problems with constraining the path to be equal in structural equations

[Eric Han]

Hi All,

I am trying to compare whether it is better to use Mcdonald Omega over Cronbach alpha to measure the reliability of concepts. For the model computing the alpha, I used the following command
to constraint all the loadings to be equal.
sem evaluation@a->BYA27*,latent(evaluation) nocapslatent stand

However, I got the following output. It seems that not all the loadings are equal, the constraints at the beginning indicate that the constraint is imposed. Could anyone tell me what goes wrong and suggest solutions?

[Clyde Schechter]

I'm not certain, but I think the problem arises due to the standardization. The constraint of equal loadings applies to the unstandardized loadings. I'm guessing that the BYA27* variables have different variances in the estimation sample, so that after standardization the coefficients you see are no longer equal.

I am one of those people who really dislikes the use of standardized variables in the first place, but I will spare you my rant on that and just assume that you actually have a good reason for doing it. In that case, I think you have to first indicate the estimation sample in your data, then standardize the variables within the estimation sample, and then run an unstandardized SEM on the standardized variables.

[Richard Williams]

I'll add that I am not sure you would want standardized coefficients constrained to be equal. As Clyde notes, variables will get standardized differently because they have different variances. Equality constraints generally don't make sense unless variables are measured the same way.

Here is my own handout on why standardized coefficients are generally the work of the devil. See the last page for a summary of the arguments.

https://www3.nd.edu/~rwilliam/stats2/l71.pdf

Also maybe see

https://www3.nd.edu/~rwilliam/stats2/l72.pdf

[Eric Han]

Thank you both for the prompt responses. Clyde, the cause for not getting equal coefficients is the same as your diagnosis. I am taking a psychometric class and has learned that Cronbach alpha calculation is based on the assumption of equal loadings with the underlying factor while Mcdonald Omega is not. It turns out that the assumption of equal loadings is not tenable in many situations. That is why I tried to test it by comparing the parallel model (same loadings) and cogeneric model (not equal loadings). It turned out that I did not specify the correct standardization for stata. I should have specified the variance of the underlying factor as one. After tinkering the code, sem evaluation@a->BYA27*,latent(evaluation) nocapslatent var(evaluation@1)*parallel vs. sem evaluation>BYA27*,latent(evaluation) nocapslatent stand *congeneric. I got the following results:

Thank you both again for the help. Richard, your handouts have been tremendously helpful for me as both a student and TA for econometrics.

[Clyde Schechter]

Glad you got it straightened out. I will just add that had I understood that you were trying to test whether all of the loadings were equal, I would have told you that the easiest way to do that is with the -test- command after -sem-.

Well, maybe that's not the easiest way, but it's the most straightforward.