Negative variance estimates
A negative variance estimate is impossible — variance cannot be below zero — so it always signals a problem with the model or data, not a real finding. This is a Heywood case, usually caused by a small sample, a factor with too few indicators, misspecification, or an outlier-driven correlation.
What it means
Variance measures spread and is mathematically bounded at zero, so a negative variance estimate is logically impossible. When CFA or SEM returns one — most often a negative residual (error) variance — the solution is inadmissible: the model as specified does not match the data. This is the same phenomenon as a Heywood case, and it often comes with a standardized loading above 1 or a non-positive-definite model-implied matrix.
Common causes
- Small sample size, which lets sampling error push a near-zero variance below zero.
- A factor with too few indicators (two, or one dominant item).
- Model misspecification — wrong number of factors, items on the wrong factor, or a missing cross-loading.
- Outliers inflating a correlation past what the model can reproduce.
- Near-zero true error variance for a very reliable item, where noise tips the estimate negative.
- Multicollinearity among indicators.
How to fix it
- Locate the parameter with the negative variance in the output.
- Re-examine whether that item belongs to its factor; check for a cross-loading or a better structure.
- Ensure each factor has at least three indicators and restructure fragile two-indicator factors.
- Screen for outliers and influential cases and re-run without them to see if the estimate moves into range.
- Increase the sample size — many negative variances vanish with more data.
- Only as a last resort, after ruling out the above, constrain the variance to a small positive value — and report that you did.
How to avoid it next time
Use at least three solid indicators per factor, screen outliers, and size the sample to the model before you analyse. Explonia reports the standardized solution and reliability, so an item heading toward a negative residual or a loading near 1 stands out and you can revise the measurement model and re-run. It flags the red flag — it does not hide an impossible estimate. See CFA and reliability.
Frequently asked questions
Is a negative variance ever a real result?
No. Variance cannot be negative, so a negative estimate is always a sign of a problem — small sample, too few indicators, misspecification or outliers — never a substantive finding. Diagnose and fix the cause rather than interpreting the value.
What is the difference between a negative variance and a Heywood case?
They describe the same situation. A Heywood case is the general term for an out-of-bounds estimate in factor analysis or SEM; a negative error variance is its most common form. A standardized loading above 1 is another version of the same problem.
Can I fix the negative variance to zero and move on?
Only as a last resort, and only after ruling out misspecification, too few indicators, outliers and small N. Constraining it to a small positive value forces an admissible solution, but if the real cause is a wrong model, the constraint just hides it and can bias other estimates. Always disclose the constraint.
How to cite this page
If this page helped your research, you can cite it (APA 7):
Szymkowiak, A. (2026). Negative Variance Estimates in CFA & SEM (Explained). Explonia. https://explonia.com/fix/negative-variance/
Last updated: July 2026
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