Heywood Case Assessment
In covariance-based structural equation modeling (CB-SEM), a Heywood case refers to a situation where an estimated parameter falls outside its admissible range, which is theoretically impossible. A common example is a negative variance estimate.
Researchers should be aware of two main potential outcomes that indicate inadmissible CB-SEM solutions and take steps to address them.
Negative Variance Estimates
A typical Heywood case occurs when a variance estimate—usually an error variance or factor variance—is negative. This is inadmissible, since variances must be zero or positive.
To detect Heywood cases in SmartPLS, check for negative variances in the model matrices Theta and Psi, which are available under Model and data in the results output:
- The Theta matrix contains indicator error variances and covariances. Variances appear on the diagonal, so examine these diagonal elements for negative values. Additionally, indicator error variances are listed under Indicator error terms -> Estimated parameter (overview). Variances correspond to entries with one indicator name, while covariances are represented by two connected indicators (indicated by “<-->”). Note that covariances can be negative, so only variances require scrutiny.
- The Psi matrix represents the variances and covariances of the latent variables or their error terms (depending on model specification). Again, check diagonal entries (variances) for negativity; off-diagonal elements (covariances) can be negative.
Performing these checks helps uncover inadmissible negative variances that might otherwise remain hidden in the standard output.
Standardized Factor Loadings Exceeding 1
Heywood cases can also be indicated by standardized factor loadings exceeding 1 in absolute value. Such values imply that an indicator’s variance is explained by more than 100%, which is theoretically impossible.
What to Do When a Heywood Case Occurs
Encountering a Heywood case in CB-SEM indicates potential problems with model specification, data quality, or estimation procedures. To address and resolve Heywood cases in SmartPLS, researchers should consider the following strategies. After implementing corrective actions, re-estimate the model in SmartPLS and carefully inspect the variance estimates and standardized loadings for any remaining Heywood cases. Repeated occurrences may point to fundamental issues with the study design or data.
Review Model Specification
- Check for Model Misspecification: Re-examine the conceptual model to ensure it accurately represents theoretical relationships. Incorrectly specified paths, omitted variables, or inappropriate measurement models can lead to inadmissible estimates.
- Simplify the Model: Complex models with many parameters relative to the sample size may cause estimation issues. Consider reducing the number of latent variables or indicators to improve model stability.
Inspect Data Quality
- Assess Sample Size: An insufficient sample size can cause unstable estimates leading to Heywood cases. Follow recommended sample size guidelines for CB-SEM and increase the sample size if necessary.
- Check for Outliers and Data Distribution: Extreme values or non-normal distributions may distort variance estimates. Conduct data screening and apply appropriate data transformations or robust estimation techniques.
- Examine Multicollinearity: High multicollinearity among latent variables can affect variance estimates. Use variance inflation factors (VIF) to identify problematic predictors and consider removing or combining variables.
- Consider Rescaling: Highly unequal variances among indicators may produce unreliable variance estimates. Rescaling or standardizing variables to similar metrics can improve estimation.
Modify Measurement Scales
- Review Indicator Reliability: Indicators with very low reliability or variance can contribute to negative variance estimates. Remove or replace indicators with poor psychometric properties.
- Adjust Indicator Sets: When theoretically justified, consider replacing reflective indicators with formative indicators or vice versa, to help mitigate Heywood cases.
Reconsider Estimation Settings
- Use Alternative Estimators: If maximum likelihood estimation is used, consider robust estimators or Bayesian methods that better handle problematic data structures.
- Set Parameter Constraints: You can restrict variance estimates to non-negative values by setting parameter constraints via the modeling window (e.g., clicking on error terms and specifying variance bounds). However, such constraints should be theoretically justified, as arbitrary limits may distort other parameter estimates, such as path coefficients or loadings.
