Reliability computed from CFA/SEM standardized loadings — the standard statistic in measurement-model tables alongside AVE.
Measure ReliabilityMultivariatealso known as: Construct reliability, Jöreskog's rho, ρc
✓ When to use
Reporting measurement quality within CFA/SEM/PLS studies (the CR column of the standard table).
Alongside AVE for the convergent-validity argument.
When reviewers of SEM-based work expect Fornell–Larcker-style reporting.
✗ When NOT to use
Without an estimated factor model.
Formative constructs.
As the sole reliability evidence when the measurement model fits poorly.
Data requirements
Dependent / outcome variable
Standardized loadings (and residuals) of a construct's indicators from CFA/SEM/PLS.
Independent / grouping variable
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Design
Same data as the measurement model.
Sample size guidance
As required by the CFA/SEM.
Assumptions
Adequately fitting reflective measurement model.
Uncorrelated indicator errors unless modeled.
Standardized solution used consistently.
Hypotheses
H₀ — — (descriptive index).
H₁ — —
The concept
CR = (Σλ)² / [(Σλ)² + Σ(1 − λ²)] — the squared sum of loadings over that plus summed error variances. It is mathematically the same congeneric-model quantity as McDonald's omega total; the 'CR' label simply reflects the SEM/PLS reporting tradition (Fornell & Larcker, 1981).
Thresholds: ≥ .70 satisfactory (≥ .60 tolerated in exploratory work); > .95 suggests redundant items. Because CR uses each item's actual loading, it is more accurate than alpha whenever loadings differ. It pairs with AVE: CR asks 'is the composite consistent?', AVE asks 'does the construct own most of its items' variance?'
Worked example
From the wellbeing CFA (loadings .62–.86): physical CR = .84, psychological CR = .89, social CR = .81 — all above .70.
Combined with AVE ≥ .52 for each construct, the measurement model shows satisfactory convergent quality.
How to run it
library(lavaan); library(semTools)
fit <- cfa(model, data = df)
reliability(fit) # row 'omega' = CR; also alpha and AVE per construct
compRelSEM(fit) # dedicated CR function (semTools ≥ 0.5)
Column of loadings λ; then =SUM(λ)^2/(SUM(λ)^2 + SUMPRODUCT(1−λ^2)).
Repeat per construct; assemble the CR/AVE table for the manuscript.
Interpreting the output
CR ≥ .70 per construct = satisfactory internal consistency in the SEM sense.
Read jointly with AVE ≥ .50; high CR with low AVE means many weak-but-numerous indicators.
CR > .95 → check for item redundancy.
State that CR was computed from standardized CFA loadings.
APA-style reporting
Composite reliabilities were satisfactory for all constructs (physical CR = .84; psychological CR = .89; social CR = .81), and all AVEs exceeded .50, supporting convergent validity.
Common mistakes
Reporting CR from an unfit measurement model.
Mixing standardized and unstandardized loadings in the formula.
Treating CR and alpha discrepancies as errors — they differ by design.