unequal group sizes in a multi-group CFA

Question

I am testing measurement invariance across several demographic categories: age (will dichotomize), race, and BMI. The problem of unequal group sizes is most evident for racial categories:

n White = 674 n Hispanic = 125 n black = 125

Reading Hox and Mass (2001) which addresses unequal sample sizes in multilevel SEM, it seems defensible to interpret path coefficients, model fit, and standard errors if the group sizes are above 100.

The estimator method I will be using is robust weighted least squares because the indicators are dichotomous and I will need to test using item thresholds. It may also be relevant that I have 11 indicators. I have three questions

1. Any evidence or dis/agreement with the rule of thumb for 100 cases per group allows for comparing groups with unequal group sizes
2. Building on what I have read in Brown, 2006 (p. 279) What should one consider if unequal sample sizes are used. For example, I believe there is an increase in Type I error rate (finding finding invariance, when the data in fact do not support invaraince)... Is this accurate and do you have other thoughts related to this.
3. Is is possible/recommended to simply take a random sample of your larger group that is balanced with your other smaller groups.

References Hox, J. J., & Maas, C. J. (2001). The accuracy of multilevel structural equation modeling with pseudobalanced groups and small samples. Structural equation modeling, 8(2), 157-174. Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York, NY: Guilford Press Publications.

Answer 1

1. I don't believe so. There's no problem here - except that it makes your analysis more sensitive to violation of the homogeneity of variance assumption. Get around that by using a robust / sandwich estimates (e.g. SB,
2. I don't know what Brown says, so I have no comment. If you are worried about this, the best thing to do is run some simulations and see if you do detect invariance when the null hypothesis is true. (Nitpick: If you find invariance, the data do support invariance. You should write "finding invariance, when the null hypothesis of no invariance is true."
3. No. Don't discard data that you have spent time, money and effort collecting.

Run your analysis and see if you have a problem first. You seem to be concerned about Type I errors. If your results aren't significant, you don't have a problem. With those sample sizes, I'd be more concerned about power.