I have a question regarding the estimation of factor means in the CFA context. I understand how factor scores for individuals are calculated, but am not clear on how factor means are estimated when one does a latent means comparison. I have looked at equations that show the link between an individual observed variable and the latent variable, but can't get my head around how this works for multiple observed variables and a single factor. What I'm ultimately interested in learning is how a latent mean changes value as one changes a single factor loading while holding everything else constant. Thanks for any insights you can provide.
It is hard to give an intuitive notion of this. First, note that factor means are not estimated by averaging factor scores. Then consider the population formulas - the factor means are related to the means for the observed variables y as
E(y) = nu + Lambda*alpha
where nu is the intercept vector, Lambda is the loading matrix and alpha is the factor mean vector to be estimated. We have sample means estimating E(y). One group sets alpha=0 which identifies nu. The second group then solves for alpha given nu and Lambda (which is identified from the covariance structure and is usually identical across the groups). If the second group changes a loading for a factor (a column in Lambda), the alpha estimate is adjusted to fit the y means. Hope that helps.
Thanks very much. So it sounds like that if I change one loading value (say for y1) out of a set of 3 y's, in order to maintain the equality above, something else would need to change as well, in addition to alpha. So as a simple example, if ybar1=2, ybar2=3 and ybar3=4, nu1=nu2=nu3=0 and alpha=1, then my loadings would need to be .2, .3 and .4. We can assume that those are invariant across groups.
Now if the loading for y1 and group 2 were changed to .4 (no longer being invariant across groups), we would have to change either alpha for group 2 or nu1. But if we change alpha, then we would also have to change the loadings for the other two variables it seems. The alternative would be for nu1 in group 2 to change as well, in which case alpha could remain the same.
So just to make sure I'm clear, it appears that if one of the loadings is different between the groups but all of the intercepts are invariant then in order for the equality to work out, other loadings must also be different for the groups. Does that sound right?
Sorry if I'm being dense, but I do want to make sure I understand this issue. Thanks so much for your help.
That's the correct reasoning. As a little additional twist, note also that if you change the y1 loading (no longer being invariant across groups), you can change alpha for that group - and you don't have to change the other loadings, but can instead compensate by changing the other nu's.
I am conducting a longitudinal analysis (two time points) in a dataset. I have fixed time 1 latent factor means to 0 for identification purposes. Is there any specific range of value for latent factor means at time 2 in a longitudinal analysis? I mean do they have a value between 0-1 or can they have a value of >1 also? Thank you!
Jason Bond posted on Monday, May 06, 2013 - 5:05 pm
Using Mplus to run a CFA on a set of poloytomous items produces a mean of a single latent factor that is not 0 (as it is for factor analyses of continuous items). Is there a reference you could provide that might explain the relationship between the factor mean and the mean of individual items? Somewhat related to this, what I'd like to be able to do is, for each item and respondent, produce the estimated value of the item from the factor score for the respondent and the factor loadings and thresholds for each item. Is there a reference you could provide that would allow me to crosswalk (i.e., invert) this relationship? Thanks,