is it possible in Mplus to examine whether the tresholds of a categorical response variable using a probit function vary a) at the within group-level between respondents (b) at the between-group level e.g. between countries? Could you provide a syntax example?
b) Yes. And this can be done in various ways. See UG ex 9.26 for one approach. Also see the papers on our website under Recent Papers:
Asparouhov, T. & Muthén, B. (2016). General random effect latent variable modeling: Random subjects, items, contexts, and parameters. In Harring, J. R., & Stapleton, L. M., & Beretvas, S. N. (Eds.), Advances in multilevel modeling for educational research: Addressing practical issues found in real-world applications (pp. 163-192). Charlotte, NC: Information Age Publishing, Inc. Mplus scripts.
Muthén, B. & Asparouhov, T. (2016). Multi-Dimensional, Multi-Level, and Multi-Timepoint Item Response Modeling. In van der Linden, W. J., Handbook of Item Response Theory. Volume One. Models, pp. 527-539. Boca Raton: CRC Press. Download output files. Download table 6 output.
Muthén, B. & Asparouhov, T. (2016). Recent methods for the study of measurement invariance with many groups: Alignment and random effects. Download Mplus scripts.