# principles and practice of sem, 5th ed.
# rex b. kline, guilford press, 2023
# chapter 14, table 14.1, analysis 1
# single-factor cfa model for the kabc-1 analyzed
# in the 10-year-old normative sample for the test
# to avoid the problem that some R packages share the same name
# for different functions, all functions are specified next as
# package::function
# which prevents masking, or the default hiding of
# a function with a redundant name from a package used next
date()
## [1] "Mon Sep 29 17:30:27 2025"
v <- R.Version()
print(paste0(v$language, " version ", v$major, ".",
v$minor, " (", v$year, "-", v$month, "-", v$day, ")"))
## [1] "R version 4.5.1 (2025-06-13)"
library(lavaan)
## This is lavaan 0.6-20
## lavaan is FREE software! Please report any bugs.
# get citation information
citation("lavaan", auto = TRUE)
## To cite package 'lavaan' in publications use:
##
## Rosseel Y, Jorgensen TD, De Wilde L (2025). _lavaan: Latent Variable
## Analysis_. doi:10.32614/CRAN.package.lavaan
## <https://doi.org/10.32614/CRAN.package.lavaan>, R package version
## 0.6-20, <https://CRAN.R-project.org/package=lavaan>.
##
## A BibTeX entry for LaTeX users is
##
## @Manual{,
## title = {lavaan: Latent Variable Analysis},
## author = {Yves Rosseel and Terrence D. Jorgensen and Luc {De Wilde}},
## year = {2025},
## note = {R package version 0.6-20},
## url = {https://CRAN.R-project.org/package=lavaan},
## doi = {10.32614/CRAN.package.lavaan},
## }
# input the correlations in lower diagnonal form
kabcLower.cor <- '
1.00
.39 1.00
.35 .67 1.00
.21 .11 .16 1.00
.32 .27 .29 .38 1.00
.40 .29 .28 .30 .47 1.00
.39 .32 .30 .31 .42 .41 1.00
.39 .29 .37 .42 .58 .51 .42 1.00 '
# name the variables and convert to full correlation matrix
# hm, hand movements; nr, number recall; wo, word order; gc, gestalt closure;
# tr, triangles; sm, spatial memory; ma, matrix analogies; ps, photo series
kabc.cor <- lavaan::getCov(kabcLower.cor, names = c("hm", "nr", "wo",
"gc", "tr", "sm", "ma", "ps"))
# display correlations
kabc.cor
## hm nr wo gc tr sm ma ps
## hm 1.00 0.39 0.35 0.21 0.32 0.40 0.39 0.39
## nr 0.39 1.00 0.67 0.11 0.27 0.29 0.32 0.29
## wo 0.35 0.67 1.00 0.16 0.29 0.28 0.30 0.37
## gc 0.21 0.11 0.16 1.00 0.38 0.30 0.31 0.42
## tr 0.32 0.27 0.29 0.38 1.00 0.47 0.42 0.58
## sm 0.40 0.29 0.28 0.30 0.47 1.00 0.41 0.51
## ma 0.39 0.32 0.30 0.31 0.42 0.41 1.00 0.42
## ps 0.39 0.29 0.37 0.42 0.58 0.51 0.42 1.00
# add the standard deviations and convert to covariances
# This gives the covariance matrix.
kabc.cov <- lavaan::cor2cov(kabc.cor, sds = c(3.40,2.40,2.90,2.70,2.70,4.20,2.80,3.00))
# display covariances
kabc.cov
## hm nr wo gc tr sm ma ps
## hm 11.5600 3.1824 3.4510 1.9278 2.9376 5.7120 3.7128 3.978
## nr 3.1824 5.7600 4.6632 0.7128 1.7496 2.9232 2.1504 2.088
## wo 3.4510 4.6632 8.4100 1.2528 2.2707 3.4104 2.4360 3.219
## gc 1.9278 0.7128 1.2528 7.2900 2.7702 3.4020 2.3436 3.402
## tr 2.9376 1.7496 2.2707 2.7702 7.2900 5.3298 3.1752 4.698
## sm 5.7120 2.9232 3.4104 3.4020 5.3298 17.6400 4.8216 6.426
## ma 3.7128 2.1504 2.4360 2.3436 3.1752 4.8216 7.8400 3.528
## ps 3.9780 2.0880 3.2190 3.4020 4.6980 6.4260 3.5280 9.000
# specify 1-factor cfa model
# single factor (general ability)
# indicator hm automatically
# specified as reference variable
kabc1.model <- '
General =~ hm + nr + wo + gc + tr + sm + ma + ps '
# fit model to data
# variances calculated with N in the denominator, not N - 1
#This uses the covariance matrix. Lavaan default is the covariance matrix.
kabc1 <- lavaan::sem(kabc1.model, sample.cov = kabc.cov, sample.nobs = 200)
lavaan::summary(kabc1, fit.measures = TRUE, standardized = TRUE, rsquare = TRUE)
## lavaan 0.6-20 ended normally after 36 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 16
##
## Number of observations 200
##
## Model Test User Model:
##
## Test statistic 105.427
## Degrees of freedom 20
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 498.336
## Degrees of freedom 28
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.818
## Tucker-Lewis Index (TLI) 0.746
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3812.592
## Loglikelihood unrestricted model (H1) -3759.878
##
## Akaike (AIC) 7657.183
## Bayesian (BIC) 7709.956
## Sample-size adjusted Bayesian (SABIC) 7659.267
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.146
## 90 Percent confidence interval - lower 0.119
## 90 Percent confidence interval - upper 0.174
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.084
##
## 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
## General =~
## hm 1.000 1.921 0.566
## nr 0.636 0.111 5.708 0.000 1.221 0.510
## wo 0.805 0.136 5.910 0.000 1.547 0.535
## gc 0.659 0.123 5.361 0.000 1.265 0.470
## tr 0.963 0.138 6.984 0.000 1.850 0.687
## sm 1.433 0.211 6.796 0.000 2.752 0.657
## ma 0.883 0.137 6.459 0.000 1.696 0.607
## ps 1.166 0.159 7.324 0.000 2.240 0.749
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .hm 7.812 0.863 9.049 0.000 7.812 0.679
## .nr 4.240 0.456 9.294 0.000 4.240 0.740
## .wo 5.975 0.650 9.195 0.000 5.975 0.714
## .gc 5.652 0.599 9.432 0.000 5.652 0.779
## .tr 3.831 0.468 8.186 0.000 3.831 0.528
## .sm 9.979 1.179 8.463 0.000 9.979 0.569
## .ma 4.925 0.558 8.822 0.000 4.925 0.631
## .ps 3.936 0.531 7.410 0.000 3.936 0.440
## General 3.690 0.921 4.008 0.000 1.000 1.000
##
## R-Square:
## Estimate
## hm 0.321
## nr 0.260
## wo 0.286
## gc 0.221
## tr 0.472
## sm 0.431
## ma 0.369
## ps 0.560
# residuals
lavaan::residuals(kabc1, type = "raw")
## $type
## [1] "raw"
##
## $cov
## hm nr wo gc tr sm ma ps
## hm 0.000
## nr 0.820 0.000
## wo 0.462 2.751 0.000
## gc -0.513 -0.836 -0.711 0.000
## tr -0.631 -0.519 -0.602 0.415 0.000
## sm 0.397 -0.452 -0.863 -0.097 0.212 0.000
## ma 0.437 0.069 -0.199 0.186 0.022 0.131 0.000
## ps -0.345 -0.659 -0.263 0.550 0.530 0.229 -0.289 0.000
lavaan::residuals(kabc1, type = "standardized.mplus")
## $type
## [1] "standardized.mplus"
##
## $cov
## hm nr wo gc tr sm ma ps
## hm 0.000
## nr 2.062 0.000
## wo 1.026 6.218 0.000
## gc -1.231 -2.727 -1.952 0.000
## tr -2.200 -2.364 -2.355 1.379 0.000
## sm 0.723 -1.188 -1.995 -0.210 0.596 0.000
## ma 1.086 0.237 -0.601 0.544 0.089 0.313 0.000
## ps -1.241 -3.422 -1.037 1.833 2.178 0.675 -1.375 0.000
lavaan::residuals(kabc1, type = "cor.bollen")
## $type
## [1] "cor.bollen"
##
## $cov
## hm nr wo gc tr sm ma ps
## hm 0.000
## nr 0.101 0.000
## wo 0.047 0.397 0.000
## gc -0.056 -0.130 -0.091 0.000
## tr -0.069 -0.080 -0.077 0.057 0.000
## sm 0.028 -0.045 -0.071 -0.009 0.019 0.000
## ma 0.046 0.010 -0.025 0.025 0.003 0.011 0.000
## ps -0.034 -0.092 -0.030 0.068 0.066 0.018 -0.035 0.000