This message typically indicates that you have one or more negative residual/error variance estimates in your model (so-called "Heywood case" or "improper solution"). Since variances cannot be negative by definition, such an estimate is considered improper or "out-of-bounds". Improper parameter estimates are typically caused by:

- model misspecification (e.g., too many or not enough factors or too many or too few other parameters)

- too few indicators per factor (factors with only 2 indicators are more likely to result in an improper solution)

- too small a sample size

- random sampling fluctuations (in other words: bad luck).

To solve this problem, usually re-specification of the model is necessary and/or the use of a larger sample (if possible).

See also my Youtube videos on this topic:

Non-positive definite theta warning in Mplus: https://youtu.be/HRZiCSfR0-s

What is a Heywood case?: https://youtu.be/bmpYqqksbtA

Top 3 reasons for improper solutions: https://youtu.be/Z_o5LOwvvxo

Avoid improper solutions: https://youtu.be/YKnXnOzW2qo

Improper solutions in Mplus: https://youtu.be/88o0drJ1VO4

Important feed, thanks! Christian Geiser I recall encountering this warning in a previous analysis (though I cannot recall exactly which one), yet the overall model fit remained acceptable. If attempts to resolve the issue are unsuccessful, would it still be justifiable to publish based on the analysis, provided the results otherwise appear plausible? Best John-Kåre

That's a good question. A model can have a decent overall fit and still be misspecified. For example, it may be overparameterized, leading to the negative error variance in question. An overparameterized model may show a very "good" fit since it overfits the data (it estimates more parameters than necessary to reproduce the observed covariance and/or mean structure). So in summary, a good overall model fit doesn't necessarily mean that the model is true/correctly specified/proper.

Since typically, a negative error variance indicates a problem with the model, I would be cautious in interpreting the results unless it is very plausible that the result was caused by random sampling error alone. For example, when dealing with very highly reliable variables (say, reliability > .95 or so), random sampling fluctuations may explain a (slightly) negative error variance, especially in smaller samples. For more guidance, see:

Chen, F., Bollen, K. A., Paxton, P., Curran, P. J., & Kirby, J. B. (2001). Improper solutions in structural equation models: Causes, consequences, and strategies. Sociological Methods & Research, 29(4), 468-508.