Structural Equation Model — SEM

Dr. R. Düsing · Osnabrück University
📋 From Measurement Model to Structural Model
The CFA checked whether the item structure (X = V1–V4 verbal, Y = S1–S4 spatial) fits the data. A structural equation model (SEM) goes a step further: it places a directed regression path between the latent factors — instead of only estimating their correlation, it estimates how strongly X affects Y, after both are cleaned of measurement error.
← Confirmatory Factor Analysis (CFA) Mediation with manifest variables → OLS Regression →
✓ The Advantage over Manifest Variables
If you instead simply formed sum scores of the items (classic test-score approach) and regressed those directly, the estimated association would be attenuated by measurement error — unreliable items pull every correlation toward 0 (see Measurement Error & Attenuation). SEM models the latent factors directly from the items and explicitly nets out measurement error — the estimated structural path targets the true, unattenuated relationship. The comparison card further below shows exactly this gap live.
① Path Diagram — Specified Model
With only 2 constructs, the arrow direction is not empirically testable. X→Y produces exactly the same implied covariance matrix (and the same fit) as Y→X or a mere correlation (CFA) — with a single free path between two factors, there is no degree of freedom that could test the direction. The arrow direction here is a substantive assumption, not a conclusion from this data. Only from 3 constructs onward (mediation model, right) does the structural model become empirically informative.
② Latent (SEM) vs. Manifest (Sum Score)
③ Fit Indices
χ² (df)
—
N = 300
CFI
—
≥ .95 good
TLI
—
≥ .95 good
RMSEA
—
≤ .06 good
SRMR
—
≤ .08 good
—
④ lavaan Syntax (R) — Matching the Specified Model
Cutoffs follow Hu & Bentler (1999), with caveats (see Help). This tool fits itself via maximum likelihood on the (known) population-implied covariance matrix — lavaan would estimate the same structure on real data and additionally deliver standard errors/p-values.
Flashcards
① CFA vs. SEM
CFA only estimates the measurement model plus an undirected factor correlation Φ. An SEM replaces Φ (partly) with directed structural equations between the factors — the same measurement model, but now with regression paths instead of (only) correlations between them.
② =~ Measures, ~~ Correlates, ~ Regresses
Three lavaan operators, three model parts: =~ defines the measurement model (factor → items), ~~ a covariance/correlation (undirected, as in CFA), ~ a regression (directed — the actual "structural" part of SEM). An SEM typically uses all three; cfa() and sem() in lavaan are nearly identical, sem() is just the conventional call once ~ lines appear.
③ Why Measurement-Error Correction Matters
A sum score is only an approximation of the latent factor — its reliability (ω) is almost never 1. Any regression on such scores attenuates the true association by the factor √(ωX·ωY). SEM sidesteps this by modeling the factor directly from the individual items — the structural path targets the relationship between the constructs, not between their error-laden indicators.
④ Indirect Effects Are Especially Vulnerable
Under mediation, attenuation multiplies: the manifest indirect effect â·b̂ is dampened by roughly ωX·ωM·ωM·ωY (measurement error in X and M acts twice). At realistic reliabilities, this can drastically underestimate the estimated indirect effect — a key argument for latent over manifest mediation analysis.
⑤ Full vs. Partial Mediation — Genuinely Testable
Unlike the simple X→Y path (flashcard "not empirically testable" above), the question "does the effect run entirely through M?" becomes a genuine model restriction (c'=0) with its own degree of freedom from 3 constructs onward. If the fit is worse at c'=0, that argues for an additional direct path — partial mediation.
⑥ What This Tool Doesn't Show
No real sampling, no standard errors/significance tests for the paths, no moderation or non-recursive (feedback) models. For manifest mediation with real significance tests (Sobel, bootstrap) see Mediation Analysis.
Structural Equation Model — Background
What This Tool Shows — and What It Doesn't

Shows: a 2- or 3-construct SEM (CFA measurement model + directed structural paths) with genuine maximum-likelihood fitting, plus a direct, closed-form comparison to the naive sum-score approach. Not here: real sampling/standard errors for the paths, free item-factor assignment, moderation, non-recursive models, multi-group SEM.

The Structural Model as a Covariance Reparameterization

For standardized factors (Var=1), a single path Y~β·X is numerically identical to Φ=[[1,β],[β,1]] from the CFA — only the interpretation (directed vs. undirected) changes. Under mediation, path tracing (Wright's rules) yields the 3×3 structure Φ=[[1,a,ab+c'],[a,1,b+ac'],[ab+c',b+ac',1]] — derived in closed form and confirmed against an independent Monte Carlo simulation (200,000 cases) to two decimal places before this tool was built.

The Manifest Comparison — Sum-Score Covariance
Cov(sum scores) = G · Σ · G' (G = grouping matrix, 1 per item→construct) r_manifest = Cov(sumX,sumY) / √(Var(sumX)·Var(sumY)) ≈ β · √(ω_X · ω_Y) (attenuation formula, exactly reproduced)

No new mechanism: the same aggregation that produces the sum-score covariance matrix is also used for the latent model covariance matrix — just fitted before instead of after the aggregation. Under mediation, the manifest indirect effect is determined as a partial standardized regression (Y on M and X, both standardized) from the same aggregated correlation matrix.

Fit Indices

The same formulas and cutoff caveats as in the CFA tool (χ², CFI, TLI, RMSEA, SRMR — explained there in detail with formulas and interpretation: → open CFA help). The structural equations only change which covariances count as "explained" — not how the overall fit is computed. An SEM with structural paths has exactly the same fit statistics as the underlying CFA with the same number of free parameters.

Advantages of SEM over Separate Analyses

Test all paths simultaneously. A classic three-step mediation approach (Baron & Kenny, 1986) estimates a, b, and c' in separate regressions one after another — each with its own measurement error, with no overall fit test. SEM estimates all parameters of the whole model in a single step (full-information ML), including all covariances between the measurement errors of the individual equations — statistically more efficient, and with a single fit test for the entire theory, not just individual paths.

Arbitrarily complex models. This tool shows at most 3 constructs, but the principle scales: multiple predictors, multiple parallel or serial mediators, multiple outcomes, higher-order (second-order factor) constructs — all in one model, with a shared fit.

Formal model comparison. "Full vs. partial mediation" (this tool) is a special case of a general principle: nested models (one model arises by fixing parameters of the other) can be formally compared via a χ² difference test or information criteria (AIC, BIC) — an explicit statistical decision between competing theories, not just "eyeballing the coefficients."

Explicit, falsifiable theory. Every fixation (loading = 0, path = 0, covariance = 0) is a publicly visible, testable assumption — the model forces you to specify precisely what the theory actually claims, instead of leaving it implicit.

Further extensible. The same logic extends to multi-group comparisons (→ Measurement Invariance), growth models (change over time), and categorical/ordinal indicators — not shown here, but the same framework.

Disadvantages & Limitations

Good fit does not prove the correct causal structure. This tool shows it itself: with 2 constructs, X→Y is indistinguishable from Y→X or a mere correlation. Even with more constructs: equally good fit is often compatible with several, sometimes opposing, causal structures (equivalent models) — fit alone never decides between them; only theory and study design (randomization, time series) can.

Larger samples needed. More free parameters require more cases for stable estimates — rules of thumb range from n ≈ 10–20 per free parameter, quickly reaching several hundred cases for more complex models.

Distributional assumptions. Standard ML estimation assumes multivariate normality; under substantial violation (e.g. categorical items, outliers), standard errors and χ² become biased — robust estimators or WLS methods are then needed, not shown here.

Specification searching is a trap. Real software shows modification indices indicating which additional fixation would improve the fit the most. Using that post hoc, data-driven (instead of justifying it theoretically in advance) drastically raises the error rate — the same logic as with Data Dredging.

Only as good as the measurement model. An incorrect fixation in the measurement model (see the CFA tool) also distorts the structural estimates — and is often not easy to distinguish from the actually interesting structural problem based on overall fit alone.

Literature

Tomarken, A. J. & Waller, N. G. (2005). Structural equation modeling: Strengths, limitations, and misconceptions. Annual Review of Clinical Psychology, 1, 31–65.
Kline, R. B. (2023). Principles and Practice of Structural Equation Modeling (5th ed.). Guilford.
Baron, R. M. & Kenny, D. A. (1986). The moderator-mediator variable distinction. Journal of Personality and Social Psychology, 51, 1173–1182.
Cole, D. A. & Preacher, K. J. (2014). Manifest variable path analysis: potentially serious and misleading consequences due to unreliability. Psychological Methods, 19(2), 300–315.
MacCallum, R. C. & Austin, J. T. (2000). Applications of structural equation modeling in psychological research. Annual Review of Psychology, 51, 201–226.
Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2).
Hoyle, R. H. (Ed.) (2023). Handbook of Structural Equation Modeling (2nd ed.). Guilford Press.