I am conducting a confirmatory factor analysis examining a three-factor model using R's lavaan package. My factors are shifting, working memory, and response inhibition. I used the modindices function to see if there was any modification index that would reduce my chi-square by 10 or more. It told me that two of my response inhibition variables highly covaried, so I redid the model and accepted the suggestion. As expected, the model improved and my loading coefficients for response inhibition evened out, but all of a sudden the standardized factor loadings for shifting became "NA." What does this mean, and why did a tweak in the response inhibition factor affect the shifting factor? In the original model, I had values for the shifting factor (though it was my worst factor). The only change I made in this new model was drawing the covariance between the two suggested response inhibition variables.
It is difficult to say without seeing the full model and/or model output. It could be that this modification resulted in an improper / inadmissible solution (Heywood case) with one or more negative residual variance estimates (which can correspond to standardized loadings > 1) or that the model became underidentified by adding this covariance parameter. I would check to see which warning message lavaan may have printed in the output.
Christian Geiser Thank you for your answer! The warning message states "some estimated lv variances are negative." So it sounds like the first possibility you mentioned occurred. I am still puzzled as to the theoretical reason why the manipulation of my response inhibition variables affected my shifting factor. But either way, this is a sign that I should accept the model without the modification index?
Again, it is impossible to say without knowing more about your model and sample size as Heywood cases can have many different causes (small sample, too few indicators per factor, model misspecification, overparameterization, etc.)
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