Structural equation modeling

Transcription

1 Structural equation modeling Rex B Kline Concordia University Montréal D ISTQL Set D CFA models

2 Resources o Bollen, K. A., & Hoyle, R. H. (202). Latent variable models in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp ). New York: Guilford. o Fabrigar, L. R., & Wegener, D. T. (202). Exploratory factor analysis. New York: Oxford University Press. o Kline, R. B. (203b). Exploratory and confirmatory factor analysis. In Y. Petscher & C. Schatsschneider (Eds.), Applied quantitative analysis in the social sciences (pp ). New York: Routledge. D2

3 EFA o Phases:. Specification 2. Extraction 3. Retention 4. Rotation D3

4 Extraction methods. Principle components analysis (PCA) 2. Principle axis factoring (PAF) 3. Alpha factoring 4. ML factoring D4

5 PCA X X2 X3 X4 X5 X6 A B D5

6 PAF E E2 E3 E4 E5 E6 X X2 X3 X4 X5 X6 A B D6

7 Indicator variance Unique Common Specific Error Systematic rxx D7

8 EFA o Retention: No need to specify But best by theory D8

9 EFA o Retention: Parallel analysis Scree plots D9

10 4 3 Eigenvalue Factor D0

11 EFA o Rotation:. Orthogonal 2. Oblique D

12 EFA o Orthogonal:. Varimax 2. Quartimax 3. Equamax D2

13 EFA o Oblique:. Promax 2. Oblimin D3

14 EFA o Rotation: Infinite Not identified D4

15 a) EFA (unrestricted; rotation) E E2 E3 E4 E5 E6 X X2 X3 X4 X5 X6 A B b) CFA (restricted; no rotation) E E2 E3 E4 E5 E6 X X2 X3 X4 X5 X6 A B D5

16 CFA after EFA o Does not confirm EFA: Restricted vs. unrestricted Items are noisy Follow EFA with EFA D6

17 CFA after EFA o Osborne, J. W., & Fitzpatrick, D. C. (202). Replication analysis in exploratory factor analysis: What it is and why it makes your analysis better. Practical Assessment, Research & Evaluation, 7. Retrieved from v7n5.pdf o van Prooijen, J.-W., & van der Kloot, W. A. (200). Confirmatory analysis of exploratively obtained factor structures. Educational and Psychological Measurement, 6, D7

18 D8

19 EG EA EE Gender Age Ethnic Background D9

20 E E2 E3 X X2 X3 + A D20

21 CFA specification o Standard model: Continuous indicators (X) A X E D2

22 Reflective measurement X = T + E σ 2 = σ 2 + σ 2 X T E r XX 2 σ = T σ 2 X D22

23 Reflective measurement r XX but rxx estimates a single source D23

24 CFA specification o Standard model: Independent E A B D24

25 CFA specification o Unidimensional: Simple indicator (A X only) No Ei Ej D25

26 CFA specification o Unidimensional: Precise test Convergent validity Discriminant validity D26

27 CFA specification o Multidimensional: Complex indicator Ei Ej D27

28 CFA specification o Ei Ej: Indicators share something Repeated measures D28

29 CFA specification o Multidimensional caution: Increases complexity Cheap way to improve fit D29

30 CFA specification o Special variations: Hierarchical CFA MTMM models D30

31 E E2 E3 E4 E5 E6 E7 E8 E9 X X2 X3 X4 X5 X6 X7 X8 X9 Verbal Visual- Spatial Memory DVe DVS DMe g D3

32 Method Method 2 Method 3 X X2 X3 X4 X5 X6 X7 X8 X9 Trait Trait 2 Trait 3 D32

33 E E2 E3 E4 E5 E6 E7 E8 E9 X X2 X3 X4 X5 X6 X7 X8 X9 Trait Trait 2 Trait 3 D33

34 CFA specification o Eid, M., Nussbeck, F. W., Geiser, C., Cole, D. A., Gollwitzer, M., & Lischetzke, T. (2008). Structural equation modeling of multitrait-multimethod data: Different models for different types of methods. Psychological Methods, 3, D34

35 CFA identification o Necessary: dfm 0 Scale each latent D35

36 Scale E ULI constraint: D36

37 Scale factor. Reference (marker) variable ULI =, unstandardized 2. Standardize factors UVI = 3. Effects coding AVE =, all same metric D37

38 E E2 E3 E4 E5 E6 X X2 X3 X4 X5 X6 A B D38

39 E E2 E3 E4 E5 E6 X X2 X3 X4 X5 X6 A B D39

40 E E2 E3 X X2 X3 λ λ2 λ3 A λ + λ + λ 2 3 = 3 D40

41 λ + λ + λ 2 3 = 3 λ = 3 λ λ 2 3 λ = 3 λ λ 2 3 λ = 3 λ λ 3 2 D4

42 CFA identification o Counting parameters:. Exog: Vars. + Covs. 2. Endog: Direct effects D42

43 CFA identification o Standard models: factor, 3 indicators 2 factors, 2 indicators But D43

44 CFA identification o Nonstandard models: No single heuristic Undecidable Ambiguous status D44

45 TABLE 6.. Identification Rule 6.6 for Nonstandard Confirmatory Factor Analysis Models with Measurement Error Correlations For a nonstandard CFA model with measurement error correlations (Rule 6.6) to be identified, all three of the conditions listed next must hold: For each factor, at least one of the following must hold: (Rule 6.6a). There are at least three indicators whose errors are uncorrelated with each other. 2. There are at least two indicators whose errors are uncorrelated and either a. the errors of both indicators are not correlated with the error term of a third indicator for a different factor, or b. an equality constraint is imposed on the loadings of the two indicators. For every pair of factors, there are at least two indicators, one from (Rule 6.6b) each factor, whose error terms are uncorrelated. For every indicator, there is at least one other indicator (not necessarily of the same factor) with which its error term is not correlated. (Rule 6.6c) D45

46 (c) (d) EX EX 2 EX 3 EX 4 EX EX 2 EX 3 EX 4 X X2 X3 X4 X X2 X3 X4 A B A B For each factor, at least one of the following must hold: (Rule 6.6a). There are at least three indicators whose errors are uncorrelated with each other. 2. There are at least two indicators whose errors are uncorrelated and either a. the errors of both indicators are not correlated with the error term of a third indicator for a different factor, or b. an equality constraint is imposed on the loadings of the two indicators. D46

47 TABLE 6.2. Identification Rule 6.7 for Multiple Loadings of Complex Indicators in Nonstandard Confirmatory Factor Analysis Models and Rule 6.8 for Error Correlations of Complex Indicators Factor loadings For every complex indicator in a nonstandard CFA model: (Rule 6.7) In order for the multiple factor loadings to be identified, both of the following must hold:. Each factor on which the complex indicator loads must satisfy Rule 6.6a for a minimum number of indicators. 2. Every pair of those factors must satisfy Rule 6.6b that each factor has an indicator that does not have an error correlation with a corresponding indicator on the other factor of that pair. Error correlations In order for error correlations that involve complex indicators (Rule 6.8) to be identified, both of the following must hold:. Rule 6.7 is satisfied. 2. For each factor on which a complex indicator loads, there must be at least one indicator with a single loading that does not have an error correlation with the complex indicator. D47

48 CFA estimates o Unstandardized:. Indicators loadings (B) 2. Factor, error variances 3. Factor, error covariances D48

49 CFA estimates o Standardized:. Indicators loadings (r, b) 2. Proportion unexplained 3. Factor, error correlations D49

50 CFA estimates o Failure to converge:. Data matrix (NPD) 2. Poor start values 3. Small N, 2 ind./factor D50

51 CFA estimates o Heywood cases (inadmissible):. Error variance < 0 2. r or R 2 >.0 3. NPD parameter matrix D5

52 EFa EMo EFM EPr EIn Father Mother Father- Mother Problems Intimacy Family of Origin Marital Adjustment D52

53 Group 2: Wives THETA-DELTA problems intimacy father mother fa_mo problems (30.844) intimacy (04.927) father (29.24).00 mother (26.870) (28.38) fa_mo (25.232) Squared Multiple Correlations for X - Variables problems intimacy father mother fa_mo D53

54 CFA estimates o Heywood causes: Identification Poor start values Small N, 2 inds./factor D54

55 CFA analysis o Testing strategy:. Fit -factor model 2. Nested under higher-order 3. Compare with χ 2 D D55

56 EHM ENR EWO EGC ETr ESM EMA EPS Hand Movements Number Recall Word Order Gestalt Closure Triangles Spatial Memory Matrix Analogies Photo Series Sequential Processing Simultaneous Processing EHM ENR EWO EGC ETr ESM EMA EPS Hand Movements Number Recall Word Order Gestalt Closure Triangles Spatial Memory Matrix Analogies Photo Series General D56

57 CFA analysis o Example: 4-factor model: 4 vs. 3 4 vs. 2 4 vs. D57

58 CFA respecify o Options:. Number of factors 2. Indicator-factor match 3. Error correlations D58

59 CFA respecify o Residual patterns: Result Correlation residuals Respecification Indicator has low standardized loading on original factor Indicator has reasonably high standardized loading on original factor High correlation residuals with indicators of another factor High correlation residuals with indicators of another factor Switch loading of indicator to other factor Allow indicator to also load on the other factor Allow measurement errors to covary D59

60 CFA respecify o Wrong number of factors: Discriminant validity Convergent validity D60

61 CFA respecify o MIs in latent variable models: Approach with caution Nonsensical respecification May not be identified D6

62 EHM ENR EWO EGC ETr ESM EMA EPS Hand Movements Number Recall Word Order Gestalt Closure Triangles Spatial Memory Matrix Analogies Photo Series Sequential Processing Simultaneous Processing Observations = v (v + )/2 = 36 Parameters = 7 dfm = 9 D62

63 Exogenous variables Direct effects on endogenous variables Variances Covariances Total Sequential NR Sequential WO Seq, Sim Seq Sim 7 Simultaneous Tr Simultaneous SM E terms (8) Simultaneous MA Simultaneous PS D63

64 Example o Amos o EQS o lavaan o LISREL o Mplus o Stata D64

65 title: principles and practice of sem (4th ed.), rex kline two-factor model of the kabc-i, figure 9.7, table 3. data: file is "kabc-mplus.dat"; type is stdeviations correlation; nobservations = 200; variable: names are handmov numbrec wordord gesclos triangle spatmem matanalg photser; analysis: type is general; model: Sequent by handmov numbrec wordord; Simul by gesclos triangle spatmem matanalg photser! first indicator in each list is automatically! specified as the reference variable output: sampstat modindices(all, 0) residual standardized tech4;! requests sample data matrix, residual diagnostics,! modification indexes > 0, all standardized! solutions (STDYX is reported), and estimated! correlation matrix for all variables D65

66 D66

67 CFA indicators o Indicators: Scale: Default ML Likert: Other method D67

68 CFA indicators o Item distributions:. Binary (e.g., T / F) 2. Likert (3-6) 3. Likert ( 7) D68

69 CFA indicators o Estimation options:. Corrected ML: a. Robust SEs b. Santorra-Bentler D69

70 CFA indicators o Estimation options: 2. Robust WLS: a. Item thresholds b. Latent response variable D70

71 CFA indicators o Threshold: Location on latent dimension Differentiates categories Estimated as z D7

72 Example: = disagree 2 = not sure 3 = agree.62.5 D72

73 X X2 X3 E X * E X * 2 E X * 3 X * X * 2 X * 3 A D73

74 CFA indicators o Latent response variables: Sample polychoric Predicted polychoric Correlation residuals D74

75 CFA indicators o Estimation options: 3. ML + numerical integration a. computation b. Markov chain Monte Carlo D75

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77 CFA indicators o Estimation options: 4. IRT, ICC a. Difficulty, discrimination b. Logit, probit link D77

78 Probability of Correct Response ICC difficulty tangent line Latent Ability (θ) D78

79 CFA indicators o Estimation options: 5. Bootstrapping: a. Very biased small N b. Not as developed D79

80 CFA indicators o Estimation options: 6. Create parcels: a. Homogenous item set b. Total score D80

81 It It 2 It 33 It 34 It 35 It 66 It 67 It 68 It 99 A B C Pr (It It ) Pr 2 (It 2 It 22) Pr 3 (It 23 It 33) Pr 4 (It 34 It 44) Pr 5 (It 45 It 55) Pr 6 (It 26 It 66) Pr 7 (It 67 It 77) Pr 8 (It 78 It 88) Pr 9 (It 89 It 99) A B C D8

82 Cautions about parcels. Assumes unidimensional 2. Ways to parcel 3. Mask multidimensionality D82

83 CFA indicators o Edwards, M. C., Wirth, R. J., Houts, C. R., & Xi, N. (202). Categorical data in the structural equation modeling framework. In R. Hoyle (Ed.), Handbook of structural equation modeling (pp ). New York: Guilford Press. o Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 2, D83

84 CFA indicators o Bernstein, I. H., & Teng, G. (989). Factoring items and factoring scales are different: Spurious evidence for multidimensionality due to item categorization. Psychological Bulletin, 05, o Bandalos, D. L., & Finney, S. J. (200). Item parceling issues in structural equation modeling. In G. A. Marcoulides and R. E. Schumaker (Eds.), New developments and techniques in structural equation modeling (pp ). Mahwah, NJ: Erlbaum. D84

85 Exploratory SEM o CFA-EFA-SR hybrid o Restricted + unrestricted o EFA part is rotated D85

86 EX X Y EY Y2 EY 2 Y3 EY 3 Y4 EY 4 Y5 EY 5 Y6 EY 6 EX 2 X2 A EX 3 X3 EX 4 X4 C DC F DF EX 5 X5 B EX 6 X6 D86

87 Exploratory SEM o Marsh, H. W., Morin, A. J. S., Parker, P. D., & Kaur, G. (204). Exploratory structural equation modeling: Integration of the best features of exploratory and confirmatory factor analysis. Annual Review of Clinical Psychology, 0, D87