Structural equation modeling

Transcription

1 Structural equation modeling Rex B Kline Concordia University Montréal E ISTQL Set E SR models

2 CFA vs. SR o Factors: CFA: Exogenous only SR: Exogenous + endogenous E2

3 CFA vs. SR o Factors & indicators: CFA: L M only SR: L M or M L E3

4 Fully latent SR EX EX 2 EY EY 2 EY 3 EY 4 X X2 Y Y2 Y3 Y4 A B C DB DC E4

5 Partially latent SR (endogenous) EX EX 2 EY 3 EY 4 X X2 Y3 Y4 A Y C DY DC E5

6 Partially latent SR (exogenous) EY EY 2 EY 3 EY 4 Y Y2 Y3 Y4 X B C DB DC E6

7 SR models o Two parts:. Measurement 2. Structural E7

8 SR models o Two steps:. Identification 2. Analysis E8

9 SR identification o Two-step rule (sufficient):. Measurement as CFA 2. Structural as PA E9

10 (a) Original SR (b) Respecified as a CFA Model EX EX 2 EY EY 2 EY 3 EY 4 EX EX 2 EY EY 2 EY 3 EY 4 X X2 Y Y2 Y3 Y4 X X2 Y Y2 Y3 Y4 A B C A B C DB DC (c) Structural Model A B C DB DC E0

11 SR analysis o One-step analysis: EX EX 2 EY EY 2 EY 3 EY 4 X X2 Y Y2 Y3 Y4 A B C DB DC E

12 SR analysis o Two-step analysis (fully latent):. CFA model 2. SR models E2

13 (a) Original SR (b) Respecified as a CFA Model EX EX 2 EY EY 2 EY 3 EY 4 EX EX 2 EY EY 2 EY 3 EY 4 X X2 Y Y2 Y3 Y4 X X2 Y Y2 Y3 Y4 A B C A B C DB DC E3

14 SR analysis o R 2 effect size: Indicators Endogenous factors E4

15 Single indicators o Partially latent (): EX EX 2 EY 3 EY 4 X X2 Y3 Y4 A Y C DY D C E5

16 Single indicators o Partially latent (2): EY EY 2 EY 3 EY 4 Y Y2 Y3 Y4 X B C D B D C E6

17 Single indicators o Requires:. Proportion error variance: ( rxx) s 2 2. Fixed parameters E7

18 .30s Y 2 EX EX 2 EY EY 3 EY 4 X X2 Y Y3 Y4 A B C DB D C E8

19 2.20s X EX EY EY 2 EY 3 EY 4 X Y Y2 Y3 Y4 A B C DB DC E9

20 X DY Y X2 E20

21 ( r ) 2 s ( r YY ) 2 s Y E EY X Y A C ( r 22 ) 2 s 2 E2 DY X2 B E2

22 Single indicators o Hayduk, L. A. & Littvay, L. (202). Should researchers use single indicators, best indicators, or multiple indicators in structural equation models? BMC Medical Research Methodology, 2(59). Retrieved from /2/59 E22

23 EAS EGS EPL EInt EJo Acculturation Scale General Status Percent Life U.S. Interpersonal Job Acculturation Stress DSt SES Depression Scale DDS Education Income EEd EInc E23

24 EAS EGS EPL EInt EJo Acculturation Scale General Status Percent Life U.S. Interpersonal Job Acculturation Stress DSt SES Depression DDe Education Income Depression Scale 2.30 s DS EEd EInc EDS E24

25 Exogenous Direct effects on endogenous Variances Covariances Total Acc GS Acc %Li Acc, SES Acc SES 20 SES Inc Str Job E terms (7) GS %Li Acc Str Str Dep D terms (2) SES Dep v = 8; 8(9)/2 = 39 dfm = = 9 E25

26 LISREL title: shen and takeuchi (200) error term for depression scale observed variables acculscl genstat perlife educ income interper job depscale latent variables: Accultur Ses Stress Depressi correlation matrix E26

27 standard deviations sample size is 983 relationships acculscl = *Accultur genstat perlife = Accultur educ = *Ses income = Ses interper = *Stress job = Stress depscale = *Depressi! depscale as single indicator Stress = Accultur Depressi = Ses Stress E27

28 set error variance of depscale to 3.06! fixes the error variance of the single indicator! rxx =.70, proportion of error variance =.30! sample variance is 0.200;.30 * = 3.06 let the errors of genstat and perlife correlate path diagram LISREL output: ND = 3 SC RS end of program E28

29 Reflective vs. formative E29

30 Reflective (L M) o Contexts:. All CFA models 2. Measurement theory E30

31 Reflective (L M) o Assumes:. Interchangeable Ms 2. High, positive rij 3. Unidimensional Ls E3

32 Reflective? o Example: Income SES Occupation Education Residence E32

33 Formative (M L) o Assumes:. M L 2. L is a composite 3. L is heterogeneous E33

34 Formative (M L) o Assumes: 4. Any pattern of rij 5. Ms not interchangeable E34

35 (a) L M block (b) M L block E E2 E3 V V2 V3 V V2 V3 F F D E35

36 Formative (M L) o Models with cause indicators:. Whole model is SR 2. Identification challenge 3. PLS path modeling E36

37 Formative (M L) o Identification: Emit 2 directs effects Downstream factors E37

38 EAS EGS EPL EInt EJo Acculturation Scale General Status Percent Life U.S. Interpersonal Job Acculturation Stress DSt SES Depression DDe Education Income Depression Scale EEd EInc EDS E38

39 Formative (M L) o Bollen, K. A., & Bauldry, S. (20). Three Cs in measurement models: Causal indicators, composite Indicators, and covariates. Psychological Methods, 6, o Diamantopoulos, A. (Ed.). (2008). Formative indicators [Special issue]. Journal of Business Research, 6(2). o Grace, J. B., & Bollen, K. A. (2008). Representing general theoretical concepts in structural equation models: The role of composite variables. Environmental and Ecological Statistics, 5, E39

40 PLS path modeling o MR and PCA o Prediction o Composites (not L) E40

41 PLS path modeling o How to combine variables o No measurement hypotheses o No identification issues E4

42 PLS path modeling o Henseler, J., & Wang, H. (200) (Eds.) Handbook of partial least squares: Concepts, methods and applications. Berlin: Springer- Verlag. E42

43 SmartPLS E43

44 Other horizons E44

45 Variations o Multiple-samples analysis o Measurement invariance E45

46 Variations o Millsap, R. E., & Olivera-Aguilar, M. (202). Investigating measurement invariance using confirmatory factor analysis. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp ). New York: Guilford Press. o Nimon, K., & Reio, T., Jr. (20). Measurement invariance: A foundational principle for quantitative theory building. Human Resource Development Review, 0, E46

47 Variations o Analysis of means o Latent growth models E47

48 Intercept Slope 0 Trial Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 E E2 E3 E4 E5 E6 E48

49 Variations o Bollen, K. A., & Curran, P. J. (2006). Latent curve models: A structural equation perspective. Hoboken, NJ: Wiley. o Preacher, K. J., Wichman, A. L., MacCallum, R. C., & Briggs, N. E. (2008). Latent growth curve modeling. Thousand Oaks, CA: Sage. E49

50 Variations o Interactive effects: Observed variables Latent variables E50

51 X DM W M DY XW Y E5

52 Variations o Aguinis, H., & Gottfredson, R. K. (200). Best-practice recommendations for estimating interaction effects using moderated multiple regression. Journal of Organizational Behavior, 3, doi: 0.002/job.686. o Klein, A. G., & Muthén, B. O. (2007). Quasi-maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects. Multivariate Behavioral Research, 42, E52