Specification (estimating in our models) \(\Rightarrow\) Measurement model specification (relationships among observed and latent variables, including variances and covariance) \(\Rightarrow\) Structural model specification (classic regression relationships among (latent) variables of substantive interest)
SEM = measurement model + structural model
- Specify # of factors and factor loadings with “BY” or
=~
- Specify regression/covariance with “ON” or ~ and “WITH” or ~~
- Specify # of factors and factor loadings with “BY” or
=~
SEM solves two regression problems:
- Simple structural models (path analysis)
- Variables measured with error (CFA)
- Simple structural models (path analysis)
First build
Quick reference of
lavaan syntax
Introduce some of the most frequently used syntax in lavaan
- ~ predict, used for regression of observed outcome
to observed predictors (e.g., y ~ x)
- =~ indicator, used for latent variable to observed
indicator in factor analysis measurement models (e.g., f =~ q + r +
s)
- ~~ covariance (e.g., x ~~ x)
- ~1 intercept or mean (e.g., x ~ 1 estimates the
mean of variable x)
- 1* fixes parameter or loading to one (e.g., f =~
1*q)
- NA* frees parameter or loading (useful to override
default marker method, (e.g., f =~ NA*q)
- a* labels the parameter ‘a’, used for model constraints (e.g., f =~ a*q)
Demonstation 1
Model in Mplus:
f1 BY y1-y3;
f2 BY y4-y6;
f2 ON f1;Model in lavaan (R):
f1 =~ y1+y2+y3
f2 =~ y4+y5+y6
f2 ~ f1
Demonstation 2
- Model in Mplus:
! Required code to estimate regression
paths
D ON A B;
E ON D C;
F ON E;
! Outcome intercepts and residual variances
estimated by default
[D E F]; D E
! To bring A, B, and C predictors into the
likelihood,
! Request their covariances
A B C WITH A B C;
! Predictor means and variances then estimated
by default
[A B C]; A B C;
- Model in lavaan (R):
D ~ A+B
E ~ D+C
F ~ E
D ~ 1; D ~~ D
A B C ~~ A B C
A ~ 1; A~~A
Be aware of 2 types of path models
- Unstandardized \(\rightarrow\)
predicts scale sensitive original variables:
- Useful for comparing across groups (whenever absolute values
matter)
- Parameters predict the variables’ means, variance s, and
covariances
- Useful for comparing across groups (whenever absolute values
matter)
- Standardized \(\rightarrow\)
predicts 𝑧 scored versions of variables
- Useful when comparing effects within a solution (are then on same
scale)
- Model parameters predict the variables’ correlations
- Useful when comparing effects within a solution (are then on same
scale)
Refer more in Web Page (2022)
Confusing terminolgy
- Predictors are known as exogenous
variables
- Outcomes are known as endogenous
variables
- Variables that are both at once are called endogenous variables
How set up in Mplus to run FIML:
- By default in Mplus , truly exogenous
predictor variables cannot have missing data, the same as in any
general(ized) linear model
- Cases with missing predictors are listwise deleted
(incomplete data then are assumed missing completely at random ), no
matter which
- Because truly exogenous predictors are not part of
likelihood function + Log likelihood (LL) contains \(\hat{y}_i\) for each person and \(\sigma^2_e\) for each outcome
- So LL can’t be calculated without the predictors that create each
\(\hat{y}_i\)
- So LL can’t be calculated without the predictors that create each
\(\hat{y}_i\)
- Cases with missing predictors are listwise deleted
(incomplete data then are assumed missing completely at random ), no
matter which
- But truly exogenous predictors also do not have assumed
distributions
- Good when you have non normally distributed predictors (e.g.,
ANOVA)!
- Good when you have non normally distributed predictors (e.g.,
ANOVA)!
- We thought full information ML allows missing data???
- NO: only endogenous outcomes can be incomplete (then assumed missing
at random, which means random only after conditioning on model
variables
- BTW, we can add other variables into the likelihood but not the
model to help (untestable) missing at random assumption using AUXILIARY
option
- Is a “saturated correlates” approach (they just covary with all
outcomes)
- Is a “saturated correlates” approach (they just covary with all
outcomes)
- NO: only endogenous outcomes can be incomplete (then assumed missing
at random, which means random only after conditioning on model
variables
- Mplus allows a work around: you can bring exogenous
predictors into the likelihood by listing their means,
variances, or covariances as parameters \(\rightarrow\) predictors then
become “outcomes”
- Even if nothing predicts the predictor (i.e., it’s not really an
outcome)
- Incomplete “endogenous predictors” can be included assuming missing
at random (MAR), but they also then have distributional assumptions
(MNV)
- Historically Mplus has not let endogenous predictors have other
distributions, so you may have to make non normal predictors an outcome
of something else
- But there may be ways to trick it in doing this that I haven’t found yet
- Historically Mplus has not let endogenous predictors have other
distributions, so you may have to make non normal predictors an outcome
of something else
- Even if nothing predicts the predictor (i.e., it’s not really an
outcome)
Steps in Model Building
Practical Steps: 2 Step
2-step (Anderson and Gerbing (1988))
- CFA with “fully-saturated” covariance structure
- Allow all latent variables to freely covary (i.e., only WITH, no
ON)
- Removes structural model from estimation of fit
- Misfit can only be due to measurement model problems
- Allow all latent variables to freely covary (i.e., only WITH, no
ON)
- Obtain fit to evaluate measurement model
- Estimate structural model with regression
- Obtain fit & contrast with previous model
- Changes in fit due to structural model misfit
Practical Steps: 4 Step
4-step (Mulaik and Millsap (2000), Hayduk and Glaser (2000), and Bollen (2000))
- Exploratory search for # of factors
- CFA with fully-saturated covariance structure
- Estimate structural model with regression
- Test nested/alternative models
An example
Mathieu and Farr (1991) article about “Further evidence for the discriminant validity of measures of organizational commitment, job involvement, and job satisfaction”
- Measured organizational commitment, performance, job involvement,
and job satisfaction and other variables
- Show discriminant validity with good-fitting CFA that specifies each as a distinct latent variable
- Then show non-huge inter-correlations among factors
- Data (Means – Stdeviations – Correlation) from engineers is used, in Appendix A
print("Mathieu & Farr 1991.txt")[1] "Mathieu & Farr 1991.txt"writeLines(readLines("Mathieu & Farr 1991.txt"))5.58 4.89 3.76 4.35 3.82 3.87 5.19 5.07 5.14 3.32 3.44 3.14 6.7 6.29 6.69 6.45 6.25 6.69 1.6 5.95 11.3 37.32
.89 1 1.11 1.1 1.31 1.11 .86 .88 .89 .43 .4 .41 .85 .92 .77 1.11 1.11 1.09 1.05 6.18 8.62 9.92
1
.76 1
.67 .68 1
.26 .2 .22 1
.28 .25 .17 .43 1
.34 .33 .26 .52 .52 1
.57 .57 .55 .09 .2 .21 1
.55 .53 .55 .16 .2 .21 .84 1
.48 .46 .49 .15 .23 .21 .79 .71 1
.36 .31 .31 .14 .13 .16 .46 .42 .48 1
.42 .36 .38 .15 .12 .14 .51 .5 .5 .7 1
.32 .34 .37 .11 .13 .13 .47 .46 .46 .65 .64 1
.21 .19 .13 .14 .18 .15 .31 .29 .31 .32 .5 .1 1
.21 .17 .1 .14 .16 .15 .25 .23 .29 .37 .25 .17 .81 1
.22 .18 .12 .15 .14 .12 .33 .29 .31 .39 .29 .15 .82 .82 1
.15 .13 .02 .23 .13 .18 .09 .07 .04 .19 .16 .08 .23 .15 .14 1
.16 .09 -.01 .17 .07 .05 .09 .09 .07 .25 .24 .16 .22 .28 .19 .59 1
.14 .09 -.01 .13 .04 .06 .07 .05 .07 .2 .17 .09 .22 .19 .23 .59 .62 1
.06 .09 .01 .07 .11 .16 .09 .1 .04 .14 .04 .05 .2 .19 .15 .15 .17 .15 1
.03 .06 .1 .17 .19 .17 .02 .01 .1 .15 .04 .09 .03 .05 .06 .1 .16 .07 .1 1
.1 .09 .1 .1 .16 .15 -.001 .02 .04 .12 .05 -.01 .03 .03 .05 .001 .18 .02 .05 .52 1
.14 .1 .12 .07 .15 .17 -.01 .01 -.001 .12 .03 .06 .1 .01 .12 .06 .18 .04 .07 .46 .8 1
Set up Mplus in Rmarkdown
knitr::opts_chunk$set(engine.path = list(
mplus = "C:/Program Files/Mplus/Mplus"
))knitr::knit_engines$set(mplus = function(options) {
code <- paste(options$code, collapse = "\n")
fileConn<-file("formplus.inp")
writeLines(code, fileConn)
close(fileConn)
out <- system2("C:/Program Files/Mplus/Mplus", "formplus.inp")
fileConnOutput <- file("formplus.out")
mplusOutput <- readLines(fileConnOutput)
knitr::engine_output(options, code, mplusOutput)
})Run Mplus in Rmarkdown
Title: CFA Example
Data: File is Mathieu & Farr 1991.txt;
Type is Means Stdeviations Correlation;
Nobservations = 483;
Variable:
Names are
OC1 OC2 OC3 ! Organizational commitment
JI1 JI2 JI3 ! Job involvement
SAT1 SAT2 SAT3 ! Job satisfaction
JS1 JS2 JS3 ! Job scope
SELF1 SELF2 SELF3 ! Self ratings of performance
SUPR1 SUPR2 SUPR3 ! Supervisor ratings of performance
Education
PostTen ! Position tenure
OrgTen ! Organizational tenure
Age;
UseVariables are OC1-OC3 SAT1-SAT3
JS1-JS3 SUPR1-SUPR3 Education;
Analysis:
Model:
OC by OC1 OC2 OC3;
SAT by SAT1 SAT2 SAT3;
JS by JS1 JS2 JS3;
PERF by SUPR1 SUPR2 SUPR3;
PERF on OC SAT JS Education;
OC on SAT;
SAT on JS;
Output: Tech1 Tech8 standardized sampstat;Mplus VERSION 8.6
MUTHEN & MUTHEN
02/25/2023 12:47 AM
INPUT INSTRUCTIONS
Title: CFA Example
Data: File is Mathieu & Farr 1991.txt;
Type is Means Stdeviations Correlation;
Nobservations = 483;
Variable:
Names are
OC1 OC2 OC3 ! Organizational commitment
JI1 JI2 JI3 ! Job involvement
SAT1 SAT2 SAT3 ! Job satisfaction
JS1 JS2 JS3 ! Job scope
SELF1 SELF2 SELF3 ! Self ratings of performance
SUPR1 SUPR2 SUPR3 ! Supervisor ratings of performance
Education
PostTen ! Position tenure
OrgTen ! Organizational tenure
Age;
UseVariables are OC1-OC3 SAT1-SAT3
JS1-JS3 SUPR1-SUPR3 Education;
Analysis:
Model:
OC by OC1 OC2 OC3;
SAT by SAT1 SAT2 SAT3;
JS by JS1 JS2 JS3;
PERF by SUPR1 SUPR2 SUPR3;
PERF on OC SAT JS Education;
OC on SAT;
SAT on JS;
Output: Tech1 Tech8 standardized sampstat;
*** WARNING in VARIABLE command
Note that only the first 8 characters of variable names are used in the output.
Shorten variable names to avoid any confusion.
*** WARNING in OUTPUT command
TECH8 option is available only with analysis types MIXTURE, RANDOM, or
TWOLEVEL with estimators ML, MLF, or MLR or ALGORITHM=INTEGRATION.
Request for TECH8 is ignored.
2 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
CFA Example
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 483
Number of dependent variables 12
Number of independent variables 1
Number of continuous latent variables 4
Observed dependent variables
Continuous
OC1 OC2 OC3 SAT1 SAT2 SAT3
JS1 JS2 JS3 SUPR1 SUPR2 SUPR3
Observed independent variables
EDUCATIO
Continuous latent variables
OC SAT JS PERF
Estimator ML
Information matrix EXPECTED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Input data file(s)
Mathieu & Farr 1991.txt
Input data format FREE
SAMPLE STATISTICS
SAMPLE STATISTICS
Means/Intercepts/Thresholds
OC1 OC2 OC3 SAT1 SAT2
________ ________ ________ ________ ________
5.580 4.890 3.760 5.190 5.070
Means/Intercepts/Thresholds
SAT3 JS1 JS2 JS3 SUPR1
________ ________ ________ ________ ________
5.140 3.320 3.440 3.140 6.450
Means/Intercepts/Thresholds
SUPR2 SUPR3 EDUCATIO
________ ________ ________
6.250 6.690 1.600
Covariances/Correlations/Residual Correlations
OC1 OC2 OC3 SAT1 SAT2
________ ________ ________ ________ ________
OC1 0.792
OC2 0.676 1.000
OC3 0.662 0.755 1.232
SAT1 0.436 0.490 0.525 0.740
SAT2 0.431 0.466 0.537 0.636 0.774
SAT3 0.380 0.409 0.484 0.605 0.556
JS1 0.138 0.133 0.148 0.170 0.159
JS2 0.150 0.144 0.169 0.175 0.176
JS3 0.117 0.139 0.168 0.166 0.166
SUPR1 0.148 0.144 0.025 0.086 0.068
SUPR2 0.158 0.100 -0.012 0.086 0.088
SUPR3 0.136 0.098 -0.012 0.066 0.048
EDUCATIO 0.056 0.095 0.012 0.081 0.092
Covariances/Correlations/Residual Correlations
SAT3 JS1 JS2 JS3 SUPR1
________ ________ ________ ________ ________
SAT3 0.792
JS1 0.184 0.185
JS2 0.178 0.120 0.160
JS3 0.168 0.115 0.105 0.168
SUPR1 0.040 0.091 0.071 0.036 1.232
SUPR2 0.069 0.119 0.107 0.073 0.727
SUPR3 0.068 0.094 0.074 0.040 0.714
EDUCATIO 0.037 0.063 0.017 0.022 0.175
Covariances/Correlations/Residual Correlations
SUPR2 SUPR3 EDUCATIO
________ ________ ________
SUPR2 1.232
SUPR3 0.750 1.188
EDUCATIO 0.198 0.172 1.103
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 42
Loglikelihood
H0 Value -5138.829
H1 Value -5075.557
Information Criteria
Akaike (AIC) 10361.657
Bayesian (BIC) 10537.218
Sample-Size Adjusted BIC 10403.913
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 126.544
Degrees of Freedom 60
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.048
90 Percent C.I. 0.036 0.060
Probability RMSEA <= .05 0.599
CFI/TLI
CFI 0.981
TLI 0.976
Chi-Square Test of Model Fit for the Baseline Model
Value 3614.872
Degrees of Freedom 78
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.040
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
OC BY
OC1 1.000 0.000 999.000 999.000
OC2 1.124 0.049 22.934 0.000
OC3 1.128 0.056 20.156 0.000
SAT BY
SAT1 1.000 0.000 999.000 999.000
SAT2 0.949 0.031 30.970 0.000
SAT3 0.900 0.034 26.719 0.000
JS BY
JS1 1.000 0.000 999.000 999.000
JS2 0.950 0.048 19.671 0.000
JS3 0.886 0.049 17.969 0.000
PERF BY
SUPR1 1.000 0.000 999.000 999.000
SUPR2 1.071 0.073 14.751 0.000
SUPR3 1.028 0.070 14.666 0.000
PERF ON
OC 0.115 0.084 1.370 0.171
SAT -0.207 0.093 -2.214 0.027
JS 0.800 0.173 4.620 0.000
OC ON
SAT 0.667 0.041 16.421 0.000
SAT ON
JS 1.471 0.107 13.733 0.000
PERF ON
EDUCATION 0.145 0.038 3.781 0.000
Intercepts
OC1 5.580 0.040 137.933 0.000
OC2 4.890 0.045 107.581 0.000
OC3 3.760 0.050 74.523 0.000
SAT1 5.190 0.039 132.768 0.000
SAT2 5.070 0.040 126.751 0.000
SAT3 5.140 0.040 127.057 0.000
JS1 3.320 0.020 169.861 0.000
JS2 3.440 0.018 189.200 0.000
JS3 3.140 0.019 168.488 0.000
SUPR1 6.219 0.079 78.778 0.000
SUPR2 6.002 0.082 73.107 0.000
SUPR3 6.452 0.080 81.098 0.000
Variances
JS 0.127 0.012 10.489 0.000
Residual Variances
OC1 0.195 0.021 9.514 0.000
OC2 0.246 0.026 9.479 0.000
OC3 0.473 0.038 12.491 0.000
SAT1 0.072 0.012 6.089 0.000
SAT2 0.173 0.015 11.389 0.000
SAT3 0.250 0.019 13.186 0.000
JS1 0.058 0.006 10.172 0.000
JS2 0.046 0.005 9.427 0.000
JS3 0.068 0.006 12.061 0.000
SUPR1 0.547 0.050 11.028 0.000
SUPR2 0.446 0.049 9.106 0.000
SUPR3 0.464 0.047 9.814 0.000
OC 0.298 0.029 10.122 0.000
SAT 0.392 0.033 11.982 0.000
PERF 0.595 0.071 8.417 0.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.496E-03
(ratio of smallest to largest eigenvalue)
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
OC BY
OC1 0.868 0.016 53.099 0.000
OC2 0.868 0.016 53.248 0.000
OC3 0.784 0.021 37.284 0.000
SAT BY
SAT1 0.950 0.009 106.178 0.000
SAT2 0.881 0.013 69.396 0.000
SAT3 0.827 0.016 50.580 0.000
JS BY
JS1 0.828 0.020 41.273 0.000
JS2 0.845 0.019 43.921 0.000
JS3 0.770 0.023 33.257 0.000
PERF BY
SUPR1 0.743 0.028 26.563 0.000
SUPR2 0.797 0.026 30.319 0.000
SUPR3 0.779 0.027 29.027 0.000
PERF ON
OC 0.108 0.078 1.377 0.169
SAT -0.205 0.092 -2.238 0.025
JS 0.346 0.070 4.912 0.000
OC ON
SAT 0.706 0.028 25.614 0.000
SAT ON
JS 0.641 0.032 19.908 0.000
PERF ON
EDUCATION 0.185 0.047 3.893 0.000
Intercepts
OC1 6.276 0.207 30.320 0.000
OC2 4.895 0.164 29.859 0.000
OC3 3.391 0.118 28.686 0.000
SAT1 6.041 0.200 30.262 0.000
SAT2 5.767 0.191 30.186 0.000
SAT3 5.781 0.191 30.190 0.000
JS1 7.729 0.253 30.573 0.000
JS2 8.609 0.281 30.669 0.000
JS3 7.666 0.251 30.565 0.000
SUPR1 5.623 0.206 27.316 0.000
SUPR2 5.429 0.202 26.917 0.000
SUPR3 5.943 0.217 27.434 0.000
Variances
JS 1.000 0.000 999.000 999.000
Residual Variances
OC1 0.247 0.028 8.723 0.000
OC2 0.246 0.028 8.694 0.000
OC3 0.385 0.033 11.653 0.000
SAT1 0.098 0.017 5.745 0.000
SAT2 0.224 0.022 10.002 0.000
SAT3 0.317 0.027 11.730 0.000
JS1 0.314 0.033 9.447 0.000
JS2 0.285 0.033 8.760 0.000
JS3 0.408 0.036 11.441 0.000
SUPR1 0.447 0.042 10.750 0.000
SUPR2 0.365 0.042 8.714 0.000
SUPR3 0.394 0.042 9.426 0.000
OC 0.501 0.039 12.872 0.000
SAT 0.589 0.041 14.243 0.000
PERF 0.881 0.034 25.794 0.000
STDY Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
OC BY
OC1 0.868 0.016 53.099 0.000
OC2 0.868 0.016 53.248 0.000
OC3 0.784 0.021 37.284 0.000
SAT BY
SAT1 0.950 0.009 106.178 0.000
SAT2 0.881 0.013 69.396 0.000
SAT3 0.827 0.016 50.580 0.000
JS BY
JS1 0.828 0.020 41.273 0.000
JS2 0.845 0.019 43.921 0.000
JS3 0.770 0.023 33.257 0.000
PERF BY
SUPR1 0.743 0.028 26.563 0.000
SUPR2 0.797 0.026 30.319 0.000
SUPR3 0.779 0.027 29.027 0.000
PERF ON
OC 0.108 0.078 1.377 0.169
SAT -0.205 0.092 -2.238 0.025
JS 0.346 0.070 4.912 0.000
OC ON
SAT 0.706 0.028 25.614 0.000
SAT ON
JS 0.641 0.032 19.908 0.000
PERF ON
EDUCATION 0.176 0.045 3.921 0.000
Intercepts
OC1 6.276 0.207 30.320 0.000
OC2 4.895 0.164 29.859 0.000
OC3 3.391 0.118 28.686 0.000
SAT1 6.041 0.200 30.262 0.000
SAT2 5.767 0.191 30.186 0.000
SAT3 5.781 0.191 30.190 0.000
JS1 7.729 0.253 30.573 0.000
JS2 8.609 0.281 30.669 0.000
JS3 7.666 0.251 30.565 0.000
SUPR1 5.623 0.206 27.316 0.000
SUPR2 5.429 0.202 26.917 0.000
SUPR3 5.943 0.217 27.434 0.000
Variances
JS 1.000 0.000 999.000 999.000
Residual Variances
OC1 0.247 0.028 8.723 0.000
OC2 0.246 0.028 8.694 0.000
OC3 0.385 0.033 11.653 0.000
SAT1 0.098 0.017 5.745 0.000
SAT2 0.224 0.022 10.002 0.000
SAT3 0.317 0.027 11.730 0.000
JS1 0.314 0.033 9.447 0.000
JS2 0.285 0.033 8.760 0.000
JS3 0.408 0.036 11.441 0.000
SUPR1 0.447 0.042 10.750 0.000
SUPR2 0.365 0.042 8.714 0.000
SUPR3 0.394 0.042 9.426 0.000
OC 0.501 0.039 12.872 0.000
SAT 0.589 0.041 14.243 0.000
PERF 0.881 0.034 25.794 0.000
STD Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
OC BY
OC1 0.771 0.034 22.930 0.000
OC2 0.867 0.038 22.957 0.000
OC3 0.870 0.044 19.799 0.000
SAT BY
SAT1 0.816 0.030 27.461 0.000
SAT2 0.775 0.032 24.210 0.000
SAT3 0.735 0.034 21.915 0.000
JS BY
JS1 0.356 0.017 20.978 0.000
JS2 0.338 0.016 21.597 0.000
JS3 0.315 0.017 18.955 0.000
PERF BY
SUPR1 0.822 0.048 17.215 0.000
SUPR2 0.881 0.047 18.653 0.000
SUPR3 0.845 0.047 18.158 0.000
PERF ON
OC 0.108 0.078 1.377 0.169
SAT -0.205 0.092 -2.238 0.025
JS 0.346 0.070 4.912 0.000
OC ON
SAT 0.706 0.028 25.614 0.000
SAT ON
JS 0.641 0.032 19.908 0.000
PERF ON
EDUCATION 0.176 0.045 3.921 0.000
Intercepts
OC1 5.580 0.040 137.933 0.000
OC2 4.890 0.045 107.581 0.000
OC3 3.760 0.050 74.523 0.000
SAT1 5.190 0.039 132.768 0.000
SAT2 5.070 0.040 126.751 0.000
SAT3 5.140 0.040 127.057 0.000
JS1 3.320 0.020 169.861 0.000
JS2 3.440 0.018 189.200 0.000
JS3 3.140 0.019 168.488 0.000
SUPR1 6.219 0.079 78.778 0.000
SUPR2 6.002 0.082 73.107 0.000
SUPR3 6.452 0.080 81.098 0.000
Variances
JS 1.000 0.000 999.000 999.000
Residual Variances
OC1 0.195 0.021 9.514 0.000
OC2 0.246 0.026 9.479 0.000
OC3 0.473 0.038 12.491 0.000
SAT1 0.072 0.012 6.089 0.000
SAT2 0.173 0.015 11.389 0.000
SAT3 0.250 0.019 13.186 0.000
JS1 0.058 0.006 10.172 0.000
JS2 0.046 0.005 9.427 0.000
JS3 0.068 0.006 12.061 0.000
SUPR1 0.547 0.050 11.028 0.000
SUPR2 0.446 0.049 9.106 0.000
SUPR3 0.464 0.047 9.814 0.000
OC 0.501 0.039 12.872 0.000
SAT 0.589 0.041 14.243 0.000
PERF 0.881 0.034 25.794 0.000
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
OC1 0.753 0.028 26.549 0.000
OC2 0.754 0.028 26.624 0.000
OC3 0.615 0.033 18.642 0.000
SAT1 0.902 0.017 53.089 0.000
SAT2 0.776 0.022 34.698 0.000
SAT3 0.683 0.027 25.290 0.000
JS1 0.686 0.033 20.636 0.000
JS2 0.715 0.033 21.961 0.000
JS3 0.592 0.036 16.629 0.000
SUPR1 0.553 0.042 13.282 0.000
SUPR2 0.635 0.042 15.159 0.000
SUPR3 0.606 0.042 14.514 0.000
Latent Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
OC 0.499 0.039 12.807 0.000
SAT 0.411 0.041 9.954 0.000
PERF 0.119 0.034 3.484 0.000
TECHNICAL 1 OUTPUT
PARAMETER SPECIFICATION
NU
OC1 OC2 OC3 SAT1 SAT2
________ ________ ________ ________ ________
1 2 3 4 5
NU
SAT3 JS1 JS2 JS3 SUPR1
________ ________ ________ ________ ________
6 7 8 9 10
NU
SUPR2 SUPR3 EDUCATIO
________ ________ ________
11 12 0
LAMBDA
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
OC1 0 0 0 0 0
OC2 13 0 0 0 0
OC3 14 0 0 0 0
SAT1 0 0 0 0 0
SAT2 0 15 0 0 0
SAT3 0 16 0 0 0
JS1 0 0 0 0 0
JS2 0 0 17 0 0
JS3 0 0 18 0 0
SUPR1 0 0 0 0 0
SUPR2 0 0 0 19 0
SUPR3 0 0 0 20 0
EDUCATIO 0 0 0 0 0
THETA
OC1 OC2 OC3 SAT1 SAT2
________ ________ ________ ________ ________
OC1 21
OC2 0 22
OC3 0 0 23
SAT1 0 0 0 24
SAT2 0 0 0 0 25
SAT3 0 0 0 0 0
JS1 0 0 0 0 0
JS2 0 0 0 0 0
JS3 0 0 0 0 0
SUPR1 0 0 0 0 0
SUPR2 0 0 0 0 0
SUPR3 0 0 0 0 0
EDUCATIO 0 0 0 0 0
THETA
SAT3 JS1 JS2 JS3 SUPR1
________ ________ ________ ________ ________
SAT3 26
JS1 0 27
JS2 0 0 28
JS3 0 0 0 29
SUPR1 0 0 0 0 30
SUPR2 0 0 0 0 0
SUPR3 0 0 0 0 0
EDUCATIO 0 0 0 0 0
THETA
SUPR2 SUPR3 EDUCATIO
________ ________ ________
SUPR2 31
SUPR3 0 32
EDUCATIO 0 0 0
ALPHA
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
0 0 0 0 0
BETA
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
OC 0 33 0 0 0
SAT 0 0 34 0 0
JS 0 0 0 0 0
PERF 35 36 37 0 38
EDUCATIO 0 0 0 0 0
PSI
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
OC 39
SAT 0 40
JS 0 0 41
PERF 0 0 0 42
EDUCATIO 0 0 0 0 0
STARTING VALUES
NU
OC1 OC2 OC3 SAT1 SAT2
________ ________ ________ ________ ________
5.580 4.890 3.760 5.190 5.070
NU
SAT3 JS1 JS2 JS3 SUPR1
________ ________ ________ ________ ________
5.140 3.320 3.440 3.140 6.450
NU
SUPR2 SUPR3 EDUCATIO
________ ________ ________
6.250 6.690 0.000
LAMBDA
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
OC1 1.000 0.000 0.000 0.000 0.000
OC2 1.000 0.000 0.000 0.000 0.000
OC3 1.000 0.000 0.000 0.000 0.000
SAT1 0.000 1.000 0.000 0.000 0.000
SAT2 0.000 1.000 0.000 0.000 0.000
SAT3 0.000 1.000 0.000 0.000 0.000
JS1 0.000 0.000 1.000 0.000 0.000
JS2 0.000 0.000 1.000 0.000 0.000
JS3 0.000 0.000 1.000 0.000 0.000
SUPR1 0.000 0.000 0.000 1.000 0.000
SUPR2 0.000 0.000 0.000 1.000 0.000
SUPR3 0.000 0.000 0.000 1.000 0.000
EDUCATIO 0.000 0.000 0.000 0.000 1.000
THETA
OC1 OC2 OC3 SAT1 SAT2
________ ________ ________ ________ ________
OC1 0.396
OC2 0.000 0.500
OC3 0.000 0.000 0.616
SAT1 0.000 0.000 0.000 0.370
SAT2 0.000 0.000 0.000 0.000 0.387
SAT3 0.000 0.000 0.000 0.000 0.000
JS1 0.000 0.000 0.000 0.000 0.000
JS2 0.000 0.000 0.000 0.000 0.000
JS3 0.000 0.000 0.000 0.000 0.000
SUPR1 0.000 0.000 0.000 0.000 0.000
SUPR2 0.000 0.000 0.000 0.000 0.000
SUPR3 0.000 0.000 0.000 0.000 0.000
EDUCATIO 0.000 0.000 0.000 0.000 0.000
THETA
SAT3 JS1 JS2 JS3 SUPR1
________ ________ ________ ________ ________
SAT3 0.396
JS1 0.000 0.092
JS2 0.000 0.000 0.080
JS3 0.000 0.000 0.000 0.084
SUPR1 0.000 0.000 0.000 0.000 0.616
SUPR2 0.000 0.000 0.000 0.000 0.000
SUPR3 0.000 0.000 0.000 0.000 0.000
EDUCATIO 0.000 0.000 0.000 0.000 0.000
THETA
SUPR2 SUPR3 EDUCATIO
________ ________ ________
SUPR2 0.616
SUPR3 0.000 0.594
EDUCATIO 0.000 0.000 0.000
ALPHA
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 1.600
BETA
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
OC 0.000 0.000 0.000 0.000 0.000
SAT 0.000 0.000 0.000 0.000 0.000
JS 0.000 0.000 0.000 0.000 0.000
PERF 0.000 0.000 0.000 0.000 0.000
EDUCATIO 0.000 0.000 0.000 0.000 0.000
PSI
OC SAT JS PERF EDUCATIO
________ ________ ________ ________ ________
OC 0.050
SAT 0.000 0.050
JS 0.000 0.000 0.050
PERF 0.000 0.000 0.000 0.050
EDUCATIO 0.000 0.000 0.000 0.000 1.100
Beginning Time: 00:47:25
Ending Time: 00:47:25
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2021 Muthen & MuthenRun lavaan (R)
We have no full dataset, but do have a sample mean, standard
deviation and correlation matrix. Due to lavaan
supports covariance matrix as input but not correlation matrix, we
need to convert correlation matrix to covariance matrix to fit your
model.
Convert correlation to covariance matrix
First, I run SAS
(proc iml) for easier manipulate the correlation matrix transfer to
covariance matrix. However, we could use
rcompanion::fullPTable() function in R to do the job.
# in SAS
PROC IML;
/** convert correlation matrix to covariance matrix **/
lR={1 . . . . . . . . . . . . . . . . . . . . .,
.76 1 . . . . . . . . . . . . . . . . . . . .,
.67 .68 1 . . . . . . . . . . . . . . . . . . .,
.26 .2 .22 1 . . . . . . . . . . . . . . . . . .,
.28 .25 .17 .43 1 . . . . . . . . . . . . . . . . .,
.34 .33 .26 .52 .52 1 . . . . . . . . . . . . . . . .,
.57 .57 .55 .09 .2 .21 1 . . . . . . . . . . . . . . .,
.55 .53 .55 .16 .2 .21 .84 1 . . . . . . . . . . . . . .,
.48 .46 .49 .15 .23 .21 .79 .71 1 . . . . . . . . . . . . .,
.36 .31 .31 .14 .13 .16 .46 .42 .48 1 . . . . . . . . . . . .,
.42 .36 .38 .15 .12 .14 .51 .5 .5 .7 1 . . . . . . . . . . .,
.32 .34 .37 .11 .13 .13 .47 .46 .46 .65 .64 1 . . . . . . . . . .,
.21 .19 .13 .14 .18 .15 .31 .29 .31 .32 .5 .1 1 . . . . . . . . .,
.21 .17 .1 .14 .16 .15 .25 .23 .29 .37 .25 .17 .81 1 . . . . . . . .,
.22 .18 .12 .15 .14 .12 .33 .29 .31 .39 .29 .15 .82 .82 1 . . . . . . .,
.15 .13 .02 .23 .13 .18 .09 .07 .04 .19 .16 .08 .23 .15 .14 1 . . . . . .,
.16 .09 -.01 .17 .07 .05 .09 .09 .07 .25 .24 .16 .22 .28 .19 .59 1 . . . . .,
.14 .09 -.01 .13 .04 .06 .07 .05 .07 .2 .17 .09 .22 .19 .23 .59 .62 1 . . . .,
.06 .09 .01 .07 .11 .16 .09 .1 .04 .14 .04 .05 .2 .19 .15 .15 .17 .15 1 . . .,
.03 .06 .1 .17 .19 .17 .02 .01 .1 .15 .04 .09 .03 .05 .06 .1 .16 .07 .1 1 . .,
.1 .09 .1 .1 .16 .15 -.001 .02 .04 .12 .05 -.01 .03 .03 .05 .001 .18 .02 .05 .52 1 .,
.14 .1 .12 .07 .15 .17 -.01 .01 -.001 .12 .03 .06 .1 .01 .12 .06 .18 .04 .07 .46 .8 1 };
p=ncol(lR); * Number of columns in the lower triangular correlation matrix;
R=J(p,p,0); * Initialize the correlation matrix;
uR=(lR`); * Transpose the lower triangular matrix to upper;
R=lR<>uR; * Select the max of the lower and upper matrices;
print R; * Complete correlation matrix;
/** standard deviations of each variable **/
c = {.89 1 1.11 1.1 1.31 1.11 .86 .88 .89 .43 .4 .41 .85 .92 .77 1.11 1.11 1.09 1.05 6.18 8.62 9.92};
D = diag(c);
S = D*R*D; /** covariance matrix **/
print S;
quit;Load lavaan
Second, load lavaan
library(lavaan)Read covariance as input
Third, we read in the lower half of the covariance matrix (including the diagonal)
options(width = 300)
lower <- '
0.7921
0.6764 1
0.661893 0.7548 1.2321
0.25454 0.22 0.26862 1.21
0.326452 0.3275 0.247197 0.61963 1.7161
0.335886 0.3663 0.320346 0.63492 0.756132 1.2321
0.436278 0.4902 0.52503 0.08514 0.22532 0.200466 0.7396
0.43076 0.4664 0.53724 0.15488 0.23056 0.205128 0.635712 0.7744
0.380208 0.4094 0.484071 0.14685 0.268157 0.207459 0.604666 0.556072 0.7921
0.137772 0.1333 0.147963 0.06622 0.073229 0.076368 0.170108 0.158928 0.183696 0.1849
0.14952 0.144 0.16872 0.066 0.06288 0.06216 0.17544 0.176 0.178 0.1204 0.16
0.116768 0.1394 0.168387 0.04961 0.069823 0.059163 0.165722 0.165968 0.167854 0.114595 0.10496 0.1681
0.158865 0.1615 0.122655 0.1309 0.20043 0.141525 0.22661 0.21692 0.234515 0.11696 0.17 0.03485 0.7225
0.171948 0.1564 0.10212 0.14168 0.192832 0.15318 0.1978 0.186208 0.237452 0.146372 0.092 0.064124 0.63342 0.8464
0.150766 0.1386 0.102564 0.12705 0.141218 0.102564 0.218526 0.196504 0.212443 0.129129 0.08932 0.047355 0.53669 0.580888 0.5929
0.148185 0.1443 0.024642 0.28083 0.189033 0.221778 0.085914 0.068376 0.039516 0.090687 0.07104 0.036408 0.217005 0.15318 0.119658 1.2321
0.158064 0.0999 -0.012321 0.20757 0.101787 0.061605 0.085914 0.087912 0.069153 0.119325 0.10656 0.072816 0.20757 0.285936 0.162393 0.726939 1.2321
0.135814 0.0981 -0.012099 0.15587 0.057116 0.072594 0.065618 0.04796 0.067907 0.09374 0.07412 0.040221 0.20383 0.190532 0.193039 0.713841 0.750138 1.1881
0.05607 0.0945 0.011655 0.08085 0.151305 0.18648 0.08127 0.0924 0.03738 0.06321 0.0168 0.021525 0.1785 0.18354 0.121275 0.174825 0.198135 0.171675 1.1025
0.165006 0.3708 0.68598 1.15566 1.538202 1.166166 0.106296 0.054384 0.55002 0.39861 0.09888 0.228042 0.15759 0.28428 0.285516 0.68598 1.097568 0.471534 0.6489 38.1924
0.76718 0.7758 0.95682 0.9482 1.806752 1.43523 -0.007413 0.151712 0.306872 0.444792 0.1724 -0.035342 0.21981 0.237912 0.33187 0.0095682 1.722276 0.187916 0.45255 27.701232 74.3044
1.236032 0.992 1.321344 0.76384 1.94928 1.871904 -0.085312 0.087296 -0.008829 0.511872 0.11904 0.244032 0.8432 0.091264 0.916608 0.660672 1.982016 0.432512 0.72912 28.200576 68.40832 98.4064 '
mathieu.cov <-
getCov(lower, names = c(
"OC1", "OC2", "OC3", # Organizational commitment
"JI1", "JI2", "JI3", # Job involvement
"SAT1", "SAT2", "SAT3", # Job satisfaction
"JS1", "JS2", "JS3", # Job scope
"SELF1", "SELF2", "SELF3", # Self ratings of performance
"SUPR1", "SUPR2", "SUPR3", # Supervisor ratings of performance
"Education",
"PostTen", # Position tenure
"OrgTen", # Organizational tenure
"Age"
))Model fit
Fourth, we fit model
options(width = 300)
mathieu.model <- '
# latent variables
OC =~ OC1 + OC2 + OC3
SAT =~ SAT1 + SAT2 + SAT3
JS =~ JS1 + JS2 + JS3
PERF=~ SUPR1+ SUPR2 + SUPR3
# regressions
PERF ~ OC + SAT + JS + Education
OC ~ SAT
SAT ~ JS
'
fit <- sem(mathieu.model,
sample.cov = mathieu.cov,
sample.nobs = 483)
summary(fit, standardized = TRUE)lavaan 0.6-10 ended normally after 47 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 30
Number of observations 483
Model Test User Model:
Test statistic 126.544
Degrees of freedom 60
P-value (Chi-square) 0.000
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
OC =~
OC1 1.000 0.771 0.868
OC2 1.124 0.049 22.934 0.000 0.867 0.868
OC3 1.128 0.056 20.156 0.000 0.870 0.784
SAT =~
SAT1 1.000 0.816 0.950
SAT2 0.949 0.031 30.970 0.000 0.775 0.881
SAT3 0.900 0.034 26.719 0.000 0.735 0.827
JS =~
JS1 1.000 0.356 0.828
JS2 0.950 0.048 19.671 0.000 0.338 0.845
JS3 0.886 0.049 17.969 0.000 0.315 0.770
PERF =~
SUPR1 1.000 0.822 0.743
SUPR2 1.071 0.073 14.751 0.000 0.881 0.797
SUPR3 1.028 0.070 14.666 0.000 0.845 0.779
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
PERF ~
OC 0.115 0.084 1.370 0.171 0.108 0.108
SAT -0.207 0.093 -2.214 0.027 -0.205 -0.205
JS 0.800 0.173 4.619 0.000 0.346 0.346
Education 0.145 0.038 3.781 0.000 0.176 0.185
OC ~
SAT 0.667 0.041 16.421 0.000 0.706 0.706
SAT ~
JS 1.471 0.107 13.733 0.000 0.641 0.641
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.OC1 0.195 0.021 9.514 0.000 0.195 0.247
.OC2 0.246 0.026 9.479 0.000 0.246 0.246
.OC3 0.473 0.038 12.491 0.000 0.473 0.385
.SAT1 0.072 0.012 6.089 0.000 0.072 0.098
.SAT2 0.173 0.015 11.389 0.000 0.173 0.224
.SAT3 0.250 0.019 13.186 0.000 0.250 0.317
.JS1 0.058 0.006 10.172 0.000 0.058 0.314
.JS2 0.046 0.005 9.427 0.000 0.046 0.285
.JS3 0.068 0.006 12.061 0.000 0.068 0.408
.SUPR1 0.547 0.050 11.028 0.000 0.547 0.447
.SUPR2 0.446 0.049 9.106 0.000 0.446 0.365
.SUPR3 0.464 0.047 9.814 0.000 0.464 0.394
.OC 0.298 0.029 10.122 0.000 0.501 0.501
.SAT 0.392 0.033 11.982 0.000 0.589 0.589
JS 0.127 0.012 10.489 0.000 1.000 1.000
.PERF 0.595 0.071 8.417 0.000 0.881 0.881Fit statistics
Fifth, fit indices
options(width = 300)
#fit statistics
summary(fit, fit.measures=TRUE)lavaan 0.6-10 ended normally after 47 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 30
Number of observations 483
Model Test User Model:
Test statistic 126.544
Degrees of freedom 60
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 3614.872
Degrees of freedom 78
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.981
Tucker-Lewis Index (TLI) 0.976
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -5138.829
Loglikelihood unrestricted model (H1) -5075.557
Akaike (AIC) 10337.657
Bayesian (BIC) 10463.058
Sample-size adjusted Bayesian (BIC) 10367.840
Root Mean Square Error of Approximation:
RMSEA 0.048
90 Percent confidence interval - lower 0.036
90 Percent confidence interval - upper 0.060
P-value RMSEA <= 0.05 0.599
Standardized Root Mean Square Residual:
SRMR 0.042
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
OC =~
OC1 1.000
OC2 1.124 0.049 22.934 0.000
OC3 1.128 0.056 20.156 0.000
SAT =~
SAT1 1.000
SAT2 0.949 0.031 30.970 0.000
SAT3 0.900 0.034 26.719 0.000
JS =~
JS1 1.000
JS2 0.950 0.048 19.671 0.000
JS3 0.886 0.049 17.969 0.000
PERF =~
SUPR1 1.000
SUPR2 1.071 0.073 14.751 0.000
SUPR3 1.028 0.070 14.666 0.000
Regressions:
Estimate Std.Err z-value P(>|z|)
PERF ~
OC 0.115 0.084 1.370 0.171
SAT -0.207 0.093 -2.214 0.027
JS 0.800 0.173 4.619 0.000
Education 0.145 0.038 3.781 0.000
OC ~
SAT 0.667 0.041 16.421 0.000
SAT ~
JS 1.471 0.107 13.733 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.OC1 0.195 0.021 9.514 0.000
.OC2 0.246 0.026 9.479 0.000
.OC3 0.473 0.038 12.491 0.000
.SAT1 0.072 0.012 6.089 0.000
.SAT2 0.173 0.015 11.389 0.000
.SAT3 0.250 0.019 13.186 0.000
.JS1 0.058 0.006 10.172 0.000
.JS2 0.046 0.005 9.427 0.000
.JS3 0.068 0.006 12.061 0.000
.SUPR1 0.547 0.050 11.028 0.000
.SUPR2 0.446 0.049 9.106 0.000
.SUPR3 0.464 0.047 9.814 0.000
.OC 0.298 0.029 10.122 0.000
.SAT 0.392 0.033 11.982 0.000
JS 0.127 0.012 10.489 0.000
.PERF 0.595 0.071 8.417 0.000Model Fit Statistics
We focus on the 5 commonly used:
1- Model chi-square is the chi-square statistic we
obtain from the maximum likelihood statistic (in lavaan,
this is known as the Test Statistic for the Model Test User Model)
2- CFI is the Comparative Fit Index – values can range
between 0 and 1 (values greater than 0.90, conservatively 0.95 indicate
good fit)
3- TLI Tucker Lewis Index which also ranges between 0
and 1 (if it’s greater than 1 it should be rounded to 1) with values
greater than 0.90 indicating good fit. If the CFI and TLI are less than
one, the CFI is always greater than the TLI.
4- RMSEA is the root mean square error of approximation
In lavaan, you also obtain a p-value of close fit, that the RMSEA <
0.05. If you reject the model, it means your model is not a close
fitting model.
5- SRMR is standardized root mean squared residual to
test close fit. Also we also obtain a p-value of close fit, that the
RMSEA < 0.05. If you reject the model, it means your model is not a
close fitting model.
In general, researchers should avoid sample sizes less than 100 when testing small degrees of freedom models. In fact, science and math education researchers should avoid reporting the RMSEA when sample sizes are smaller than 200, particularly when combined with small degrees of freedom. Small degrees of freedom do not tend to result in rejection of correctly specified models for the TLI, CFI, and SRMR, particularly if they tested using larger sample sizes (Taasoobshirazi and Wang (2016)).
(be continued)
Model chi-square
CFI, TLI
RMSEA, SRMR
Summary
- SEM was running both in Mplus and lavaan produced the identical
results. Fantastic!
- SEM is simply path analysis + CFA
- Model-based procedure: a means to make almost any idea an empirical
question
- Model comparisons allow testing theory
- Measurement models create latent constructs (= random effects) that
better represent trait individual differences than any one outcome
- Structural models test relations involving those latent
constructs
- Measurement models create latent constructs (= random effects) that
better represent trait individual differences than any one outcome
- It’s easy to setup, but with complicated models
- ML MAY BREAK when your models get too complicated (or
realistic)
- You have named your factors, but it doesn’t mean you are right!
(Validity)
- Distributional assumptions matter, but so do linear model
assumptions (nonlinear measurement and structural models may be
- Factor scores are not perfectly determined (and neither are sum scores), so make sure to represent their uncertainty in any SEM alternative
- ML MAY BREAK when your models get too complicated (or
realistic)
- Model-based procedure: a means to make almost any idea an empirical
question
Anderson, James C., and David W. Gerbing. 1988. “Structural
Equation Modeling in Practice: A Review and Recommended Two-Step
Approach.” Journal Article. Psychological Bulletin 103
(3): 411–23. https://doi.org/10.1037/0033-2909.103.3.411.
Bollen, Kenneth A. 2000. “Modeling Strategies: In Search of the
Holy Grail.” Journal Article. Structural Equation Modeling: A
Multidisciplinary Journal 7 (1): 74–81. https://doi.org/10.1207/S15328007SEM0701_03.
Hayduk, Leslie A., and Dale N. Glaser. 2000. “Doing the Four-Step,
Right–2–3, Wrong–2–3: A Brief Reply to Mulaik and Millsap; Bollen;
Bentler; and Herting and Costner.” Journal Article.
Structural Equation Modeling 7: 111–23. https://doi.org/10.1207/S15328007SEM0701_06.
Mathieu, John E., and James L. Farr. 1991. “Further Evidence for
the Discriminant Validity of Measures of Organizational Commitment, Job
Involvement, and Job Satisfaction.” Journal Article. Journal
of Applied Psychology 76 (1): 127–33. https://doi.org/10.1037/0021-9010.76.1.127.
Mulaik, Stanley A., and Roger E. Millsap. 2000. “Doing the
Four-Step Right.” Journal Article. Structural Equation
Modeling: A Multidisciplinary Journal 7 (1): 36–73. https://doi.org/10.1207/S15328007SEM0701_02.
Taasoobshirazi, Gita, and Shanshan Wang. 2016. “The Performance of
the SRMR, RMSEA, CFI, and TLI: An Examination of Sample Size, Path Size,
and Degrees of Freedom.” Journal Article. Journal of Applied
Quantitative Methods 11 (3): 31.
Web Page. 2019. https://melbourne.figshare.com/articles/media/Mplus_Workshop_at_The_University_of_Melbourne_February_4-8_2019_5_Days_/7797620.
———. 2021. https://stats.oarc.ucla.edu/r/seminars/rsem/.
———. 2022. https://www.lesahoffman.com/PSQF6249/index.html.