1. PLS Group Comparison: A Proposal for Relaxing Measurement Invariance Assumptions Lucian Visinescu University of North Texas, Denton, USA

2. Presentation Agenda Framing the Question The Simulation Conclusions

3. Framing the Question Understanding and generalizing phenomena involves comparison of frameworks or models in different settings Multi-group analysis studies require adequate equivalence of the instruments used to assess the theoretical constructs under investigation The study of the equivalence of instruments in cross- national/multiple-group studies is known as the problem of measurement invariance

4. SEM Measurement Invariance “Whether or not, under different conditions of observing and studying phenomena, measurement operations yield measures of the same attribute” (Horn and McArdle 1992, p. 117) Measurement invariance hypotheses verify: - configural variance - construct-level metric invariance - item-level metric invariance - residual variance invariance - intercept invariance - equivalence of construct variance - equivalence of construct covariance - equivalence of latent means (Cheung & Rensvold, 2002).

5. SEM Measurement Invariance “ hypothesis H Λ,Φ(jj) states that the variance of constructs (i.e., latent variables) are invariant across groups…whereas hypothesis H Λ,ν,Κ posits that the latent means are invariant across groups.” (Cheung & Rensvold, 2002 p.238) There have been calls and discussions for measurement invariance relaxation, known as partial measurement invariance (Byrne, Shavelson, & Muthen, 1989; Cheung and Rensvold, 2002).

6. PLS Measurement Invariance Group Comparison Construct measures are invariant among groups Moderator variable effect is restricted to the path coefficient Sarstedt et al. (2011)

7. Imaginary Model LV1 X1 X3 X2 LV2 Y1 Y3 Y2 m Framework From Sarstedt et al. (2011)

8. Group 1 LV 1 1 X1 X3 X2 LV 2 1 Y1 Y3 Y2 LV 1 2 X1 X3 X2 LV 2 2 Y1 Y3 Y2 Group 2

9. Latent Variable Means LV 1 1,2 LV 2 1 LV 2 2 When latent variables are not significantly different Sarstedt et al. (2011), path coefficients can be directly compared using Chin (2000)

10. Latent Variable Means Easy Scenario

11. Latent Variable Means Scenario LV 1 1,2 LV 2 1 LV 2 2

12. Latent Variable Means Scenario LV 1 1 LV 1 2 LV 2 1 LV 2 2

13. If Latent Variable Means are different… What to do? How might we approach this?

14. “ΠΑ ΒΩ ΚΑΙ ΧΑΡΙΣΤΙΩΝΙ ΤΑΝ ΓΑΝ ΚΙΝΗΣΩ ΠΑΣΑΝ.” Archimedes “Give me a place to stand and with a lever I will move the whole world.”

15. Latent Variable Means LV 1 1,2 LV 2 1 LV 2 2

16. Latent Variable Means L 1 1,2 L21L21 L22L22

17. The Simulation

19. Conclusion Before comparing path coefficients in multi-group analysis, checking the latent variable means may remove a potential for bias from our analysis. If latent variable means are different, one might adopt the suggested approach in a similar situation.

20. Questions and suggestions, please.