ADVANCED SEM REX B KLINE CONCORDIA A1 A. POWER, ORDINAL CFA
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topics power ordinal cfa means A4
topics latent growth cfa invariance moderation A5
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power proper a priori (planning) improper A8
power applications model level effect level A9
Input (1) H0: parameter0, α, N, dfm H1: parameter1 A10
Output (1) p (reject H0 H1) 1 β A11
Input (2) Target power (e.g.,.85) H0, α, statistic, dfm, H1 A12
Output (2) Target N E.g., if power.85, then N 500 A13
MacCallum et al. RMSEA, 0 1 Type of H0, H1 A14
ˆ df M ˆ ( N 1) ˆ max (0, 2 df ) M M A15
ˆ, 90% CI [ ˆ, ˆ ] L U E.g., ˆ =.02 [0,.15] A16
H0 Accept support Reject support Exact fit Close fit Not close fit A17
Accept-support Logically weak Power, model A18
Reject-support Conventional logic Power, model A19
Low power Exact fit, close fit p (reject false model) A20
Low power Not close fit p (detect close model) A21
Test Null Exact fit H0: = 0 0 Close fit H0:.05 0 Not close fit H0: >.05 0 * A22
Close fit * Given ˆ, 90% CI [ ˆ, ˆ ] L U ˆ >.05, reject H0:.05 L 0 A23
Test H0 H1 Close fit.05 =.08 0 1 Not close fit >.05 =.01 0 1 A24
N = 373, dfm = 5 A25
Goodness of Fit Statistics Degrees of Freedom for (C1)-(C2) 5 Maximum Likelihood Ratio Chi-Square (C1) 11.107 (P = 0.0493) Browne's (1984) ADF Chi-Square (C2_NT) 11.103 (P = 0.0494) Estimated Non-centrality Parameter (NCP) 6.107 90 Percent Confidence Interval for NCP (0.0167 ; 19.837) Minimum Fit Function Value 0.0298 Population Discrepancy Function Value (F0) 0.0164 90 Percent Confidence Interval for F0 (0.000 ; 0.0532) Root Mean Square Error of Approximation (RMSEA) 0.0572 90 Percent Confidence Interval for RMSEA (0.00299 ; 0.103) P-Value for Test of Close Fit (RMSEA < 0.05) 0.336 A26
Close but failing Exact-fit H0 rejected Close-fit H0 retained Inspect residuals A27
semtools for R http://cran.r-project.org/web/packages/semtools/index.html A28
semtools for R A29
date() library(semtools) # power for test of close fit hypothesis for N = 373 findrmseapower(.05,.08, 5, 373,.05, 1) # sample size for target power =.80 for close fit hypothesis findrmseasamplesize(.05,.08, 5,.80,.05, 1) # power for test of not close fit hypothesis for N = 373 findrmseapower(.05,.01, 5, 373,.05, 1) # sample size for target power =.80 for not close fit hypothesis findrmseasamplesize(.05,.01, 5,.80,.05, 1) A30
Statistic N 373 dfm 5 Power Close fit a.317 Not close fit b.229 a H0:.05, 0 =.08, α =.05 1 b H0: >.05, 0 =.01, α =.05 1 A31
Target power.80 Target N Close fit a 1,464 Not close fit b 1,216 a H0:.05, 0 =.08, α =.05 1 b H0: >.05, 0 =.01, α =.05 1 A32
STATISTICA Power Analysis http://www.statsoft.com Generate SPSS, R syntax http://timo.gnambs.at/en/scripts/powerforsem SAS/STAT syntax http://www.datavis.ca/sasmac/csmpower.html A33
.90.80.70 Power.60.50.40.30.20 200 400 600 800 1000 1200 1400 1600 1800 Sample Size (N) A34
Minimum N for power.80 dfm 2 6 10 14 16 18 20 25 30 40 N 1,926 910 651 525 483 449 421 368 329 277 A35
Bandalos, D. L., & Leite, W. (2013). Use of Monte Carlo studies in structural equation modeling. In G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A second course (2nd ed.) (pp. 625 666). Charlotte, NC: IAP. A36
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ordinal likert scale 5 levels skewed A38
ordinal robust wls thresholds polychoric A39
ordinal global fit stats interpretation? residuals A40
(a) Histogram of observed item X responses with cumulative probabilities.40.60 1.0.30.25 Proportion.20.10 1 Disagree 2 Neutral 3 Agree Response Category A41
(b) Latent response variable X* with threshold estimates 25% 35% 40% 3.0 2.0 1.0 0 1.0 2.0 3.0 ˆ 1=.67 ˆ 2=.25 X* A42
.45 Probability.30.15 0 * X1 * X2 A43
X1 X2 X3 1 E X * 1 1 E X * 2 1 E X * 3 X * 1 X * 2 X * 3 A A44
Delta scaling Var (X*) = 1.0 Correlation metric A45
Delta standardized Simple indicator, r Threshold, z ~ND (0, 1) A46
Theta scaling Var (EX*) = 1.0 Probit metric A47
Theta unstandardized Indicators, probit z Thresholds, z ~ND (0, 1) A48
ordinal delta vs. theta 1 sample simplicity A49
ordinal delta vs. theta 2 samples error testing A50
Mplus WLSM Mean-adjusted A51
Mplus WLSMV Mean- and variance-adjusted Estimated dfm A52
LISREL RDWLS Robust diagonally-weighted A53
LISREL PRELIS: Thresholds Polychoric r LISREL Asymptotic cov A54
Example 5 items, CES-D 0 = < 1 day 1 = 1 2 days 2 = 3 4 days 3 = 5 7 days A55
Example N = 2,004 White men A56
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X2 X2 X3 X4 X5 τ 11 τ 13 τ 21 τ 23 τ 31 τ 33 τ 41 τ 43 τ 51 τ 53 X * 1 X * 2 X * 3 X * 4 X * 5 1 λ 2 λ 3 λ 4 λ 5 φ A A58
Observations v = 5, 5(4)/2 = 10 polychoric 5 3 = 15 thresholds A59
Parameters 15 thresholds (τ) 4 loadings (λ), 1 variance (φ) dfm = 25 20 = 5 A60
PRELIS, LISREL Sorry, SIMPLIS Mplus A61
title: principles and practice of sem (4th ed.), rex kline single-factor model of depression, white sample, figure 13.6 data: file is radloff-white-mplus.dat; variable: names are x1-x5; categorical are x1-x5;! variables correspond to, respectively,! CES Depression scale items 1, 2, 7, 11, and 20 analysis: parameterization is delta;! total variance of latent response variables fixed to 1 model: Conflict by x1-x5; output: sampstat residual standardized tech1; A62
SUMMARY OF ANALYSIS Number of groups 1 Number of observations 2004 Number of dependent variables 5 Number of independent variables 0 Number of continuous latent variables 1 Observed dependent variables Binary and ordered categorical (ordinal) X1 X2 X3 X4 X5 A63
Continuous latent variables CONFLICT Estimator WLSMV Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Parameterization DELTA A64
UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES X1 Category 1 0.780 1563.000 Category 2 0.142 285.000 Category 3 0.047 95.000 Category 4 0.030 61.000 X2 Category 1 0.852 1707.000 Category 2 0.087 174.000 Category 3 0.031 62.000 Category 4 0.030 61.000 A65
X3 Category 1 0.706 1414.000 Category 2 0.170 340.000 Category 3 0.058 117.000 Category 4 0.066 133.000 X4 Category 1 0.613 1229.000 Category 2 0.228 457.000 Category 3 0.092 184.000 Category 4 0.067 134.000 X5 Category 1 0.712 1426.000 Category 2 0.183 367.000 Category 3 0.062 124.000 Category 4 0.043 87.000 A66
ESTIMATED SAMPLE STATISTICS MEANS/INTERCEPTS/THRESHOLDS X1$1 X1$2 X1$3 X2$1 X2$2 0.772 1.420 1.874 1.044 1.543 MEANS/INTERCEPTS/THRESHOLDS X2$3 X3$1 X3$2 X3$3 X4$1 1.874 0.541 1.152 1.503 0.288 MEANS/INTERCEPTS/THRESHOLDS X4$2 X4$3 X5$1 X5$2 X5$3 1.000 1.500 0.558 1.252 1.712 A67
CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL) X1 X2 X3 X4 X5 X1 X2 0.437 X3 0.471 0.480 X4 0.401 0.418 0.454 X5 0.423 0.489 0.627 0.465 A68
MODEL FIT INFORMATION Number of Free Parameters 20 Chi-Square Test of Model Fit Value 17.904* Degrees of Freedom 5 P-Value 0.0031 The chi-square value for MLM, MLMV, MLR, ULSMV, WLSM and WLSMV cannot be used for chi-square difference testing in the regular way. MLM, MLR and WLSM chi-square difference testing is described on the Mplus website. MLMV, WLSMV, and ULSMV difference testing is done using the DIFFTEST option. A69
RMSEA (Root Mean Square Error Of Approximation) Estimate 0.036 90 Percent C.I. 0.019 0.055 Probability RMSEA <=.05 0.887 CFI/TLI CFI 0.994 TLI 0.989 A70
MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value CONFLICT BY X1 1.000 0.000 999.000 999.000 X2 1.070 0.065 16.576 0.000 X3 1.285 0.065 19.820 0.000 X4 1.004 0.056 17.929 0.000 X5 1.266 0.065 19.396 0.000 Variances CONFLICT 0.370 0.034 10.940 0.000 A71
Two-Tailed Estimate S.E. Est./S.E. P-Value Thresholds X1$1 0.772 0.031 24.703 0.000 X1$2 1.420 0.041 34.543 0.000 X1$3 1.874 0.056 33.636 0.000 X2$1 1.044 0.034 30.428 0.000 X2$2 1.543 0.044 34.903 0.000 X2$3 1.874 0.056 33.636 0.000 X3$1 0.541 0.030 18.302 0.000 X3$2 1.152 0.036 32.070 0.000 X3$3 1.503 0.043 34.839 0.000 X4$1 0.288 0.028 10.128 0.000 X4$2 1.000 0.034 29.646 0.000 X4$3 1.500 0.043 34.830 0.000 X5$1 0.558 0.030 18.826 0.000 X5$2 1.252 0.038 33.270 0.000 X5$3 1.712 0.049 34.638 0.000 A72
STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value CONFLICT BY X1 0.609 0.028 21.879 0.000 X2 0.651 0.029 22.142 0.000 X3 0.782 0.020 38.609 0.000 X4 0.611 0.023 26.941 0.000 X5 0.771 0.021 35.928 0.000 Variances CONFLICT 1.000 0.000 999.000 999.000 A73
Thresholds X1$1 0.772 0.031 24.703 0.000 X1$2 1.420 0.041 34.543 0.000 X1$3 1.874 0.056 33.636 0.000 X2$1 1.044 0.034 30.428 0.000 X2$2 1.543 0.044 34.903 0.000 X2$3 1.874 0.056 33.636 0.000 X3$1 0.541 0.030 18.302 0.000 X3$2 1.152 0.036 32.070 0.000 X3$3 1.503 0.043 34.839 0.000 X4$1 0.288 0.028 10.128 0.000 X4$2 1.000 0.034 29.646 0.000 X4$3 1.500 0.043 34.830 0.000 X5$1 0.558 0.030 18.826 0.000 X5$2 1.252 0.038 33.270 0.000 X5$3 1.712 0.049 34.638 0.000 A74
R-SQUARE Observed Two-Tailed Residual Variable Estimate S.E. Est./S.E. P-Value Variance X1 0.370 0.034 10.940 0.000 0.630 X2 0.424 0.038 11.071 0.000 0.576 X3 0.612 0.032 19.304 0.000 0.388 X4 0.373 0.028 13.471 0.000 0.627 X5 0.594 0.033 17.964 0.000 0.406 A75
RESIDUAL OUTPUT ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) Residuals for Means/Intercepts/Thresholds X1$1 X1$2 X1$3 X2$1 X2$2 0.000 0.000 0.000 0.000 0.000 Residuals for Means/Intercepts/Thresholds X2$3 X3$1 X3$2 X3$3 X4$1 0.000 0.000 0.000 0.000 0.000 Residuals for Means/Intercepts/Thresholds X4$2 X4$3 X5$1 X5$2 X5$3 0.000 0.000 0.000 0.000 0.000 A76
Model Estimated Covariances/Correlations/Residual Correlations X1 X2 X3 X4 X5 X1 X2 0.396 X3 0.476 0.509 X4 0.372 0.398 0.478 X5 0.469 0.502 0.603 0.471 Residuals for Covariances/Correlations/Residual Correlations X1 X2 X3 X4 X5 X1 X2 0.041 X3-0.005-0.029 X4 0.030 0.020-0.024 X5-0.046-0.013 0.024-0.005 A77
Unstandardized Standardized Parameter Estimate SE Estimate SE R 2 Pattern coefficients A X1* 1.000.609.028.370 A X2* 1.070.065.651.029.424 A X3* 1.285.065.782.020.612 A X4* 1.004.056.611.023.373 A X5* 1.266.065.771.021.594 Factor variance A (Depression).370.034 1.000 Note. Thresholds: X1,.772, 1.420, 1.874; X2, 1.044, 1.543, 1.874; X3,.541, 1.152, 1.503; X4,.288, 1.000, 1.500; X5,.558, 1.252, 1.712. All results were computed with Mplus in delta parameterization and STDYX standardization. A78
Indicator X1* X2* X3* X4* X5* Correlation residuals X1* X2*.041 X3*.005.029 X4*.030.020.024 X5*.046.013.024.005 Note. The correlation residuals were computed by Mplus. A79
Indicator X1* X2* X3* X4* X5* Standardized residuals X1* X2* 1.331 X3*.213 1.193 X4* 1.110.679 1.230 X5* 1.935.511 2.370.282 Note. The standardized residuals were computed by LISREL. A80
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