1. Measurement Bias Detection Through Factor Analysis Barendse, M. T., Oort, F. J. Werner, C. S., Ligtvoet, R., Schermelleh-Engel, K.

2. Defining measurement bias Violation of measurement invariance Where V is violator If V is grouping variable, then MGFA is suitable Intercepts – uniform bias Factor loadings – non-uniform bias (vary with t)

3. Restricted Factor Analysis (RFA) Advantages of RFA over MGFA: V can be continuous or discrete, observed or latent Investigate measurement bias with multiple Vs. More precise parameter estimates and larger power Disadvantage of RFA: Not suited for nonuniform bias (interaction term)

4. Approaches for non-uniform bias RFA with latent moderated structural equations (LMS) ---- Simulation (categorical V) showed at least as good as MGFA RFA with random regression coefficients in structural equation modeling (RSP) ---- performance unknown

5. This paper… Compared methods: MGFA RFA with LMS RFA with RSP Measurement bias Uniform Nonuniform Violator Dichotomous Continous

6. Data generation (RFA) True model: Uniform bias:. Nonuniform bias: T and v are bivariate standard normal distributed with correlation r e is standard normal distributed u is null vector

7. Simulation Design For continuous V: Type of bias (only on item 1): No bias (b=c=0), uniform bias(b=0.3,c=0), nonuniform bias (b=0,c=0.3), mixed bias (b=c=0.3) Relationship between T and V Independent (r=0), dependent (r=0.5)

8. Simulation Design For dichotomous V: V=-1 for group 1 and v=1 for group 2 Model can be rewritten into Relationship between T and V: Correlation varies!

9. The MGFA method When v is dichotomous, regular MGFA When v is continuous, dichotomize x by V Using chi-square difference test with df=2 Uniform : intercepts Nonuniform: loadings

10. The RFA/LMS method V is modeled as latent variable: Single indicator Fix residual variance (0.01) Fix factor loading Three-factor model: T, V, T*V Robust ML estimation Chi-square test with S-B correction: : uniform bias : nonuniform bias

11. RFA/RSP method Replacing with, where is a random slope. Robust ML estimation Chi-square test with S-B correction: : uniform bias : nonuniform bias

12. Single & iterative procedures Single run procedure: test once for each item Iterative procedure: 1)Locate the item with the largest chi-square difference 2)Free constrains on intercepts and factor loadings for this item and test others 3)Locate the item with the largest chi-sqaure difference 4)… 5)Stops when no significant results exist or half are detected as biased

13. Results of MGFA – single run Shown in Table 2. Conclusion: 1.better with dichotomous than with continuous V; 2.non-uniform bias is more difficult to detect than uniform bias; 3.Type I error inflated.

14. Results of MGFA – iterative run Shown in Table 3. Conclusion: 1.Iterative procedure produces close power as single run does. 2.Iterative procedure produces better controlled Type I error rate.

15. Results of RFA/LMS & RFA/RSP - single run Shown in Table 4 and Table 5. Conclusion: 1.LMS and RSP produce almost equivalent results. 2.larger power than MGFA with continuous V. 3.More severely inflated Type I error rates

16. Results of RFA/LMS & RFA/RSP - iterative run Shown in Table 6. Conclusion: 1.Power is close to the single run 2.Type I error rates are improved

17. Results of estimation bias - MGFA Shown in Table 7. Conclusion: 1.Bias in estimates is small 2.Bias in SD is non-ignorable 3.Smaller bias in estimates for dichotomous V (dependent T&V)

18. Results of estimation bias - RFA Shown in Table 8 & 9 Conclusion: 1.Similar results for LMS and RSP 2.Small bias in estimates 3.Non-ignorable bias in SD 4.Smaller SE than MGFA 5.Smaller bias in estimates than MGFA with dependent T&V, continuous V.

19. Discussion Nonconvergence occurs with RFA/LMS

20. Non-convergence Summary: