Robustness Testing in SEM: Building a Robustness-Test Matrix

1. What does robustness testing ask?

A conclusion is robust when its direction, magnitude and substantive meaning do not change severely under plausible analytical alternatives. Each variation must represent a credible design, measurement or estimation risk.

Robust standard errors, robustness testing and replication differ. Robust corrections address certain distributional violations; robustness testing compares specifications; replication tests new data. None replaces the others.

Vaithilingam et al. (2024) review PLS-SEM robustness practice and emphasize matching techniques to threats. This article extends threat-based checking to CB-SEM while identifying PLS-specific tools.

• Specify the primary model before inspecting alternatives.
• Link each check to a specific threat.
• Compare direction, magnitude and uncertainty.
• Do not report only favorable alternatives.
• Use tools appropriate to CB-SEM or PLS-SEM.
• Disclose weakened conclusions.

2. Fix the primary model before robustness testing

The primary specification follows theory, design and the analysis plan. It defines variables, relationships, controls, estimator, missing-data treatment, scales and exclusions. Changing it after every result turns robustness into specification searching. Kline (2023) connects specification and assessment with theory, diagnostics and assessable comparisons.

Create a version table first. Each row should identify the analysis, one main change, the threat and the result being compared. This clarifies every alternative’s purpose.

Robustness cannot rescue an unidentified causal model. Cross-sectional, self-reported, nonrandom data still limit causal claims despite stable coefficients. Antonakis et al. (2010) warn that endogeneity can invalidate causal interpretation even with good fit.

3. Map threats before selecting techniques

A long test list does not automatically establish credibility. Ask which assumptions may fail, through what mechanism, which conclusions would be affected and how far the available data can investigate.

Not every study needs every check. Same-source cross-sectional studies should prioritize measurement quality, common method and causal limitations. Group comparisons need invariance. Prediction-oriented PLS-SEM also requires out-of-sample evaluation.

The following matrix selects checks by risk rather than available software.

4. Sensitivity to distribution and estimator

In CB-SEM, comparing conventional ML with robust corrections or bootstrap assesses sensitivity of SEs, CIs and fit to nonnormality. Curran, West and Finch (1996) show interactions with misspecification; Savalei (2014) explains that robust corrections adjust inference, not incorrect models. Yuan and Bentler (2000) discuss likelihood methods for nonnormal missing data. Explain which estimator best fits assumptions rather than selecting the favorable p-value.

For AMOS, Bollen and Stine (1992) provide a bootstrap goodness-of-fit approach under nonnormality, while bootstrap CIs support some parameter inference. Few-category ordinal data or scaled statistics may require software with suitable categorical or robust estimators.

In PLS-SEM, bootstrap is common for inference but does not check every threat. Investigate endogeneity, nonlinearity, heterogeneity and prediction separately when justified.

5. Outliers, influential cases and missing data

Unusual observations can strongly influence covariances and paths. Verify data, assess whether cases belong to the population, then compare retention and treatment under prespecified criteria. Do not delete simply because a p-value changes.

Describe missingness rates, patterns and assumed mechanisms. Complete-case analysis can reduce power and bias results when missingness relates to study variables. Consider FIML in CB-SEM under appropriate assumptions, or multiple imputation preserving the analytical structure.

Focus sensitivity on target parameters. If a path changes sign or its CI widens substantially under another plausible missing-data treatment, reduce certainty.

6. Specification sensitivity and endogeneity

Alternatives must be theoretically plausible, not every imaginable configuration. Examples include models with and without a prespecified control or competing causal orders the design cannot distinguish. Identify prespecified and post hoc variations.

Endogeneity can arise from omitted variables, measurement error, reverse causality or selection. SEM fit does not establish exogeneity. Responses require better design, temporal data, instruments with defensible identification or specialized techniques with explicit assumptions.

Gaussian copula may investigate a form of PLS-SEM endogeneity when the regressor is nonnormal and other conditions hold. It cannot replace identification arguments. IV or simultaneous CB-SEM models also need explicit specification and identification.

7. Heterogeneity, invariance and nonlinearity

For theoretically meaningful groups, assess invariance before comparing paths. CB-SEM multigroup testing usually proceeds from configuration through loading, intercept and residual constraints according to objectives. PLS-SEM uses MICOM to assess composite invariance before MGA.

Unobserved heterogeneity can make average effects conceal opposing mechanisms. Segmentation or mixtures may help, but solutions need stability, sufficient group sizes and substantive interpretation. Do not choose group count merely for favorable p-values.

Investigate nonlinearity when theory or residuals suggest it. Quadratic or interaction terms need predicted-value plots and supported data ranges. A nonsignificant linear coefficient does not establish no relationship when the true shape is curved.

8. A reproducible robustness workflow

Use this as an analysis appendix. The number of variations depends on risks, but readers should see the primary model before its alternatives.

9. Hypothetical example: how strong is a conclusion?

This example is illustrative. Primary CB-SEM on N = 438 estimates Information Quality → Usage Intention at beta = .24, 95% CI [.10, .38]. A robust estimator yields beta = .23, 95% CI [.08, .38]. FIML instead of complete cases yields beta = .19, 95% CI [.04, .34].

Adding a prespecified control reduces beta to .14, 95% CI [-.01, .29]. Direction remains positive but uncertainty includes 0. Evidence is therefore sensitive to controls rather than fully robust.

An honest report states stability to estimator and missing-data treatment but weaker evidence under a plausible control specification. Do not report only the first two alternatives.

10. APA 7 reporting template and checklist

This template uses hypothetical values. Include a robustness table in the article or appendix, including alternatives that weaken conclusions.

• Prespecify the estimand and primary model before sensitivity checks.
• Link every check to a specific threat.
• Distinguish robust correction from robustness testing.
• Use tools appropriate for CB-SEM or PLS-SEM.
• Compare estimates and CIs, not p-values alone.
• Inspect original data before treating outliers.
• State missing-data assumptions.
• Consider misspecification and endogeneity in the design.
• Assess invariance before group comparisons.
• Report alternatives that weaken conclusions.
• Do not equate stability with causal evidence.
• Preserve code, output and version tables for reproduction.
• Identify hypothetical examples.
• Match citations and references.