Structural equation modelling · South Africa
StatForge helps Master's, MBA and PhD researchers choose, run and interpret structural equation models. Led by Dr Michael Pietersen, we provide measurement-model assessment, hypothesis testing, report-ready tables, analysis code and explanations you can discuss with your supervisor. Support is scoped to your study, from a CFA review to a full SEM analysis.
Reviewed by Dr Michael Pietersen, quantitative research methodologist · Last reviewed · Supporting individual candidates from WBS, GIBS, UCT GSB, Stellenbosch, Henley, Milpark and UNISA SBL
What you receive
Structural equation modelling has two parts. A confirmatory factor analysis (CFA) assesses whether your proposed measurement structure, meaning which questionnaire items measure which construct, is consistent with the data. The structural model then estimates the hypothesised relationships between those constructs. Formative constructs, where the indicators together form the construct, follow a different assessment approach.
A standard Analysis Execution engagement includes the following. Your agreed scope covers the stages relevant to your research questions and model. All support is delivered remotely, to researchers anywhere in South Africa.
Why CB-SEM or PLS-SEM, how each construct is measured, and which estimator suits your data.
Loadings, reliability and validity for reflective constructs, or weights and collinearity for formative ones.
Path estimates with confidence intervals, explained variance, and an assessment of the evidence for each hypothesis, including effect size and uncertainty.
Report-ready tables formatted to your institution’s requirements, and a model diagram.
The R code behind your results, and plain-English notes on what each result does and does not show.
Analysis is run in R: lavaan (Rosseel, 2012) for CB-SEM and SEMinR or cSEM for PLS-SEM, so every result can be re-run. Tables follow the conventions examiners know from AMOS and SmartPLS output. We can also review and troubleshoot output you have already produced in SmartPLS, AMOS or SPSS. If your programme requires native SmartPLS or AMOS project files, raise it in the scoping call.
The free 15-minute scoping call establishes what you need and whether we are a good fit. The paid Risk Review (60–90 minutes plus a written memo) is a full senior review of your model, questionnaire and analysis plan, best done before you collect data or run the analysis.
Why researchers get in touch
Each problem has a likely cause and a decision to make. These are the first checks we run.
Fit indices outside their usual guidelines are a symptom, not a diagnosis.
What we check: Check coding and the specification, then residuals and loadings. We separate theory-led corrections from post hoc changes, and document any exploratory change. Freeing parameters suggested by modification indices without a theoretical reason is a red flag for examiners.
Some items measure their construct weakly.
What we check: Before deleting anything, we check what each item contributes to the content of the construct. We show you how keeping or removing an item changes the interpretation, so the decision can be defended.
Two constructs look statistically too similar.
What we check: Check coding, overlapping item wording and how the constructs are defined, before merging constructs or deleting items. HTMT detects this more reliably than the Fornell–Larcker criterion alone.
Negative error variances (Heywood cases), models that will not converge, or estimates that make no sense.
What we check: Possible causes include coding errors, very different variable scales, near-redundant items, misspecification, identification problems and a small sample. We work through them in that order.
Five- or seven-point items are ordered categories, not truly continuous.
What we check: We inspect the number of categories, their distributions, sample size and missing data. WLSMV models ordered categories explicitly; MLR can be appropriate when treating items as approximately continuous is defensible.
The Baron and Kenny causal-steps approach is still common in drafts.
What we check: We test the indirect effect directly, usually with a bootstrapped confidence interval, and word the conclusion to match what the design can support.
The standard of work
An illustrative reporting layout showing the format and depth of what we deliver: the measurement model, the discriminant validity check, the structural results, and the interpretation an examiner will look for.
Illustrative example with a synthetic topic and figures. It is not client work.
| Construct | Items | Loadings | α | CR | AVE |
|---|---|---|---|---|---|
| Transformational leadership | 6 | .72–.86 | .91 | .91 | .63 |
| Psychological safety | 5 | .70–.84 | .87 | .88 | .59 |
| Work engagement | 5 | .74–.88 | .90 | .90 | .65 |
| Turnover intention | 4 | .79–.89 | .89 | .90 | .69 |
N = 412; seven-point items treated as approximately continuous; robust maximum likelihood (MLR) with full-information missing-data handling. Robust fit: χ²(164) = 298.4, CFI = .962, TLI = .956, RMSEA = .045 [90% CI .037, .053], SRMR = .041. Loadings are standardised; α = Cronbach's alpha; CR = composite reliability; AVE = average variance extracted.
| TL | PS | WE | |
|---|---|---|---|
| Psychological safety (PS) | .58 | ||
| Work engagement (WE) | .49 | .62 | |
| Turnover intention | .41 | .47 | .66 |
TL = transformational leadership. Original HTMT (Henseler, Ringle and Sarstedt, 2015); all ratios below the conservative .85 guideline.
# CFA: robust ML, full-information ML for missing data
library(lavaan); library(semTools)
cfa_model <- '
TL =~ tl1 + tl2 + tl3 + tl4 + tl5 + tl6
PS =~ ps1 + ps2 + ps3 + ps4 + ps5
WE =~ we1 + we2 + we3 + we4 + we5
TI =~ ti1 + ti2 + ti3 + ti4
'
fit <- cfa(cfa_model, data = survey, estimator = "MLR", missing = "fiml")
fitMeasures(fit, c("chisq.scaled", "df", "cfi.robust", "tli.robust",
"rmsea.robust", "rmsea.ci.lower.robust",
"rmsea.ci.upper.robust", "srmr"))
standardizedSolution(fit) # standardised loadings
compRelSEM(fit) # composite reliability
AVE(fit) # average variance extracted
htmt(cfa_model, data = survey,
missing = "ml", htmt2 = FALSE) # original HTMT, same missing-data handlingInterpretation note
The four-factor measurement model shows acceptable approximate fit (robust CFI = .962, RMSEA = .045, SRMR = .041). Composite reliability exceeds .70 and AVE exceeds .50 for every construct, supporting convergent validity. The highest HTMT ratio is .66, below the conservative .85 guideline, supporting discriminant validity. These are statistical checks; whether the items capture each construct's meaning is argued from the item content and theory.
Why it is reported this way
RMSEA is given with its confidence interval because the point estimate alone can mislead. The estimator, how the items were treated and how missing data were handled are stated, because the fit statistics depend on them. Robust fit indices are labelled as robust so an examiner knows which version is reported.
Anticipated examiner question
“Why use HTMT rather than only the Fornell–Larcker criterion?” Answer prepared: simulation evidence shows Fornell–Larcker often fails to detect discriminant validity problems (Henseler, Ringle and Sarstedt, 2015). Both are reported, with HTMT as the primary check.
| Hypothesised relationship | β | 95% CI | Interpretation |
|---|---|---|---|
| H1: Leadership → Psychological safety | .52 | [.42, .61] | Supported; a large positive association. |
| H2: Psychological safety → Engagement | .44 | [.33, .54] | Supported; a moderate-to-large positive association. |
| H3: Engagement → Turnover intention | −.38 | [−.49, −.27] | Supported; a moderate negative association. |
| H4: Leadership → Turnover intention (direct) | −.07 | [−.19, .05] | Inconclusive; the interval includes zero. |
| Indirect: Leadership → Safety → Engagement → Turnover | −.09 | [−.12, −.06] | Non-zero indirect effect under the fitted model. |
Standardised estimates; percentile bootstrap confidence intervals (5,000 resamples) for the indirect effect.
Interpretation note
H1 to H3 are supported, with confidence intervals well away from zero. The direct association between leadership and turnover intention (H4) is small and its interval includes zero, so the data neither support nor rule out a modest direct relationship. The relationship appears to run mainly through psychological safety and engagement, but with cross-sectional data this describes a pattern of associations, not a demonstrated causal sequence.
When a hypothesis is not supported
An inconclusive result is reported as it is, with its estimate and interval, not hidden or explained away. We check for legitimate specification issues, and help you discuss what the result means for your theory, your sample and future research.
Choosing a method
Choose the SEM approach to match your research question and measurement model. CB-SEM is commonly used to evaluate common-factor models and how well they reproduce the observed covariances. PLS-SEM can suit composite models and prediction-focused studies (Hair et al., 2019). A small sample, a complex model or poor CB-SEM fit is not, on its own, a sufficient reason to choose PLS-SEM.
| Aspect | Covariance-based SEM (CB-SEM) | Partial least squares SEM (PLS-SEM) |
|---|---|---|
| Starting point | Theory testing with constructs modelled as common factors | Constructs modelled as composites; explanation and prediction of key outcomes |
| Measurement | Reflective measurement, assessed with CFA | Reflective or formative measurement, each with its own criteria |
| Model assessment | Chi-square test and approximate fit indices (CFI, TLI, RMSEA, SRMR), plus residuals | No universally accepted global fit test; measurement quality, path estimates and explained variance (R²) |
| Prediction | Not the primary purpose | R² is in-sample only; out-of-sample predictive power is assessed with PLSpredict against benchmarks (Shmueli et al., 2019) |
| Sample size planning | Power analysis or Monte Carlo simulation for the specific model (Wolf et al., 2013) | Inverse square root method as an initial approximation (Kock and Hadaya, 2018) |
| Typical software | lavaan, AMOS, Mplus; recent SmartPLS versions also estimate CB-SEM | SmartPLS, SEMinR, cSEM |
The workflow
A typical analysis starts with data screening, followed by measurement assessment and structural modelling. Your agreed scope includes the stages relevant to your research questions.
Missing data, outliers, distributions and the number of response categories. The estimator follows from these: WLSMV explicitly models ordered categorical items; MLR adjusts inference for non-normality when treating items as approximately continuous is defensible (Rhemtulla, Brosseau-Liard and Savalei, 2012).
Assessed before interpreting the structural relationships; in CB-SEM this is the two-step approach (Anderson and Gerbing, 1988). The criteria depend on how each construct is measured.
Reflective constructs
Items are interchangeable expressions of the construct. Checks: indicator loadings, reliability, convergent validity (AVE) and discriminant validity (HTMT).
Formative constructs
Indicators each form part of the construct. Checks: content coverage, indicator collinearity, weights and their significance, indicator relevance, and convergent validity through redundancy analysis (Hair et al., 2019).
Hypothesised paths with standardised estimates and confidence intervals, explained variance (R²), and model assessment suited to CB-SEM or PLS-SEM.
Indirect effects tested directly, usually with bootstrapped confidence intervals (Preacher and Hayes, 2008); moderation through latent interactions or multi-group models.
Common method bias assessment, competing models, and measurement invariance where groups are compared.
Reference guide
These commonly cited guidelines help interpret results. Their suitability depends on the estimator, model and data, so we assess them alongside parameter estimates, residuals, uncertainty and the substantive meaning of the measures, not as pass-or-fail rules.
| Criterion | Applies to | Common guideline | How to read it | Source |
|---|---|---|---|---|
| Comparative fit index (CFI) and Tucker–Lewis index (TLI) | CB-SEM model fit | Close to .95 or higher | Derived from simulations of maximum-likelihood fit indices; judge in combination with other indices and residuals. | Hu and Bentler (1999) |
| Root mean square error of approximation (RMSEA) | CB-SEM model fit | Close to .06 or lower | Report with its 90% confidence interval; can be unreliable in models with few degrees of freedom. | Hu and Bentler (1999) |
| Standardised root mean square residual (SRMR) | CB-SEM model fit | Close to .08 or lower | The root mean square of standardised residuals: how far, on average, the model misses the observed correlations. | Hu and Bentler (1999) |
| Indicator loadings | Reflective measurement | .70 or higher preferred | Lower loadings are judged against the content the item adds before it is removed. | Hair et al. (2019) |
| Composite reliability | Reflective measurement | .70 to .95 | Values approaching or above .95 can indicate redundant, near-identical items rather than better measurement. | Hair et al. (2019) |
| Average variance extracted (AVE) | Reflective measurement | .50 or higher | Convergent validity; not used for formative constructs. | Fornell and Larcker (1981) |
| Heterotrait–monotrait ratio (HTMT) | Reflective measurement | Below .85, or .90 for conceptually similar constructs | Discriminant validity; a high value prompts a check of wording and construct definitions. | Henseler, Ringle and Sarstedt (2015) |
| Variance inflation factor (VIF) | Formative measurement | Ideally close to 3 or lower; 5 or above is critical | Collinearity between formative indicators distorts their weights. | Hair et al. (2019) |
| Indicator weights | Formative measurement | Significant weight, or loading of .50 or higher if the weight is not significant | Indicator relevance; removal also requires a content argument. | Hair et al. (2019) |
Before comparing groups, for example respondents who completed a questionnaire in English and in isiZulu, you need evidence the constructs are measured the same way in each. Configural invariance means the same structure; metric invariance (equal loadings) allows relationships to be compared; scalar invariance (equal intercepts) allows latent means to be compared (Putnick and Bornstein, 2016). Ordinal models also require attention to thresholds.
Change scores are calculated as the more constrained model minus the less constrained model. A commonly used guideline treats a drop in CFI larger than .010, with a rise in RMSEA above .015, as evidence against invariance; the study behind it examined specific conditions (Chen, 2007). For PLS composite models, the MICOM procedure is used instead (Henseler, Ringle and Sarstedt, 2016).
We test the indirect effect directly, usually with a bootstrapped confidence interval where the estimator and data structure support it (Preacher and Hayes, 2008). Your deliverables state the indirect effect, the interval method and the assumptions behind it.
An interval that excludes zero supports a non-zero indirect effect under the fitted model. It does not on its own establish a causal mechanism, and cross-sectional mediation can misrepresent a process that unfolds over time (Maxwell and Cole, 2007). We word your conclusions to match what the design can support.
When predictors and outcomes come from the same respondents in the same questionnaire, shared method variance can inflate relationships. The strongest remedies are in the design: separating measures, varying response formats or using different sources (Podsakoff et al., 2003).
Statistical checks, such as a marker variable or a latent method factor, help gauge the risk. No single diagnostic certifies that bias is absent, and passing Harman's single-factor test does not show it is.
Client feedback
“I appreciated the detailed and in-depth engagement [from Michael]. [Michael] didn't just run the statistical analysis for me, [he] handled everything with such care that I had complete confidence in the results. What really stood out for me though was how [he] caught a few critical weaknesses in my survey. Not only did [he] flag the issues, but [he] also identified possible solutions that I would've never considered. I would recommend [him] without hesitation to anyone doing high-level postgraduate research.”
Aaron
Master of Business Administration, SBS
“I really appreciated the guidance [Michael] provided on the steps I needed to take to ensure that I present and communicate my research and results properly. No amount of reading prepared me for the complexity of what I had to do. I appreciated the assistance in showing me how to set everything up for success.”
“At proposal stage, if you are in doubt about quantitative analysis or mixed-methods, Michael is able to guide you to make the right decision. The research methodology you choose will support a high-performance dissertation and may have a meaningful impact on your future career.”
Deborah
Master’s in Organisational Psychology, UNICAF
Who does the work
Every engagement is led personally by Dr Pietersen, a quantitative researcher who has been through doctoral examination himself. Your analysis is never passed on to a junior analyst.
View publications on Google ScholarPhD, University of Pretoria
Doctorate completed in 2022, with a strong quantitative focus. MPhil awarded cum laude.
Peer-reviewed researcher
Four peer-reviewed publications, two of them in ABDC A-rated journals.
Advanced methods
SEM/CFA, PLS-SEM, panel data and GMM, ARDL and GARCH, fsQCA, and applied machine learning.
Reproducible by default
Analysis in documented R or Python code, so every number in your thesis can be traced and re-run.
Transparent pricing
Fixed scope, price and revision rounds agreed before any work starts.
R4,000–R8,500
A senior review of your conceptual model, questionnaire and SEM plan, before you collect data or run the analysis.
From R9,500
Measurement-model assessment and structural analysis for a survey-based study, using CB-SEM or PLS-SEM as appropriate.
From R20,000
Complex models and extended supervisor or examiner feedback.
Multi-group comparisons, invariance testing, higher-order models or several mediators move a project to Research Pro. Already have results and need supervisor-requested changes? Extensions & Revisions start at R3,500.
We advise on methods, run the analysis, and deliver tables, code and explanations so you can defend your results. We do not write chapters, ghostwrite or make your interpretive decisions. Your data is treated as confidential and used only for your project.
A few lines are enough. After the scoping call we'll ask for your proposal, questionnaire and data description.
Choose the SEM approach to match your research question and measurement model. CB-SEM is commonly used to evaluate common-factor models and how well they reproduce the observed covariances. PLS-SEM can suit composite models and prediction-focused studies. A small sample, a complex model or poor CB-SEM fit is not, on its own, a sufficient reason to choose PLS-SEM. We help you make the choice and justify it in your methodology chapter.
It depends on your study. The sample needed is driven by the size of the effects you expect, the strength of the factor loadings, the number of parameters, the precision and power you need, missing data and the size of any subgroups you compare. Rules of thumb such as 200 cases are a rough starting point at best. For CB-SEM we plan with a power analysis or Monte Carlo simulation; for PLS-SEM the inverse square root method gives an initial approximation, not a guarantee.
First we find out why: check the coding and the model specification, then the residuals and loadings, to locate a weak item, an unmodelled cross-loading, correlated errors with a substantive reason, or a structural path the data do not support. Changes are made only where they can be justified theoretically and are reported transparently as post hoc. Freeing parameters suggested by modification indices without a theoretical reason is something examiners recognise and challenge.
For latent-variable CB-SEM with reflective measures, we normally assess the measurement model with a confirmatory factor analysis first, then test the structural paths. In PLS-SEM the equivalent first step is assessing the measurement model, with different criteria for reflective and formative constructs. A path model built only from observed variables does not necessarily have a CFA stage.
Yes, in two ways. We run analyses in R, using lavaan (Rosseel, 2012) for CB-SEM and SEMinR or cSEM for PLS-SEM, because scripted analysis is fully reproducible, and we report the results in the format SmartPLS and AMOS users and examiners expect. We can also review and troubleshoot output you have already produced in SmartPLS, AMOS or SPSS. If your programme requires native SmartPLS or AMOS project files, raise it in the scoping call. Estimates from different packages only match when the model, estimator, settings and missing-data treatment match.
Yes. Indirect effects are tested directly, usually with bootstrapped confidence intervals where the estimator supports it. A confidence interval that excludes zero supports a non-zero indirect effect under the fitted model; it does not on its own establish a causal mechanism, especially with cross-sectional data. Moderation is tested with latent interaction terms or multi-group models, with measurement invariance checked before groups are compared.
A Research Design & Analysis Risk Review costs R4,000–R8,500. Measurement-model assessment and structural analysis for a survey-based study, using CB-SEM or PLS-SEM as appropriate, is covered by Analysis Execution, from R9,500. Measurement invariance, multi-group models, higher-order constructs, mediation chains and extended feedback cycles are quoted under Research Pro, from R20,000. You receive a fixed quote, including the agreed revision rounds, before work starts.
Yes. If you already have output from SmartPLS, AMOS, SPSS, R or another package, we can review the model specification, estimator, measurement assessment and reporting, and work through your supervisor's comments. Depending on what the review finds, we quote for a focused correction under Extensions & Revisions, from R3,500, or a re-analysis.
Non-significant results are reported transparently, with their estimates and confidence intervals, not hidden or forced. We check for legitimate issues such as coding errors, measurement problems or a misspecified model, and correct them only where there is a defensible reason. We then help you understand what the findings mean, so you can discuss them in your thesis.
No. We deliver the analysis, report-ready tables and figures, analysis code and interpretation notes that explain each result and how an examiner is likely to read it. You write the chapter in your own words and make the final interpretive decisions.
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