Structural Equation Modelling (SEM) for Beginners

Structural equation modelling (SEM) has become the default analysis for many MS and PhD theses in management sciences, education, psychology and public health. It looks intimidating because of its path diagrams and long output, but it rests on ideas you already know: factor analysis and regression. This guide explains the logic, then works through the calculations you are most likely to report.

What SEM Does

SEM combines two models into one analysis:

1. The measurement model links observed questionnaire items to latent constructs. This is a confirmatory factor analysis (CFA). For example, four items measure "Training Quality".
2. The structural model specifies the hypothesised relationships among the latent constructs, like a set of regressions estimated at the same time. For example, Training Quality → Job Satisfaction → Job Performance.

Because constructs are modelled as latent variables, SEM separates measurement error from the true relationships, which ordinary regression on averaged scale scores cannot do.

SEM grew out of path analysis, invented by the geneticist Sewall Wright around 1918–1921, and was turned into a general framework by Karl Jöreskog and colleagues in the 1970s, leading to the LISREL program.

Key Vocabulary

• Latent variable: an unobserved construct (drawn as an oval).
• Indicator: an observed item (drawn as a rectangle).
• Exogenous variable: not predicted by anything in the model (an independent variable).
• Endogenous variable: predicted by at least one other variable (a mediator or outcome).
• Path coefficient: a regression weight between constructs, usually reported in standardised form (β).
• Disturbance: the unexplained part of an endogenous construct.

Covariance-Based SEM vs PLS-SEM

• CB-SEM (AMOS, R's lavaan, Stata's sem, Mplus) tests how well the whole model reproduces the observed covariance matrix. It needs larger samples (often 200+) and reports global fit indices.
• PLS-SEM (SmartPLS, R's seminr) maximises explained variance in the outcomes. It is popular for prediction-oriented research, smaller samples and complex models, but it does not provide the same global fit tests.

Choose based on your research aim and justify the choice with a citation in your methodology chapter.

The Six Steps of an SEM Study

1. Specify the model from theory and draw the path diagram.
2. Identify it: make sure there is enough information to estimate every parameter.
3. Estimate it, usually by maximum likelihood.
4. Evaluate fit of the measurement model first, then the structural model.
5. Modify only with strong theoretical justification.
6. Report paths, R², fit indices and indirect effects.

Worked Example 1: Counting Degrees of Freedom

A researcher studies bank employees in Karachi with three constructs: Training Quality (4 items), Job Satisfaction (4 items) and Job Performance (3 items), giving \( p = 11 \) indicators.

Step 1: Available information. The number of unique variances and covariances is

$$ \frac{p(p+1)}{2} = \frac{11 \times 12}{2} = 66 $$

Step 2: Free parameters.

• Factor loadings: 11 minus 3 fixed to 1 for scaling = 8
• Indicator error variances: 11
• Variance of the exogenous construct (Training): 1
• Disturbance variances of the two endogenous constructs: 2
• Structural paths (Training → Satisfaction, Satisfaction → Performance, Training → Performance): 3

Total free parameters: \( 8 + 11 + 1 + 2 + 3 = 25 \).

Step 3: Degrees of freedom.

$$ df = 66 - 25 = 41 $$

Since \( df > 0 \), the model is over-identified and its fit can be tested.

Worked Example 2: Evaluating Model Fit

With \( N = 300 \), the software reports \( \chi^2 = 85.4 \) with \( df = 41 \). The baseline (independence) model has \( \chi^2_b = 1250.0 \) with \( df_b = 55 \).

χ²/df:

$$ \frac{85.4}{41} = 2.08 $$

This is below 3, which is acceptable.

RMSEA:

$$ RMSEA = \sqrt{\frac{\chi^2 - df}{df(N-1)}} = \sqrt{\frac{85.4 - 41}{41 \times 299}} = \sqrt{\frac{44.4}{12259}} = \sqrt{0.003622} = 0.060 $$

This meets the commonly used "good fit" threshold of about .06.

CFI:

$$ CFI = 1 - \frac{\chi^2 - df}{\chi^2_b - df_b} = 1 - \frac{44.4}{1195} = 1 - 0.037 = 0.963 $$

TLI:

$$ TLI = \frac{\chi^2_b/df_b - \chi^2/df}{\chi^2_b/df_b - 1} = \frac{22.727 - 2.083}{22.727 - 1} = \frac{20.644}{21.727} = 0.950 $$

All indices meet common benchmarks (CFI and TLI ≥ .95, RMSEA ≤ .06), so the model fits the data well. The raw chi-square test is significant (p < .001), but with 300 respondents even trivial misfit produces a significant chi-square, which is why the other indices are relied upon.

Worked Example 3: Testing Mediation

The standardised structural paths are:

• Training → Satisfaction: \( a = 0.45 \)
• Satisfaction → Performance: \( b = 0.40 \)
• Training → Performance (direct): \( c' = 0.15 \)

Step 1: Indirect effect.

$$ ab = 0.45 \times 0.40 = 0.18 $$

Step 2: Total effect.

$$ c = c' + ab = 0.15 + 0.18 = 0.33 $$

Step 3: Proportion mediated. \( 0.18/0.33 = 0.545 \), so about 55% of Training's effect on Performance travels through Satisfaction.

Step 4: Significance. Suppose the standard errors are \( SE_a = 0.06 \) and \( SE_b = 0.07 \). The Sobel test gives

$$ z = \frac{ab}{\sqrt{b^2SE_a^2 + a^2SE_b^2}} = \frac{0.18}{\sqrt{(0.16)(0.0036) + (0.2025)(0.0049)}} = \frac{0.18}{\sqrt{0.001568}} = \frac{0.18}{0.0396} = 4.55 $$

so p < .001. Modern practice prefers bias-corrected bootstrap confidence intervals (for example 5,000 resamples): if the 95% interval for \( ab \) excludes zero, the indirect effect is significant. Because the direct path is also significant in this example, we would describe partial mediation.

Reporting in APA Style

The structural model fitted the data well, χ²(41, N = 300) = 85.40, p < .001, χ²/df = 2.08, CFI = .96, TLI = .95, RMSEA = .06. Training quality positively predicted job satisfaction (β = .45, p < .001), which in turn predicted job performance (β = .40, p < .001). The indirect effect of training on performance through satisfaction was significant (β = .18, 95% bootstrap CI excluding zero), indicating partial mediation.

A Minimal lavaan Model in R

r library(lavaan) model <- ' Train =~ t1 + t2 + t3 + t4 Sat =~ s1 + s2 + s3 + s4 Perf =~ p1 + p2 + p3 Sat ~ a*Train Perf ~ b*Sat + c*Train indirect := a*b ' fit <- sem(model, data = df, se = "bootstrap", bootstrap = 5000) summary(fit, fit.measures = TRUE, standardized = TRUE)

Common Beginner Mistakes

• Skipping the measurement model and testing paths on poorly measured constructs.
• Adding error covariances suggested by modification indices just to improve fit.
• Claiming causal effects from cross-sectional survey data; SEM tests whether data are consistent with a causal theory, it does not prove one.
• Ignoring sample size: very small samples give unstable estimates and unreliable fit indices.

Did You Know?

• Sewall Wright applied his early path analysis to guinea pigs, untangling how heredity and environment shaped traits such as coat pattern and birth weight.
• Different models can fit the data equally well while implying different causal directions; these are called equivalent models.
• LISREL, one of the first SEM programs, stands for LInear Structural RELations.

Key Takeaways

• SEM combines a measurement model (CFA) with a structural model (simultaneous regressions).
• Count degrees of freedom: \( p(p+1)/2 \) minus free parameters must be positive.
• Judge fit with χ²/df, CFI, TLI, RMSEA and SRMR rather than the raw chi-square alone.
• Test mediation through the indirect effect \( ab \), ideally with bootstrap confidence intervals.

If SEM is part of your thesis and you need help with AMOS, SmartPLS or lavaan, Amathify's experts and PhD scholars offer step-by-step guidance and analysis; contact us on WhatsApp or use the Ask a Question form.