Structural equation modeling
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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
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