Ben Kite, Center for Research Methods and Data Analysis, University of Kansas <bakite@ku.edu>
Chong Xing, Center for Research Methods and Data Analysis, University of Kansas <cxing@ku.edu>
Please visit http://crmda.ku.edu/guides
Keywords: SEM, CFA, Measurement Invariance, R, lavaan
Abstract
This guide outlines how to conduct measurement invariance testing in the R package lavaan. The manifest variables used in this example are continuous in scale and tested for invariance across gender. This guide is intended for researchers familiar with structural equation modeling. Model syntax is compared to the popular SEM software package Mplus.
Load the lavaan package at the beginning of the session
library(lavaan)This is lavaan 0.6-3lavaan is BETA software! Please report any bugs.The data file is read in, columns are named, and missing values are specified.
dat <- read.csv("../../data/job_placement.csv", header = FALSE)
colnames(dat) <- c("id", "wjcalc", "wjspl", "wratspl", "wratcalc",
"waiscalc", "waisspl", "edlevel", "newschl",
"suspend", "expelled", "haveld", "gender", "age")
dat[dat == 99999] <- NA ## convert the original missing data value "99999" to NA
dat$gender <- as.factor(dat$gender) ## change "gender" to a factor variable
levels(dat$gender) ## check the levels of the factor variable [1] "0" "1"table(dat$gender) ## request a frequency summary
0 1
221 101 levels(dat$gender) <- c("Male", "Female") ## assign level labels
table(dat$gender) ## request a frequency summary to verify assigned labels
Male Female
221 101 Now the CFA model that is to be tested needs to be specified as an R object. In lavaan the “=~” operator is the exact same as the BY operator in Mplus. In the statements below the variables on the right of the “=~” are indicators of the variable to the left of the “=~”. Also notice that there are “+” signs that separate the variables on the right side of the equation. The entire model statement needs to be wrapped in quotation marks.
ConfigModel <- 'MATH =~ wratcalc + wjcalc + waiscalc
SPELL =~ wratspl + wjspl + waisspl'The cfa function is used to actually fit the model. The grouping variable must be provided to the group argument. The std.lv = TRUE argument tells lavaan to used fixed factor model identification where the latent variables are standardized with a mean of 0 and a variance of 1.
ConfigOutput <- cfa(model = ConfigModel, data = dat, std.lv = TRUE,
missing = "fiml", mimic = "Mplus", group = "gender")
summary(ConfigOutput, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-3 ended normally after 67 iterations
Optimization method NLMINB
Number of free parameters 38
Number of observations per group
Female 101
Male 221
Number of missing patterns per group
Female 4
Male 2
Estimator ML
Model Fit Test Statistic 19.215
Degrees of freedom 16
P-value (Chi-square) 0.258
Chi-square for each group:
Female 11.555
Male 7.660
Model test baseline model:
Minimum Function Test Statistic 1905.113
Degrees of freedom 30
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.998
Tucker-Lewis Index (TLI) 0.997
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -5112.407
Loglikelihood unrestricted model (H1) -5102.800
Number of free parameters 38
Akaike (AIC) 10300.815
Bayesian (BIC) 10444.248
Sample-size adjusted Bayesian (BIC) 10323.717
Root Mean Square Error of Approximation:
RMSEA 0.035
90 Percent Confidence Interval 0.000 0.085
P-value RMSEA <= 0.05 0.631
Standardized Root Mean Square Residual:
SRMR 0.029
Parameter Estimates:
Information Observed
Observed information based on Hessian
Standard Errors Standard
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratcalc 6.545 0.496 13.187 0.000 6.545 0.978
wjcalc 4.215 0.366 11.530 0.000 4.215 0.904
waiscalc 2.290 0.276 8.306 0.000 2.290 0.724
SPELL =~
wratspl 7.010 0.547 12.817 0.000 7.010 0.950
wjspl 6.833 0.518 13.182 0.000 6.833 0.965
waisspl 6.638 0.520 12.763 0.000 6.638 0.949
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.634 0.064 9.914 0.000 0.634 0.634
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.wratcalc 38.267 0.666 57.464 0.000 38.267 5.718
.wjcalc 23.603 0.465 50.781 0.000 23.603 5.064
.waiscalc 10.266 0.316 32.528 0.000 10.266 3.245
.wratspl 37.171 0.735 50.590 0.000 37.171 5.037
.wjspl 42.337 0.705 60.073 0.000 42.337 5.978
.waisspl 38.030 0.697 54.579 0.000 38.030 5.435
MATH 0.000 0.000 0.000
SPELL 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.wratcalc 1.957 1.621 1.208 0.227 1.957 0.044
.wjcalc 3.960 0.848 4.671 0.000 3.960 0.182
.waiscalc 4.765 0.714 6.678 0.000 4.765 0.476
.wratspl 5.307 1.105 4.804 0.000 5.307 0.097
.wjspl 3.474 0.910 3.818 0.000 3.474 0.069
.waisspl 4.900 1.007 4.865 0.000 4.900 0.100
MATH 1.000 1.000 1.000
SPELL 1.000 1.000 1.000
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratcalc 5.764 0.330 17.492 0.000 5.764 0.929
wjcalc 4.123 0.244 16.926 0.000 4.123 0.910
waiscalc 2.446 0.200 12.212 0.000 2.446 0.729
SPELL =~
wratspl 6.278 0.337 18.646 0.000 6.278 0.943
wjspl 6.788 0.359 18.929 0.000 6.788 0.951
waisspl 6.177 0.335 18.441 0.000 6.177 0.937
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.520 0.053 9.762 0.000 0.520 0.520
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.wratcalc 39.222 0.417 93.962 0.000 39.222 6.321
.wjcalc 23.910 0.305 78.420 0.000 23.910 5.275
.waiscalc 11.367 0.226 50.320 0.000 11.367 3.389
.wratspl 36.172 0.448 80.762 0.000 36.172 5.433
.wjspl 41.371 0.480 86.157 0.000 41.371 5.796
.waisspl 36.767 0.444 82.899 0.000 36.767 5.579
MATH 0.000 0.000 0.000
SPELL 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.wratcalc 5.280 1.232 4.285 0.000 5.280 0.137
.wjcalc 3.549 0.664 5.342 0.000 3.549 0.173
.waiscalc 5.269 0.557 9.463 0.000 5.269 0.468
.wratspl 4.915 0.726 6.772 0.000 4.915 0.111
.wjspl 4.881 0.795 6.138 0.000 4.881 0.096
.waisspl 5.285 0.736 7.176 0.000 5.285 0.122
MATH 1.000 1.000 1.000
SPELL 1.000 1.000 1.000Now we fit a model where the factor loadings are constrained to equality across males and females.
In the model we define we are specifying the measurement structure and the latent variable variances. Because we are constraining the factor loadings, we can estimate the latent variable variances for the second group (males in this case). We can constrain the factor loadings using the group.equal = “loadings” argument in the “cfa” function.
metricModel <-
' MATH =~ c(NA, NA)*wratcalc + wjcalc + waiscalc
## freely estimate the first factor loading in both groups
MATH ~~ c(NA, 1)*MATH
## fix the factor variance of MATH to 1 in male group, and freely estimate in female group
MATH ~ c(0, 0)*1
SPELL =~ c(NA, NA)*wratspl + wjspl + waisspl
## freely estimate the first factor loading in both groups
SPELL ~~ c(NA, 1)*SPELL
## fix the factor variance of SPELL to 1 in male group, and freely estimate in female group
SPELL ~ c(0, 0)*1
'metricOutput <- cfa(model = metricModel, data = dat,
missing = "fiml", mimic = "Mplus",
group = "gender", group.equal = "loadings")
summary(metricOutput, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-3 ended normally after 69 iterations
Optimization method NLMINB
Number of free parameters 40
Number of equality constraints 6
Number of observations per group
Female 101
Male 221
Number of missing patterns per group
Female 4
Male 2
Estimator ML
Model Fit Test Statistic 25.871
Degrees of freedom 20
P-value (Chi-square) 0.170
Chi-square for each group:
Female 15.581
Male 10.290
Model test baseline model:
Minimum Function Test Statistic 1905.113
Degrees of freedom 30
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.997
Tucker-Lewis Index (TLI) 0.995
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -5115.735
Loglikelihood unrestricted model (H1) -5102.800
Number of free parameters 34
Akaike (AIC) 10299.471
Bayesian (BIC) 10427.805
Sample-size adjusted Bayesian (BIC) 10319.962
Root Mean Square Error of Approximation:
RMSEA 0.043
90 Percent Confidence Interval 0.000 0.085
P-value RMSEA <= 0.05 0.566
Standardized Root Mean Square Residual:
SRMR 0.053
Parameter Estimates:
Information Observed
Observed information based on Hessian
Standard Errors Standard
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratclc (.p1.) 5.898 0.317 18.615 0.000 6.359 0.964
wjcalc (.p2.) 4.039 0.232 17.414 0.000 4.354 0.917
waisclc (.p3.) 2.321 0.173 13.389 0.000 2.502 0.756
SPELL =~
wratspl (.p6.) 6.387 0.332 19.245 0.000 6.751 0.943
wjspl (.p7.) 6.640 0.346 19.186 0.000 7.020 0.969
waisspl (.p8.) 6.213 0.326 19.054 0.000 6.568 0.947
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.731 0.150 4.857 0.000 0.641 0.641
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 0.000 0.000 0.000
SPELL 0.000 0.000 0.000
.wratcalc 38.267 0.656 58.319 0.000 38.267 5.803
.wjcalc 23.602 0.474 49.832 0.000 23.602 4.969
.waiscalc 10.263 0.330 31.085 0.000 10.263 3.100
.wratspl 37.172 0.713 52.136 0.000 37.172 5.192
.wjspl 42.337 0.721 58.719 0.000 42.337 5.843
.waisspl 38.031 0.691 55.052 0.000 38.031 5.482
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 1.162 0.211 5.501 0.000 1.000 1.000
SPELL 1.117 0.197 5.676 0.000 1.000 1.000
.wratcalc 3.047 1.413 2.157 0.031 3.047 0.070
.wjcalc 3.603 0.778 4.632 0.000 3.603 0.160
.waiscalc 4.700 0.714 6.586 0.000 4.700 0.429
.wratspl 5.686 1.123 5.061 0.000 5.686 0.111
.wjspl 3.230 0.929 3.478 0.001 3.230 0.062
.waisspl 4.995 1.023 4.883 0.000 4.995 0.104
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratclc (.p1.) 5.898 0.317 18.615 0.000 5.898 0.940
wjcalc (.p2.) 4.039 0.232 17.414 0.000 4.039 0.900
waisclc (.p3.) 2.321 0.173 13.389 0.000 2.321 0.708
SPELL =~
wratspl (.p6.) 6.387 0.332 19.245 0.000 6.387 0.946
wjspl (.p7.) 6.640 0.346 19.186 0.000 6.640 0.946
waisspl (.p8.) 6.213 0.326 19.054 0.000 6.213 0.938
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.519 0.053 9.733 0.000 0.519 0.519
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 0.000 0.000 0.000
SPELL 0.000 0.000 0.000
.wratcalc 39.222 0.422 92.880 0.000 39.222 6.248
.wjcalc 23.910 0.302 79.192 0.000 23.910 5.327
.waiscalc 11.367 0.221 51.462 0.000 11.367 3.466
.wratspl 36.172 0.454 79.673 0.000 36.172 5.359
.wjspl 41.371 0.472 87.637 0.000 41.371 5.895
.waisspl 36.767 0.446 82.523 0.000 36.767 5.553
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 1.000 1.000 1.000
SPELL 1.000 1.000 1.000
.wratcalc 4.619 1.205 3.834 0.000 4.619 0.117
.wjcalc 3.835 0.654 5.864 0.000 3.835 0.190
.waiscalc 5.370 0.564 9.525 0.000 5.370 0.499
.wratspl 4.764 0.721 6.607 0.000 4.764 0.105
.wjspl 5.154 0.790 6.526 0.000 5.154 0.105
.waisspl 5.233 0.732 7.153 0.000 5.233 0.119In the scalar invariance model we want to constrain the factor loadings and intercepts. The latent variable means and variances are freely estimated for males. The loadings and intercepts are constrained with the group.equal argument.
scalarModel <- 'MATH =~ c(NA, NA)*wratcalc + wjcalc + waiscalc
MATH ~~ c(NA, 1)*MATH
MATH ~ c(NA, 0)*1
SPELL =~ c(NA, NA)*wratspl + wjspl + waisspl
SPELL ~~ c(NA, 1)*SPELL
SPELL ~ c(NA, 0)*1'scalarOutput <- cfa(model = scalarModel, data = dat,
missing = "fiml", mimic = "Mplus",
group = "gender",
group.equal = c("loadings", "intercepts"))
summary(scalarOutput, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-3 ended normally after 65 iterations
Optimization method NLMINB
Number of free parameters 42
Number of equality constraints 12
Number of observations per group
Female 101
Male 221
Number of missing patterns per group
Female 4
Male 2
Estimator ML
Model Fit Test Statistic 36.362
Degrees of freedom 24
P-value (Chi-square) 0.051
Chi-square for each group:
Female 22.546
Male 13.815
Model test baseline model:
Minimum Function Test Statistic 1905.113
Degrees of freedom 30
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.993
Tucker-Lewis Index (TLI) 0.992
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -5120.981
Loglikelihood unrestricted model (H1) -5102.800
Number of free parameters 30
Akaike (AIC) 10301.961
Bayesian (BIC) 10415.198
Sample-size adjusted Bayesian (BIC) 10320.042
Root Mean Square Error of Approximation:
RMSEA 0.057
90 Percent Confidence Interval 0.000 0.092
P-value RMSEA <= 0.05 0.356
Standardized Root Mean Square Residual:
SRMR 0.053
Parameter Estimates:
Information Observed
Observed information based on Hessian
Standard Errors Standard
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratclc (.p1.) 5.907 0.317 18.658 0.000 6.381 0.967
wjcalc (.p2.) 4.017 0.232 17.352 0.000 4.340 0.913
waisclc (.p3.) 2.341 0.175 13.388 0.000 2.529 0.750
SPELL =~
wratspl (.p6.) 6.385 0.332 19.249 0.000 6.751 0.943
wjspl (.p7.) 6.634 0.346 19.183 0.000 7.015 0.969
waisspl (.p8.) 6.221 0.326 19.064 0.000 6.578 0.946
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.731 0.151 4.855 0.000 0.640 0.640
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH -0.160 0.131 -1.223 0.221 -0.148 -0.148
SPELL 0.165 0.127 1.296 0.195 0.156 0.156
.wratclc (.18.) 39.217 0.419 93.583 0.000 39.217 5.942
.wjcalc (.19.) 24.016 0.295 81.470 0.000 24.016 5.054
.waisclc (.20.) 11.125 0.207 53.742 0.000 11.125 3.299
.wratspl (.21.) 36.157 0.449 80.484 0.000 36.157 5.050
.wjspl (.22.) 41.317 0.466 88.688 0.000 41.317 5.705
.waisspl (.23.) 36.843 0.440 83.721 0.000 36.843 5.301
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 1.167 0.212 5.502 0.000 1.000 1.000
SPELL 1.118 0.197 5.676 0.000 1.000 1.000
.wratcalc 2.841 1.424 1.995 0.046 2.841 0.065
.wjcalc 3.743 0.793 4.723 0.000 3.743 0.166
.waiscalc 4.977 0.766 6.496 0.000 4.977 0.438
.wratspl 5.681 1.123 5.056 0.000 5.681 0.111
.wjspl 3.237 0.930 3.481 0.000 3.237 0.062
.waisspl 5.035 1.032 4.880 0.000 5.035 0.104
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratclc (.p1.) 5.907 0.317 18.658 0.000 5.907 0.941
wjcalc (.p2.) 4.017 0.232 17.352 0.000 4.017 0.897
waisclc (.p3.) 2.341 0.175 13.388 0.000 2.341 0.709
SPELL =~
wratspl (.p6.) 6.385 0.332 19.249 0.000 6.385 0.946
wjspl (.p7.) 6.634 0.346 19.183 0.000 6.634 0.946
waisspl (.p8.) 6.221 0.326 19.064 0.000 6.221 0.939
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.519 0.053 9.732 0.000 0.519 0.519
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 0.000 0.000 0.000
SPELL 0.000 0.000 0.000
.wratclc (.18.) 39.217 0.419 93.583 0.000 39.217 6.247
.wjcalc (.19.) 24.016 0.295 81.470 0.000 24.016 5.363
.waisclc (.20.) 11.125 0.207 53.742 0.000 11.125 3.368
.wratspl (.21.) 36.157 0.449 80.484 0.000 36.157 5.359
.wjspl (.22.) 41.317 0.466 88.688 0.000 41.317 5.892
.waisspl (.23.) 36.843 0.440 83.721 0.000 36.843 5.558
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 1.000 1.000 1.000
SPELL 1.000 1.000 1.000
.wratcalc 4.518 1.204 3.753 0.000 4.518 0.115
.wjcalc 3.911 0.655 5.970 0.000 3.911 0.195
.waiscalc 5.431 0.576 9.435 0.000 5.431 0.498
.wratspl 4.761 0.721 6.603 0.000 4.761 0.105
.wjspl 5.173 0.791 6.541 0.000 5.173 0.105
.waisspl 5.235 0.734 7.134 0.000 5.235 0.119Now the models can be compared via chi-square difference testing.
anova(ConfigOutput, metricOutput)Chi Square Difference Test
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
ConfigOutput 16 10301 10444 19.215
metricOutput 20 10300 10428 25.871 6.6558 4 0.1552Here the metric model does not fit significantly worse than the configural model, so metric invariance is supported.
Now we can test for scalar invariance.
anova(metricOutput, scalarOutput)Chi Square Difference Test
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricOutput 20 10300 10428 25.871
scalarOutput 24 10302 10415 36.361 10.491 4 0.03293 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Here the scalar model fits significantly worse, so scalar invariance is not supported.
In addition to the constraints specified for the scalar invariance model (i.e., euqal factor loadings, equal indicator intercepts), indicator residual variances can also be equated between groups. However, strict factorial invariance is an optional test - this level of invariance is not necessary for conducting latent parameter comparisons.
strictModel <- 'MATH =~ c(NA, NA)*wratcalc + wjcalc + waiscalc
MATH ~~ c(NA, 1)*MATH
MATH ~ c(NA, 0)*1
SPELL =~ c(NA, NA)*wratspl + wjspl + waisspl
SPELL ~~ c(NA, 1)*SPELL
SPELL ~ c(NA, 0)*1'strictOutput <- cfa(model = strictModel, data = dat,
missing = "fiml", mimic = "Mplus",
group = "gender",
group.equal = c("loadings", "intercepts", "residuals"))
summary(strictOutput, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-3 ended normally after 68 iterations
Optimization method NLMINB
Number of free parameters 42
Number of equality constraints 18
Number of observations per group
Female 101
Male 221
Number of missing patterns per group
Female 4
Male 2
Estimator ML
Model Fit Test Statistic 40.717
Degrees of freedom 30
P-value (Chi-square) 0.092
Chi-square for each group:
Female 26.601
Male 14.115
Model test baseline model:
Minimum Function Test Statistic 1905.113
Degrees of freedom 30
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.994
Tucker-Lewis Index (TLI) 0.994
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -5123.158
Loglikelihood unrestricted model (H1) -5102.800
Number of free parameters 24
Akaike (AIC) 10294.316
Bayesian (BIC) 10384.905
Sample-size adjusted Bayesian (BIC) 10308.781
Root Mean Square Error of Approximation:
RMSEA 0.047
90 Percent Confidence Interval 0.000 0.081
P-value RMSEA <= 0.05 0.521
Standardized Root Mean Square Residual:
SRMR 0.056
Parameter Estimates:
Information Observed
Observed information based on Hessian
Standard Errors Standard
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratclc (.p1.) 5.905 0.316 18.706 0.000 6.300 0.952
wjcalc (.p2.) 4.040 0.231 17.469 0.000 4.310 0.911
waisclc (.p3.) 2.358 0.175 13.463 0.000 2.516 0.738
SPELL =~
wratspl (.p6.) 6.392 0.333 19.208 0.000 6.770 0.949
wjspl (.p7.) 6.663 0.345 19.316 0.000 7.057 0.957
waisspl (.p8.) 6.219 0.326 19.073 0.000 6.586 0.946
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.730 0.150 4.860 0.000 0.646 0.646
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH -0.159 0.130 -1.221 0.222 -0.149 -0.149
SPELL 0.166 0.128 1.298 0.194 0.156 0.156
.wratclc (.18.) 39.217 0.417 93.947 0.000 39.217 5.926
.wjcalc (.19.) 24.012 0.296 81.184 0.000 24.012 5.073
.waisclc (.20.) 11.139 0.206 54.146 0.000 11.139 3.269
.wratspl (.21.) 36.153 0.451 80.244 0.000 36.153 5.068
.wjspl (.22.) 41.328 0.466 88.600 0.000 41.328 5.606
.waisspl (.23.) 36.840 0.440 83.781 0.000 36.840 5.289
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 1.138 0.206 5.512 0.000 1.000 1.000
SPELL 1.122 0.198 5.672 0.000 1.000 1.000
.wratclc (.11.) 4.104 0.998 4.111 0.000 4.104 0.094
.wjcalc (.12.) 3.827 0.530 7.224 0.000 3.827 0.171
.waisclc (.13.) 5.283 0.457 11.562 0.000 5.283 0.455
.wratspl (.14.) 5.057 0.612 8.266 0.000 5.057 0.099
.wjspl (.15.) 4.554 0.616 7.388 0.000 4.554 0.084
.waisspl (.16.) 5.140 0.598 8.588 0.000 5.140 0.106
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH =~
wratclc (.p1.) 5.905 0.316 18.706 0.000 5.905 0.946
wjcalc (.p2.) 4.040 0.231 17.469 0.000 4.040 0.900
waisclc (.p3.) 2.358 0.175 13.463 0.000 2.358 0.716
SPELL =~
wratspl (.p6.) 6.392 0.333 19.208 0.000 6.392 0.943
wjspl (.p7.) 6.663 0.345 19.316 0.000 6.663 0.952
waisspl (.p8.) 6.219 0.326 19.073 0.000 6.219 0.940
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH ~~
SPELL 0.517 0.053 9.710 0.000 0.517 0.517
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 0.000 0.000 0.000
SPELL 0.000 0.000 0.000
.wratclc (.18.) 39.217 0.417 93.947 0.000 39.217 6.282
.wjcalc (.19.) 24.012 0.296 81.184 0.000 24.012 5.350
.waisclc (.20.) 11.139 0.206 54.146 0.000 11.139 3.383
.wratspl (.21.) 36.153 0.451 80.244 0.000 36.153 5.335
.wjspl (.22.) 41.328 0.466 88.600 0.000 41.328 5.907
.waisspl (.23.) 36.840 0.440 83.781 0.000 36.840 5.565
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
MATH 1.000 1.000 1.000
SPELL 1.000 1.000 1.000
.wratclc (.11.) 4.104 0.998 4.111 0.000 4.104 0.105
.wjcalc (.12.) 3.827 0.530 7.224 0.000 3.827 0.190
.waisclc (.13.) 5.283 0.457 11.562 0.000 5.283 0.487
.wratspl (.14.) 5.057 0.612 8.266 0.000 5.057 0.110
.wjspl (.15.) 4.554 0.616 7.388 0.000 4.554 0.093
.waisspl (.16.) 5.140 0.598 8.588 0.000 5.140 0.117anova(scalarOutput, strictOutput)Chi Square Difference Test
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
scalarOutput 24 10302 10415 36.361
strictOutput 30 10294 10385 40.717 4.3551 6 0.6287R version 3.5.1 (2018-07-02)
Platform: x86_64-redhat-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)
Matrix products: default
BLAS/LAPACK: /usr/lib64/R/lib/libRblas.so
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lavaan_0.6-3 stationery_0.98.5.7
loaded via a namespace (and not attached):
[1] Rcpp_1.0.0 digest_0.6.18 MASS_7.3-51.1 plyr_1.8.4
[5] xtable_1.8-3 magrittr_1.5 stats4_3.5.1 evaluate_0.12
[9] zip_1.0.0 stringi_1.2.4 pbivnorm_0.6.0 openxlsx_4.1.0
[13] rmarkdown_1.11 tools_3.5.1 stringr_1.3.1 foreign_0.8-71
[17] kutils_1.59 yaml_2.2.0 xfun_0.4 compiler_3.5.1
[21] mnormt_1.5-5 htmltools_0.3.6 knitr_1.21 Available under Created Commons license 3.0 