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REX B KLINE CONCORDIA A. POWER, ORDINAL CFA
SEM ADVANCED
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power
ordinal cfa
meanstop
ics
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latent growth
cfa invariance
moderationtop
ics
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proper
a priori (planning)
improper
po
we
r
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applications
model level
effect level
po
we
r
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Input (1)
H0: parameter0, α, N, dfM
H1: parameter1
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Output (1)
p (reject H0|H1)
1 – β
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Input (2)
Target power (e.g., ≥ .85)
H0, α, statistic, dfM, H1
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Output (2)
Target N
E.g., if power ≥ .85, then N ≥ 500
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MacCallum et al.
RMSEA
0 ,
1
Type of H0, H1
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M
ˆˆ
( 1)df N
2
M Mˆ max (0, )df
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, 90% CI [L , U
]
E.g., = .02 [0, .15]
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H0 Accept–
support
Reject–
support
Exact fit ×
Close fit ×
Not close fit ×
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Accept-support
Logically weak
Power ↓, model ↑
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Reject-support
Conventional logic
Power ↓, model ↓
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Low power
Exact fit, close fit
p (reject false model) ↓
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Low power
Not close fit
p (detect close model) ↓
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Test Null
Exact fit H0: 0 = 0
Close fit H0: 0 ≤ .05
Not close fit H0: 0 > .05
* *
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Close fit
Given , 90% CI [L , U
]
L > .05, reject H0: 0
≤ .05
*
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Test H0 H1
Close fit 0 ≤ .05
1 = .08
Not close fit 0 > .05
1 = .01
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N = 373, dfM = 5
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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
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Close but failing
Exact-fit H0 rejected
Close-fit H0 retained
Inspect residuals
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semTools for R
http://cran.r-project.org/web/packages/semTools/index.html
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semTools for R
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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)
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Statistic
N 373
dfM 5
Power
Close fita .317
Not close fitb .229
aH0: 0 ≤ .05,
1 = .08, α = .05
bH0: 0 > .05,
1 = .01, α = .05
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Target power ≥ .80 Target N
Close fita 1,464
Not close fitb 1,216
aH0: 0 ≤ .05,
1 = .08, α = .05
bH0: 0 > .05,
1 = .01, α = .05
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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
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1200 1600 1000 1800 800 600 400 200
Sample Size (N)
1400
Po
we
r
.90
.80
.70
.60
.50
.40
.30
.20
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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
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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.
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likert scale
≤ 5 levels
skewed
ord
ina
l
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robust wls
thresholds
polychoric
ord
ina
l
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global fit stats
interpretation?
residuals
ord
ina
l
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(a) Histogram of observed item X responses with cumulative probabilities
Pro
po
rtio
n
.30
.10
.40
.20
.25
.60
1.0
Response Category
1
Disagree
2
Neutral
3
Agree
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25% 35% 40%
(b) Latent response variable X* with threshold estimates
X*
1 = −.67 2 = .25
1.0 3.0 2.0 0 −1.0 −2.0 −3.0
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1
*X2
*X
Pro
ba
bili
ty
.45
.30
.15
0
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A
1X *
1 1X
E *
2X *
1 2X
E *
3X *
1 3X
E *
X1 X2 X3
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Delta scaling
Var (X*) = 1.0
Correlation metric
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Delta standardized
Simple indicator, r
Threshold, z ~ND (0, 1)
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Theta scaling
Var (EX*) = 1.0
Probit metric
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Theta unstandardized
Indicators, probit z
Thresholds, z ~ND (0, ≠1)
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delta vs. theta
1 sample
simplicity
ord
ina
l
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delta vs. theta
2 samples
error testing
ord
ina
l
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Mplus
WLSM
Mean-adjusted
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Mplus
WLSMV
Mean- and variance-adjusted
Estimated dfM
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LISREL
RDWLS
Robust diagonally-weighted
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LISREL
PRELIS: Thresholds
Polychoric r LISREL
Asymptotic cov
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Example
5 items, CES-D
0 = < 1 day 1 = 1–2 days
2 = 3–4 days 3 = 5–7 days
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Example
N = 2,004
White men
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1 λ2
λ3 λ4 λ5
A φ
5X *
X5
τ51–τ53
4X *
X4
τ41–τ43
3X *
X3
τ31–τ33
2X *
X2
τ21–τ23
1X *
X2
τ11–τ13
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Observations
v = 5, 5(4)/2 = 10 polychoric
5 × 3 = 15 thresholds
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Parameters
15 thresholds (τ)
4 loadings (λ), 1 variance (φ)
dfM = 25 – 20 = 5
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PRELIS, LISREL
Sorry, SIMPLIS
Mplus
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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;
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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Unstandardized
Standardized
Parameter Estimate SE Estimate SE R2
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.
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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.
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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.
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