Structural equation modeling

Transcription

1 Structural equation modeling Rex B Kline Concordia University Montréal ISTQL Set B B1 Data, path models

2 Data o N o Form o Screening B2

3 B3

4 Sample size o N needed: Complexity Estimation method Distributions True model B4

5 Sample size o Levels: 1. Minimum (precision) 2. Power B5

6 Sample size o Default method (ML): Continuous outcomes Normal distributions B6

7 Sample size o Default method (ML): Cases : parameters 20 : 1 10 : 1, but B7

8 Sample size o Other outcomes: Continuous but nonnormal Ordinal or nominal B8

9 Minimum N for power.80 dfm N 1, B9

10 Roth et al. (1989), N = DFi Fitness Exercise 1 DIl Illness Hardiness Stress 1 DSt dfm = 5, Power =.317, Minimum N = 1,465 B10

11 Sample size o Loehlin, J. C. (2004). Latent variable models: An introduction to factor, path, and structural equation analysis (4th ed.). Mahwah, NJ: Erlbaum. B11

12 Nor should one feel compelled to drop every path that is nonsignificant, especially when the sample size is small. With small samples, a path that is numerically appreciable may not exceed twice its standard error, yet its removal may materially affect the solution. If the path was theoretically justified in the first place, it is often wiser to leave it in the model until cross-validation confirms that it is trivial and can be safely dropped (p. 59). B12

13 Sample size o Westland, C. J. (2010). Lower bounds on sample size in structural equation modeling. Electronic Commerce Research and Applications, 9, o Wolf, E. J., Harrington, K. M., Clark, S. L., & Miller, M. W. (2013). Sample size requirements for structural equation models: An evaluation of power, bias, and solution propriety. (2013). Educational and Psychological Measurement, 73, B13

14 Data form o Submit: Raw data file Summary matrix B14

15 Data form o Considerations: Continuous? Normal? Missing? B15

16 Data form o Raw data needed: 1. Continuous but nonnormal 2. Categorical outcomes 3. Missing data B16

17 Data form o Matrix summary: 1. Continuous and normal 2. No missing data 3. Secondary analysis B17

18 Case X W Y A B C D E M SD s B18

19 Covariance r, SD cov covxy = rxy SDX SDY =.470 (6.205) (14.577) B19

20 Covariance + means r, SD, M cov, M B20

21 B21

22 Data screening o Missing data o Distribution shape o Outliers o Extreme collinearity o Psychometrics B22

23 Data screening o Special issues: 1. Normality 2. Missing data B23

24 Normality o Assumed by default ML o Take seriously o Measure skew, kurtosis B24

25 3 S ( S ) γ = and ˆ S γ ˆ 2 = ( S ) 2 ( X M ) S = N 2 3 ( X M ) S = N 4 ( X M ) S = N 3 4 B25

26 Normality o ˆγ > 3 1 o ˆγ > 10, 2 ˆγ > 20 2 o Just say no to * B26

27 B27

28 Normality o Little need to transform: Normality not expected Lose meaningful metric Use corrected method B28

29 Normality o Robust ML estimation o Robust SEs, corrected o Raw data needed B29

30 Missing data o Diagnose patterns: SPSS (MVA) LISREL (PRELIS) B30

31 Missing data o Old (obsolete) methods: 1. Case deletion 2. Single imputation B31

32 Missing data o Case deletion: Listwise Pairwise B32

33 Case X W Y A B C D 14 8 E F G B33

34 X W Y X W Y B34

35 covwy = rwy SDW SDY = r = WY r (6.530) = WY r WY = = B35

36 X W Y X W Y Nonpositive definitive Negative eigenvalue B36

37 B37

38 B38

39 Missing data o Single imputation: Mean substitution Regression substitution B39

40 Missing data o Obsolete: Assumes MCAR Distorts distribution B40

41 Missing data o Newer methods: 3. Special ML method 4. E M multiple imputation B41

42 Missing data o Newer methods: Assume MAR More accurate if not MCAR B42

43 Missing data o Special ML method: Partition by missing pattern Extract information, pool No deletion, imputation B43

44 Missing data o E M method: Relies on auxiliary variables Multiple imputed scores Integrate results B44

45 Missing data o If MNAR: No cure Understand pattern Qualify results B45

46 Missing data o Allison, P. D. (2003). Missing data techniques for structural equation modeling. Journal of Abnormal Psychology, 112, o Enders, C. K. (2010). Applied missing data analysis. New York: Guilford Press. o Graham, J. W., & Coffman, D. L. (2012). Structural equation modeling with missing data. In R. Hoyle (Ed.), Handbook of structural equation modeling (pp ). New York: Guilford Press. B46

47 B47

48 Resources o Bollen, K. A., & Pearl, J. (2013). Eight myths about causality and structural equation models. In S.L. Morgan (Ed.), Handbook of causal analysis for social research (pp ). New York: Springer. o Kline, R, B. (2012). Assumptions of structural equation modeling. In R. Hoyle (Ed.), Handbook of structural equation modeling (pp ). New York: Guilford B48

49 Context o Origins: Sewall Wright, genetics Causal model known Estimate sizes in SEM B49

50 Context o Behavioral sciences: True model unknown We guess (model) Fit model to data B50

51 Assumes model is correct SEM does not discover causality Estimates it, given correct model B51

52 Model fits data, no replication 1. Model is true 2. Equivalent to true model 3. Fit in unrepresentative sample 4. So complex it must fit B52

53 1 DY X Y X1 a 1 D Y X2 b 1 D1 X a Y1 c b 1 D2 Y2 B53

54 X a Y 1 DY X Y Omitted causes X rxx = 1.0 B54

55 Systematic Measurement error (random) Omitted causes Explained 1 ryy R 2 Disturbance (residual) 1 R 2 B55

56 X a Y 1 DY Y b X 1 DX cov = r SD SD XY XY X Y a r SD Y = XY SDX b r SD X = XY SDY B56

57 X a Y 1 DY Y b X 1 DX? X 1 D DX 1 1 DY Y X Y? B57

58 Design support for X Y Manipulate X, measure Y Isolation over replications Validity threats B58

59 Design support for X Y Measure X, then later Y Correct lag Resources B59

60 Assume X Y Measure X before Y Regress Y on X BYX is large Proved? B60

61 Truth Y X Regress Y3 on X2 BY 3 X 2 is large because Y1 X2 B61

62 No design support for X Y Cross-sectional design Concurrent measurement Rationale only B62

63 X1 a 1 D Y X2 b X1 Y, X2 Y Omitted causes X1, X2 r11 = r22 = 1.0 B63

64 Unreliability o Cole, D. A., & Preacher, K. J. (2014). Manifest variable path analysis: Potentially serious and misleading consequences due to uncorrected measurement error. Psychological Methods, 19, B64

65 1 D1 X a Y1 c b 1 D2 Y2 X Y1, X Y2, Y1 Y2 rxx = 1.0 B65

66 1 D1 X a Y1 c b 1 D2 Y2 Omitted causes Y1, Y2 X Omitted Y1 omitted Y2 B66

67 1 D1 X a Y1 c b 1 D2 Y2 No omitted cause between any pair among X, Y1, Y2 B67

68 1 D1 X a Y1 c b 1 D2 Y2 Continuous variables No interaction B68

69 1 D1 X a Y1 c b 1 D2 Y2 Indirect effect = ab The M word B69

70 1 D1 X a Y1 X c b 1 D2 c 1 D2 Y2 Y2 1. c 0 (inconsistent) 2. a 0 3. b 0 4. c = 0 (full) B70

71 Mediation then o Cross-sectional design o No interaction o ab is mediation B71

72 Mediation now o Time precedence o No? Indirect effect o Allow interaction B72

73 Mediation o Kline, R. B. (in press). The mediation myth. Basic and Applied Social Psychology. o Little, T. D. (2013). Longitudinal structural equation modeling. New York: Guilford. B73

74 Mediation refers to the causal hypothesis that one variable causes changes in another variable, which in turn leads to changes in the outcome variable. Little (2013) B74

75 Mediation o Experimental: X manipulated o Measurement of mediation o Manipulation of mediation B75

76 Half longitudinal mediation X1 a 1 D12 M1 b M2 1 D22 O1 O2 B76

77 Full longitudinal mediation X1 a X2 X3 M1 M2 b M3 O1 O2 O3 B77

78 Mediation o Cross-sectional o No clear rationale o Equivalent models B78

79 B79 X M Y X Y M Y X M X M Y Y M X Y M X Y X M X Y M X M Y Y M X Y M X X M Y Y X M X Y M X M Y Y M X Y M X X M Y Figure 2. Equivalent models for three variables in cross-sectional designs without temporal precedence and without a strong rationale for directionality specification. Reciprocal direct effects are constrained to equality.

80 Mediation o Bullock, J. G., Green, D. P., & Ha, S. E. (2010). Yes, but what s the mechanism? (Don t expect an easy answer). Journal of Personality and Social Psychology, 98, o Cole, D. A., & Maxwell, S. E. (2003). Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology, 112, B80

81 Mediation o Selig, J. P., & Preacher, K. J. (2009). Mediation models for longitudinal data in developmental research. Research in Human Development, 6, o Stone Romero, E. F., & Rosopa, P. J. (2011). Experimental tests of mediation models: Prospects, problems, and some solutions. Organizational Research Methods, 14, B81

82 Path model types o Recursive: Unidirectional effects Bow-free, D1 D2 B82

83 1 D1 X1 Y1 1 D2 X2 Y2 B83

84 Path model types o Nonrecursive: Feedback loop Bow-pattern, D1 D2 B84

85 1 D1 1 D1 X1 Y1 X1 Y1 1 D2 1 D2 X2 Y2 X2 Y2 B85

86 Path model types o Recursive vs. nonrecursive: Identification Analysis problems B86

87 (a) Direct feedback 1 D1 X1 Y1 1 D2 X2 Y2 (b) Indirect feedback 1 D1 X1 Y1 X2 Y2 1 D2 X3 Y3 B87 1 D3

88 Instruments o Bollen, K. A. (2012). Instrumental variables in sociology and the social sciences. Annual Review of Sociology, 38, B88

89 Identification B89

90 Identification o General: Every latent is scaled dfm 0 B90

91 dfm = no. obs no. free param. Observations = no. vars. + no. covs = v( v + 1) 2 B91

92 Example: v = 4 4(5) No. obs = = B92

93 Identification o Parameter status: Free Fixed Constrained B93

94 Identification o Counting parameters: 1. Exog: Vars. + Covs. 2. Endog: Direct effects B94

95 1 D1 X1 Y1 1 D2 X2 Y2 B95

96 Exogenous Model Direct effects on endogenous Variances Covariances Total X1 Y1 X1, X2 X1 X2 9 X2 Y2 D1, D2 D1 D2 Y1 Y2 B96

97 Identification o How analysis can fail: Poor data screening Extreme sample Bad start values B97