Longitudinal Measurement Invariance with Binary Data: Identification Strategies

[alexander starlinger]

Dear Prof. Rosseel, dear lavaan-team, dear group members,

This is not per se a lavaan-specific question but is generally more concerned with longitudinal MI with binary data - I hope that's okay!

Im currently involved in a project for which longitudinal measurement invariance of an ADHD symptoms measure needs to be established. The key characteristics of the data are:

- 1 common factor

- 5 indicators 
- 4 measurement occasions (spanning 5 years)

- dichotomous format 

The data is not naturally dichotomous but had to be collapsed due to high skewness in some items. 

So far we followed the identification strategy described by Liu et al (2017) (who adapt the framework from Millsap & Tein, 2004 for the longitudinal case) which is however only concerned with the polytomous case. The identification setup is basically

a) All intercepts of latent responses are fixed to 0

b) The common factor means get 1) fixed to 0 for the reference measurement and 2) are freely estimated for all other occasions

c) The unique factor covariance matrix is 1) constrained to identity for the reference measurement and b) a diagonal matrix for all other occasions (i.e., no correlated errors between different indicators, variances are constrained to 1 for the reference, and variances are estimated freely for all others)

d) For all measurement occasions, the factor loading of the marker item is constrained to 1

e) One threshold for each indicator is constrained to equality across measurement occasions; a second threshold is additionally constrained for the marker item

The problem lies with the last constrain. As the data is binary, only one threshold per item exists. Liu et al (2017) note that the binary case comes with additional identification constrains, and these are spelled out in Millsap & Tein (2004) (under 5d), however as far as I understand they only apply for the multigroup case, not the longitudinal one.

On top of not being able to constrain a second threshold, our baseline model additionally shows that the threshold constrain has to be released for one indicator. Naturally, the resulting model is not identified (as indicated by a singular information matrix).

My question is the following: How is identification for longitudinal MI achieved when dealing with binary responses? Are there alternative identification strategies for this? I can not seem to find literature covering this case. 

Also, how does that impact subsequent model comparisons? Is it correct that establishing the assumptions for latent mean comparisons (i.e., metric/scalar invariance) would then only make one model comparison necessary (as threshold invariance can not be tested and is part of the configural model)?

The models so far have been run with WLSMV and theta parametrization using lavaan version 0.6.21.

Unfortunately, i can not make the data available - I hope this description is sufficient.

Thanks in advance!

Best
Alex