# Library
library(tidyverse)
library(readxl)
library(janitor)
library(lavaan)
library(psych)
library(MVN)
library(semTools)
library(lavaanPlot)Covariance-based SEM (CB-SEM)
1 Sample study
- Journal article: Young people’s perceived service quality and environmental performance of hybrid electric bus.
- Author: Zial Haque and Tehjeeb Noor
- Article link: DOI link
- Download the dataset here.
2 Libraries
3 Data
## data
case_data <- read_excel("00_data/e_bus_customer_satisfaction.xlsx") %>%
clean_names()
case_data_items <- case_data %>%
select(bt1:bt7, bd1:bd4, emp1:emp5, cs1:cs3, ep1:ep4, ls1:ls5)
## datatable
DT::datatable(case_data)4 Exploratory factor analysis
4.1 Scree plot
We can use a scree test to identify the optimum number of factors that can be extracted. In the example below, a parallel analysis was used to identify the potential number of factors that can be extracted.
The test will generate simulated dataset with random values for the same number of variables and sample size. The test involved the following procedures (see Hair, 2019):
Then each of these simulated datasets is then factor analyzed, either with PCA or CFA and the eigen values are averaged for each factor across all the data sets.
The results is the average eigenvalues for the first, second and so on across the set of simulated dataset.
These values are then compared to the eigen values extracted for the original data.
All factors with eigen values above those average eigen values are retained.
Result of the parallel analysis shows that there are 6 potential factors (i.e., six triangles are above the red broken line or the simulated data). At this stage, you compare your expected number of factors to be extracted against the scree test. Note that the test can be used as one of your bases for factor extraction.
## Scree plot using parallel analysis
fa.parallel(case_data_items, fa = "fa")
Parallel analysis suggests that the number of factors = 7 and the number of components = NA 4.2 Factor extraction
An EFA with a varimax rotation was used to identify which items load to their expected construct. Items with high cross-loadings (i.e., items which load highly to more than one construct) and loadings below 0.30 were excluded.
High cross-loadings pose difficulty in establishing a separate concept of each variable when factors are apparently shared variables. The goal is to identify the pattern where each item associates only with one factor.
Some guides for factor loadings (Hair, 2019):
Factor loadings less than ±0.10 can be considered equivalent to zero for purposes of assessing simple structure.
Factor loadings in the range of ±0.30 and ±0.40 are considered to meet the minimal level for interpretation of the structure.
Loadings ±0.50 or greater are considered practically significant
Loadings exceeding ±0.70 are considered indicative of well-defined structure and are the goal of any factor analysis.
Extremely high loadings such as ±0.90 and above are not typical, and the practical significance of the loadings is an important criterion.
## Factor loading
bus_fa <- fa(r = case_data_items,
nfactors = 6,
rotate = "varimax")
print(bus_fa$loadings, sort = TRUE, cutoff = 0.4)
Loadings:
MR2 MR1 MR3 MR4 MR6 MR5
ls1 0.820
ls2 0.891
ls3 0.828
ls4 0.806
ls5 0.599
bt1 0.673
bt2 0.666
bt4 0.549
bt5 0.680
bt6 0.578
bt7 0.550
ep1 0.864
ep2 0.900
ep3 0.690
ep4 0.705
emp1 0.688
emp2 0.662
emp3 0.636
emp4 0.697
emp5 0.502
bd1 0.679
bd2 0.640
bd3 0.676
bd4 0.629
cs1 0.774
cs2 0.817
cs3 0.768
bt3 0.476
MR2 MR1 MR3 MR4 MR6 MR5
SS loadings 3.477 3.363 3.081 2.658 2.429 2.297
Proportion Var 0.124 0.120 0.110 0.095 0.087 0.082
Cumulative Var 0.124 0.244 0.354 0.449 0.536 0.6185 Confirmatory factor analysis
5.1 Multivariate normality test
One of the assumptions in CFA and CB-SEM is multivariate normality. Following the test employed in the sample study, the mvn function of the MVN package (Korkmaz et al., 2014) allows us to run Mardia’s test for multivariate normality.
Using the Mardia test wherein p-value < 0.5 indicate rejection of the null hypothesis of multivariate and univariate normality. Result shows that the null hypothesis of multivariate normality of all the multivariate normality was rejected. Therefore, the study use the maximum likelihood robust (MLR) estimator to estimate the measurement model instead of the usual maximum likelihood estimator (MLR) (Rosseel, 2012).
MVN::mvn(data = case_data_items, mvnTest = "mardia", desc = F)$multivariateNormality
Test Statistic p value Result
1 Mardia Skewness 7270.01984441523 4.08329952017666e-186 NO
2 Mardia Kurtosis 31.3657300202103 0 NO
3 MVN <NA> <NA> NO
$univariateNormality
Test Variable Statistic p value Normality
1 Anderson-Darling bt1 7.2262 <0.001 NO
2 Anderson-Darling bt2 8.0671 <0.001 NO
3 Anderson-Darling bt3 6.6800 <0.001 NO
4 Anderson-Darling bt4 6.4183 <0.001 NO
5 Anderson-Darling bt5 6.1728 <0.001 NO
6 Anderson-Darling bt6 8.6698 <0.001 NO
7 Anderson-Darling bt7 7.5770 <0.001 NO
8 Anderson-Darling bd1 5.9732 <0.001 NO
9 Anderson-Darling bd2 8.7052 <0.001 NO
10 Anderson-Darling bd3 5.9169 <0.001 NO
11 Anderson-Darling bd4 6.8989 <0.001 NO
12 Anderson-Darling emp1 5.2108 <0.001 NO
13 Anderson-Darling emp2 6.4303 <0.001 NO
14 Anderson-Darling emp3 4.7131 <0.001 NO
15 Anderson-Darling emp4 7.8380 <0.001 NO
16 Anderson-Darling emp5 5.5762 <0.001 NO
17 Anderson-Darling cs1 6.9341 <0.001 NO
18 Anderson-Darling cs2 7.1469 <0.001 NO
19 Anderson-Darling cs3 6.2791 <0.001 NO
20 Anderson-Darling ep1 7.5913 <0.001 NO
21 Anderson-Darling ep2 7.3530 <0.001 NO
22 Anderson-Darling ep3 7.2375 <0.001 NO
23 Anderson-Darling ep4 6.5793 <0.001 NO
24 Anderson-Darling ls1 9.0564 <0.001 NO
25 Anderson-Darling ls2 8.2974 <0.001 NO
26 Anderson-Darling ls3 8.5914 <0.001 NO
27 Anderson-Darling ls4 6.9281 <0.001 NO
28 Anderson-Darling ls5 5.8779 <0.001 NO 5.2 Specifying the measurement model
Below is the syntax used in defining our factor. For example, the drivers_quality factor is specified using the syntax drivers_quality =~ bd1 + bd2 + bd3 + bd4, which means this factor is measured using bd1 to bd4.
Note that we know which indicators to use to represent the factor based on our exploratory factor analysis in the previous sections.
## Specifying CFA model
cfa_model <- "tangible =~ bt1 + bt2 + bt4 + bt5 + bt6 + bt7
drivers_quality =~ bd1 + bd2 + bd3 + bd4
empathy =~ emp1 + emp2 + emp3 + emp4 + emp5
env_perf =~ ep1 + ep2 + ep3 + ep4
customer_sat =~ cs1 + cs2 + cs3
life_sat =~ ls1 + ls2 + ls3 + ls4 + ls5"5.3 Fitting the model
Given the result of the normality test, a maximum likelihood robust (MLR) was used to estimate the initial measurement model. Based on the assessment criteria, the model shows an acceptable and adequate model fit.
The fit of the measurement model was assessed using the ratio of the Chi-square to the degrees of freedom, Tucker-Lewis Index (TLI), Comparative Fit Index (CFI), and the Root Mean Square Error of Approximation (RMSEA).
For the recommended values of the fit measures, see Table 3 below taken from the study of W. Shiau & M. Luo (2013).
Sample GOF results from W. Shiau & M. Luo (2013). Continuance intention of blog users: The impact of perceived enjoyment, habit, user involvement and blogging time
## Fitting CFA model
cfa_fit <- cfa(model = cfa_model,
data = case_data_items,
estimator = "MLR")
## Summary results
cfa_fit %>% summary(standardized = TRUE,
fit.measures = TRUE)lavaan 0.6.17 ended normally after 57 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 69
Number of observations 272
Model Test User Model:
Standard Scaled
Test Statistic 546.100 483.499
Degrees of freedom 309 309
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.129
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 5050.919 4114.132
Degrees of freedom 351 351
P-value 0.000 0.000
Scaling correction factor 1.228
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.950 0.954
Tucker-Lewis Index (TLI) 0.943 0.947
Robust Comparative Fit Index (CFI) 0.957
Robust Tucker-Lewis Index (TLI) 0.952
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -10679.787 -10679.787
Scaling correction factor 1.510
for the MLR correction
Loglikelihood unrestricted model (H1) -10406.737 -10406.737
Scaling correction factor 1.199
for the MLR correction
Akaike (AIC) 21497.575 21497.575
Bayesian (BIC) 21746.375 21746.375
Sample-size adjusted Bayesian (SABIC) 21527.595 21527.595
Root Mean Square Error of Approximation:
RMSEA 0.053 0.046
90 Percent confidence interval - lower 0.046 0.038
90 Percent confidence interval - upper 0.060 0.053
P-value H_0: RMSEA <= 0.050 0.236 0.839
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.048
90 Percent confidence interval - lower 0.040
90 Percent confidence interval - upper 0.057
P-value H_0: Robust RMSEA <= 0.050 0.615
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.053 0.053
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
tangible =~
bt1 1.000 0.913 0.699
bt2 1.022 0.081 12.634 0.000 0.932 0.696
bt4 0.983 0.098 10.003 0.000 0.897 0.655
bt5 1.070 0.102 10.542 0.000 0.977 0.691
bt6 1.126 0.120 9.354 0.000 1.028 0.717
bt7 1.138 0.128 8.890 0.000 1.039 0.694
drivers_quality =~
bd1 1.000 1.145 0.762
bd2 0.942 0.071 13.188 0.000 1.079 0.762
bd3 1.039 0.065 15.858 0.000 1.189 0.801
bd4 0.897 0.068 13.217 0.000 1.027 0.767
empathy =~
emp1 1.000 1.145 0.705
emp2 0.923 0.078 11.875 0.000 1.057 0.731
emp3 0.922 0.091 10.179 0.000 1.056 0.627
emp4 0.867 0.081 10.696 0.000 0.993 0.739
emp5 0.877 0.093 9.428 0.000 1.004 0.681
env_perf =~
ep1 1.000 1.260 0.909
ep2 1.065 0.035 30.476 0.000 1.342 0.982
ep3 0.851 0.061 13.892 0.000 1.072 0.774
ep4 0.893 0.050 17.777 0.000 1.125 0.782
customer_sat =~
cs1 1.000 1.212 0.910
cs2 1.022 0.040 25.743 0.000 1.239 0.955
cs3 1.045 0.043 24.120 0.000 1.268 0.889
life_sat =~
ls1 1.000 1.058 0.855
ls2 1.091 0.055 19.807 0.000 1.154 0.926
ls3 1.051 0.096 10.991 0.000 1.112 0.851
ls4 1.102 0.069 15.894 0.000 1.166 0.799
ls5 0.876 0.090 9.678 0.000 0.927 0.597
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
tangible ~~
drivers_qualty 0.721 0.100 7.196 0.000 0.690 0.690
empathy 0.492 0.080 6.144 0.000 0.471 0.471
env_perf 0.596 0.097 6.166 0.000 0.518 0.518
customer_sat 0.625 0.094 6.680 0.000 0.565 0.565
life_sat 0.353 0.072 4.895 0.000 0.366 0.366
drivers_quality ~~
empathy 0.798 0.112 7.142 0.000 0.609 0.609
env_perf 0.641 0.105 6.074 0.000 0.444 0.444
customer_sat 0.789 0.112 7.041 0.000 0.568 0.568
life_sat 0.344 0.088 3.923 0.000 0.284 0.284
empathy ~~
env_perf 0.596 0.099 6.000 0.000 0.413 0.413
customer_sat 0.831 0.114 7.278 0.000 0.599 0.599
life_sat 0.316 0.098 3.239 0.001 0.261 0.261
env_perf ~~
customer_sat 0.733 0.109 6.743 0.000 0.480 0.480
life_sat 0.368 0.094 3.919 0.000 0.276 0.276
customer_sat ~~
life_sat 0.385 0.077 5.022 0.000 0.300 0.300
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.bt1 0.870 0.126 6.918 0.000 0.870 0.511
.bt2 0.927 0.111 8.366 0.000 0.927 0.516
.bt4 1.070 0.127 8.427 0.000 1.070 0.571
.bt5 1.043 0.126 8.311 0.000 1.043 0.522
.bt6 0.999 0.112 8.904 0.000 0.999 0.486
.bt7 1.164 0.139 8.396 0.000 1.164 0.519
.bd1 0.946 0.114 8.271 0.000 0.946 0.419
.bd2 0.842 0.114 7.374 0.000 0.842 0.420
.bd3 0.791 0.103 7.659 0.000 0.791 0.359
.bd4 0.737 0.095 7.751 0.000 0.737 0.411
.emp1 1.330 0.146 9.121 0.000 1.330 0.504
.emp2 0.971 0.115 8.432 0.000 0.971 0.465
.emp3 1.722 0.180 9.574 0.000 1.722 0.607
.emp4 0.818 0.099 8.296 0.000 0.818 0.453
.emp5 1.163 0.142 8.186 0.000 1.163 0.536
.ep1 0.334 0.059 5.632 0.000 0.334 0.174
.ep2 0.068 0.043 1.566 0.117 0.068 0.036
.ep3 0.772 0.142 5.421 0.000 0.772 0.402
.ep4 0.803 0.152 5.299 0.000 0.803 0.388
.cs1 0.307 0.078 3.950 0.000 0.307 0.173
.cs2 0.148 0.041 3.631 0.000 0.148 0.088
.cs3 0.425 0.101 4.202 0.000 0.425 0.209
.ls1 0.411 0.078 5.276 0.000 0.411 0.269
.ls2 0.222 0.049 4.563 0.000 0.222 0.143
.ls3 0.472 0.125 3.786 0.000 0.472 0.276
.ls4 0.770 0.099 7.807 0.000 0.770 0.362
.ls5 1.548 0.180 8.579 0.000 1.548 0.643
tangible 0.833 0.134 6.237 0.000 1.000 1.000
drivers_qualty 1.311 0.175 7.494 0.000 1.000 1.000
empathy 1.311 0.188 6.980 0.000 1.000 1.000
env_perf 1.588 0.138 11.538 0.000 1.000 1.000
customer_sat 1.470 0.159 9.267 0.000 1.000 1.000
life_sat 1.120 0.149 7.519 0.000 1.000 1.000## plotting cfa model
lavaanPlot(model = cfa_fit, coefs = TRUE, covs = TRUE)5.4 Relibility and validity tests
## Reliability and validity
reliability(cfa_fit) %>% round(2) tangible drivers_quality empathy env_perf customer_sat life_sat
alpha 0.85 0.86 0.82 0.92 0.94 0.90
omega 0.85 0.86 0.82 0.92 0.94 0.90
omega2 0.85 0.86 0.82 0.92 0.94 0.90
omega3 0.85 0.86 0.82 0.92 0.94 0.89
avevar 0.48 0.60 0.48 0.75 0.84 0.636 Structural equation modelling
6.1 Specifying structural model
## Specifying structural model
ebus_model <- "tangible =~ bt1 + bt2 + bt4 + bt5 + bt6 + bt7
drivers_quality =~ bd1 + bd2 + bd3 + bd4
empathy =~ emp1 + emp2 + emp3 + emp4 + emp5
env_perf =~ ep1 + ep2 + ep3 + ep4
customer_sat =~ cs1 + cs2 + cs3
life_sat =~ ls1 + ls2 + ls3 + ls4 + ls5
# structural model
customer_sat ~ tangible + drivers_quality + empathy + env_perf
life_sat ~ customer_sat"6.2 Fitting the structural model
In the previous section, we only defined our model. Here we fit our specified model using the Lavaan package (Rosseel, 2012) for SEM analysis. The output composes mainly of the fit indices and parameter estimates. The parameter estimates are shown under the Latent variables, Regressions, and Variances section.
Latent variables
The Estimates column is the non-standardized factor loadings coefficient accompanied by its Std Error (standard error). The most important to look at in the Estimates column is the presence of negative values since variance should not be negative. Thus, a negative value is a sign of a problem (see Heywood case for more information).
For interpretation, for example, the non-standardized factor loading of bt2, that is, 1.021, indicates an increase of one unit in the latent variable, tangible factor, is associated with an increase of 1.021 in bt2 item.
The last column std.lv and std.all present a standardized solution. The std.lv presents a factor loading of a solution that only latent variables (factors) were standardized. Whereas the std.all presents the factor loading of a solution where latent variables and indicators were standardized. We want a value to be positive and greater than 0.40 as a required minimum to evaluate the relevance of the indicators and the factor. However, this set required minimum is not unanimously accepted, but the larger, the better.
… in progress.
## Fitting structural model
ebus_fit <- sem(model = ebus_model,
data = case_data,
estimator = "MLR")
## Summary results
ebus_fit %>% summary(standardized = TRUE,
fit.measures = TRUE,
rsq = TRUE)lavaan 0.6.17 ended normally after 46 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 65
Number of observations 272
Model Test User Model:
Standard Scaled
Test Statistic 561.166 496.106
Degrees of freedom 313 313
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.131
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 5050.919 4114.132
Degrees of freedom 351 351
P-value 0.000 0.000
Scaling correction factor 1.228
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.947 0.951
Tucker-Lewis Index (TLI) 0.941 0.945
Robust Comparative Fit Index (CFI) 0.955
Robust Tucker-Lewis Index (TLI) 0.950
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -10687.320 -10687.320
Scaling correction factor 1.525
for the MLR correction
Loglikelihood unrestricted model (H1) -10406.737 -10406.737
Scaling correction factor 1.199
for the MLR correction
Akaike (AIC) 21504.640 21504.640
Bayesian (BIC) 21739.017 21739.017
Sample-size adjusted Bayesian (SABIC) 21532.920 21532.920
Root Mean Square Error of Approximation:
RMSEA 0.054 0.046
90 Percent confidence interval - lower 0.047 0.039
90 Percent confidence interval - upper 0.061 0.053
P-value H_0: RMSEA <= 0.050 0.178 0.793
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.049
90 Percent confidence interval - lower 0.041
90 Percent confidence interval - upper 0.057
P-value H_0: Robust RMSEA <= 0.050 0.545
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.069 0.069
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
tangible =~
bt1 1.000 0.912 0.699
bt2 1.021 0.081 12.564 0.000 0.931 0.695
bt4 0.986 0.098 10.030 0.000 0.900 0.657
bt5 1.079 0.103 10.493 0.000 0.984 0.697
bt6 1.124 0.121 9.299 0.000 1.026 0.716
bt7 1.132 0.128 8.862 0.000 1.033 0.689
drivers_quality =~
bd1 1.000 1.145 0.762
bd2 0.941 0.071 13.196 0.000 1.078 0.761
bd3 1.038 0.065 15.982 0.000 1.189 0.801
bd4 0.897 0.068 13.220 0.000 1.027 0.767
empathy =~
emp1 1.000 1.144 0.704
emp2 0.922 0.078 11.823 0.000 1.055 0.730
emp3 0.923 0.091 10.159 0.000 1.056 0.627
emp4 0.870 0.080 10.805 0.000 0.995 0.741
emp5 0.878 0.093 9.403 0.000 1.004 0.681
env_perf =~
ep1 1.000 1.260 0.909
ep2 1.065 0.035 30.266 0.000 1.342 0.982
ep3 0.851 0.061 13.865 0.000 1.072 0.773
ep4 0.893 0.050 17.778 0.000 1.125 0.782
customer_sat =~
cs1 1.000 1.212 0.909
cs2 1.023 0.040 25.878 0.000 1.240 0.955
cs3 1.045 0.043 24.227 0.000 1.266 0.888
life_sat =~
ls1 1.000 1.058 0.855
ls2 1.092 0.056 19.402 0.000 1.156 0.927
ls3 1.050 0.096 10.969 0.000 1.111 0.850
ls4 1.102 0.070 15.713 0.000 1.166 0.799
ls5 0.874 0.091 9.609 0.000 0.925 0.596
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
customer_sat ~
tangible 0.313 0.127 2.457 0.014 0.235 0.235
drivers_qualty 0.132 0.122 1.087 0.277 0.125 0.125
empathy 0.367 0.110 3.347 0.001 0.346 0.346
env_perf 0.155 0.060 2.604 0.009 0.162 0.162
life_sat ~
customer_sat 0.270 0.055 4.908 0.000 0.309 0.309
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
tangible ~~
drivers_qualty 0.721 0.100 7.180 0.000 0.690 0.690
empathy 0.492 0.080 6.144 0.000 0.471 0.471
env_perf 0.595 0.097 6.132 0.000 0.517 0.517
drivers_quality ~~
empathy 0.798 0.112 7.139 0.000 0.609 0.609
env_perf 0.641 0.105 6.074 0.000 0.444 0.444
empathy ~~
env_perf 0.595 0.099 5.997 0.000 0.412 0.412
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.bt1 0.870 0.125 6.942 0.000 0.870 0.511
.bt2 0.929 0.111 8.372 0.000 0.929 0.517
.bt4 1.065 0.125 8.490 0.000 1.065 0.568
.bt5 1.029 0.124 8.300 0.000 1.029 0.515
.bt6 1.002 0.114 8.809 0.000 1.002 0.488
.bt7 1.177 0.139 8.464 0.000 1.177 0.525
.bd1 0.945 0.114 8.276 0.000 0.945 0.419
.bd2 0.843 0.114 7.376 0.000 0.843 0.420
.bd3 0.792 0.103 7.675 0.000 0.792 0.359
.bd4 0.738 0.095 7.748 0.000 0.738 0.411
.emp1 1.332 0.146 9.134 0.000 1.332 0.504
.emp2 0.975 0.116 8.420 0.000 0.975 0.467
.emp3 1.721 0.180 9.566 0.000 1.721 0.607
.emp4 0.814 0.098 8.350 0.000 0.814 0.451
.emp5 1.163 0.142 8.181 0.000 1.163 0.536
.ep1 0.335 0.059 5.646 0.000 0.335 0.174
.ep2 0.066 0.043 1.518 0.129 0.066 0.035
.ep3 0.773 0.143 5.417 0.000 0.773 0.402
.ep4 0.804 0.152 5.307 0.000 0.804 0.389
.cs1 0.307 0.077 3.971 0.000 0.307 0.173
.cs2 0.148 0.040 3.713 0.000 0.148 0.088
.cs3 0.428 0.100 4.262 0.000 0.428 0.211
.ls1 0.412 0.080 5.147 0.000 0.412 0.269
.ls2 0.219 0.049 4.503 0.000 0.219 0.141
.ls3 0.475 0.125 3.815 0.000 0.475 0.278
.ls4 0.769 0.098 7.818 0.000 0.769 0.361
.ls5 1.551 0.180 8.608 0.000 1.551 0.644
tangible 0.832 0.134 6.206 0.000 1.000 1.000
drivers_qualty 1.312 0.174 7.521 0.000 1.000 1.000
empathy 1.309 0.189 6.942 0.000 1.000 1.000
env_perf 1.587 0.138 11.533 0.000 1.000 1.000
.customer_sat 0.748 0.090 8.314 0.000 0.509 0.509
.life_sat 1.012 0.143 7.089 0.000 0.904 0.904
R-Square:
Estimate
bt1 0.489
bt2 0.483
bt4 0.432
bt5 0.485
bt6 0.512
bt7 0.475
bd1 0.581
bd2 0.580
bd3 0.641
bd4 0.589
emp1 0.496
emp2 0.533
emp3 0.393
emp4 0.549
emp5 0.464
ep1 0.826
ep2 0.965
ep3 0.598
ep4 0.611
cs1 0.827
cs2 0.912
cs3 0.789
ls1 0.731
ls2 0.859
ls3 0.722
ls4 0.639
ls5 0.356
customer_sat 0.491
life_sat 0.096## plotting sem model using LavaanPlot package
lavaanPlot(model = ebus_fit, coefs = TRUE, covs = TRUE, stars = TRUE, digits = 2)## plotting sem model using semPlot package
semPlot::semPaths(object = ebus_fit,
what = "std",
layout = "tree",
rotation = 2,
sizeMan = 3,
sizeLat = 5,
intercepts = FALSE,
residuals = FALSE,
exoCov = F)
6.3 Estimating indirect effects
6.3.1 Specifying model with indirect effects
### SEM model with mediation
ebus_model_ie <- "tangible =~ bt1 + bt2 + bt4 + bt5 + bt6 + bt7
drivers_quality =~ bd1 + bd2 + bd3 + bd4
empathy =~ emp1 + emp2 + emp3 + emp4 + emp5
env_perf =~ ep1 + ep2 + ep3 + ep4
customer_sat =~ cs1 + cs2 + cs3
life_sat =~ ls1 + ls2 + ls3 + ls4 + ls5
# structural model
customer_sat ~ a*tangible + b*drivers_quality + c*empathy + d*env_perf
life_sat ~ e*customer_sat
# indirect effects
ie_tangible := a*e
ie_drivers_qual := b*e
ie_empathy := c*e
ie_en_perf := d*e"6.3.2 Fitting the model with indirect effects
### Fitting structural model with mediation
ebus_fit_ie <- sem(model = ebus_model_ie,
data = case_data,
estimator = "MLR")
### Summary results
ebus_fit_ie %>% summary(standardized = TRUE,
fit.measures = TRUE,
rsq = TRUE)lavaan 0.6.17 ended normally after 46 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 65
Number of observations 272
Model Test User Model:
Standard Scaled
Test Statistic 561.166 496.106
Degrees of freedom 313 313
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.131
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 5050.919 4114.132
Degrees of freedom 351 351
P-value 0.000 0.000
Scaling correction factor 1.228
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.947 0.951
Tucker-Lewis Index (TLI) 0.941 0.945
Robust Comparative Fit Index (CFI) 0.955
Robust Tucker-Lewis Index (TLI) 0.950
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -10687.320 -10687.320
Scaling correction factor 1.525
for the MLR correction
Loglikelihood unrestricted model (H1) -10406.737 -10406.737
Scaling correction factor 1.199
for the MLR correction
Akaike (AIC) 21504.640 21504.640
Bayesian (BIC) 21739.017 21739.017
Sample-size adjusted Bayesian (SABIC) 21532.920 21532.920
Root Mean Square Error of Approximation:
RMSEA 0.054 0.046
90 Percent confidence interval - lower 0.047 0.039
90 Percent confidence interval - upper 0.061 0.053
P-value H_0: RMSEA <= 0.050 0.178 0.793
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.049
90 Percent confidence interval - lower 0.041
90 Percent confidence interval - upper 0.057
P-value H_0: Robust RMSEA <= 0.050 0.545
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.069 0.069
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
tangible =~
bt1 1.000 0.912 0.699
bt2 1.021 0.081 12.564 0.000 0.931 0.695
bt4 0.986 0.098 10.030 0.000 0.900 0.657
bt5 1.079 0.103 10.493 0.000 0.984 0.697
bt6 1.124 0.121 9.299 0.000 1.026 0.716
bt7 1.132 0.128 8.862 0.000 1.033 0.689
drivers_quality =~
bd1 1.000 1.145 0.762
bd2 0.941 0.071 13.196 0.000 1.078 0.761
bd3 1.038 0.065 15.982 0.000 1.189 0.801
bd4 0.897 0.068 13.220 0.000 1.027 0.767
empathy =~
emp1 1.000 1.144 0.704
emp2 0.922 0.078 11.823 0.000 1.055 0.730
emp3 0.923 0.091 10.159 0.000 1.056 0.627
emp4 0.870 0.080 10.805 0.000 0.995 0.741
emp5 0.878 0.093 9.403 0.000 1.004 0.681
env_perf =~
ep1 1.000 1.260 0.909
ep2 1.065 0.035 30.266 0.000 1.342 0.982
ep3 0.851 0.061 13.865 0.000 1.072 0.773
ep4 0.893 0.050 17.778 0.000 1.125 0.782
customer_sat =~
cs1 1.000 1.212 0.909
cs2 1.023 0.040 25.878 0.000 1.240 0.955
cs3 1.045 0.043 24.227 0.000 1.266 0.888
life_sat =~
ls1 1.000 1.058 0.855
ls2 1.092 0.056 19.402 0.000 1.156 0.927
ls3 1.050 0.096 10.969 0.000 1.111 0.850
ls4 1.102 0.070 15.713 0.000 1.166 0.799
ls5 0.874 0.091 9.609 0.000 0.925 0.596
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
customer_sat ~
tangible (a) 0.313 0.127 2.457 0.014 0.235 0.235
drvrs_qlty (b) 0.132 0.122 1.087 0.277 0.125 0.125
empathy (c) 0.367 0.110 3.347 0.001 0.346 0.346
env_perf (d) 0.155 0.060 2.604 0.009 0.162 0.162
life_sat ~
customr_st (e) 0.270 0.055 4.908 0.000 0.309 0.309
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
tangible ~~
drivers_qualty 0.721 0.100 7.180 0.000 0.690 0.690
empathy 0.492 0.080 6.144 0.000 0.471 0.471
env_perf 0.595 0.097 6.132 0.000 0.517 0.517
drivers_quality ~~
empathy 0.798 0.112 7.139 0.000 0.609 0.609
env_perf 0.641 0.105 6.074 0.000 0.444 0.444
empathy ~~
env_perf 0.595 0.099 5.997 0.000 0.412 0.412
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.bt1 0.870 0.125 6.942 0.000 0.870 0.511
.bt2 0.929 0.111 8.372 0.000 0.929 0.517
.bt4 1.065 0.125 8.490 0.000 1.065 0.568
.bt5 1.029 0.124 8.300 0.000 1.029 0.515
.bt6 1.002 0.114 8.809 0.000 1.002 0.488
.bt7 1.177 0.139 8.464 0.000 1.177 0.525
.bd1 0.945 0.114 8.276 0.000 0.945 0.419
.bd2 0.843 0.114 7.376 0.000 0.843 0.420
.bd3 0.792 0.103 7.675 0.000 0.792 0.359
.bd4 0.738 0.095 7.748 0.000 0.738 0.411
.emp1 1.332 0.146 9.134 0.000 1.332 0.504
.emp2 0.975 0.116 8.420 0.000 0.975 0.467
.emp3 1.721 0.180 9.566 0.000 1.721 0.607
.emp4 0.814 0.098 8.350 0.000 0.814 0.451
.emp5 1.163 0.142 8.181 0.000 1.163 0.536
.ep1 0.335 0.059 5.646 0.000 0.335 0.174
.ep2 0.066 0.043 1.518 0.129 0.066 0.035
.ep3 0.773 0.143 5.417 0.000 0.773 0.402
.ep4 0.804 0.152 5.307 0.000 0.804 0.389
.cs1 0.307 0.077 3.971 0.000 0.307 0.173
.cs2 0.148 0.040 3.713 0.000 0.148 0.088
.cs3 0.428 0.100 4.262 0.000 0.428 0.211
.ls1 0.412 0.080 5.147 0.000 0.412 0.269
.ls2 0.219 0.049 4.503 0.000 0.219 0.141
.ls3 0.475 0.125 3.815 0.000 0.475 0.278
.ls4 0.769 0.098 7.818 0.000 0.769 0.361
.ls5 1.551 0.180 8.608 0.000 1.551 0.644
tangible 0.832 0.134 6.206 0.000 1.000 1.000
drivers_qualty 1.312 0.174 7.521 0.000 1.000 1.000
empathy 1.309 0.189 6.942 0.000 1.000 1.000
env_perf 1.587 0.138 11.533 0.000 1.000 1.000
.customer_sat 0.748 0.090 8.314 0.000 0.509 0.509
.life_sat 1.012 0.143 7.089 0.000 0.904 0.904
R-Square:
Estimate
bt1 0.489
bt2 0.483
bt4 0.432
bt5 0.485
bt6 0.512
bt7 0.475
bd1 0.581
bd2 0.580
bd3 0.641
bd4 0.589
emp1 0.496
emp2 0.533
emp3 0.393
emp4 0.549
emp5 0.464
ep1 0.826
ep2 0.965
ep3 0.598
ep4 0.611
cs1 0.827
cs2 0.912
cs3 0.789
ls1 0.731
ls2 0.859
ls3 0.722
ls4 0.639
ls5 0.356
customer_sat 0.491
life_sat 0.096
Defined Parameters:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
ie_tangible 0.084 0.039 2.144 0.032 0.073 0.073
ie_drivers_qul 0.036 0.034 1.058 0.290 0.039 0.039
ie_empathy 0.099 0.032 3.051 0.002 0.107 0.107
ie_en_perf 0.042 0.018 2.292 0.022 0.050 0.050References
Hair, J. F. (2019). Multivariate data analysis (Eighth edition). Cengage.
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Rosseel, Y. (2012). Lavaan: An r package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. https://doi.org/10.18637/jss.v048.i02
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