1. Factor-based SEM building on consistent PLS
Ned Kock
Texas A&M International University

2. Outline of presentation
Path analysis and SEM: History
Factor and indicators
Measurement errors in factors
True composites
The PLSF method
Finite population test
Monte Carlo test
Derivation of key equations

3. Links Available from warppls.blogspot.com:
This .pptx file, linked in 8 June 2017 blog post titled “WarpPLS 6.0 now available …”
Available from warppls.com:
Kock, N. (2017). Factor-based SEM building on consistent PLS: An information systems illustration. Laredo, TX: ScriptWarp Systems.
Kock, N., & Moqbel, M. (2016). A six-stage framework for evolutionary IS research using path models: Conceptual development and a social networking illustration. Journal of Systems and Information Technology, 18(1),

4. Darwin’s theory: Missing genetics

5. Mathematics of evolution + genes
Sewall Wright
J. B. S. Haldane
- Ronald Fisher

6. Path analysis: Sewall Wright
total effect of trait on reproductive success

7. SEM: Path analysis with latent variables
Latent variables (factors) are estimated.
A path analysis is conducted.
This is what our factor-based SEM method aims to do. This is not what covariance-based SEM does.

8. Factor and indicators
indicator error terms
loadings

9. Factor causes indicators?
Yes, because the construct’s mental representation, used by a researcher to develop the question-statements in a questionnaire, exists before the question-statements.

10. Regress factor on indicators
measurement error
The measurement error term accounts for the variance in the factor that is not explained by the factor’s indicators.

11. Factor’s true reliability
measurement error
The factor’s true reliability is the percentage of the variance explained in the factor by its indicators.

12. True composite

13. Composites and factors
θini
θj1
θj2
θjnj
θi1
θi2
θini
θi ε
θj1
θj2
θjnj
θj ε
xi1
xi2
...
xini
xj1
xj2
...
xjnj
xi1
xi2
...
xini εi
xj1
xj2
...
xjnj εj
λi2
λini
λj2
λjnj
λi2
λini
λj2
λjnj
λi1
λj1
λi1
λiε
λj1
λjε Ci Cj Fi Fj
ΣCiCj
ΣFiFj
xi1
xi2
...
xini
xj1
xj2
...
xjnj
xi1
xi2
...
xini εi
xj1
xj2
...
xjnj εj
ωjnj
ωi2
ωini
ωj2
ωi2
ωini
ωj2
ωjnj
ωi1
ωj1
ωi1
ωj1
ωiε
ωjε Ci Cj Fi Fj
ΣCiCj
ΣFiFj

14. PLSF in a nutshell
True reliabilities and true loadings are estimated via consistent PLS.
True weights and true composites are estimated.
The true factor correlation matrix is estimated.
True factor estimates are obtained from true composite estimates via variation sharing (composite-factor correlation matrix fitting).
A path analysis is conducted.

15. Consistent PLS (1-4)

16. Consistent PLS (5-8)

17. True composites (1-3)

18. True composites (4-7)

19. Variation sharing

20. True factors (1-4)

21. True factors (5-9)

22. PLSF test: Illustrative model

23. PLSF test: Finite population
A normal finite population (N=10,000) was created, based on the illustrative model described earlier, to demonstrate the performance of the PLSF method against other methods:
Full-information maximum likelihood (FIML).
Ordinary least squares regression with summed indicators (OLS).
PLS Mode A employing the path weighting scheme (PLS).
Note: Factor scores for FIML estimated via the Thurstone method.

24. Finite pop.: Paths and VIFs

25. Finite pop.: Loadings and weights

26. Finite pop.: Graphical summary

27. PLSF test: Monte Carlo
- While the analyses of the finite population provide an idea of the comparative performance of the four methods, a full Monte Carlo experiment is needed to assess performance in terms of statistical power and percentages of false positives; as well as in terms of estimation of path coefficients with respect to an infinite population, where the distorting effect of sampling error is minimized.
- We generated 1,000 samples of normal and non-normal data with the following sample sizes: 100, 300 and 500.

28. MC: Power and false positives

29. MC: Power graphs

30. True composite estimation
matrix of covariances among indicators and indicator errors
matrix of indicators for a factor
matrix of covariances among indicators

31. Derivation of A.3

32. True factor estimation
estimated matrix of correlations among factors
matrix of correlations among estimated factors

33. Derivation of A.4

34. Thank you

35. Fi Fi Fi Fi Ci θi1 θi2 θini θi1 θi2 θini θi ε xi1 xi2 ... xini εi xi1