What each invariance result licenses
Use the invariance label to constrain the comparison sentence, not to award a generic quality score.
| Result or method | What is constrained | Comparison it can support in the specified model | Stronger claim not licensed by the label alone |
|---|---|---|---|
| Configural invariance | Factor pattern and basic construct representation. | A claim that the same structural pattern is being examined across groups. | Equal item meaning, equal relations, or equal latent means. |
| Metric invariance | Factor loadings across groups. | Comparisons involving specified relations such as covariances or unstandardized regressions. | A higher or lower latent mean. |
| Scalar invariance | Loadings plus intercepts or thresholds as specified. | Latent-mean comparisons under the declared model and sample. | Permanence across samples, modes, translations, or instruments. |
| Partial invariance | Selected parameters are released while others remain equal. | The narrower comparison justified by the retained constraints and identification. | Treating every parameter as equal. |
| Approximate invariance | Small differences are allowed under an explicit tolerance or prior. | The comparison justified by that approximate model and its assumptions. | Calling the result exact or applying its tolerance universally. |
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Reference & Evidence
Source: Author's comparison framework grounded in Cieciuch et al. (2014). The rows state methodological boundaries, not results for a new instrument or sample.
Each line is a claim from the register this journal publishes against, resolved from the register at build time.
- A Measurement invariance concerns whether the same construct is measured in the same way across groups; metric and scalar invariance license different comparisons cieciuch-davidov-2014-exact-approximate-measurement-invariance ·
CDAS14-C1 - A Configural invariance fixes the loading pattern, metric invariance constrains loadings, and scalar invariance additionally constrains indicator intercepts cieciuch-davidov-2014-exact-approximate-measurement-invariance ·
CDAS14-C2 - A In the cited eight-country PVQ-5X illustration, exact scalar invariance covered 10 of 19 values in the earlier analysis, while the specified approximate analysis covered all 19 cieciuch-davidov-2014-exact-approximate-measurement-invariance ·
CDAS14-C4 - A Approximate invariance allows small parameter differences through a prior distribution and is not the same as partial invariance cieciuch-davidov-2014-exact-approximate-measurement-invariance ·
CDAS14-C5
Grades: A, verified against the printed page of the primary source · B, primary source, text layer only · C, authoritative secondary · D, reported.
Related exhibits
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The bilingual measurement-release gate
From the essay Measurement invariance before comparing English and German scores
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From translation version to comparison sentence
From the essay Measurement invariance before comparing English and German scores
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The synthetic selection-bias eligibility ledger
From the essay What is selection bias? The sample can change the answer