When testing for model invariance across two groups, what is the impact of different sample sizes in the two groups? If we compare one group of n = 328 to another group of n = 1150, which is what we have in mind, is there any bias involved, perhaps due to the greater precision of estimation in the larger sample? We have considered an alternative in which we select a subsample from the 1150 group matched to the 328 group on relevant variables. If we do that, we can compare two groups of 328 to each other. Are there ways in which the latter is better?
We get that question often. I don't know that there necessarily is a great disadvantage of having a smaller sample in one of the groups - as long as there is a sufficient sample size in each group. n=328 seems sufficient for most models. It is true of course that the results for invariant parameters will be geared towards the larger sample.