ADVANCED C. MEASUREMENT INVARIANCE SEM REX B KLINE CONCORDIA
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1 ADVANCED SEM C. MEASUREMENT INVARIANCE REX B KLINE CONCORDIA C
2 C2
3 multiple model 2 data sets simultaneous C3
4 multiple 2 populations 2 occasions 2 methods C4
5 multiple unstandardized constrain to equal fit to data C5
6 multiple fit, yes conclude, equal no? release C6
7 multiple χ 2 D n n C7
8 Levels Dimensional Configural Weak (Metric) Strong (Scalar) Strict (Error) C8
9 Dimensional E E2 E3 E4 E5 E6 E E2 E3 E4 E5 E6 X X2 X3 X4 X5 X6 X X2 X3 X4 X5 X6 A B A B C9
10 Configural Same no. factors, match No other constraints Different scoring systems C0
11 Weak Assumes configural Equal pattern coefficients Same scoring system C
12 Strong Assumes weak Equal intercepts, thresholds Same response level C2
13 Strict Assumes strong Same error variance Identical measurement C3
14 DeShon, R. P. (2004). Measures are not invariant across groups without error variance homogeneity. Psychology Science, 46, Wu, A. D., Li, Z., & Zumbo, B. D. (2007). Decoding the meaning of factorial Invariance and updating the practice of multi-group confirmatory factor analysis: A demonstration with TIMSS data. Practical Assessment Research & Evaluation, 2(3). Retrieved from C4
15 Partial invariance () Configural retained Weak: Some, not all DIF () C5
16 DX A X C6
17 Partial invariance () Different relative meaning Extreme response sets C7
18 Ryder, A. G., Yang J., Zhu, X., Yao, S., Yi, J., Heine, S. J., & Bagby, R. M. (2009). The cultural shaping of depression: Somatic symptoms in China, psychological symptoms in North America? Journal of Abnormal Psychology, 7, C8
19 Cheung, G. W., & Rensvold, R. B. (2000). Assessing extreme and acquiescence response sets in cross-cultural research using structural equations modeling. Journal of Cross-Cultural Psychology, 3, C9
20 Partial invariance (2) Configural retained Strong: Some, not all DIF (2) C20
21 DX A X C2
22 Partial invariance (2) Differential additive response Cultural, gender, procedural C22
23 Gregorich, S. E. (2006). Do self-report instruments allow meaningful comparisons across diverse population groups? Testing measurement invariance using the confirmatory factor analysis framework. Medical Care, 44 (Suppl. 3), S78 S94. C23
24 χ 2 D Low power, n < 400 Very large n... CFI, Δ.0 rule C24
25 Chen, F. F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling, 4, C25
26 Sequence () Configural Weak Strong Strict Free baseline approach Model trimming C26
27 Sequence (2) Strict...? Constrained baseline approach Model building C27
28 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. C28
29 Example Hispanic with teenagers English, n = 93 Spanish, n2 = 257 C29
30 English Item fc fc4 fc5 fc6 fc9 M SD fc fc fc fc fc Spanish M SD Note. English-speaking (above diagonal; n = 93), Spanish-speaking (below diagonal; n 2 = 257). C30
31 E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 Conflict C3
32 Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 3, C32
33 Reference group E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 0 Conflict C33
34 Marker variable E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 0 Conflict C34
35 Effects coding () E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 λ λ2 λ3 λ4 λ5 Conflict C35
36 C36
37 Effects coding (2) E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 τ τ2 τ3 τ4 τ5 Conflict C37
38 C38
39 Models. Configural (M) 2. M with error correlation (M2) 3. M2 with full metric (M3) 4. M3 with full error (M4) 5. M3 with full scalar (M5) 6. M3 with partial scalar (M6) C39
40 Model (No) 2 C40
41 Model 2 (Yes) C4
42 Model 3 (Yes) C42
43 Model 4 (No) C43
44 Model 5 (No) C44
45 Model 6 (Yes) C45
46 Model 2 χ M dfm 2 χ D dfd Comparison RMSEA 90% CI (N) (Y) ** 2 vs (Y) * 4 3 vs (N) 87.57** ** 5 4 vs (N) vs (Y) vs C46
47 Model E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 Conflict Observations v (v + 3)/2 no. groups 5(8)/2 2 = 40 C47
48 E E4 E5 E6 E9 fc fc4 fc5 fc6 fc9 Conflict Parameters Variances: (5 + ) 2 = 2 Loadings: (4) 2 = 8 Intercepts = (4) 2 = 8 Means = 2 dfm = = 0 C48
49 C49
50 LISREL classic Sorry, SIMPLIS Mplus C50
51 C5
52 Mplus defaults st indicator, loading = Other loadings free but equal Intercepts free but equal C52
53 Mplus defaults Factor, error vars. & covs. free st group, factor mean = 0 Otherwise free and unequal C53
54 title: dillman data model configural invariance data: file is dillman.dat; type is means std corr; nobservations = ; ngroups = 2;! group is english, group 2 is spanish variable: names = fc fc4 fc5 fc6 fc9; analysis: type = general; estimator = ml; C54
55 model:! names group factor loadings: Conflict by fc* (g_load) fc4 (g_load2) fc5 (g_load3) fc6 (g_load4) fc9 (g_load5);! names group intercepts: [fc] (g_int); [fc4] (g_int2); [fc5] (g_int3); [fc6] (g_int4); [fc9] (g_int5); model g:! group factor mean is free parameter: [Conflict]; C55
56 model g2:! names group 2 factor loadings! separate loadings estimated in group 2: Conflict by fc* (g2_load) fc4 (g2_load2) fc5 (g2_load3) fc6 (g2_load4) fc9 (g2_load5);! names group 2 factor loadings! separate intercepts estimated in group 2: [fc] (g2_int); [fc4] (g2_int2); [fc5] (g2_int3); [fc6] (g2_int4); [fc9] (g2_int5); C56
57 ! separate factor mean, variance in group 2: Conflict; [Conflict];! by default, measurement errors are freely! estimated in each group model constraint:! effects-coding method for scaling! and identifying factor! average loading constrained to.0 and! average intercept constrained to 0 C57
58 ! group : g_load = 5 - g_load2 - g_load3 - g_load4 - g_load5; g_int = -g_int2 - g_int3 - g_int4 - g_int5;! group 2: g2_load = 5 - g2_load2 - g2_load3 - g2_load4 - g2_load5; g2_int = -g2_int2 - g2_int3 - g2_int4 - g2_int5; output: samp stdyx res; C58
59 THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom 0 P-Value 0.94 Chi-Square Contributions From Each Group G 8.99 G C59
60 CFI/TLI CFI 0.99 TLI RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I SRMR (Standardized Root Mean Square Residual) Value C60
61 RESIDUAL OUTPUT ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G Model Estimated Covariances/Correlations/Residual Correlations FC FC4 FC5 FC6 FC9 FC FC FC FC FC Residuals for Covariances/Correlations/Residual Correlations FC FC4 FC5 FC6 FC9 FC FC FC FC FC C6
62 ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G Standardized Residuals (z-scores) for Covariances/Correlations/Residual Corr FC FC4 FC5 FC6 FC9 FC 0.04 FC FC FC FC C62
63 ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G2 Model Estimated Covariances/Correlations/Residual Correlations FC FC4 FC5 FC6 FC9 FC.065 FC FC FC FC Residuals for Covariances/Correlations/Residual Correlations FC FC4 FC5 FC6 FC9 FC FC FC FC FC C63
64 ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G2 Standardized Residuals (z-scores) for Covariances/Correlations/Residual Corr FC FC4 FC5 FC6 FC9 FC FC FC FC FC C64
65 PAGE : 6 EQS TITLE: Group : English MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V4, V2.9 V4, V V4, V V2, V V5, V V3, V V3, V V5, V V4, V V999,V V3, V.07 6 V999,V V5, V V999,V V5, V V999,V V5, V.00 9 V999,V V2, V V, V.000 C65
66 PAGE : 2 EQS TITLE: Group 2: Spanish MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP 2 MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V3, V.054 V5, V V2, V V4, V V3, V V3, V V4, V V2, V V4, V V999,V V4, V V999,V V5, V V999,V V5, V V999,V V5, V V999,V V5, V V, V.000 C66
67 Model 2 (Yes) C67
68 title: dillman-carpentier model 2 configural invariance with error correlation group, fc4 and fc6... model g:! group factor mean is free parameter: [Conflict];! ** new to model 2 **! error covariance group only: fc4 with fc6; C68
69 TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom 9 P-Value Chi-Square Contributions From Each Group G.478 G C69
70 CFI/TLI CFI.000 TLI.004 RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I SRMR (Standardized Root Mean Square Residual) Value 0.08 C70
71 RESIDUAL OUTPUT ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G Standardized Residuals (z-scores) for Covariances/Correlations/Residual Corr FC FC4 FC5 FC6 FC9 FC FC FC FC FC C7
72 ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G2 Standardized Residuals (z-scores) for Covariances/Correlations/Residual Corr FC FC4 FC5 FC6 FC9 FC FC FC FC FC C72
73 PAGE : 6 EQS TITLE: Group : English MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V4, V V4, V V5, V V4, V V2, V.08 3 V3, V V3, V.0 4 V2, V V4, V V, V V5, V V999,V V3, V V999,V V5, V V999,V V5, V V999,V V5, V V999,V3.000 C73
74 PAGE : 4 EQS TITLE: Group 2: Spanish MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP 2 MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V3, V.054 V5, V V2, V V4, V V3, V V3, V V4, V V2, V V4, V V999,V V4, V V999,V V5, V V999,V V5, V V999,V V5, V V999,V V5, V V, V.000 C74
75 Model 3 (Yes) C75
76 title: dillman-carpentier model 3 metric invariance with error correlation group, fc4 and fc6... C76
77 model g2:! names group 2 factor loadings! ** new to model 3 **! pairwise equality constraints imposed on loadings! by commenting out the following group 2 syntax:! Conflict by fc* (g2_load)! fc4 (g2_load2)! fc5 (g2_load3)! fc6 (g2_load4)! fc9 (g2_load5);... C77
78 model constraint:! effects-coding method for scaling! and identifying factor! average loading constrained to.0 and! average intercept constrained to 0! group : g_load = 5 - g_load2 - g_load3 - g_load4 - g_load5; g_int = -g_int2 - g_int3 - g_int4 - g_int5;! group 2:! ** new to model 3 **! original group 2 constraint on loadings not needed! and is commented out:! g2_load = 5 - g2_load2 - g2_load3 - g2_load4 - g2_load5; C78
79 TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 3.36 Degrees of Freedom 3 P-Value Chi-Square Contributions From Each Group G 3.67 G C79
80 CFI/TLI CFI TLI RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I SRMR (Standardized Root Mean Square Residual) Value C80
81 RESIDUAL OUTPUT ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G Standardized Residuals (z-scores) for Covariances/Correlations/Residual Corr FC FC4 FC5 FC6 FC9 FC FC FC FC FC C8
82 ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G2 Standardized Residuals (z-scores) for Covariances/Correlations/Residual Corr FC FC4 FC5 FC6 FC9 FC.205 FC FC FC FC C82
83 PAGE : 6 EQS TITLE: Group : English MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V4, V V4, V V3, V V3, V.03 3 V, V V5, V V3, V V5, V V5, V V4, V V2, V.04 6 V999,V V2, V V999,V V5, V V999,V V5, V V999,V V4, V V999,V3.000 C83
84 PAGE : 4 EQS TITLE: Group 2: Spanish MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP 2 MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V5, V.068 V4, V V3, V V3, V V3, V V4, V V, V V4, V V5, V V5, V V4, V V999,V V5, V V999,V V5, V V999,V V2, V V999,V V2, V V999,V5.000 C84
85 Model 4 (No) C85
86 title: dillman-carpentier model 4 metric invariance with error correlation group, fc4 and fc6 equality of error variances... C86
87 model:! names group factor loadings: Conflict by fc* (g_load) fc4 (g_load2) fc5 (g_load3) fc6 (g_load4) fc9 (g_load5);! names group intercepts: [fc] (g_int); [fc4] (g_int2); [fc5] (g_int3); [fc6] (g_int4); [fc9] (g_int5);! ** new to model 4 **! names group error variances: fc (g_err); fc4 (g_err2); fc5 (g_err3); fc6 (g_err4); fc9 (g_err5); C87
88 model g2:! names group 2 factor loadings! pairwise equality constraints imposed on loadings! by commenting out the following group 2 syntax:! Conflict by fc* (g2_load)! fc4 (g2_load2)! fc5 (g2_load3)! fc6 (g2_load4)! fc9 (g2_load5);! names group 2 factor loadings! separate intercepts estimated in group 2: [fc] (g2_int); [fc4] (g2_int2); [fc5] (g2_int3); [fc6] (g2_int4); [fc9] (g2_int5);! ** new to model 4 **! names group 2 error variances: fc (g2_err); fc4 (g2_err2); fc5 (g2_err3); fc6 (g2_err4); fc9 (g2_err5); C88
89 model constraint:! effects-coding method for scaling! and identifying factor! average loading constrained to.0 and! average intercept constrained to 0! group : g_load = 5 - g_load2 - g_load3 - g_load4 - g_load5; g_int = -g_int2 - g_int3 - g_int4 - g_int5;! group 2:! original group 2 constraint on loadings not needed! and is commented out:! g2_load = 5 - g2_load2 - g2_load3 - g2_load4 - g2_load5; g2_int = -g2_int2 - g2_int3 - g2_int4 - g2_int5;! ** new to model 4 **! constrain error variances: g_err = g2_err; g_err2 = g2_err2; g_err3 = g2_err3; g_err4 = g2_err4; g_err5 = g2_err5; C89
90 Chi-Square Test of Model Fit Value Degrees of Freedom 8 P-Value Chi-Square Contributions From Each Group G G C90
91 CFI/TLI CFI TLI RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I SRMR (Standardized Root Mean Square Residual) Value 0.22 C9
92 C92
93 Model 3 (Yes) C93
94 MODEL RESULTS Group G Model 3 Two-Tailed Estimate S.E. Est./S.E. P-Value Residual Variances FC FC FC FC FC Group G2 Residual Variances FC FC FC FC FC C94
95 Model 5 (metric + scalar invariance) C95
96 title: dillman-carpentier model 5 metric invariance with error correlation group, fc4 and fc6 scalar invariance... C96
97 model g2:! names group 2 factor loadings! pairwise equality constraints imposed on loadings! by commenting out the following group 2 syntax:! Conflict by fc* (g2_load)! fc4 (g2_load2)! fc5 (g2_load3)! fc6 (g2_load4)! fc9 (g2_load5);! names group 2 factor loadings! separate intercepts estimated in group 2:! ** new to model 5 **! comment out naming of constraints in group 2! now pairwise constrained to equal group values:![fc] (g2_int);![fc4] (g2_int2);![fc5] (g2_int3);![fc6] (g2_int4);![fc9] (g2_int5); C97
98 model constraint:! effects-coding method for scaling! and identifying factor! average loading constrained to.0 and! average intercept constrained to 0! group : g_load = 5 - g_load2 - g_load3 - g_load4 - g_load5; g_int = -g_int2 - g_int3 - g_int4 - g_int5;! group 2:! original group 2 constraint on loadings not needed! and is commented out:! g2_load = 5 - g2_load2 - g2_load3 - g2_load4 - g2_load5;! ** new to model 5 **! original group 2 constraint on loadings not needed! g2_int = -g2_int2 - g2_int3 - g2_int4 - g2_int5; C98
99 TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom 7 P-Value Chi-Square Contributions From Each Group G G C99
100 CFI/TLI CFI 0.99 TLI RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I SRMR (Standardized Root Mean Square Residual) Value C00
101 RESIDUAL OUTPUT ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G Model Estimated Means/Intercepts/Thresholds FC FC4 FC5 FC6 FC Residuals for Means/Intercepts/Thresholds FC FC4 FC5 FC6 FC Standardized Residuals (z-scores) for Means/Intercepts/Thresholds FC FC4 FC5 FC6 FC C0
102 ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) FOR G2 Model Estimated Means/Intercepts/Thresholds FC FC4 FC5 FC6 FC Residuals for Means/Intercepts/Thresholds FC FC4 FC5 FC6 FC Standardized Residuals (z-scores) for Means/Intercepts/Thresholds FC FC4 FC5 FC6 FC C02
103 PAGE : 6 EQS TITLE: Group : English MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V999,V2.7 V, V V999,V V5, V V4, V V5, V V3, V V5, V V3, V V999,V V2, V V5, V V4, V V4, V V3, V.03 8 V2, V V5, V V4, V V999,V V999,V3.000 C03
104 PAGE : 4 EQS TITLE: Group 2: Spanish MULTIPLE POPULATION ANALYSIS, INFORMATION IN GROUP 2 MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION THEORY) LARGEST STANDARDIZED RESIDUALS: NO. PARAMETER ESTIMATE NO. PARAMETER ESTIMATE V5, V.088 V5, V V3, V V3, V V999,V V4, V V5, V V2, V V, V V4, V V2, V V5, V V999,V V999,V V3, V V999,V V5, V V999,V V4, V V4, V2.000 C04
105 Model 6 (metric + partial scalar) C05
106 title: dillman-carpentier model 6 metric invariance with error correlation group, fc4 and fc6 partial scalar invariance... C06
107 model g2:! names group 2 factor loadings! pairwise equality constraints imposed on loadings! by commenting out the following group 2 syntax:! Conflict by fc* (g2_load)! fc4 (g2_load2)! fc5 (g2_load3)! fc6 (g2_load4)! fc9 (g2_load5);! names group 2 factor loadings! separate intercepts estimated in group 2:! ** new to model 6 **! comment out naming of constraints in group 2! for only 3 indicators: [fc] (g2_int); [fc4] (g2_int2);![fc5] (g2_int3);![fc6] (g2_int4);![fc9] (g2_int5); C07
108 TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom 5 P-Value Chi-Square Contributions From Each Group G G C08
109 CFI/TLI CFI.000 TLI.003 RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I SRMR (Standardized Root Mean Square Residual) Value C09
110 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value Group G CONFLICT BY FC FC FC FC FC FC4 WITH FC Group G2 CONFLICT BY FC FC FC FC FC C0
111 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value Group G Residual Variances FC FC FC FC FC Group G2 Residual Variances FC FC FC FC FC C
112 MODEL RESULTS Group G Two-Tailed Estimate S.E. Est./S.E. P-Value Means CONFLICT Intercepts FC FC FC FC FC Group G2 Means CONFLICT Intercepts FC FC FC FC FC C2
113 MODEL RESULTS Group G Two-Tailed Estimate S.E. Est./S.E. P-Value Means CONFLICT Variances CONFLICT Group G2 Means CONFLICT Variances CONFLICT C3
114 C4
115 full metric tally partial scalar (3/5) error, none C5
116 C6
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