Structural Equation Modeling
Note updated May 09, 2023. Not for sale :-)
Wan Nor Arifin (wnarifin@usm.my),Biostatistics and Research Methodology Unit, Universiti Sains MalaysiaWebsite: wnarifin.github.io
This
work is licensed under a
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Commons Attribution-ShareAlike 4.0 International License.
1 Libraries
We are going to use psych (Revelle, 2023),
lavaan version 0.6.15 (Rosseel, Jorgensen, &
Rockwood, 2023), semTools version 0.5.6 (Jorgensen, Pornprasertmanit, Schoemann, &
Rosseel, 2022) and semPlot version 1.1.6 (Epskamp,
2022) in our analysis. Load the packages.
library(foreign)
library(psych)
library(lavaan)
library(semTools)
library(semPlot)2 Load data
data = read.spss("Attitude_Statistics v3.sav", F, T)
dim(data)## [1] 150 13names(data)## [1] "ID" "Q1" "Q2" "Q3" "Q4" "Q5" "Q6" "Q7" "Q8" "Q9" "Q10" "Q11" "Q12"describe(data[-1])| vars | n | mean | sd | median | trimmed | mad | min | max | range | skew | kurtosis | se | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Q1 | 1 | 150 | 3.126667 | 1.1009250 | 3 | 3.125000 | 1.4826 | 1 | 5 | 4 | -0.0995748 | -0.7324083 | 0.0898901 |
| Q2 | 2 | 150 | 3.506667 | 1.0346135 | 3 | 3.550000 | 1.4826 | 1 | 5 | 4 | -0.1441109 | -0.4692084 | 0.0844758 |
| Q3 | 3 | 150 | 3.180000 | 1.0302147 | 3 | 3.166667 | 1.4826 | 1 | 5 | 4 | -0.0321343 | -0.4202388 | 0.0841167 |
| Q4 | 4 | 150 | 2.813333 | 1.1723130 | 3 | 2.775000 | 1.4826 | 1 | 5 | 4 | 0.1888890 | -0.8077519 | 0.0957190 |
| Q5 | 5 | 150 | 3.313333 | 1.0109247 | 3 | 3.316667 | 1.4826 | 1 | 5 | 4 | -0.2242453 | -0.4760474 | 0.0825417 |
| Q6 | 6 | 150 | 3.053333 | 1.0916810 | 3 | 3.050000 | 1.4826 | 1 | 5 | 4 | -0.0432196 | -0.7058165 | 0.0891354 |
| Q7 | 7 | 150 | 2.920000 | 1.1901159 | 3 | 2.925000 | 1.4826 | 1 | 5 | 4 | -0.0366766 | -1.0562073 | 0.0971726 |
| Q8 | 8 | 150 | 3.326667 | 0.9999776 | 3 | 3.341667 | 1.4826 | 1 | 5 | 4 | -0.0816209 | -0.1204552 | 0.0816478 |
| Q9 | 9 | 150 | 3.440000 | 1.0457970 | 3 | 3.483333 | 1.4826 | 1 | 5 | 4 | -0.2091593 | -0.3234321 | 0.0853890 |
| Q10 | 10 | 150 | 3.313333 | 1.0999492 | 3 | 3.358333 | 1.4826 | 1 | 5 | 4 | -0.2157639 | -0.3888402 | 0.0898105 |
| Q11 | 11 | 150 | 3.353333 | 0.9350496 | 3 | 3.366667 | 1.4826 | 1 | 5 | 4 | -0.3074829 | -0.3281186 | 0.0763465 |
| Q12 | 12 | 150 | 2.826667 | 0.9813473 | 3 | 2.833333 | 1.4826 | 1 | 5 | 4 | 0.0945084 | -0.6761637 | 0.0801267 |
3 Correlational & causal
3.1 Correlational
3.1.1 Observed variables
# Q4 & Q11
model.c = "
Q4 ~~ Q11
"
corr.c = sem(model.c, data = data, meanstructure = T) # meanstructure -> display mean
summary(corr.c, fit.measures = T, standardized = T)## lavaan 0.6.15 ended normally after 13 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 5
##
## Number of observations 150
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Model Test Baseline Model:
##
## Test statistic 32.305
## Degrees of freedom 1
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.000
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -422.299
## Loglikelihood unrestricted model (H1) -422.299
##
## Akaike (AIC) 854.599
## Bayesian (BIC) 869.652
## Sample-size adjusted Bayesian (SABIC) 853.828
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.000
## P-value H_0: RMSEA <= 0.050 NA
## P-value H_0: RMSEA >= 0.080 NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## Q4 ~~
## Q11 0.479 0.097 4.934 0.000 0.479 0.440
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## Q4 2.813 0.095 29.490 0.000 2.813 2.408
## Q11 3.353 0.076 44.070 0.000 3.353 3.598
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## Q4 1.365 0.158 8.660 0.000 1.365 1.000
## Q11 0.868 0.100 8.660 0.000 0.868 1.000semPaths(corr.c, what = "path", whatLabels = "par", edge.color = "black",
edge.label.cex = 1, residuals = F, sizeInt = 4)
cor(data$Q4, data$Q11)## [1] 0.4401738
3.1.2 Latent variables
model.cl = "
F1 =~ Q4 + Q6 + Q7 + Q11
F2 =~ Q8 + Q9 + Q10
"
corr.cl = sem(model.cl, data = data)
summary(corr.cl, fit.measures = T, standardized = T)## lavaan 0.6.15 ended normally after 22 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 15
##
## Number of observations 150
##
## Model Test User Model:
##
## Test statistic 20.451
## Degrees of freedom 13
## P-value (Chi-square) 0.085
##
## Model Test Baseline Model:
##
## Test statistic 380.263
## Degrees of freedom 21
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.979
## Tucker-Lewis Index (TLI) 0.966
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1380.510
## Loglikelihood unrestricted model (H1) -1370.284
##
## Akaike (AIC) 2791.020
## Bayesian (BIC) 2836.179
## Sample-size adjusted Bayesian (SABIC) 2788.707
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.062
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.111
## P-value H_0: RMSEA <= 0.050 0.314
## P-value H_0: RMSEA >= 0.080 0.306
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.063
##
## 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
## F1 =~
## Q4 1.000 0.941 0.806
## Q6 0.830 0.103 8.040 0.000 0.781 0.718
## Q7 0.960 0.115 8.348 0.000 0.904 0.762
## Q11 0.504 0.088 5.742 0.000 0.474 0.509
## F2 =~
## Q8 1.000 0.651 0.653
## Q9 1.351 0.170 7.951 0.000 0.880 0.844
## Q10 1.444 0.182 7.927 0.000 0.940 0.858
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## F1 ~~
## F2 0.048 0.060 0.800 0.424 0.078 0.078
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .Q4 0.479 0.097 4.932 0.000 0.479 0.351
## .Q6 0.574 0.088 6.530 0.000 0.574 0.485
## .Q7 0.591 0.101 5.825 0.000 0.591 0.420
## .Q11 0.644 0.081 7.997 0.000 0.644 0.741
## .Q8 0.569 0.076 7.526 0.000 0.569 0.573
## .Q9 0.313 0.077 4.054 0.000 0.313 0.288
## .Q10 0.318 0.086 3.693 0.000 0.318 0.264
## F1 0.886 0.168 5.279 0.000 1.000 1.000
## F2 0.424 0.101 4.191 0.000 1.000 1.000semPaths(corr.cl, what = "path", whatLabels = "par", edge.color = "black",
layout = "tree2", edge.label.cex = 1, residuals = F)
3.2 Causal effects
3.2.1 Observed variables
# Q4 & Q11
model.cs = "
Q4 ~ Q11
"
cause.cs = sem(model.cs, data = data)
summary(cause.cs, fit.measures = T, standardized = T)## lavaan 0.6.15 ended normally after 1 iteration
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 2
##
## Number of observations 150
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Model Test Baseline Model:
##
## Test statistic 32.305
## Degrees of freedom 1
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.000
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -220.034
## Loglikelihood unrestricted model (H1) -220.034
##
## Akaike (AIC) 444.067
## Bayesian (BIC) 450.088
## Sample-size adjusted Bayesian (SABIC) 443.759
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.000
## P-value H_0: RMSEA <= 0.050 NA
## P-value H_0: RMSEA >= 0.080 NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## Q4 ~
## Q11 0.552 0.092 6.004 0.000 0.552 0.440
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .Q4 1.101 0.127 8.660 0.000 1.101 0.806summary(lm(formula = Q4 ~ Q11, data = data)) # compare with SLR##
## Call:
## lm(formula = Q4 ~ Q11, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.7221 -0.6183 -0.1702 0.8298 2.9335
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.96274 0.32207 2.989 0.00328 **
## Q11 0.55187 0.09254 5.964 1.74e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.056 on 148 degrees of freedom
## Multiple R-squared: 0.1938, Adjusted R-squared: 0.1883
## F-statistic: 35.57 on 1 and 148 DF, p-value: 1.737e-08semPaths(cause.cs, what = "path", whatLabels = "par", edge.color = "black",
rotation = 2, edge.label.cex = 1, residuals = F)