a structure you have already justified from the literature. Many theses legitimately use both — EFA on a pilot sample, CFA on the main sample.

Figure 1 — The measurement model. The latent construct is inferred from its observed items; each item carries its own error term.

At the CFA stage you establish four things:

Factor loadings — each item’s standardised loading on its construct should generally be 0.7 or above, though 0.5–0.7 may be retained if other criteria hold.
Composite Reliability (CR) — internal consistency of the construct, conventionally 0.7 or above.
Convergent validity — assessed by Average Variance Extracted (AVE), conventionally 0.5 or above, meaning the construct explains more than half the variance in its own indicators.
Discriminant validity — evidence that constructs are genuinely distinct from one another. The traditional Fornell-Larcker criterion requires the square root of each construct’s AVE to exceed its correlations with other constructs; the HTMT ratio (heterotrait-monotrait), now widely preferred, should fall below 0.85 or 0.90 depending on how conceptually similar the constructs are.

Only when the measurement model is sound do you proceed. Testing structural hypotheses on constructs that have not been validated produces results that cannot be interpreted, however impressive the path coefficients look.

“The most common failure I see in SEM chapters is not statistical — it is sequential. Researchers rush to the path diagram because that is where the hypotheses live, and treat the measurement model as a formality to be reported afterwards. But if your constructs are not measuring what you claim, every arrow in that diagram is measuring something you cannot name.”

Dr. Madhuri Kanojiya — Empire Research Press

Stage Two: The Structural Model

With validated constructs in place, the structural model tests the relationships between them — your actual hypotheses. Each arrow represents a hypothesised directional relationship, and the output gives you a path coefficient (β), a significance level, and the variance explained in each endogenous construct (R²).

Some vocabulary that recurs constantly in SEM writing:

Exogenous variables — constructs with no arrows pointing into them; they are explained by factors outside the model.
Endogenous variables — constructs that receive at least one arrow; the model attempts to explain them.
Direct effect — the influence of one construct on another along a single path.
Indirect effect — influence transmitted through a mediating construct.
Total effect — the sum of direct and indirect effects.

Figure 2 — A full structural equation model. The measurement model validates the constructs; the structural path tests the hypothesis.

This is where SEM’s simultaneous estimation earns its reputation. Mediation, moderation, and multiple outcome variables can all be specified within one model rather than approximated through a series of separate regressions.

Model Fit: What the Indices Mean and What to Report

Fit indices answer one question: how closely does the covariance matrix implied by your model resemble the covariance matrix actually observed in your data? Good fit does not prove your theory is correct — it means your data do not contradict it.

The conventional thresholds, drawn largely from Hu and Bentler’s widely cited work:

CMIN/df (chi-square divided by degrees of freedom) — below 3 is generally acceptable, below 5 tolerable in large samples. The raw chi-square is reported but rarely interpreted alone, because it is almost always significant with large samples regardless of model quality.
CFI (Comparative Fit Index) — 0.90 acceptable, 0.95 or above good.
TLI (Tucker-Lewis Index) — same thresholds as CFI.
RMSEA (Root Mean Square Error of Approximation) — below 0.08 acceptable, below 0.06 good. Report the 90% confidence interval alongside it.
SRMR (Standardised Root Mean Square Residual) — below 0.08.

Report several indices from different families, not a hand-picked selection of the ones that happened to pass. Examiners and reviewers notice selective reporting immediately, and it damages credibility more than a marginal index ever would.

A word of caution about modification indices. Software will readily suggest correlating error terms or adding paths to improve fit. Each such change should be theoretically defensible and reported transparently. Chasing fit statistics by applying every suggested modification produces a model that fits your particular sample beautifully and generalises to nothing.

Sample Size

SEM is a large-sample technique, and inadequate sample size is the most common reason a study cannot support the model it proposes.

Several rules of thumb circulate. The most widely used is 10 to 20 respondents per estimated parameter, or alternatively 10 per observed indicator. A practical floor of around 200 is often cited for covariance-based SEM with a model of moderate complexity, with 300 to 400 more comfortable for complex models. PLS-SEM tolerates smaller samples, but “tolerates” is not “prefers” — the appropriate size still depends on model complexity and the required statistical power.

The essential point is that sample size must be planned before data collection, not justified afterwards. A model that is under-identified relative to its sample cannot be rescued at the analysis stage.

Choosing Your Approach: CB-SEM or PLS-SEM

The choice between covariance-based SEM and partial least squares SEM is a genuine methodological decision, not a software preference, and your methodology chapter should justify it explicitly.

Covariance-based SEM (AMOS, Lisrel, Mplus, R’s lavaan) estimates model parameters by minimising the difference between observed and implied covariance matrices. Choose it when your purpose is theory testing or confirmation, your model is grounded in established theory, your sample is large, and your data approximate multivariate normality. It provides the full battery of global fit indices, which is why journals in established disciplines often expect it.

PLS-SEM (SmartPLS, R’s seminr) is variance-based, maximising explained variance in the endogenous constructs. Choose it when your purpose is prediction or theory development, your model is exploratory or complex, your sample is modest, your data depart from normality, or your model includes formative constructs. PLS-SEM does not produce the same global fit indices; evaluation rests on measurement quality, R², f², Q², and SRMR.

Neither is superior in the abstract. The wrong choice is choosing on the basis of which software your department has a licence for, and then writing the justification backwards.

A Practical Workflow

Specify the model from theory — constructs, indicators, and hypothesised paths — before you collect data.
Calculate the required sample size for your model’s complexity.
Screen the data: missing values, outliers, normality, and common method bias.
Run EFA if you are developing or adapting a scale.
Run CFA and establish reliability, convergent validity, and discriminant validity.
Assess measurement model fit, refining only where theoretically defensible.
Estimate the structural model and test the hypothesised paths.
Test mediation and moderation using bootstrapping where relevant.
Report fit indices, path coefficients with significance, R², and effect sizes.
Interpret in relation to your theory — not simply as a list of supported and unsupported hypotheses.

Common Mistakes

Skipping the measurement model. Path coefficients between unvalidated constructs are uninterpretable.
Reporting only the fit indices that passed. Selective reporting is transparent to any experienced reviewer.
Applying every modification index the software suggests. This fits the sample, not the phenomenon.
Using SEM because it looks sophisticated. If your research question involves one outcome and a few directly measured predictors, regression is the honest and appropriate choice.
Justifying the software choice after the fact. The CB-SEM versus PLS-SEM decision follows from your research purpose and data, and belongs in the methodology chapter as a reasoned argument.
Treating good fit as proof. Fit indicates consistency between model and data; it does not establish causation, and equivalent models may fit equally well.

Quick-Reference Checklist

Model specified from theory before data collection
Sample size calculated for model complexity
Data screened for missing values, outliers, normality, common method bias
Measurement model validated: loadings, CR, AVE, discriminant validity
Fit indices reported across families: CMIN/df, CFI, TLI, RMSEA, SRMR
Structural paths reported with β, significance, and R²
Mediation and moderation tested with bootstrapping where applicable
CB-SEM versus PLS-SEM choice justified in the methodology chapter
Findings interpreted theoretically, not merely tabulated

Sources

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis. Structural Equation Modeling, 6(1), 1–55.
Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of Marketing Research, 18(1), 39–50.
Henseler, J., Ringle, C. M., & Sarstedt, M. (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.
Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM). SAGE.

Empire Research Press provides methodology consultation for researchers designing and analysing structural models — including measurement model validation, fit assessment, and reporting standards for thesis and journal submission.

About the Author

Dr. Madhuri Kanojiya is a researcher, author and educator with a PhD in Computer Science and Management. She is the Founder and Director of Empire Research Press — an independent international publisher and research consultancy based in Goa, India. She writes on research methodology, AI adoption, cloud computing, organisational systems and academic publishing.

Google Scholar: https://scholar.google.com/citations?user=67en4MwAAAAJ&hl=en
ORCID: https://orcid.org/0000-0003-1499-6112
ResearchGate: https://www.researchgate.net/profile/Madhuri-Kanojiya