Hi, I am posting to bring a topic here to the group that Ed and I discussed previously (thanks Ed!) regarding
how blavaan's wiggle priors computationally loosen constraints, as it applies to Bayesian approximate measurement invariance testing.
I have another follow-up question in hopes of clarifying the technical level of computation in the context of cross-group approximate constraints. I'll refer to the factor loading example provided previously:
loading1 ~ normal(0, 10)
loading2 ~ normal(loading1, .01)
loading3 ~ normal(loading1, .01)
My question regards how parameters (e.g., loadings 2 and 3 in your example) that are being estimated at the same time as their prior-defining reference (i.e., loading 1) are able to receive a prior that relies upon the reference parameter's estimate.
Here is my understanding of your explanation: This example of three loading parameters and their assigned priors correspond to each of three respective groups measured on the given indicator (manifest) variable related to the factor by said loading. Specifically, I interpreted this example to be illustrating the case where the 3 groups—which in traditional measurement invariance testing would all be treated as exactly equal to each other when constrained—are assigned priors such
that
- the first group (G1) on the loading is assigned a diffuse normal prior and once estimated,
- the G1 parameter for the loading then populates the mean hyperparameters of G2 and G3 for the same loading, and
- the variance for G2 and G3 priors remains small to allow for wiggle room around the parameter estimate of G1 to effectively constrain each group's estimate for that loading to be very similar.
Is this interpretation correct? And if so, how does this strategy account for the loading2 and loading3 parameters in computing the estimate for loading1 that must then populate the prior means which become assigned to loading2 and loading3?
Thank you so much!