1 Preparing Data

1.1 Load Libraries

## Necessary Libraries loaded.

1.2 Load Data

2 Overview of Raw Data

2.1 Histograms of all IWB for all Organisations

library(ggplot2)
ggplot(DFvar, aes(x = Org, y = IWB, fill = Sector)) + ylim(1, 7) +
    geom_violin(trim = FALSE) + ggtitle("Individual Innovative Work Behaviour (IWB) by Organisation") + 
    scale_fill_manual(values = c("lightblue", "pink")) +
    xlab("") + ylab("") + 
    geom_boxplot(width = 0.2, fill = "lightgrey") +
    geom_jitter(shape = 16, position = position_jitter(0.2)) +
    stat_summary(fun.y = mean, geom = "point", shape = 15, size = 5, color = "red") +
    theme(
    plot.title   = element_text(color = "black", size = 35, face = "bold", hjust = 0.5, margin = margin(30,0,30,0)),
    axis.title.x = element_blank(),
    axis.title.y = element_blank(),
    axis.text.x  = element_text(color = "darkblue", size = 30, margin = margin(30,0,0,0)),
    axis.text.y  = element_text(color = "darkblue", size = 30, margin = margin(0,30,0,0)),
    panel.background = element_rect(fill = "white", colour = "grey", size = 0.5, linetype = "solid"),
    panel.grid.major = element_line(size = 0.5, linetype = 'solid', colour = "lightblue"),
    plot.margin = margin(2, 2, 2, 2, "cm"),
    legend.position = "top",
    legend.title =element_blank(), 
    legend.text = element_text(size = 25))

2.2 Descriptive Statistics of Individual Innovative Work Behaviour

library(RcmdrMisc)
## Loading required package: sandwich
## 
## Attaching package: 'RcmdrMisc'
## The following object is masked from 'package:psych':
## 
##     reliability
numSummary(DFvar$IWB, 
           groups = DFvar$Org, 
           statistics = c("mean", "sd", "skewness", "kurtosis", "quantiles"), 
                          quantiles = c(0.0, 0.25, 0.5, 0.75, 1.00))
##           mean     sd skewness kurtosis    0%   25%   50%   75%  100% data:n
## Private1 5.091 1.1254  -0.5235  -0.2731 2.111 4.472 5.111 6.000 7.000     90
## Private2 5.006 0.9753  -0.5676  -0.2870 1.778 4.222 5.167 5.889 6.444    128
## Private3 4.771 1.0767  -0.1944   0.5547 2.333 4.167 4.667 5.500 7.000     29
## Public1  4.731 1.1847  -0.6806  -0.4105 1.889 4.028 5.000 5.750 6.222     26
## Public2  4.453 1.0426  -0.2478   0.1286 1.778 3.889 4.556 5.111 7.000    155
## Public3  4.632 1.1316  -0.7535   1.0568 1.000 3.889 4.833 5.333 6.889     54

3 Studying Data

3.1 Running EFA for Pax Data (DFs)

# Conduct a single-factor EFA
# install.packages("lavaan")
EFA_Pax <- fa(DFs, )

# View the results
# print(EFA_Pax)

# View the factor loadings
# EFA_Pax$loadings

# Plot factor scores
plot(density(EFA_Pax$scores, na.rm = TRUE), main = "Factor Scores", size = 2, col = "blue")
grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted")

The plot of factor scores indicates the distribution of all survey ratings assuming they all belong to a common factor.

3.2 Evaluating data with factor analysis

Evaluating Model Structure with Indices: RMSR < 0.08, SRMR < 0.08, TLI > 0.90, CFI > 0.90, i.e. model fit is acceptable with seven factors. Scree Plot hints to eight factors instead of seven proposed by our variables. Therefore, we may need to split one less consistently structured factor into two.

# Calculate the correlation matrix first
DFS_EFA_corr <- cor(DFs, use = "pairwise.complete.obs")

# Then use that correlation matrix to calculate eigenvalues
eigenvals <- eigen(DFS_EFA_corr)

# Look at the eigenvalues returned
eigenvals$values
##  [1] 28.4010  7.9071  3.9078  2.7397  1.4387  1.3085  1.1872  1.0916  1.0194
## [10]  1.0049  0.9391  0.9017  0.8229  0.7906  0.7537  0.7240  0.7033  0.6908
## [19]  0.6654  0.6556  0.6103  0.5983  0.5674  0.5527  0.5388  0.5356  0.5236
## [28]  0.5214  0.5022  0.4846  0.4771  0.4585  0.4528  0.4485  0.4445  0.4196
## [37]  0.4082  0.4014  0.3893  0.3809  0.3692  0.3630  0.3566  0.3505  0.3305
## [46]  0.3293  0.3212  0.3121  0.3103  0.2980  0.2890  0.2825  0.2779  0.2687
## [55]  0.2636  0.2573  0.2459  0.2381  0.2308  0.2226  0.2171  0.2110  0.2066
## [64]  0.1938  0.1889  0.1836  0.1830  0.1738  0.1686  0.1618  0.1549  0.1511
## [73]  0.1445  0.1378  0.1217  0.1158
# Prepare PDF
# pdf("ScreePlot.pdf", width = 25, height = 16)
# par(mai=c(2, 2, 2, 2))
# par(oma=c(2, 2, 2, 2))

# Plot Scree Plot
plot(eigenvals$values, log = "y", 
     main = "Scree Plot for Eigenvalues",
     xlab = "Factor", xlim = c(0, 20), 
     ylab = "Eigenvalue of Factor", ylim = c(0.5, 25),
     type  = "o" # connect dots with line.
      )
abline(h = 1, col = "red", lty = 5)
text(x = 12,
     y = 1,
     labels = "Eigenvalue of 1", cex = 1.0,
     pos = 3,
     col  = "red")
grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted")

# dev.off()

# Use the correlation matrix to create the scree plot
# scree(DFS_EFA_corr, factors = TRUE)

# Run the EFA with 7 factors (as indicated by scree plot)
library(psych)
EFA_model <- fa(DFs, nfactors = 7)
## Loading required namespace: GPArotation
print(EFA_model)
## Factor Analysis using method =  minres
## Call: fa(r = DFs, nfactors = 7)
## Standardized loadings (pattern matrix) based upon correlation matrix
##       MR1   MR2   MR3   MR4   MR5   MR6   MR7   h2   u2 com
## Q001 0.64  0.24  0.10  0.08 -0.19 -0.17  0.07 0.55 0.45 1.7
## Q002 0.66  0.25  0.07  0.00 -0.15 -0.12  0.00 0.54 0.46 1.5
## Q003 0.56  0.31  0.13  0.13 -0.17 -0.11  0.12 0.50 0.50 2.3
## Q004 0.62  0.19  0.14  0.04  0.01 -0.08 -0.08 0.46 0.54 1.4
## Q005 0.59  0.28  0.11  0.03  0.12 -0.19 -0.08 0.49 0.51 1.9
## Q006 0.67  0.29  0.07  0.05  0.02 -0.08 -0.15 0.57 0.43 1.6
## Q007 0.55  0.34  0.19  0.08  0.28 -0.14  0.05 0.56 0.44 2.8
## Q008 0.44  0.23  0.17  0.19 -0.04  0.02  0.20 0.35 0.65 2.8
## Q009 0.51  0.21  0.15  0.09 -0.02 -0.14 -0.01 0.36 0.64 1.8
## Q010 0.46  0.34  0.25  0.26  0.03 -0.13  0.00 0.47 0.53 3.4
## Q011 0.66  0.17  0.24  0.06 -0.03 -0.06 -0.14 0.55 0.45 1.5
## Q012 0.61  0.12  0.22  0.06 -0.06 -0.09  0.00 0.45 0.55 1.4
## Q013 0.64 -0.47 -0.07  0.10  0.08 -0.12  0.01 0.66 0.34 2.1
## Q014 0.58 -0.47  0.03  0.18 -0.08  0.05 -0.01 0.60 0.40 2.2
## Q015 0.62 -0.53  0.02  0.17  0.07 -0.07 -0.07 0.70 0.30 2.2
## Q016 0.61 -0.45  0.02  0.15  0.09  0.06 -0.03 0.61 0.39 2.1
## Q017 0.65 -0.45 -0.11 -0.01  0.03 -0.17 -0.03 0.67 0.33 2.0
## Q018 0.67 -0.50 -0.04  0.08 -0.03  0.04  0.01 0.71 0.29 1.9
## Q019 0.63 -0.43  0.02  0.22 -0.09  0.03 -0.06 0.64 0.36 2.1
## Q020 0.68 -0.42  0.00  0.19 -0.06 -0.02 -0.03 0.68 0.32 1.9
## Q021 0.66 -0.48 -0.03  0.09 -0.03 -0.06 -0.10 0.69 0.31 1.9
## Q022 0.65 -0.49  0.01  0.12 -0.06  0.01 -0.06 0.68 0.32 2.0
## Q023 0.66 -0.52 -0.02  0.06  0.03 -0.08 -0.07 0.71 0.29 2.0
## Q024 0.70 -0.48 -0.02  0.13 -0.04 -0.04  0.01 0.75 0.25 1.9
## Q025 0.56 -0.51 -0.05  0.09 -0.02  0.04  0.03 0.59 0.41 2.1
## Q026 0.59 -0.55 -0.02  0.07 -0.01  0.12 -0.04 0.68 0.32 2.1
## Q027 0.64 -0.49 -0.03  0.08 -0.02  0.00 -0.06 0.66 0.34 1.9
## Q028 0.64 -0.52 -0.10  0.05  0.06  0.00 -0.01 0.69 0.31 2.0
## Q029 0.51 -0.58 -0.10 -0.03  0.07 -0.10  0.03 0.62 0.38 2.2
## Q030 0.38 -0.18 -0.08  0.14 -0.02  0.06  0.11 0.22 0.78 2.1
## Q031 0.63 -0.21 -0.20  0.10 -0.01 -0.03  0.05 0.50 0.50 1.5
## Q032 0.66 -0.54 -0.03  0.01  0.11 -0.11 -0.03 0.76 0.24 2.1
## Q033 0.65 -0.06  0.33 -0.21 -0.03  0.25  0.04 0.64 0.36 2.1
## Q034 0.62 -0.11  0.35 -0.24 -0.07  0.18  0.05 0.62 0.38 2.3
## Q035 0.61 -0.20  0.34 -0.26 -0.03  0.23  0.04 0.65 0.35 2.6
## Q037 0.39 -0.17  0.23 -0.39  0.04 -0.10  0.02 0.40 0.60 3.2
## Q038 0.57 -0.14  0.33 -0.31 -0.01  0.01  0.07 0.56 0.44 2.4
## Q044 0.55 -0.15  0.32 -0.29 -0.08  0.07  0.15 0.55 0.45 2.7
## Q045 0.57 -0.07  0.37 -0.17  0.04  0.20 -0.09 0.55 0.45 2.3
## Q049 0.59 -0.12  0.23 -0.41  0.04  0.03  0.04 0.59 0.41 2.3
## Q050 0.43 -0.09  0.18 -0.48  0.15  0.00  0.02 0.48 0.52 2.6
## Q051 0.54 -0.04  0.18 -0.43  0.10 -0.12 -0.11 0.55 0.45 2.5
## Q054 0.53 -0.03  0.22 -0.31  0.02 -0.18 -0.03 0.46 0.54 2.3
## Q055 0.62  0.30 -0.03  0.00 -0.20  0.07 -0.12 0.54 0.46 1.8
## Q057 0.60  0.30  0.00 -0.04 -0.22  0.03 -0.15 0.52 0.48 2.0
## Q058 0.60  0.26 -0.03 -0.08 -0.06  0.02 -0.21 0.48 0.52 1.7
## Q059 0.65  0.39  0.05  0.07 -0.15  0.09 -0.15 0.63 0.37 2.0
## Q060 0.62  0.39 -0.09 -0.22 -0.04 -0.04 -0.14 0.61 0.39 2.1
## Q061 0.59  0.42 -0.09 -0.09 -0.06 -0.12 -0.07 0.56 0.44 2.1
## Q062 0.69  0.28 -0.18  0.08  0.08  0.18 -0.14 0.66 0.34 1.8
## Q063 0.51  0.17 -0.37 -0.12  0.12  0.07 -0.14 0.48 0.52 2.6
## Q064 0.65  0.25 -0.35  0.06  0.19  0.15 -0.12 0.69 0.31 2.4
## Q066 0.61  0.28 -0.26  0.03  0.15  0.19 -0.10 0.58 0.42 2.3
## Q067 0.68  0.27 -0.32 -0.08  0.11  0.16 -0.03 0.69 0.31 2.0
## Q068 0.67  0.29 -0.43 -0.02  0.08  0.06 -0.08 0.73 0.27 2.2
## Q069 0.45  0.11 -0.27 -0.02  0.05  0.13 -0.09 0.31 0.69 2.2
## Q070 0.58  0.17  0.18  0.11 -0.13  0.11  0.08 0.45 0.55 1.7
## Q071 0.67  0.08  0.18  0.04 -0.16  0.09 -0.02 0.52 0.48 1.4
## Q072 0.50  0.29  0.20  0.25 -0.18  0.07  0.09 0.48 0.52 3.1
## Q075 0.65  0.37  0.07  0.00 -0.12 -0.17 -0.07 0.61 0.39 1.9
## Q076 0.66  0.37 -0.02 -0.06  0.03 -0.09 -0.10 0.60 0.40 1.7
## Q077 0.55  0.22  0.15  0.23 -0.06 -0.07  0.03 0.44 0.56 2.0
## Q081 0.44  0.21  0.16  0.24 -0.18  0.17  0.07 0.39 0.61 3.2
## Q083 0.50  0.28  0.31  0.26  0.35  0.05  0.06 0.62 0.38 3.9
## Q084 0.53  0.34  0.23  0.22  0.32  0.01  0.12 0.62 0.38 3.5
## Q085 0.57  0.30  0.32  0.20  0.31  0.01  0.08 0.66 0.34 3.2
## Q086 0.47  0.23  0.31  0.29  0.14  0.10  0.09 0.50 0.50 3.6
## Q087 0.74  0.18 -0.34 -0.05 -0.05  0.02  0.11 0.70 0.30 1.6
## Q088 0.71  0.11 -0.33 -0.10 -0.04 -0.01  0.15 0.67 0.33 1.6
## Q089 0.72  0.17 -0.30  0.02 -0.09  0.05  0.07 0.65 0.35 1.5
## Q090 0.59  0.03 -0.21 -0.07  0.08  0.11  0.17 0.44 0.56 1.6
## Q091 0.72  0.15 -0.37 -0.16  0.00 -0.05  0.15 0.74 0.26 1.8
## Q092 0.72  0.13 -0.27 -0.05 -0.01 -0.05  0.09 0.62 0.38 1.4
## Q093 0.71  0.17 -0.37 -0.15  0.01 -0.09  0.22 0.74 0.26 2.0
## Q094 0.71  0.11 -0.33 -0.21 -0.03 -0.10  0.15 0.70 0.30 1.8
## Q095 0.67  0.08 -0.30 -0.05 -0.05  0.03  0.27 0.62 0.38 1.8
## 
##                         MR1  MR2  MR3  MR4  MR5  MR6  MR7
## SS loadings           28.00 7.53 3.50 2.28 1.01 0.86 0.74
## Proportion Var         0.37 0.10 0.05 0.03 0.01 0.01 0.01
## Cumulative Var         0.37 0.47 0.51 0.54 0.56 0.57 0.58
## Proportion Explained   0.64 0.17 0.08 0.05 0.02 0.02 0.02
## Cumulative Proportion  0.64 0.81 0.89 0.94 0.96 0.98 1.00
## 
## Mean item complexity =  2.1
## Test of the hypothesis that 7 factors are sufficient.
## 
## The degrees of freedom for the null model are  2850  and the objective function was  61.91 with Chi Square of  28161
## The degrees of freedom for the model are 2339  and the objective function was  8.77 
## 
## The root mean square of the residuals (RMSR) is  0.02 
## The df corrected root mean square of the residuals is  0.03 
## 
## The harmonic number of observations is  482 with the empirical chi square  1538  with prob <  1 
## The total number of observations was  482  with Likelihood Chi Square =  3947  with prob <  6.2e-86 
## 
## Tucker Lewis Index of factoring reliability =  0.922
## RMSEA index =  0.038  and the 90 % confidence intervals are  0.036 0.04
## BIC =  -10503
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    MR1  MR2  MR3  MR4  MR5  MR6
## Correlation of (regression) scores with factors   0.99 0.98 0.95 0.92 0.84 0.82
## Multiple R square of scores with factors          0.99 0.95 0.90 0.84 0.71 0.68
## Minimum correlation of possible factor scores     0.97 0.91 0.80 0.68 0.43 0.36
##                                                    MR7
## Correlation of (regression) scores with factors   0.81
## Multiple R square of scores with factors          0.65
## Minimum correlation of possible factor scores     0.30

3.3 Error Plot of Data for all Items (Questions)

Error plots give an idea of potential misfitting items (questions/statements) per factor (variable.)

# Create mean scores and confidence intervals
library(psych)
error.dots(Creativ, eyes = TRUE, sort = FALSE, main = "Creative Role Identity - CI Around Mean", xlim = c(1, 7), lcolor = "azure2")

error.dots(InnoReady, eyes = TRUE, sort = FALSE, main = "Individual Innovation Readyness - CI Around Mean", xlim = c(1, 7), lcolor = "pink")

error.dots(IRJP, eyes = TRUE, sort = FALSE, main = "In-Role Job Performance - CI Around Mean", xlim = c(1, 7), lcolor = "darkgoldenrod1")

error.dots(PsyCap, eyes = TRUE, sort = FALSE, main = "PsyCap - CI Around Mean", xlim = c(1, 7), lcolor = "lightblue")

error.dots(Support, eyes = TRUE, sort = FALSE, main = "Support for Innovation - CI Around Mean", xlim = c(1, 7), lcolor = "orange")

error.dots(Lead, eyes = TRUE, sort = FALSE, main = "Transformational Leadership - CI Around Mean", xlim = c(1, 7), lcolor = "lightgreen")

error.dots(IWB, eyes = TRUE, sort = FALSE, main = "Individual Innovative Work Behaviour - CI Around Mean", xlim = c(1, 7), lcolor = "cyan")

Support for innovation seems to have a less consistent item structure.

3.4 Cronbach’s Alpha for all Latent Variables

Cronbach’s Alpha is an indicator for scale reliability. Raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances. A level of 0.7 and above is acceptable, whereas 0.8 should be achieved in research. When making important data driven decisions, 0.9 marks a minimum. If Guttman’s Lambda 6 (G6) is higher than Alpha, factor shows unequal item loadings.

# Cronbach Alpha
library(psych)
library(data.table)
## 
## Attaching package: 'data.table'
## The following objects are masked from 'package:dplyr':
## 
##     between, first, last
# omega(Creativ, digits = 3)
cAlpha <- psych::alpha(x = Creativ)
## Number of categories should be increased  in order to count frequencies.
alphaDF <- data.table(variable = "Individual Creative Role Identity", id = "ICR", cAlpha$total)

cAlpha <- psych::alpha(x = InnoReady)
## Number of categories should be increased  in order to count frequencies.
alphaDF2 <- data.table(variable = "Individual Innovation Readiness", id = "IIR", cAlpha$total)
alphaDF  <- rbind(alphaDF, alphaDF2)

cAlpha <- psych::alpha(x = IRJP)
alphaDF2 <- data.table(variable = "In-Role Job Performance", id = "IRJP", cAlpha$total)
alphaDF  <- rbind(alphaDF, alphaDF2)

cAlpha <- psych::alpha(x = PsyCap)
## Number of categories should be increased  in order to count frequencies.
alphaDF2 <- data.table(variable = "Psychological Capital", id = "PSY", cAlpha$total)
alphaDF  <- rbind(alphaDF, alphaDF2)

cAlpha <- psych::alpha(x = Support)
## Number of categories should be increased  in order to count frequencies.
alphaDF2 <- data.table(variable = "Support for Innovation", id = "SFI", cAlpha$total)
alphaDF  <- rbind(alphaDF, alphaDF2)

cAlpha <- psych::alpha(x = Lead)
## Number of categories should be increased  in order to count frequencies.
alphaDF2 <- data.table(variable = "Transformational Leadership", id = "TFL", cAlpha$total)
alphaDF  <- rbind(alphaDF, alphaDF2)

cAlpha <- psych::alpha(x = IWB)
## Number of categories should be increased  in order to count frequencies.
alphaDF2 <- data.table(variable = "Individual Innovative Work Behaviour", id = "IWB", cAlpha$total)
alphaDF  <- rbind(alphaDF, alphaDF2)
alphaDF
##                                variable   id raw_alpha std.alpha G6(smc)
## 1:    Individual Creative Role Identity  ICR    0.9286    0.9297  0.9361
## 2:      Individual Innovation Readiness  IIR    0.8526    0.8532  0.8433
## 3:              In-Role Job Performance IRJP    0.8491    0.8526  0.8167
## 4:                Psychological Capital  PSY    0.9054    0.9061  0.9072
## 5:               Support for Innovation  SFI    0.9128    0.9145  0.9179
## 6:          Transformational Leadership  TFL    0.9678    0.9685  0.9714
## 7: Individual Innovative Work Behaviour  IWB    0.9402    0.9403  0.9365
##    average_r    S/N      ase  mean     sd median_r
## 1:    0.5041 13.215 0.004776 4.914 0.8973   0.5020
## 2:    0.4537  5.813 0.010227 5.506 0.7324   0.4487
## 3:    0.5911  5.783 0.011328 5.762 0.7654   0.5911
## 4:    0.4458  9.652 0.006361 5.360 0.7716   0.4469
## 5:    0.4930 10.698 0.005939 4.923 0.9511   0.4698
## 6:    0.6057 30.717 0.002109 5.047 1.0886   0.6360
## 7:    0.6365 15.757 0.004046 4.773 1.0880   0.6556

Individual Creative Role Identity, Support for Innovation and Transformational Leadership show indication of unequal factor loading by items.

4 Overview of Variable Data

4.1 Histograms of all Variables

par(mfrow=c(3, 3))
hist(DFvar$ICR, prob = T, xlim = c(1, 7), col = "azure", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)
hist(DFvar$IIR, prob = T, xlim = c(1, 7), col = "pink", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)
hist(DFvar$IRJ, prob = T, xlim = c(1, 7), col = "yellow", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)
hist(DFvar$PSY, prob = T, xlim = c(1, 7), col = "lightblue", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)
hist(DFvar$SFI, prob = T, xlim = c(1, 7), col = "orange", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)
hist(DFvar$TFL, prob = T, xlim = c(1, 7), col = "lightgreen", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)
hist(DFvar$IWB, prob = T, xlim = c(1, 7), col = "cyan", xlab= "", ylab="", , cex.lab = 2.5, cex.axis = 2.5, cex.main = 3.0)

4.2 Descriptive Statistics of all Variables

library(RcmdrMisc)
numSummary(DFvar[,c("PSY", "TFL", "SFI", "ICR", "IIR", "IRJ", "IWB")], 
           groups = DFvar$Org, 
           statistics = c("mean", "sd", "skewness", "kurtosis", "quantiles"), 
                          quantiles = c(0.0, 0.25, 0.5, 0.75, 1.00))
## 
## Variable: PSY 
##           mean     sd skewness kurtosis    0%   25%   50%   75%  100%   n
## Private1 5.395 0.7982 -0.80944   0.7569 3.250 5.083 5.542 5.896 7.000  90
## Private2 5.693 0.6389 -0.87454   1.1036 3.333 5.333 5.750 6.104 6.750 128
## Private3 5.545 0.6299  0.27186   0.2756 4.167 5.333 5.500 5.750 6.833  29
## Public1  5.516 0.5916 -0.17174   1.3777 3.917 5.250 5.458 5.875 6.833  26
## Public2  4.990 0.6846 -0.08597   0.2481 3.083 4.500 5.000 5.500 7.000 155
## Public3  5.404 0.9571 -1.93557   4.9623 2.000 5.021 5.583 6.000 6.833  54
## 
## Variable: TFL 
##           mean     sd skewness kurtosis   0%   25%   50%  75% 100%   n
## Private1 5.364 1.0871  -1.7026   4.1699 1.00 4.850 5.650 6.00 7.00  90
## Private2 5.002 1.1130  -0.6791   0.2834 1.65 4.375 5.100 5.90 6.90 128
## Private3 5.029 1.0490  -0.3276  -0.7175 2.85 4.050 5.200 5.80 6.90  29
## Public1  5.317 0.9582  -1.9878   5.0109 1.95 5.025 5.600 5.90 6.35  26
## Public2  4.859 1.0458  -0.6597   0.6068 1.95 4.350 5.000 5.55 7.00 155
## Public3  5.044 1.1381  -1.0379   2.3655 1.00 4.513 5.175 5.75 6.90  54
## 
## Variable: SFI 
##           mean     sd skewness kurtosis    0%   25%   50%   75%  100%   n
## Private1 5.107 0.9159  -1.1674   2.2492 1.545 4.750 5.273 5.795 6.545  90
## Private2 5.096 0.9435  -1.4131   2.9715 1.364 4.727 5.273 5.727 6.727 128
## Private3 4.851 0.9306  -0.9161   0.8079 2.500 4.364 5.182 5.455 6.455  29
## Public1  5.287 0.5579  -0.6276   1.3488 3.727 5.091 5.273 5.523 6.182  26
## Public2  4.681 0.8980  -0.6190   0.8620 1.273 4.091 4.727 5.364 7.000 155
## Public3  4.763 1.1513  -0.9790   1.3020 1.000 4.205 5.000 5.545 6.636  54
## 
## Variable: ICR 
##           mean     sd skewness kurtosis    0%   25%   50%   75%  100%   n
## Private1 5.199 0.8605 -0.47540  -0.2484 2.846 4.500 5.346 5.769 7.000  90
## Private2 5.276 0.7548 -0.35706  -0.3440 3.308 4.769 5.346 5.846 6.846 128
## Private3 4.729 0.9577  0.51991   0.3866 2.769 4.154 4.538 5.269 7.000  29
## Public1  4.947 0.8885 -0.61054  -0.4811 3.154 4.481 5.038 5.596 6.308  26
## Public2  4.513 0.8315  0.01644   0.4962 2.077 4.000 4.462 5.077 7.000 155
## Public3  4.812 0.9367 -0.70387   1.1393 2.000 4.231 4.885 5.462 6.538  54
## 
## Variable: IIR 
##           mean     sd  skewness kurtosis    0%   25%   50%   75%  100%   n
## Private1 5.613 0.6857 -0.692813   1.3332 3.143 5.286 5.714 6.000 7.000  90
## Private2 5.765 0.6821 -0.981684   1.8367 3.286 5.429 5.857 6.143 7.000 128
## Private3 5.396 0.5212  0.237384  -0.3181 4.286 5.071 5.400 5.714 6.429  29
## Public1  5.714 0.5525  0.359473  -0.4587 4.857 5.286 5.714 6.107 7.000  26
## Public2  5.183 0.7060 -0.004028  -0.3551 3.571 4.643 5.171 5.643 7.000 155
## Public3  5.601 0.8401 -1.811214   5.7027 2.000 5.179 5.714 6.143 6.857  54
## 
## Variable: IRJ 
##           mean     sd skewness kurtosis   0%   25% 50%   75% 100%   n
## Private1 5.747 0.7749  -0.5938   0.3166 3.50 5.250 6.0 6.188    7  90
## Private2 5.963 0.7044  -0.6636   0.2451 3.75 5.500 6.0 6.500    7 128
## Private3 6.048 0.5968   0.0304  -0.6618 4.75 5.500 6.0 6.500    7  29
## Public1  5.942 0.7947  -0.5974  -0.1182 4.25 5.562 6.0 6.438    7  26
## Public2  5.468 0.7121  -0.2693  -0.3093 3.50 5.000 5.5 6.000    7 155
## Public3  5.917 0.8509  -2.0373   7.7973 2.00 5.500 6.0 6.500    7  54
## 
## Variable: IWB 
##           mean     sd skewness kurtosis    0%   25%   50%   75%  100%   n
## Private1 5.091 1.1254  -0.5235  -0.2731 2.111 4.472 5.111 6.000 7.000  90
## Private2 5.006 0.9753  -0.5676  -0.2870 1.778 4.222 5.167 5.889 6.444 128
## Private3 4.771 1.0767  -0.1944   0.5547 2.333 4.167 4.667 5.500 7.000  29
## Public1  4.731 1.1847  -0.6806  -0.4105 1.889 4.028 5.000 5.750 6.222  26
## Public2  4.453 1.0426  -0.2478   0.1286 1.778 3.889 4.556 5.111 7.000 155
## Public3  4.632 1.1316  -0.7535   1.0568 1.000 3.889 4.833 5.333 6.889  54

5 Plotting Correlation Matrices for all Latent Variables (Pearson)

All items (questions/statements) should be correlated with a p-value < 0.01. ## Correlation Matrix - Individual Creative Role Identity

# Plot Correlation Matrix
library(corrplot)

Mat <- cor(Creativ)
Res <- cor.mtest(Creativ, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

## Correlation Matrix - Individual Innovation Readyness

Mat <- cor(InnoReady)
Res <- cor.mtest(InnoReady, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

## Correlation Matrix - In-Role Job Performance

Mat <- cor(IRJP)
Res <- cor.mtest(IRJP, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

## Correlation Matrix - PsyCap

Mat <- cor(PsyCap)
Res <- cor.mtest(PsyCap, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

## Correlation Matrix - Support for Innovation

Mat <- cor(Support)
Res <- cor.mtest(Support, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

## Correlation Matrix - Transformational Leadership

Mat <- cor(Lead)
Res <- cor.mtest(Lead, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

## Correlation Matrix - Individdual Innovative Work Behaviour

Mat <- cor(IWB)
Res <- cor.mtest(IWB, conf.level = .95)
corrplot.mixed(Mat, lower.col = "black", number.cex = 2.0, upper = "pie", tl.col = "blue", tl.cex = 2.0, p.mat = Res$p, insig = "p-value", cl.ratio = 0.2, cl.align = "r", cl.lim = c(0, 1), cl.cex = 2.0)

5.1 Analysis of Measurement Reproducability using Intraclass Correlation (ICC)

Transformational Leadership

library(irr)
## Loading required package: lpSolve
ICC(Lead)
## Call: ICC(x = Lead)
## 
## Intraclass correlation coefficients 
##                          type  ICC  F df1  df2 p lower bound upper bound
## Single_raters_absolute   ICC1 0.59 30 481 9158 0        0.56        0.62
## Single_random_raters     ICC2 0.59 31 481 9139 0        0.56        0.62
## Single_fixed_raters      ICC3 0.60 31 481 9139 0        0.57        0.63
## Average_raters_absolute ICC1k 0.97 30 481 9158 0        0.96        0.97
## Average_random_raters   ICC2k 0.97 31 481 9139 0        0.96        0.97
## Average_fixed_raters    ICC3k 0.97 31 481 9139 0        0.96        0.97
## 
##  Number of subjects = 482     Number of Judges =  20
## See the help file for a discussion of the other 4 McGraw and Wong estimates,

Support for Innovation

ICC(Support)
## Call: ICC(x = Support)
## 
## Intraclass correlation coefficients 
##                          type  ICC    F df1  df2 p lower bound upper bound
## Single_raters_absolute   ICC1 0.44  9.7 481 4820 0        0.41        0.48
## Single_random_raters     ICC2 0.45 11.5 481 4810 0        0.40        0.49
## Single_fixed_raters      ICC3 0.49 11.5 481 4810 0        0.45        0.52
## Average_raters_absolute ICC1k 0.90  9.7 481 4820 0        0.88        0.91
## Average_random_raters   ICC2k 0.90 11.5 481 4810 0        0.88        0.91
## Average_fixed_raters    ICC3k 0.91 11.5 481 4810 0        0.90        0.92
## 
##  Number of subjects = 482     Number of Judges =  11
## See the help file for a discussion of the other 4 McGraw and Wong estimates,

Individual Innovation Readiness

ICC(InnoReady)
## Call: ICC(x = InnoReady)
## 
## Intraclass correlation coefficients 
##                          type  ICC   F df1  df2        p lower bound
## Single_raters_absolute   ICC1 0.42 6.1 481 2892 2.3e-212        0.38
## Single_random_raters     ICC2 0.42 6.8 481 2886 2.9e-243        0.38
## Single_fixed_raters      ICC3 0.45 6.8 481 2886 2.9e-243        0.41
## Average_raters_absolute ICC1k 0.83 6.1 481 2892 2.3e-212        0.81
## Average_random_raters   ICC2k 0.84 6.8 481 2886 2.9e-243        0.81
## Average_fixed_raters    ICC3k 0.85 6.8 481 2886 2.9e-243        0.83
##                         upper bound
## Single_raters_absolute         0.46
## Single_random_raters           0.47
## Single_fixed_raters            0.49
## Average_raters_absolute        0.86
## Average_random_raters          0.86
## Average_fixed_raters           0.87
## 
##  Number of subjects = 482     Number of Judges =  7
## See the help file for a discussion of the other 4 McGraw and Wong estimates,

Individual Innovative Work Behaviour

ICC(IWB)
## Call: ICC(x = IWB)
## 
## Intraclass correlation coefficients 
##                          type  ICC  F df1  df2 p lower bound upper bound
## Single_raters_absolute   ICC1 0.63 16 481 3856 0        0.59        0.66
## Single_random_raters     ICC2 0.63 17 481 3848 0        0.59        0.66
## Single_fixed_raters      ICC3 0.64 17 481 3848 0        0.60        0.67
## Average_raters_absolute ICC1k 0.94 16 481 3856 0        0.93        0.95
## Average_random_raters   ICC2k 0.94 17 481 3848 0        0.93        0.95
## Average_fixed_raters    ICC3k 0.94 17 481 3848 0        0.93        0.95
## 
##  Number of subjects = 482     Number of Judges =  9
## See the help file for a discussion of the other 4 McGraw and Wong estimates,

Intraclass correlation is evaluated against the following thresholds (Koo and Li, 2016): * below 0.50: poor * between 0.50 and 0.75: moderate * between 0.75 and 0.90: good * above 0.90: excellent

6 Building Linear Regression Model with all Factors

Variance Inflation Factors (VIF) have been studied.

lmModel <- lm(IWB ~ ICR + IIR + IRJ + PSY + SFI + TFL, data = DFvar)
# Check variance inflation factors
library(car)
vif <- (vif(lmModel))
vif
##   ICR   IIR   IRJ   PSY   SFI   TFL 
## 2.513 3.724 2.190 4.447 1.929 1.609

All VIF are found to be lower than 5, i.e. this model is not inflated. It is ok.

IIR and SFI are removed step by step.

# TFI on IWB
summary(lmModel)
## 
## Call:
## lm(formula = IWB ~ ICR + IIR + IRJ + PSY + SFI + TFL, data = DFvar)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9692 -0.2752  0.0537  0.3222  1.7225 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.65526    0.20942   -3.13   0.0019 ** 
## ICR          0.83829    0.04384   19.12   <2e-16 ***
## IIR          0.00961    0.06539    0.15   0.8832    
## IRJ         -0.14944    0.04798   -3.11   0.0020 ** 
## PSY          0.14326    0.06782    2.11   0.0352 *  
## SFI          0.01273    0.03624    0.35   0.7255    
## TFL          0.25501    0.02891    8.82   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.544 on 475 degrees of freedom
## Multiple R-squared:  0.753,  Adjusted R-squared:  0.75 
## F-statistic:  241 on 6 and 475 DF,  p-value: <2e-16
# Remove IIR due to insignificance
lmModel <- lm(IWB ~ ICR + IRJ + PSY + SFI + TFL, data = DFvar)
summary(lmModel)
## 
## Call:
## lm(formula = IWB ~ ICR + IRJ + PSY + SFI + TFL, data = DFvar)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9677 -0.2744  0.0511  0.3201  1.7275 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.6470     0.2016   -3.21   0.0014 ** 
## ICR           0.8403     0.0416   20.22   <2e-16 ***
## IRJ          -0.1479     0.0468   -3.16   0.0017 ** 
## PSY           0.1475     0.0612    2.41   0.0163 *  
## SFI           0.0131     0.0361    0.36   0.7174    
## TFL           0.2552     0.0288    8.85   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.544 on 476 degrees of freedom
## Multiple R-squared:  0.753,  Adjusted R-squared:  0.75 
## F-statistic:  290 on 5 and 476 DF,  p-value: <2e-16
# Remove SFI due to insignificance
lmModel <- lm(IWB ~ ICR + IRJ + PSY + TFL, data = DFvar)
summary(lmModel)
## 
## Call:
## lm(formula = IWB ~ ICR + IRJ + PSY + TFL, data = DFvar)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9727 -0.2768  0.0481  0.3214  1.6920 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.6426     0.2011   -3.20   0.0015 ** 
## ICR           0.8407     0.0415   20.25   <2e-16 ***
## IRJ          -0.1472     0.0467   -3.15   0.0017 ** 
## PSY           0.1532     0.0591    2.59   0.0098 ** 
## TFL           0.2599     0.0258   10.07   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.543 on 477 degrees of freedom
## Multiple R-squared:  0.753,  Adjusted R-squared:  0.751 
## F-statistic:  363 on 4 and 477 DF,  p-value: <2e-16
# Remove SFI due to insignificance
lmModel <- lm(IWB ~ IIR + SFI + TFL, data = DFvar)
summary(lmModel)
## 
## Call:
## lm(formula = IWB ~ IIR + SFI + TFL, data = DFvar)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -2.533 -0.423  0.103  0.509  2.653 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -1.0673     0.2664   -4.01  7.1e-05 ***
## IIR           0.6854     0.0567   12.10  < 2e-16 ***
## SFI           0.0770     0.0486    1.58     0.11    
## TFL           0.3345     0.0396    8.44  3.7e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.753 on 478 degrees of freedom
## Multiple R-squared:  0.524,  Adjusted R-squared:  0.521 
## F-statistic:  176 on 3 and 478 DF,  p-value: <2e-16
# Prepare PDF
# pdf("Model01.pdf", width = 25, height = 16)
par(mar=c(7, 1, 7, 1))
par(oma=c(1, 1, 1, 1))

# Plot model
library(semPlot)
semPaths(lmModel, intAtSide = FALSE, what = "stand", whatLabels = "stand", intercepts = FALSE, residuals = FALSE, edge.label.cex = 0.9, edge.color = "blue", layout = "tree", rotation = 2, curve = 1, curvePivot = TRUE, mar = c(3,2,7,2), sizeLat = 10, sizeMan = 10, color = colorlist)
title(main = "Model of ICR, IRJ, PSY, TFL Influencing IWB - Model 01\nIIR and SFI are not significant", cex.main = 2.5, col.main = "darkblue", font.main = 3, cex.sub = 1.4)

# # dev.off()

Strongest driver seems to be Individual Creative Role Identity. In the following, the impact of TFI on IWB under different models with SFI and IIR are studied.

6.1 Group Variables “Sector” and “Org” are Tested as Moderators

# Add Sector as Moderator
lmModel <- lm(IWB ~ ICR + IRJ + PSY + TFL + Sector, data = DFvar)
summary(lmModel)
## 
## Call:
## lm(formula = IWB ~ ICR + IRJ + PSY + TFL + Sector, data = DFvar)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9403 -0.2805  0.0503  0.3278  1.6949 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -0.7251     0.2133   -3.40  0.00073 ***
## ICR                    0.8492     0.0421   20.15  < 2e-16 ***
## IRJ                   -0.1491     0.0467   -3.19  0.00150 ** 
## PSY                    0.1595     0.0593    2.69  0.00739 ** 
## TFL                    0.2577     0.0259    9.96  < 2e-16 ***
## SectorPublic Service   0.0605     0.0523    1.16  0.24825    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.543 on 476 degrees of freedom
## Multiple R-squared:  0.754,  Adjusted R-squared:  0.751 
## F-statistic:  291 on 5 and 476 DF,  p-value: <2e-16
# Add Org as Moderator
lmModel <- lm(IWB ~ ICR + IRJ + PSY + TFL + Org, data = DFvar)
summary(lmModel)
## 
## Call:
## lm(formula = IWB ~ ICR + IRJ + PSY + TFL + Org, data = DFvar)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8687 -0.2697  0.0522  0.3348  1.6292 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.8199     0.2185   -3.75   0.0002 ***
## ICR           0.8629     0.0427   20.19   <2e-16 ***
## IRJ          -0.1505     0.0467   -3.22   0.0014 ** 
## PSY           0.1730     0.0600    2.88   0.0041 ** 
## TFL           0.2529     0.0261    9.68   <2e-16 ***
## OrgPrivate2  -0.0789     0.0764   -1.03   0.3024    
## OrgPrivate3   0.1895     0.1185    1.60   0.1106    
## OrgPublic1   -0.1225     0.1211   -1.01   0.3124    
## OrgPublic2    0.1095     0.0743    1.47   0.1414    
## OrgPublic3   -0.0206     0.0948   -0.22   0.8283    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.539 on 472 degrees of freedom
## Multiple R-squared:  0.759,  Adjusted R-squared:  0.755 
## F-statistic:  165 on 9 and 472 DF,  p-value: <2e-16

Both, Sector and Org do not have significant moderating effect on the multiple linear model.

7 Confirmatory Factor Analysis (CFA) with Different Models

7.1 How to Evaluate CFA Output?

Chisq: The model Chi-squared assesses overall fit and the discrepancy between the sample and fitted covariance matrices. Its p-value should be > .05 (i.e., the hypothesis of a perfect fit cannot be rejected). However, it is quite sensitive to sample size. Commonly, Chisq/DF < 3 is used as criteria.

GFI/AGFI: The (Adjusted) Goodness of Fit is the proportion of variance accounted for by the estimated population covariance. Analogous to R2. The GFI and the AGFI should be > .95 and > .90, respectively.

NFI/NNFI/TLI: The (Non) Normed Fit Index. An NFI of 0.95, indicates the model of interest improves the fit by 95% relative to the null model. The NNFI (also called the Tucker Lewis index; TLI) is preferable for smaller samples. They should be > .90 (Byrne, 1994) or > .95 (Schumacker & Lomax, 2004). TLI > .90. TLI: < .85(poor); .85-.90 (mediocre); .90-.95 (acceptable); .95-.99 (close); 1.00 (perfect)

CFI: The Comparative Fit Index is a revised form of NFI. Not very sensitive to sample size (Fan, Thompson, & Wang, 1999). Compares the fit of a target model to the fit of an independent, or null, model. CFI: < .85(poor); .85-.90 (mediocre); .90-.95 (acceptable); .95-.99 (close); 1.00 (perfect)

RMSEA: The Root Mean Square Error of Approximation is a parsimony-adjusted index. Values closer to 0 represent a good fit. It should be < .08 or < .05. The p-value printed with it tests the hypothesis that RMSEA is less than or equal to .05 (a cutoff sometimes used for good fit), and thus should be not significant. RMSEA: > .10 (poor); .08-.10 (mediocre); .05-.08 (acceptable); .01-.05 (close); .00 (perfect)

RMR/SRMR: the (Standardized) Root Mean Square Residual represents the square-root of the difference between the residuals of the sample covariance matrix and the hypothesized model. As the RMR can be sometimes hard to interpret, better to use SRMR. SRMR: > .10 (poor); .08-.10 (mediocre); .05-.08 (acceptable); .01-.05 (close); .00 (perfect)

RFI: the Relative Fit Index, also known as RHO1, is not guaranteed to vary from 0 to 1. However, RFI close to 1 indicates a good fit.

IFI: the Incremental Fit Index (IFI) adjusts the Normed Fit Index (NFI) for sample size and degrees of freedom (Bollen’s, 1989). Over 0.90 is a good fit, but the index can exceed 1.

PNFI: the Parsimony-Adjusted Measures Index. There is no commonly agreed-upon cutoff value for an acceptable model for this index. Should be > 0.50.

To Report: chi-square, RMSEA, CFI, RMSEA and SRMR.

7.2 Model 11: Direct effect only

# Building model for linear regression
library(lavaan)
## This is lavaan 0.6-12
## lavaan is FREE software! Please report any bugs.
## 
## Attaching package: 'lavaan'
## The following object is masked from 'package:psych':
## 
##     cor2cov
model11 <- '

# latent variables
 TFL =~ Q013 + Q014 + Q015 + Q016 + Q017 + Q018 + Q019 + Q020 + Q021 + Q022 + Q023 + Q024 + Q025 + Q026 +           Q027 + Q028 + Q029 + Q030 + Q031 + Q032
 IWB =~ Q087 + Q088 + Q089 + Q090 + Q091 + Q092 + Q093 + Q094 + Q095
 
# direct effect
 IWB ~ a * TFL
 DirEff := a
 TotEff := a

'

# Calculate fit
fit11 <- sem(model11, data = DFs)

# Print results (fit indices, parameters, hypothesis tests)
summary(fit11, fit.measures = TRUE, rsquare = TRUE)
## lavaan 0.6-12 ended normally after 33 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        59
## 
##   Number of observations                           482
## 
## Model Test User Model:
##                                                       
##   Test statistic                              1040.564
##   Degrees of freedom                               376
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             12232.254
##   Degrees of freedom                               406
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.944
##   Tucker-Lewis Index (TLI)                       0.939
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -18516.795
##   Loglikelihood unrestricted model (H1)     -17996.513
##                                                       
##   Akaike (AIC)                               37151.589
##   Bayesian (BIC)                             37398.088
##   Sample-size adjusted Bayesian (BIC)        37210.827
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.061
##   90 Percent confidence interval - lower         0.056
##   90 Percent confidence interval - upper         0.065
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.044
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   TFL =~                                              
##     Q013              1.000                           
##     Q014              0.903    0.048   18.823    0.000
##     Q015              0.939    0.044   21.270    0.000
##     Q016              0.858    0.045   19.226    0.000
##     Q017              0.989    0.049   20.076    0.000
##     Q018              0.976    0.045   21.763    0.000
##     Q019              0.842    0.043   19.539    0.000
##     Q020              0.957    0.046   20.971    0.000
##     Q021              0.952    0.044   21.412    0.000
##     Q022              0.918    0.043   21.115    0.000
##     Q023              1.006    0.046   21.952    0.000
##     Q024              0.931    0.041   22.774    0.000
##     Q025              0.843    0.045   18.897    0.000
##     Q026              0.914    0.045   20.468    0.000
##     Q027              0.937    0.045   20.702    0.000
##     Q028              0.941    0.044   21.380    0.000
##     Q029              1.002    0.053   18.814    0.000
##     Q030              0.542    0.057    9.497    0.000
##     Q031              0.744    0.048   15.399    0.000
##     Q032              1.040    0.047   22.254    0.000
##   IWB =~                                              
##     Q087              1.000                           
##     Q088              1.043    0.047   22.228    0.000
##     Q089              0.933    0.044   21.123    0.000
##     Q090              0.776    0.049   15.669    0.000
##     Q091              1.089    0.045   24.408    0.000
##     Q092              0.919    0.043   21.153    0.000
##     Q093              1.086    0.045   24.095    0.000
##     Q094              1.104    0.050   22.224    0.000
##     Q095              0.939    0.046   20.516    0.000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   IWB ~                                               
##     TFL        (a)    0.530    0.043   12.413    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Q013              0.770    0.052   14.753    0.000
##    .Q014              0.856    0.057   14.959    0.000
##    .Q015              0.577    0.039   14.617    0.000
##    .Q016              0.717    0.048   14.916    0.000
##    .Q017              0.811    0.055   14.809    0.000
##    .Q018              0.562    0.039   14.519    0.000
##    .Q019              0.651    0.044   14.879    0.000
##    .Q020              0.637    0.043   14.671    0.000
##    .Q021              0.576    0.040   14.590    0.000
##    .Q022              0.569    0.039   14.646    0.000
##    .Q023              0.574    0.040   14.477    0.000
##    .Q024              0.408    0.029   14.266    0.000
##    .Q025              0.736    0.049   14.952    0.000
##    .Q026              0.642    0.044   14.752    0.000
##    .Q027              0.645    0.044   14.716    0.000
##    .Q028              0.566    0.039   14.597    0.000
##    .Q029              1.055    0.071   14.960    0.000
##    .Q030              1.875    0.122   15.431    0.000
##    .Q031              1.079    0.071   15.220    0.000
##    .Q032              0.573    0.040   14.406    0.000
##    .Q087              0.462    0.035   13.382    0.000
##    .Q088              0.626    0.045   13.808    0.000
##    .Q089              0.600    0.043   14.092    0.000
##    .Q090              0.990    0.066   14.927    0.000
##    .Q091              0.468    0.036   13.012    0.000
##    .Q092              0.580    0.041   14.086    0.000
##    .Q093              0.492    0.037   13.152    0.000
##    .Q094              0.702    0.051   13.809    0.000
##    .Q095              0.670    0.047   14.226    0.000
##     TFL               1.378    0.130   10.586    0.000
##    .IWB               0.755    0.068   11.087    0.000
## 
## R-Square:
##                    Estimate
##     Q013              0.642
##     Q014              0.568
##     Q015              0.678
##     Q016              0.586
##     Q017              0.624
##     Q018              0.700
##     Q019              0.600
##     Q020              0.665
##     Q021              0.684
##     Q022              0.671
##     Q023              0.708
##     Q024              0.745
##     Q025              0.571
##     Q026              0.642
##     Q027              0.653
##     Q028              0.683
##     Q029              0.567
##     Q030              0.177
##     Q031              0.414
##     Q032              0.722
##     Q087              0.712
##     Q088              0.665
##     Q089              0.624
##     Q090              0.410
##     Q091              0.743
##     Q092              0.625
##     Q093              0.732
##     Q094              0.665
##     Q095              0.600
##     IWB               0.339
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     DirEff            0.530    0.043   12.413    0.000
##     TotEff            0.530    0.043   12.413    0.000
modInd <- modificationIndices(fit11, minimum.value = 10)
# head(modInd[order(modInd$mi, decreasing=TRUE), ], 10)
subset(modInd[order(modInd$mi, decreasing=TRUE), ], mi > 5)
##      lhs op  rhs    mi    epc sepc.lv sepc.all sepc.nox
## 91   IWB =~ Q031 53.41  0.422   0.451    0.332    0.332
## 423 Q029 ~~ Q032 45.83  0.255   0.255    0.328    0.328
## 105 Q013 ~~ Q026 23.26 -0.163  -0.163   -0.232   -0.232
## 224 Q018 ~~ Q020 22.90  0.140   0.140    0.233    0.233
## 290 Q021 ~~ Q023 21.90  0.132   0.132    0.229    0.229
## 276 Q020 ~~ Q029 20.34 -0.177  -0.177   -0.216   -0.216
## 111 Q013 ~~ Q032 19.01  0.141   0.141    0.213    0.213
## 106 Q013 ~~ Q027 18.80 -0.147  -0.147   -0.209   -0.209
## 261 Q019 ~~ Q089 17.61  0.128   0.128    0.204    0.204
## 221 Q017 ~~ Q094 17.47  0.155   0.155    0.206    0.206
## 141 Q014 ~~ Q089 15.47  0.137   0.137    0.191    0.191
## 94  Q013 ~~ Q015 15.03  0.125   0.125    0.188    0.188
## 255 Q019 ~~ Q029 14.41 -0.149  -0.149   -0.180   -0.180
## 433 Q030 ~~ Q031 13.91  0.245   0.245    0.172    0.172
## 258 Q019 ~~ Q032 13.86 -0.110  -0.110   -0.181   -0.181
## 125 Q014 ~~ Q019 13.71  0.131   0.131    0.176    0.176
## 456 Q032 ~~ Q089 13.58 -0.107  -0.107   -0.182   -0.182
## 254 Q019 ~~ Q028 13.56 -0.108  -0.108   -0.177   -0.177
## 259 Q019 ~~ Q087 13.55  0.101   0.101    0.183    0.183
## 379 Q026 ~~ Q027 12.74  0.111   0.111    0.172    0.172
## 391 Q026 ~~ Q093 12.29 -0.099  -0.099   -0.177   -0.177
## 471 Q088 ~~ Q089 12.09 -0.110  -0.110   -0.180   -0.180
## 121 Q014 ~~ Q015 11.60  0.115   0.115    0.163    0.163
## 108 Q013 ~~ Q029 11.45  0.146   0.146    0.162    0.162
## 164 Q015 ~~ Q032 11.24  0.095   0.095    0.165    0.165
## 429 Q029 ~~ Q092 11.01 -0.126  -0.126   -0.161   -0.161
## 212 Q017 ~~ Q031 10.71  0.144   0.144    0.154    0.154
## 75   IWB =~ Q015 10.13 -0.137  -0.147   -0.110   -0.110

Composite Reliability must be > 0.7

# Composite reliability
sl <- standardizedSolution(fit11)
sl <- sl$est.std[sl$op == "=~"]
re <- 1 - sl^2

cat(paste("Composite Reliability of Model is: ", round(sum(sl)^2 / (sum(sl)^2 + sum(re)), 5), "\n", sep = " ", collapse = NULL))
## Composite Reliability of Model is:  0.97934

7.3 Model 12: Direct effect with Covariances

# Amending model 
model12 <- '

# latent variables
 TFL =~ Q013 + Q014 + Q016 + Q017 + Q018 + Q019 + Q020 + Q021 + Q022 + Q023 + Q024 + Q025 + Q026 + Q027 + Q028 + Q029 + Q030 + Q032
 IWB =~ Q087 + Q088 + Q089 + Q090 + Q091 + Q092 + Q093 + Q094 + Q095
 
# direct effect
 IWB ~ b * TFL
 DirEff := b
 TotEff := b

 
# covariances
 Q013 ~~ Q026
 Q013 ~~ Q027
 Q013 ~~ Q028
 Q013 ~~ Q032
 Q018 ~~ Q020
 Q018 ~~ Q029
 Q020 ~~ Q029
 Q021 ~~ Q023
 Q029 ~~ Q032
'

# Calculate fit
fit12 <- sem(model12, data = DFs)

# Print results (fit indices, parameters, hypothesis tests)
summary(fit12, fit.measures = TRUE, rsquare = TRUE)
## lavaan 0.6-12 ended normally after 37 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        64
## 
##   Number of observations                           482
## 
## Model Test User Model:
##                                                       
##   Test statistic                               699.086
##   Degrees of freedom                               314
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             11289.321
##   Degrees of freedom                               351
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.965
##   Tucker-Lewis Index (TLI)                       0.961
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -17161.883
##   Loglikelihood unrestricted model (H1)     -16812.339
##                                                       
##   Akaike (AIC)                               34451.765
##   Bayesian (BIC)                             34719.154
##   Sample-size adjusted Bayesian (BIC)        34516.023
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.050
##   90 Percent confidence interval - lower         0.045
##   90 Percent confidence interval - upper         0.055
##   P-value RMSEA <= 0.05                          0.435
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.035
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   TFL =~                                              
##     Q013              1.000                           
##     Q014              0.899    0.048   18.757    0.000
##     Q016              0.855    0.045   19.201    0.000
##     Q017              0.984    0.049   20.008    0.000
##     Q018              0.971    0.045   21.658    0.000
##     Q019              0.839    0.043   19.495    0.000
##     Q020              0.953    0.046   20.892    0.000
##     Q021              0.947    0.044   21.284    0.000
##     Q022              0.920    0.043   21.237    0.000
##     Q023              1.003    0.046   21.919    0.000
##     Q024              0.928    0.041   22.774    0.000
##     Q025              0.843    0.045   18.928    0.000
##     Q026              0.926    0.049   19.090    0.000
##     Q027              0.943    0.049   19.251    0.000
##     Q028              0.944    0.046   20.632    0.000
##     Q029              0.993    0.053   18.599    0.000
##     Q030              0.540    0.057    9.478    0.000
##     Q032              1.022    0.043   23.916    0.000
##   IWB =~                                              
##     Q087              1.000                           
##     Q088              1.043    0.047   22.229    0.000
##     Q089              0.933    0.044   21.118    0.000
##     Q090              0.776    0.050   15.671    0.000
##     Q091              1.089    0.045   24.404    0.000
##     Q092              0.919    0.043   21.151    0.000
##     Q093              1.086    0.045   24.092    0.000
##     Q094              1.104    0.050   22.222    0.000
##     Q095              0.939    0.046   20.522    0.000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   IWB ~                                               
##     TFL        (b)    0.527    0.043   12.362    0.000
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##  .Q013 ~~                                             
##    .Q026             -0.137    0.033   -4.190    0.000
##    .Q027             -0.127    0.033   -3.834    0.000
##    .Q028             -0.061    0.031   -1.966    0.049
##    .Q032              0.113    0.033    3.461    0.001
##  .Q018 ~~                                             
##    .Q020              0.129    0.031    4.214    0.000
##    .Q029              0.079    0.036    2.180    0.029
##  .Q020 ~~                                             
##    .Q029             -0.138    0.038   -3.611    0.000
##  .Q021 ~~                                             
##    .Q023              0.121    0.030    4.100    0.000
##  .Q029 ~~                                             
##    .Q032              0.236    0.039    5.979    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Q013              0.753    0.053   14.312    0.000
##    .Q014              0.863    0.058   14.911    0.000
##    .Q016              0.719    0.048   14.859    0.000
##    .Q017              0.819    0.056   14.749    0.000
##    .Q018              0.571    0.040   14.369    0.000
##    .Q019              0.655    0.044   14.821    0.000
##    .Q020              0.643    0.044   14.526    0.000
##    .Q021              0.587    0.041   14.470    0.000
##    .Q022              0.561    0.039   14.535    0.000
##    .Q023              0.577    0.040   14.331    0.000
##    .Q024              0.410    0.029   14.142    0.000
##    .Q025              0.734    0.049   14.891    0.000
##    .Q026              0.608    0.042   14.484    0.000
##    .Q027              0.628    0.043   14.480    0.000
##    .Q028              0.554    0.039   14.337    0.000
##    .Q029              1.077    0.072   14.905    0.000
##    .Q030              1.876    0.122   15.423    0.000
##    .Q032              0.610    0.043   14.349    0.000
##    .Q087              0.462    0.035   13.381    0.000
##    .Q088              0.626    0.045   13.806    0.000
##    .Q089              0.601    0.043   14.093    0.000
##    .Q090              0.990    0.066   14.926    0.000
##    .Q091              0.468    0.036   13.012    0.000
##    .Q092              0.580    0.041   14.085    0.000
##    .Q093              0.492    0.037   13.151    0.000
##    .Q094              0.702    0.051   13.808    0.000
##    .Q095              0.669    0.047   14.224    0.000
##     TFL               1.382    0.130   10.610    0.000
##    .IWB               0.758    0.068   11.080    0.000
## 
## R-Square:
##                    Estimate
##     Q013              0.648
##     Q014              0.564
##     Q016              0.584
##     Q017              0.621
##     Q018              0.695
##     Q019              0.598
##     Q020              0.661
##     Q021              0.679
##     Q022              0.676
##     Q023              0.707
##     Q024              0.744
##     Q025              0.572
##     Q026              0.661
##     Q027              0.662
##     Q028              0.690
##     Q029              0.559
##     Q030              0.177
##     Q032              0.703
##     Q087              0.712
##     Q088              0.665
##     Q089              0.623
##     Q090              0.410
##     Q091              0.743
##     Q092              0.625
##     Q093              0.732
##     Q094              0.665
##     Q095              0.601
##     IWB               0.337
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     DirEff            0.527    0.043   12.362    0.000
##     TotEff            0.527    0.043   12.362    0.000
modInd <- modificationIndices(fit12, minimum.value = 10)
# head(modInd[order(modInd$mi, decreasing=TRUE), ], 10)
subset(modInd[order(modInd$mi, decreasing=TRUE), ], mi > 5)
##      lhs op  rhs    mi    epc sepc.lv sepc.all sepc.nox
## 224 Q019 ~~ Q089 18.29  0.131   0.131    0.208    0.208
## 188 Q017 ~~ Q094 17.63  0.157   0.157    0.207    0.207
## 218 Q019 ~~ Q028 17.23 -0.122  -0.122   -0.203   -0.203
## 136 Q014 ~~ Q089 16.05  0.140   0.140    0.195    0.195
## 121 Q014 ~~ Q019 14.81  0.138   0.138    0.184    0.184
## 222 Q019 ~~ Q087 13.79  0.102   0.102    0.185    0.185
## 107 Q013 ~~ Q029 13.35  0.159   0.159    0.177    0.177
## 410 Q088 ~~ Q089 12.07 -0.110  -0.110   -0.180   -0.180
## 379 Q029 ~~ Q092 11.26 -0.119  -0.119   -0.151   -0.151
## 312 Q024 ~~ Q089 10.29  0.079   0.079    0.160    0.160
## 81   IWB =~ Q017 10.25  0.163   0.175    0.119    0.119
## 345 Q026 ~~ Q093 10.16 -0.088  -0.088   -0.161   -0.161
# Show model performance
# library(performance)
# model_performance(fit12, metrics = "all", verbose = TRUE)

7.4 Model 13: Direct effect with Covariances and Estimation Method DWLS

Commonly used estimation method Maximum Likelihood (ML) has been developed for continuous data following the normal-theory. This method is used to estimate all fit indices like RMSEA, CFI, and TLI. Since our data is generated using a 7-point Likert Scale, i.e. resulting in ordered discrete/categorical data, this method may not be the best to determine model fit (Yan Xia & Yanyun Yang, 2018).Therefore, methods based on Unweighted Least Squares (ULS) and Diagonally Weighted Least Squares (DWLS) based on polychoric correlation matrices have been considered in previous studies.

# Amending model 
model13 <- '

# latent variables
 TFL =~ Q013 + Q014 + Q016 + Q017 + Q018 + Q019 + Q020 + Q021 + Q022 + Q023 + Q024 + Q025 + Q026 + Q027 + Q028 + Q029 + Q030 + Q032
 IWB =~ Q087 + Q088 + Q089 + Q090 + Q091 + Q092 + Q093 + Q094 + Q095
 
# direct effect
 IWB ~ b * TFL
 DirEff := b
 TotEff := b

 
# covariances
 Q013 ~~ Q026
 Q013 ~~ Q027
 Q013 ~~ Q028
 Q013 ~~ Q032
 Q018 ~~ Q020
 Q018 ~~ Q029
 Q020 ~~ Q029
 Q021 ~~ Q023
 Q029 ~~ Q032
'

# Calculate fit
fit13 <- sem(model13, data = DFs, estimator = "WLSMV")

# Print results (fit indices, parameters, hypothesis tests)
summary(fit13, fit.measures = TRUE, rsquare = TRUE)
## lavaan 0.6-12 ended normally after 55 iterations
## 
##   Estimator                                       DWLS
##   Optimization method                           NLMINB
##   Number of model parameters                        64
## 
##   Number of observations                           482
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                               119.906     408.982
##   Degrees of freedom                               314         314
##   P-value (Chi-square)                           1.000       0.000
##   Scaling correction factor                                  0.544
##   Shift parameter                                          188.558
##     simple second-order correction                                
## 
## Model Test Baseline Model:
## 
##   Test statistic                             27290.560    3428.274
##   Degrees of freedom                               351         351
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  8.754
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000       0.969
##   Tucker-Lewis Index (TLI)                       1.008       0.965
##                                                                   
##   Robust Comparative Fit Index (CFI)                            NA
##   Robust Tucker-Lewis Index (TLI)                               NA
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000       0.025
##   90 Percent confidence interval - lower         0.000       0.018
##   90 Percent confidence interval - upper         0.000       0.032
##   P-value RMSEA <= 0.05                          1.000       1.000
##                                                                   
##   Robust RMSEA                                                  NA
##   90 Percent confidence interval - lower                        NA
##   90 Percent confidence interval - upper                        NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.032       0.032
## 
## Parameter Estimates:
## 
##   Standard errors                           Robust.sem
##   Information                                 Expected
##   Information saturated (h1) model        Unstructured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   TFL =~                                              
##     Q013              1.000                           
##     Q014              0.867    0.047   18.269    0.000
##     Q016              0.831    0.044   18.763    0.000
##     Q017              0.994    0.039   25.206    0.000
##     Q018              0.962    0.039   24.720    0.000
##     Q019              0.812    0.046   17.509    0.000
##     Q020              0.946    0.042   22.719    0.000
##     Q021              0.927    0.042   22.066    0.000
##     Q022              0.894    0.050   18.006    0.000
##     Q023              0.978    0.036   27.301    0.000
##     Q024              0.918    0.038   23.904    0.000
##     Q025              0.818    0.045   18.185    0.000
##     Q026              0.886    0.045   19.868    0.000
##     Q027              0.921    0.045   20.384    0.000
##     Q028              0.936    0.041   22.579    0.000
##     Q029              0.955    0.042   22.989    0.000
##     Q030              0.561    0.060    9.395    0.000
##     Q032              0.999    0.037   27.136    0.000
##   IWB =~                                              
##     Q087              1.000                           
##     Q088              1.088    0.081   13.445    0.000
##     Q089              0.963    0.056   17.273    0.000
##     Q090              0.855    0.074   11.583    0.000
##     Q091              1.084    0.066   16.380    0.000
##     Q092              0.966    0.053   18.121    0.000
##     Q093              1.060    0.050   21.025    0.000
##     Q094              1.137    0.057   19.897    0.000
##     Q095              0.990    0.065   15.251    0.000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   IWB ~                                               
##     TFL        (b)    0.513    0.052    9.902    0.000
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##  .Q013 ~~                                             
##    .Q026             -0.158    0.046   -3.430    0.001
##    .Q027             -0.162    0.040   -4.045    0.000
##    .Q028             -0.116    0.044   -2.657    0.008
##    .Q032              0.122    0.056    2.186    0.029
##  .Q018 ~~                                             
##    .Q020              0.103    0.047    2.181    0.029
##    .Q029              0.079    0.051    1.554    0.120
##  .Q020 ~~                                             
##    .Q029             -0.136    0.050   -2.701    0.007
##  .Q021 ~~                                             
##    .Q023              0.133    0.046    2.886    0.004
##  .Q029 ~~                                             
##    .Q032              0.292    0.060    4.844    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Q013              0.714    0.088    8.102    0.000
##    .Q014              0.904    0.134    6.726    0.000
##    .Q016              0.741    0.077    9.628    0.000
##    .Q017              0.742    0.073   10.112    0.000
##    .Q018              0.547    0.057    9.555    0.000
##    .Q019              0.681    0.059   11.511    0.000
##    .Q020              0.617    0.059   10.502    0.000
##    .Q021              0.595    0.060    9.876    0.000
##    .Q022              0.584    0.081    7.243    0.000
##    .Q023              0.597    0.055   10.901    0.000
##    .Q024              0.393    0.041    9.620    0.000
##    .Q025              0.758    0.100    7.620    0.000
##    .Q026              0.669    0.083    8.036    0.000
##    .Q027              0.639    0.095    6.721    0.000
##    .Q028              0.530    0.046   11.546    0.000
##    .Q029              1.130    0.097   11.707    0.000
##    .Q030              1.832    0.163   11.270    0.000
##    .Q032              0.632    0.065    9.679    0.000
##    .Q087              0.525    0.062    8.457    0.000
##    .Q088              0.591    0.112    5.267    0.000
##    .Q089              0.593    0.073    8.178    0.000
##    .Q090              0.889    0.109    8.156    0.000
##    .Q091              0.554    0.084    6.581    0.000
##    .Q092              0.538    0.074    7.284    0.000
##    .Q093              0.624    0.066    9.456    0.000
##    .Q094              0.700    0.089    7.867    0.000
##    .Q095              0.619    0.082    7.577    0.000
##     TFL               1.438    0.130   11.073    0.000
##    .IWB               0.704    0.089    7.894    0.000
## 
## R-Square:
##                    Estimate
##     Q013              0.668
##     Q014              0.544
##     Q016              0.573
##     Q017              0.657
##     Q018              0.709
##     Q019              0.582
##     Q020              0.676
##     Q021              0.675
##     Q022              0.663
##     Q023              0.697
##     Q024              0.755
##     Q025              0.559
##     Q026              0.628
##     Q027              0.656
##     Q028              0.704
##     Q029              0.537
##     Q030              0.198
##     Q032              0.694
##     Q087              0.673
##     Q088              0.684
##     Q089              0.629
##     Q090              0.471
##     Q091              0.696
##     Q092              0.652
##     Q093              0.661
##     Q094              0.666
##     Q095              0.631
##     IWB               0.350
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     DirEff            0.513    0.052    9.902    0.000
##     TotEff            0.513    0.052    9.902    0.000
modInd <- modificationIndices(fit12, minimum.value = 10)
# head(modInd[order(modInd$mi, decreasing=TRUE), ], 10)
subset(modInd[order(modInd$mi, decreasing=TRUE), ], mi > 5)
##      lhs op  rhs    mi    epc sepc.lv sepc.all sepc.nox
## 224 Q019 ~~ Q089 18.29  0.131   0.131    0.208    0.208
## 188 Q017 ~~ Q094 17.63  0.157   0.157    0.207    0.207
## 218 Q019 ~~ Q028 17.23 -0.122  -0.122   -0.203   -0.203
## 136 Q014 ~~ Q089 16.05  0.140   0.140    0.195    0.195
## 121 Q014 ~~ Q019 14.81  0.138   0.138    0.184    0.184
## 222 Q019 ~~ Q087 13.79  0.102   0.102    0.185    0.185
## 107 Q013 ~~ Q029 13.35  0.159   0.159    0.177    0.177
## 410 Q088 ~~ Q089 12.07 -0.110  -0.110   -0.180   -0.180
## 379 Q029 ~~ Q092 11.26 -0.119  -0.119   -0.151   -0.151
## 312 Q024 ~~ Q089 10.29  0.079   0.079    0.160    0.160
## 81   IWB =~ Q017 10.25  0.163   0.175    0.119    0.119
## 345 Q026 ~~ Q093 10.16 -0.088  -0.088   -0.161   -0.161
# Show model performance
# library(performance)
# model_performance(fit12, metrics = "all", verbose = TRUE)

7.5 Model 14: Direct effect with Covariances and Estimation Method ULS

An alternative for models based on ordered discrete/categorical data for fit estimation is the Unweighted Least Squares (ULS) method.

# Amending model 
model14 <- '

# latent variables
 TFL =~ Q013 + Q014 + Q016 + Q017 + Q018 + Q019 + Q020 + Q021 + Q022 + Q023 + Q024 + Q025 + Q026 + Q027 + Q028 + Q029 + Q030 + Q032
 IWB =~ Q087 + Q088 + Q089 + Q090 + Q091 + Q092 + Q093 + Q094 + Q095
 
# direct effect
 IWB ~ b * TFL
 DirEff := b
 TotEff := b

 
# covariances
 Q013 ~~ Q026
 Q013 ~~ Q027
 Q013 ~~ Q028
 Q013 ~~ Q032
 Q018 ~~ Q020
 Q018 ~~ Q029
 Q020 ~~ Q029
 Q021 ~~ Q023
 Q029 ~~ Q032
'

# Calculate fit
fit14 <- sem(model14, data = DFs, estimator = "ULSMV")

# Print results (fit indices, parameters, hypothesis tests)
summary(fit14, fit.measures = TRUE, rsquare = TRUE)
## lavaan 0.6-12 ended normally after 85 iterations
## 
##   Estimator                                        ULS
##   Optimization method                           NLMINB
##   Number of model parameters                        64
## 
##   Number of observations                           482
## 
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                               663.522     413.518
##   Degrees of freedom                               314         314
##   P-value (Unknown)                                 NA       0.000
##   Scaling correction factor                                  2.918
##   Shift parameter                                          186.098
##     simple second-order correction                                
## 
## Model Test Baseline Model:
## 
##   Test statistic                            161341.604    3452.942
##   Degrees of freedom                               351         351
##   P-value                                           NA       0.000
##   Scaling correction factor                                 51.366
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.998       0.968
##   Tucker-Lewis Index (TLI)                       0.998       0.964
##                                                                   
##   Robust Comparative Fit Index (CFI)                            NA
##   Robust Tucker-Lewis Index (TLI)                               NA
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.048       0.026
##   90 Percent confidence interval - lower         0.043       0.018
##   90 Percent confidence interval - upper         0.053       0.032
##   P-value RMSEA <= 0.05                          0.724       1.000
##                                                                   
##   Robust RMSEA                                                  NA
##   90 Percent confidence interval - lower                        NA
##   90 Percent confidence interval - upper                        NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.032       0.032
## 
## Parameter Estimates:
## 
##   Standard errors                           Robust.sem
##   Information                                 Expected
##   Information saturated (h1) model        Unstructured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   TFL =~                                              
##     Q013              1.000                           
##     Q014              0.869    0.047   18.500    0.000
##     Q016              0.832    0.044   18.794    0.000
##     Q017              0.990    0.039   25.533    0.000
##     Q018              0.959    0.038   24.998    0.000
##     Q019              0.814    0.046   17.690    0.000
##     Q020              0.942    0.040   23.312    0.000
##     Q021              0.926    0.041   22.413    0.000
##     Q022              0.895    0.049   18.191    0.000
##     Q023              0.978    0.035   27.827    0.000
##     Q024              0.915    0.038   24.044    0.000
##     Q025              0.820    0.044   18.502    0.000
##     Q026              0.889    0.044   20.119    0.000
##     Q027              0.921    0.044   20.726    0.000
##     Q028              0.933    0.041   22.738    0.000
##     Q029              0.957    0.041   23.101    0.000
##     Q030              0.553    0.060    9.209    0.000
##     Q032              0.998    0.037   27.315    0.000
##   IWB =~                                              
##     Q087              1.000                           
##     Q088              1.092    0.082   13.358    0.000
##     Q089              0.964    0.057   17.026    0.000
##     Q090              0.858    0.074   11.602    0.000
##     Q091              1.085    0.067   16.314    0.000
##     Q092              0.968    0.054   17.983    0.000
##     Q093              1.061    0.051   20.764    0.000
##     Q094              1.142    0.058   19.717    0.000
##     Q095              0.992    0.065   15.190    0.000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   IWB ~                                               
##     TFL        (b)    0.511    0.051    9.965    0.000
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##  .Q013 ~~                                             
##    .Q026             -0.165    0.045   -3.676    0.000
##    .Q027             -0.165    0.039   -4.229    0.000
##    .Q028             -0.114    0.042   -2.701    0.007
##    .Q032              0.120    0.053    2.266    0.023
##  .Q018 ~~                                             
##    .Q020              0.110    0.046    2.363    0.018
##    .Q029              0.078    0.049    1.580    0.114
##  .Q020 ~~                                             
##    .Q029             -0.136    0.049   -2.768    0.006
##  .Q021 ~~                                             
##    .Q023              0.132    0.044    2.994    0.003
##  .Q029 ~~                                             
##    .Q032              0.288    0.057    5.013    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Q013              0.711    0.084    8.416    0.000
##    .Q014              0.895    0.133    6.712    0.000
##    .Q016              0.738    0.077    9.585    0.000
##    .Q017              0.750    0.072   10.420    0.000
##    .Q018              0.553    0.056    9.821    0.000
##    .Q019              0.675    0.058   11.571    0.000
##    .Q020              0.624    0.057   10.975    0.000
##    .Q021              0.596    0.059   10.141    0.000
##    .Q022              0.580    0.080    7.218    0.000
##    .Q023              0.594    0.053   11.305    0.000
##    .Q024              0.399    0.040    9.999    0.000
##    .Q025              0.752    0.098    7.668    0.000
##    .Q026              0.658    0.081    8.150    0.000
##    .Q027              0.636    0.092    6.898    0.000
##    .Q028              0.535    0.045   11.897    0.000
##    .Q029              1.123    0.094   11.987    0.000
##    .Q030              1.843    0.163   11.296    0.000
##    .Q032              0.631    0.063   10.074    0.000
##    .Q087              0.529    0.063    8.381    0.000
##    .Q088              0.585    0.113    5.182    0.000
##    .Q089              0.595    0.075    7.959    0.000
##    .Q090              0.888    0.109    8.154    0.000
##    .Q091              0.555    0.085    6.567    0.000
##    .Q092              0.537    0.075    7.151    0.000
##    .Q093              0.628    0.067    9.319    0.000
##    .Q094              0.694    0.090    7.670    0.000
##    .Q095              0.617    0.082    7.559    0.000
##     TFL               1.441    0.128   11.221    0.000
##    .IWB               0.701    0.089    7.874    0.000
## 
## R-Square:
##                    Estimate
##     Q013              0.670
##     Q014              0.549
##     Q016              0.575
##     Q017              0.653
##     Q018              0.706
##     Q019              0.586
##     Q020              0.672
##     Q021              0.675
##     Q022              0.665
##     Q023              0.699
##     Q024              0.751
##     Q025              0.563
##     Q026              0.634
##     Q027              0.658
##     Q028              0.701
##     Q029              0.540
##     Q030              0.193
##     Q032              0.695
##     Q087              0.671
##     Q088              0.687
##     Q089              0.628
##     Q090              0.472
##     Q091              0.696
##     Q092              0.653
##     Q093              0.659
##     Q094              0.669
##     Q095              0.632
##     IWB               0.350
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     DirEff            0.511    0.051    9.965    0.000
##     TotEff            0.511    0.051    9.965    0.000
modInd <- modificationIndices(fit14, minimum.value = 10)
# head(modInd[order(modInd$mi, decreasing=TRUE), ], 10)
subset(modInd[order(modInd$mi, decreasing=TRUE), ], mi > 5)
##      lhs op  rhs    mi    epc sepc.lv sepc.all sepc.nox
## 94   IWB =~ Q030 81.55  0.210   0.218    0.144    0.144
## 81   IWB =~ Q017 46.34  0.163   0.170    0.115    0.115
## 72   TFL =~ Q090 40.41  0.115   0.138    0.106    0.106
## 75   TFL =~ Q093 35.24 -0.112  -0.135   -0.099   -0.099
## 188 Q017 ~~ Q094 26.72  0.243   0.243    0.336    0.336
## 90   IWB =~ Q026 24.46 -0.118  -0.123   -0.092   -0.092
## 345 Q026 ~~ Q093 18.35 -0.200  -0.200   -0.311   -0.311
## 373 Q029 ~~ Q030 17.24 -0.197  -0.197   -0.137   -0.137
## 79   IWB =~ Q014 16.77 -0.097  -0.101   -0.072   -0.072
## 388 Q030 ~~ Q091 15.22  0.180   0.180    0.178    0.178
## 389 Q030 ~~ Q092 15.14  0.180   0.180    0.180    0.180
## 73   TFL =~ Q091 14.67 -0.073  -0.087   -0.065   -0.065
## 386 Q030 ~~ Q089 14.28  0.174   0.174    0.166    0.166
## 140 Q014 ~~ Q093 14.20 -0.176  -0.176   -0.234   -0.234
## 426 Q090 ~~ Q094 12.68 -0.175  -0.175   -0.223   -0.223
## 384 Q030 ~~ Q087 12.62  0.164   0.164    0.166    0.166
## 93   IWB =~ Q029 12.31 -0.085  -0.089   -0.057   -0.057
## 121 Q014 ~~ Q019 11.86  0.164   0.164    0.211    0.211
## 107 Q013 ~~ Q029 10.52  0.160   0.160    0.179    0.179
## 182 Q017 ~~ Q088 10.46  0.152   0.152    0.229    0.229
# Show model performance
# library(performance)
# model_performance(fit12, metrics = "all", verbose = TRUE)
# Comparing model fits
# anova(fit11, fit12, fit13) # Lavaan ERROR: some models (but not all) have scaled test statistics


# Prepare PDF for Model 11
# pdf("Model11.pdf", width = 25, height = 16)
par(mar=c(7, 1, 7, 1))
par(oma=c(1, 1, 1, 1))

# Prepare fitmeasures
ParDF <- as.data.frame(do.call(cbind, fit11@ParTable))
DirEff <- round(as.numeric(ParDF[ParDF$label == "DirEff", 14]), 4)
TotEff <- round(as.numeric(ParDF[ParDF$label == "TotEff", 14]), 4)
n      <- fit11@SampleStats@ntotal

fitMeas <- toString(c("ChiSq: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "chisq"))), 4),
            " - ChiSq-p: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "pvalue"))), 4),
            " - DF: ", gsub("[^0-9.]", "", fitMeasures(fit11, "df")),
            " - ChiSq/DF: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "chisq")))/ as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "df"))), 4),
            " - CFI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "cfi"))), 4),
            " - TLI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "tli"))), 4),
            " - RMSEA: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "rmsea"))), 4),
            " - SRMR: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit11, "srmr"))), 4),
            " - n: ", n,
            " - DirEff: ", DirEff,
            " - TotalEff: ", TotEff
            ), 
            width = 255)
fitMeas <- gsub(",", "", fitMeas)

# Plot model
library(semPlot)
semPaths(fit11, what = "stand", whatLabels = "stand", intercepts = FALSE, residuals = FALSE, edge.label.cex = 0.5, edge.color = "blue", layout = "tree", rotation = 3, curve = 1, curvePivot = TRUE, mar = c(3,1,5,1), sizeLat = 7, sizeMan = 3, color = colorlist)
title(main = "Model Showing Only Direct Effect of TFL on IWB - Model 11", cex.main = 2.5, col.main = "darkblue", font.main = 3, sub = fitMeas, cex.sub = 1.4)

# dev.off()


# Prepare PDF for Model 12
# pdf("Model12.pdf", width = 25, height = 16)
par(mar=c(7, 1, 7, 1))
par(oma=c(1, 1, 1, 1))

# Prepare fitmeasures
ParDF  <- as.data.frame(do.call(cbind, fit12@ParTable))
DirEff <- round(as.numeric(ParDF[ParDF$label == "DirEff", 14]), 4)
TotEff <- round(as.numeric(ParDF[ParDF$label == "TotEff", 14]), 4)
n      <- fit12@SampleStats@ntotal

fitMeas <- toString(c("ChiSq: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "chisq"))), 4),
            " - ChiSq-p: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "pvalue"))), 4),
            " - DF: ", gsub("[^0-9.]", "", fitMeasures(fit12, "df")),
            " - ChiSq/DF: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "chisq")))/ as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "df"))), 4),
            " - CFI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "cfi"))), 4),
            " - TLI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "tli"))), 4),
            " - RMSEA: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "rmsea"))), 4),
            " - SRMR: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit12, "srmr"))), 4),
            " - n: ", n,
            " - DirEff: ", DirEff,
            " - TotalEff: ", TotEff
            ), 
            width = 255)
fitMeas <- gsub(",", "", fitMeas)

# Plot model
library(semPlot)
semPaths(fit12, what = "est", whatLabels = "est", intercepts = FALSE, residuals = FALSE, edge.label.cex = 0.5, edge.color = "blue", layout = "tree", rotation = 3, curve = 1, curvePivot = TRUE, mar = c(3,1,5,1), sizeLat = 7, sizeMan = 3, color = colorlist)
title(main = "Model Showing Only Direct Effect of TFL on IWB - Model 12\nSome Modification Indices Applied", cex.main = 2.5, col.main = "darkblue", font.main = 3, sub = fitMeas, cex.sub = 1.4)

# dev.off()


# Prepare PDF for Model 13
# pdf("Model13.pdf", width = 25, height = 16)
par(mar=c(7, 1, 7, 1))
par(oma=c(1, 1, 1, 1))

# Prepare fitmeasures
ParDF <- as.data.frame(do.call(cbind, fit13@ParTable))
DirEff <- round(as.numeric(ParDF[ParDF$label == "DirEff", 14]), 4)
TotEff <- round(as.numeric(ParDF[ParDF$label == "TotEff", 14]), 4)
n      <- fit13@SampleStats@ntotal

fitMeas <- toString(c("ChiSq: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "chisq"))), 4),
            " - ChiSq-p: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "pvalue"))), 4),
            " - DF: ", gsub("[^0-9.]", "", fitMeasures(fit13, "df")),
            " - ChiSq/DF: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "chisq")))/ as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "df"))), 4),
            " - CFI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "cfi"))), 4),
            " - TLI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "tli"))), 4),
            " - RMSEA: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "rmsea"))), 4),
            " - SRMR: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit13, "srmr"))), 4),
            " - n: ", n,
            " - DirEff: ", DirEff,
            " - TotalEff: ", TotEff
            ), 
            width = 255)
fitMeas <- gsub(",", "", fitMeas)

# Plot model
library(semPlot)
semPaths(fit13, what = "stand", whatLabels = "stand", intercepts = FALSE, residuals = FALSE, edge.label.cex = 0.5, edge.color = "blue", layout = "tree", rotation = 3, curve = 1, curvePivot = TRUE, mar = c(3,1,5,1), sizeLat = 7, sizeMan = 3, color = colorlist)
title(main = "Model Showing Only Direct Effect of TFL on IWB - Model 13\nEstimation Method DWLS", cex.main = 2.5, col.main = "darkblue", font.main = 3, sub = fitMeas, cex.sub = 1.4)

# dev.off()


# Prepare PDF for Model 14
# pdf("Model14.pdf", width = 25, height = 16)
par(mar=c(7, 1, 7, 1))
par(oma=c(1, 1, 1, 1))

# Prepare fitmeasures
ParDF <- as.data.frame(do.call(cbind, fit14@ParTable))
DirEff <- round(as.numeric(ParDF[ParDF$label == "DirEff", 14]), 4)
TotEff <- round(as.numeric(ParDF[ParDF$label == "TotEff", 14]), 4)
n      <- fit14@SampleStats@ntotal

fitMeas <- toString(c("ChiSq: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "chisq"))), 4),
            " - ChiSq-p: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "pvalue"))), 4),
            " - DF: ", gsub("[^0-9.]", "", fitMeasures(fit14, "df")),
            " - ChiSq/DF: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "chisq")))/ as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "df"))), 4),
            " - CFI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "cfi"))), 4),
            " - TLI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "tli"))), 4),
            " - RMSEA: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "rmsea"))), 4),
            " - SRMR: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit14, "srmr"))), 4),
            " - n: ", n,
            " - DirEff: ", DirEff,
            " - TotalEff: ", TotEff
            ), 
            width = 255)
fitMeas <- gsub(",", "", fitMeas)

# Plot model
library(semPlot)
semPaths(fit14, what = "stand", whatLabels = "stand", intercepts = FALSE, residuals = FALSE, edge.label.cex = 0.5, edge.color = "blue", layout = "tree", rotation = 3, curve = 1, curvePivot = TRUE, mar = c(3,1,5,1), sizeLat = 7, sizeMan = 3, color = colorlist)
title(main = "Model Showing Only Direct Effect of TFL on IWB - Model 14\nEstimation Method ULS", cex.main = 2.5, col.main = "darkblue", font.main = 3, sub = fitMeas, cex.sub = 1.4)

# dev.off()

After modification from Model 11 to Model 12, all indicators have improved. A significant difference between models was found.

7.6 Comparing Fit Measures for Simple Models using Different Estimation Methods

# Compare fit measures
cbind("Model 11"               = round(inspect(fit11, 'fit.measures'), 4), 
      "Model 12 (Cov)"         = round(inspect(fit12, 'fit.measures'), 4),
      "Model 13 (Cov + DWLS)"  = round(inspect(fit13, 'fit.measures'), 4),
      "Model 14 (Cov + ULS)"   = round(inspect(fit14, 'fit.measures'), 4)
      )
##                                 Model 11 Model 12 (Cov) Model 13 (Cov + DWLS)
## npar                           5.900e+01      6.400e+01             6.400e+01
## fmin                           1.079e+00      7.252e-01             1.244e-01
## chisq                          1.041e+03      6.991e+02             1.199e+02
## df                             3.760e+02      3.140e+02             3.140e+02
## pvalue                         0.000e+00      0.000e+00             1.000e+00
## chisq.scaled                   1.223e+04      1.129e+04             4.090e+02
## df.scaled                      4.060e+02      3.510e+02             3.140e+02
## pvalue.scaled                  0.000e+00      0.000e+00             2.000e-04
## chisq.scaling.factor           9.438e-01      9.648e-01             5.440e-01
## baseline.chisq                 9.393e-01      9.606e-01             2.729e+04
## baseline.df                    9.393e-01      9.606e-01             3.510e+02
## baseline.pvalue                9.081e-01      9.308e-01             0.000e+00
## baseline.chisq.scaled          9.149e-01      9.381e-01             3.428e+03
## baseline.df.scaled             8.473e-01      8.392e-01             3.510e+02
## baseline.pvalue.scaled         9.439e-01      9.649e-01             0.000e+00
## baseline.chisq.scaling.factor  9.438e-01      9.648e-01             8.754e+00
## cfi                           -1.852e+04     -1.716e+04             1.000e+00
## tli                           -1.800e+04     -1.681e+04             1.008e+00
## nnfi                           3.715e+04      3.445e+04             1.008e+00
## rfi                            3.740e+04      3.472e+04             9.951e-01
## nfi                            4.820e+02      4.820e+02             9.956e-01
## pnfi                           3.721e+04      3.452e+04             8.907e-01
## ifi                            6.060e-02      5.040e-02             1.007e+00
## rni                            5.620e-02      4.540e-02             1.007e+00
## cfi.scaled                     6.490e-02      5.550e-02             9.691e-01
## tli.scaled                     0.000e+00      4.348e-01             9.655e-01
## cfi.robust                     8.190e-02      6.610e-02                    NA
## tli.robust                     8.190e-02      6.610e-02                    NA
## nnfi.scaled                    4.390e-02      3.510e-02             9.655e-01
## nnfi.robust                    4.390e-02      3.510e-02                    NA
## rfi.scaled                     4.390e-02      3.510e-02             8.666e-01
## nfi.scaled                     4.550e-02      3.640e-02             8.807e-01
## ifi.scaled                     4.550e-02      3.640e-02             9.695e-01
## rni.scaled                     4.390e-02      3.510e-02             9.691e-01
## rni.robust                     4.390e-02      3.510e-02                    NA
## rmsea                          1.966e+02      2.467e+02             0.000e+00
## rmsea.ci.lower                 2.061e+02      2.597e+02             0.000e+00
## rmsea.ci.upper                 8.629e-01      9.012e-01             0.000e+00
## rmsea.pvalue                   8.413e-01      8.810e-01             1.000e+00
## rmsea.scaled                   7.458e-01      7.486e-01             2.510e-02
## rmsea.ci.lower.scaled          5.019e-01      6.707e-01             1.760e-02
## rmsea.ci.upper.scaled          2.404e+00      1.716e+00             3.170e-02
## rmsea.pvalue.scaled            5.900e+01      6.400e+01             1.000e+00
## rmsea.robust                   1.079e+00      7.252e-01                    NA
## rmsea.ci.lower.robust          1.041e+03      6.991e+02                    NA
## rmsea.ci.upper.robust          3.760e+02      3.140e+02                    NA
## rmsea.pvalue.robust            0.000e+00      0.000e+00                    NA
## rmr                            1.223e+04      1.129e+04             6.050e-02
## rmr_nomean                     4.060e+02      3.510e+02             6.050e-02
## srmr                           0.000e+00      0.000e+00             3.210e-02
## srmr_bentler                   9.438e-01      9.648e-01             3.210e-02
## srmr_bentler_nomean            9.393e-01      9.606e-01             3.210e-02
## crmr                           9.393e-01      9.606e-01             3.340e-02
## crmr_nomean                    9.081e-01      9.308e-01             3.340e-02
## srmr_mplus                     9.149e-01      9.381e-01             3.210e-02
## srmr_mplus_nomean              8.473e-01      8.392e-01             3.210e-02
## cn_05                          9.439e-01      9.649e-01             1.430e+03
## cn_01                          9.438e-01      9.648e-01             1.506e+03
## gfi                           -1.852e+04     -1.716e+04             9.964e-01
## agfi                          -1.800e+04     -1.681e+04             9.957e-01
## pgfi                           3.715e+04      3.445e+04             8.277e-01
## mfi                            3.740e+04      3.472e+04             1.224e+00
## ecvi                           4.820e+02      4.820e+02             5.154e-01
##                               Model 14 (Cov + ULS)
## npar                                     6.400e+01
## fmin                                     6.883e-01
## chisq                                    6.635e+02
## df                                       3.140e+02
## pvalue                                          NA
## chisq.scaled                             4.135e+02
## df.scaled                                3.140e+02
## pvalue.scaled                            1.000e-04
## chisq.scaling.factor                     2.918e+00
## baseline.chisq                           1.613e+05
## baseline.df                              3.510e+02
## baseline.pvalue                                 NA
## baseline.chisq.scaled                    3.453e+03
## baseline.df.scaled                       3.510e+02
## baseline.pvalue.scaled                   0.000e+00
## baseline.chisq.scaling.factor            5.137e+01
## cfi                                      9.978e-01
## tli                                      9.976e-01
## nnfi                                     9.976e-01
## rfi                                      9.954e-01
## nfi                                      9.959e-01
## pnfi                                     8.909e-01
## ifi                                      9.978e-01
## rni                                      9.978e-01
## cfi.scaled                               9.679e-01
## tli.scaled                               9.641e-01
## cfi.robust                                      NA
## tli.robust                                      NA
## nnfi.scaled                              9.641e-01
## nnfi.robust                                     NA
## rfi.scaled                               8.661e-01
## nfi.scaled                               8.802e-01
## ifi.scaled                               9.683e-01
## rni.scaled                               9.679e-01
## rni.robust                                      NA
## rmsea                                    4.810e-02
## rmsea.ci.lower                           4.300e-02
## rmsea.ci.upper                           5.320e-02
## rmsea.pvalue                             7.239e-01
## rmsea.scaled                             2.570e-02
## rmsea.ci.lower.scaled                    1.830e-02
## rmsea.ci.upper.scaled                    3.220e-02
## rmsea.pvalue.scaled                      1.000e+00
## rmsea.robust                                    NA
## rmsea.ci.lower.robust                           NA
## rmsea.ci.upper.robust                           NA
## rmsea.pvalue.robust                             NA
## rmr                                      6.040e-02
## rmr_nomean                               6.040e-02
## srmr                                     3.210e-02
## srmr_bentler                             3.210e-02
## srmr_bentler_nomean                      3.210e-02
## crmr                                     3.330e-02
## crmr_nomean                              3.330e-02
## srmr_mplus                               3.210e-02
## srmr_mplus_nomean                        3.210e-02
## cn_05                                    2.593e+02
## cn_01                                    2.730e+02
## gfi                                      9.968e-01
## agfi                                     9.961e-01
## pgfi                                     8.280e-01
## mfi                                      6.954e-01
## ecvi                                     1.646e+00

TLI and CFI show higher (better fit) values when using estimation methods DWLS and ULS whereas RMSEA shows lower (better fit) values when using those methods. Since there are no commony used thresholds available for both methods, we continue using the ML estimation method for model fit knowing that it might not be fair to our 7-point Likert Scale data, i.e. ordered discrete/categorical data.

8 Adding Mediators:

8.1 Model 22: IIR Mediates the Relationship Between TFL and Employee IWB

# Amending model for Mediation by IRR 
model22 <- '

# latent variables
 IWB =~ Q088 + Q089 + Q090 + Q091 + Q092 + Q093 + Q094 + Q095
 TFL =~ Q013 + Q014 + Q015 + Q016 + Q017 + Q018 + Q019 + Q020 + Q021 + Q022 + Q023 + Q024 + Q025 + Q026 +         Q027 + Q028 + Q029 + Q031 + Q032
 IIR =~ Q071 + Q072 + Q075 + Q076 + Q077 + Q081
 
# direct effect
 IWB ~ c * TFL
 DirEff := c

# mediator
 IIR ~ a1 * TFL
 IWB ~ b1 * IIR

# indirect effect
 ab1 := a1 * b1

# total effect
 TotEff := c + (a1 * b1)

# covariances
 Q013 ~~ Q026
 Q013 ~~ Q032
 Q013 ~~ Q027
 Q013 ~~ Q015
 Q018 ~~ Q020
 Q020 ~~ Q029
 Q021 ~~ Q023
 Q029 ~~ Q032
 Q075 ~~ Q076
 
'

# Calculate fit22
fit22 <- sem(model22, data = DF)

# Print results (fit22 indices, parameters, hypothesis tests)
summary(fit22, fit.measures = TRUE, rsquare = TRUE)
## lavaan 0.6-12 ended normally after 40 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        78
## 
##   Number of observations                           482
## 
## Model Test User Model:
##                                                       
##   Test statistic                              1095.749
##   Degrees of freedom                               483
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             12996.947
##   Degrees of freedom                               528
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.951
##   Tucker-Lewis Index (TLI)                       0.946
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -20614.225
##   Loglikelihood unrestricted model (H1)     -20066.351
##                                                       
##   Akaike (AIC)                               41384.450
##   Bayesian (BIC)                             41710.330
##   Sample-size adjusted Bayesian (BIC)        41462.765
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.051
##   90 Percent confidence interval - lower         0.047
##   90 Percent confidence interval - upper         0.055
##   P-value RMSEA <= 0.05                          0.292
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.052
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   IWB =~                                              
##     Q088              1.000                           
##     Q089              0.890    0.044   20.036    0.000
##     Q090              0.742    0.049   15.225    0.000
##     Q091              1.043    0.045   22.942    0.000
##     Q092              0.878    0.044   20.096    0.000
##     Q093              1.033    0.046   22.463    0.000
##     Q094              1.059    0.050   21.133    0.000
##     Q095              0.904    0.046   19.761    0.000
##   TFL =~                                              
##     Q013              1.000                           
##     Q014              0.904    0.048   18.847    0.000
##     Q015              0.936    0.041   22.912    0.000
##     Q016              0.859    0.045   19.242    0.000
##     Q017              0.989    0.049   20.066    0.000
##     Q018              0.971    0.045   21.600    0.000
##     Q019              0.843    0.043   19.563    0.000
##     Q020              0.954    0.046   20.810    0.000
##     Q021              0.948    0.045   21.237    0.000
##     Q022              0.920    0.043   21.173    0.000
##     Q023              1.003    0.046   21.839    0.000
##     Q024              0.932    0.041   22.814    0.000
##     Q025              0.845    0.045   18.931    0.000
##     Q026              0.924    0.048   19.073    0.000
##     Q027              0.947    0.049   19.247    0.000
##     Q028              0.941    0.044   21.390    0.000
##     Q029              1.000    0.053   18.766    0.000
##     Q031              0.743    0.048   15.364    0.000
##     Q032              1.032    0.044   23.645    0.000
##   IIR =~                                              
##     Q071              1.000                           
##     Q072              0.836    0.067   12.391    0.000
##     Q075              1.069    0.078   13.681    0.000
##     Q076              0.965    0.076   12.777    0.000
##     Q077              0.808    0.065   12.450    0.000
##     Q081              0.715    0.066   10.793    0.000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   IWB ~                                               
##     TFL        (c)    0.297    0.040    7.333    0.000
##   IIR ~                                               
##     TFL       (a1)    0.301    0.035    8.714    0.000
##   IWB ~                                               
##     IIR       (b1)    0.865    0.083   10.454    0.000
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##  .Q013 ~~                                             
##    .Q026             -0.128    0.032   -3.973    0.000
##    .Q032              0.087    0.031    2.843    0.004
##    .Q027             -0.131    0.032   -4.050    0.000
##    .Q015              0.098    0.032    3.021    0.003
##  .Q018 ~~                                             
##    .Q020              0.146    0.031    4.735    0.000
##  .Q020 ~~                                             
##    .Q029             -0.158    0.037   -4.298    0.000
##  .Q021 ~~                                             
##    .Q023              0.126    0.030    4.252    0.000
##  .Q029 ~~                                             
##    .Q032              0.218    0.039    5.578    0.000
##  .Q075 ~~                                             
##    .Q076              0.127    0.035    3.650    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Q088              0.622    0.046   13.591    0.000
##    .Q089              0.606    0.043   13.956    0.000
##    .Q090              0.990    0.067   14.860    0.000
##    .Q091              0.465    0.037   12.708    0.000
##    .Q092              0.583    0.042   13.938    0.000
##    .Q093              0.507    0.039   12.988    0.000
##    .Q094              0.697    0.051   13.591    0.000
##    .Q095              0.657    0.047   14.033    0.000
##    .Q013              0.756    0.052   14.503    0.000
##    .Q014              0.855    0.057   14.930    0.000
##    .Q015              0.588    0.040   14.527    0.000
##    .Q016              0.716    0.048   14.885    0.000
##    .Q017              0.813    0.055   14.777    0.000
##    .Q018              0.576    0.040   14.457    0.000
##    .Q019              0.649    0.044   14.845    0.000
##    .Q020              0.652    0.045   14.616    0.000
##    .Q021              0.591    0.041   14.533    0.000
##    .Q022              0.566    0.039   14.593    0.000
##    .Q023              0.584    0.041   14.410    0.000
##    .Q024              0.407    0.029   14.194    0.000
##    .Q025              0.734    0.049   14.921    0.000
##    .Q026              0.619    0.042   14.601    0.000
##    .Q027              0.622    0.043   14.559    0.000
##    .Q028              0.566    0.039   14.551    0.000
##    .Q029              1.057    0.071   14.919    0.000
##    .Q031              1.082    0.071   15.208    0.000
##    .Q032              0.591    0.041   14.350    0.000
##    .Q071              0.634    0.049   12.988    0.000
##    .Q072              0.536    0.040   13.438    0.000
##    .Q075              0.524    0.045   11.667    0.000
##    .Q076              0.583    0.047   12.514    0.000
##    .Q077              0.492    0.037   13.398    0.000
##    .Q081              0.637    0.045   14.250    0.000
##    .IWB               0.494    0.053    9.264    0.000
##     TFL               1.376    0.130   10.600    0.000
##    .IIR               0.435    0.056    7.762    0.000
## 
## R-Square:
##                    Estimate
##     Q088              0.667
##     Q089              0.620
##     Q090              0.409
##     Q091              0.744
##     Q092              0.622
##     Q093              0.724
##     Q094              0.667
##     Q095              0.608
##     Q013              0.645
##     Q014              0.568
##     Q015              0.672
##     Q016              0.586
##     Q017              0.623
##     Q018              0.693
##     Q019              0.601
##     Q020              0.658
##     Q021              0.677
##     Q022              0.673
##     Q023              0.703
##     Q024              0.746
##     Q025              0.572
##     Q026              0.655
##     Q027              0.665
##     Q028              0.683
##     Q029              0.565
##     Q031              0.412
##     Q032              0.713
##     Q071              0.469
##     Q072              0.422
##     Q075              0.550
##     Q076              0.472
##     Q077              0.426
##     Q081              0.310
##     IWB               0.604
##     IIR               0.223
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     DirEff            0.297    0.040    7.333    0.000
##     ab1               0.260    0.034    7.580    0.000
##     TotEff            0.557    0.046   12.245    0.000
modInd <- modificationIndices(fit22, minimum.value = 10)
# head(modInd[order(modInd$mi, decreasing=TRUE), ], 10)
subset(modInd[order(modInd$mi, decreasing=TRUE), ], mi > 5)
##      lhs op  rhs    mi    epc sepc.lv sepc.all sepc.nox
## 102  IWB =~ Q031 50.84  0.398   0.444    0.327    0.327
## 107  IWB =~ Q076 33.30  0.340   0.380    0.361    0.361
## 118  TFL =~ Q071 30.81  0.231   0.270    0.248    0.248
## 149  IIR =~ Q031 27.86  0.430   0.321    0.237    0.237
## 105  IWB =~ Q072 24.10 -0.289  -0.322   -0.335   -0.335
## 664 Q072 ~~ Q081 23.23  0.147   0.147    0.252    0.252
## 195 Q089 ~~ Q019 19.30  0.135   0.135    0.215    0.215
## 333 Q094 ~~ Q017 16.76  0.153   0.153    0.203    0.203
## 190 Q089 ~~ Q014 14.81  0.135   0.135    0.188    0.188
## 511 Q019 ~~ Q028 13.87 -0.109  -0.109   -0.180   -0.180
## 437 Q015 ~~ Q032 13.75  0.102   0.102    0.173    0.173
## 403 Q014 ~~ Q019 13.49  0.130   0.130    0.175    0.175
## 663 Q072 ~~ Q077 12.33  0.099   0.099    0.193    0.193
## 151 Q088 ~~ Q089 12.09 -0.112  -0.112   -0.183   -0.183
## 148  IIR =~ Q029 11.66 -0.260  -0.194   -0.125   -0.125
## 268 Q091 ~~ Q072 11.53 -0.090  -0.090   -0.180   -0.180
## 399 Q014 ~~ Q015 11.27  0.112   0.112    0.158    0.158
## 292 Q092 ~~ Q029 11.12 -0.120  -0.120   -0.153   -0.153
## 477 Q017 ~~ Q031 11.10  0.148   0.148    0.157    0.157
## 145  IIR =~ Q026 10.95 -0.207  -0.155   -0.115   -0.115
## 87   IWB =~ Q015 10.91 -0.136  -0.152   -0.113   -0.113
## 125  IIR =~ Q089 10.70  0.299   0.224    0.177    0.177
## 391 Q013 ~~ Q029 10.59  0.136   0.136    0.152    0.152
## 285 Q092 ~~ Q022 10.59  0.092   0.092    0.161    0.161
## 89   IWB =~ Q017 10.54  0.159   0.178    0.121    0.121
## 109  IWB =~ Q081 10.40 -0.195  -0.218   -0.227   -0.227
## 647 Q031 ~~ Q076 10.27  0.119   0.119    0.149    0.149
# Show model performance
# library(performance)
# model_performance(fit22, metrics = "all", verbose = TRUE)

# Prepare PDF
# pdf("Model22.pdf", width = 25, height = 16)
par(mar=c(7, 1, 7, 1))
par(oma=c(1, 1, 1, 1))

# Prepare fitmeasures
ParDF  <- as.data.frame(do.call(cbind, fit22@ParTable))
DirEff <- round(as.numeric(ParDF[ParDF$label == "DirEff", 14]), 4)
TotEff <- round(as.numeric(ParDF[ParDF$label == "TotEff", 14]), 4)
n      <- fit22@SampleStats@ntotal

fitMeas <- toString(c("ChiSq: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "chisq"))), 4),
                      " - DF: ", gsub("[^0-9.]", "", fitMeasures(fit22, "df")),
                      " - ChiSq/DF: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "chisq")))/ as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "df"))), 4),
                      " - CFI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "cfi"))), 4),
                      " - TLI: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "tli"))), 4),
                      " - RMSEA: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "rmsea"))), 4),
                      " - SRMR: ", round(as.numeric(gsub("[^0-9.]", "", fitMeasures(fit22, "srmr"))), 4),
                      " - n: ", n,
                      " - DirEff: ", DirEff,
                      " - TotalEff: ", TotEff
                      ), 
                    width = 255)
fitMeas <- gsub(",", "", fitMeas)

# Plot model
library(semPlot)
semPaths(fit22, what = "stand", whatLabels = "stand", intercepts = FALSE, residuals = FALSE, edge.label.cex = 0.5, edge.color = "blue", layout = "tree", rotation = 3, curve = 1, curvePivot = TRUE, mar = c(3,1,5,1), sizeLat = 7, sizeMan = 3, color = colorlist)
title(main = "IIR Mediates Relationship of TFL on IWB - Model 22", cex.main = 2.5, col.main = "darkblue", font.main = 3, sub = fitMeas, cex.sub = 1.4)