Factor loadings change direction after changing from fixing variance to fixing factor loading

[Dai Duong]

Hello everyone,

I face a weird situation:

When I change from fixing variance to fixing factor loading, the standardized factor loadings change their direction:

Other paramaters are the same in both fixed variance fitting and fixed factor loading fitting.

Why it happen like that?

Thank you very much, especially Dr. Yves Rossel for previous answers,

Dai

[Edward Rigdon]

This is not surprising. When you establish identification by fixing the factor variance, the loadings are indeterminate with respect to sign. Remember, to create the model-implied covariance matrix, pairs of loadings will be multiplied together--if both are positive or both are negative, the result is a positive covariance either way. If one is positive an one is negative, the result is negative, and it does not matter which one was positive and which one was negative. By contrast, when you establish identification by fixing a factor loading, the sign of that loading provides an orientation for the signs of the other loadings.

If you prefer one set of signs or the other, and you want to fix factor variances, use starting values for some loadings. Once you get the optimization process started on one side of the mountain or the other, it will very likely stay there. 

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[Dai Duong]

Mr. Edward Rigdon is correct.

THe issue occurs because one of indicators (named "treated") is supposed to be negative while other indicators are positive. So, when fixing variance of the latent variable, factor loadings are indeterminate, to guarantee a positive variance, other factor loadings will change to negative (while they are supposed to be positive). In case of fixing factor loading, the default sets a positive loading, so it is suitable.

I fix the issue by reversing "treated" to "free from treated", then all loadings are positive. Two scaling methods produce the same standardized factor loadings.
Thank Mr. Edward Rigdon for the great help.