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I have two latent factors, F1 and F2, with 3 indicators each. I have two groups, males and females so I'm doing multi-group CFA.

I easily achieved configural and metric equivalence (equal factor loadings).

I also got partial scalar equivalence (all but one of the 6 indicator variable intercepts are set equal).

I understand everything pretty well except the latent factor means. For identification, they are set to zero for one group and estimated for the other group.

So for the first group (males) they are both zero, for the 2nd group (females):

F1 latent mean = -.15, standardized latent mean = -.7 (p-value .00)

F2 latent mean = -.17, standardized latent mean = -.2 (p-value .06)

I'm not sure how to interpret these. I'm pretty sure F1's mean is significant because the indicator variable for F1 who's loading is fixed to 1 is on a scale of 1-2 (yes or no), whereas for F2 the variable who's loading is fixed is on a scale of 1-5. So it makes sense that -.15 is significant for F1 but that -.17 is not significant for F2, because the scale of F1 and F2 are 1-2 and 1-5 respectively...but I'm still not sure how to interpret this.

Does it mean that for females, they are at a "baseline"/"expected value" of .7 standard deviations below men for F1? Does it mean that even if males and females have the same factor loading and same intercept (which is true for 5/6 of my variables) that females are still expected to have a lower value for the item?

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The mean of the free latent is the difference between the male and female mean. So the female mean is 0.15 points lower than the male for F1, and 0.2 points lower for F2. These are on the scale of the units that were used to identify the variances - so they are in units of a 5 point scale and a 2 point scale.

Standardization can be weird. I don't know what program you are using, or how it standardizes. But does that make sense in terms of the units? That is, is 0.15 (the difference) about 0.7 SDs on that 5 point scale (it seems high, which is why I wouldn't trust it). Did you constrain variances to be equal across groups?

I have never tried to interpret the standardized means.

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    $\begingroup$ It's the -.15 that is on the 1-2 scale and the -.17 on the 1-5 scale. Therefore I guess I can see how -.15 might be almost a SD lower on a 1-2 scale, (I'll check later to be sure). I used the lavaan package in R, and I did not constrain the variances to be equal. I think I was trying to over-interpret the means - as you mentioned, all it really says is that females on average have lower F1 and F2 scores by the amounts shown, as determined by how they answered F1s indicator questions in the survey data. $\endgroup$ Jul 29, 2017 at 0:28

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