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REX B KLINE CONCORDIA A. POWER, ORDINAL CFA

SEM ADVANCED

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A3

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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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