Modified Partial Least Square Structural Equation Model with Multivariate Adaptive Regression Spline : Parameter Estimation Technique and Applications
Resource
Data will be made available on request
Endogenous Laten Variables
η Behavioral Intention
Endogenous Laten Variables
ξ1 Performance expectations
ξ2 Business expectations
ξ3 Social influence
ξ4 Facility conditions
Library Packages
Before we start the analysis, here are some packages that are
required to perform data analysis using Multivariate Adaptive Regression
Spline (MARSPLS). However, we must ensure that all packages are
installed so we can call the packages as follows.
library(ggplot2)
library(lmtest)
library(car)
library(earth)
#Call syntax of related functions
source("Score Latent Variable Estimation Function.R")Input Data
#load data
data_utaut = read.csv("Hasil Kuisioner E-Wallet.csv", sep=";")
n = nrow(data_utaut)
head(data_utaut)## EK1 EK2 EK3 EK4 EU1 EU2 EU3 EU4 PS1 PS2 PS3 PS4 KF1 KF2 KF3 KF4 NB1 NB2 NB3
## 1 5 5 5 5 5 4 5 5 5 5 5 3 5 5 5 5 5 5 4
## 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
## 3 5 4 4 4 5 5 4 5 3 3 4 3 4 5 4 4 3 4 4
## 4 5 5 5 4 5 5 5 5 2 3 4 4 5 5 5 4 3 3 2
## 5 4 4 3 4 5 4 3 4 2 3 3 1 5 4 5 4 3 4 3
## 6 5 5 4 5 5 5 5 5 3 5 5 3 5 5 5 3 5 5 5
## NB4
## 1 4
## 2 5
## 3 4
## 4 4
## 5 3
## 6 5Estimating latent variable score
# Score Latent Variable
est=est_LVS(data_utaut)
xi=est$ksi
eta=est$eta
ksi1 = xi[,1]
ksi2 = xi[,2]
ksi3 = xi[,3]
ksi4 = xi[,4]
eta = eta
cat("\nScore Latent Variable\n")##
## Score Latent Variablescore = cbind(xi,eta)
colnames(score) = c(variable.names(xi),"η")
head(score)## ξ1 ξ2 ξ3 ξ4 η
## [1,] 0.9211893 0.4783715 1.1280358 0.9581119 0.5632644
## [2,] 0.9211893 0.9168385 1.6398722 0.9581119 1.2796490
## [3,] -0.4053724 0.4418537 -0.2245001 -0.3625242 -0.4204520
## [4,] 0.6128363 0.9168385 -0.1705984 0.6078286 -1.4607742
## [5,] -1.3326322 -0.7801609 -1.2953708 0.0730969 -1.1368366
## [6,] 0.4533420 0.9168385 0.7240028 0.2575452 1.2796490cat("\n\nFactor Loading\n")##
##
## Factor Loadingload.fak = est$loading
print(load.fak)## ξ1 ξ2 ξ3 ξ4 η
## λ1 0.8010378 0.8122458 0.7590981 0.7801763 0.8435768
## λ2 0.8204748 0.8568206 0.8591629 0.8516156 0.8152430
## λ3 0.7889583 0.8637763 0.8217405 0.8631266 0.8653736
## λ4 0.6014878 0.7177231 0.8353661 0.6545402 0.8275223Linearity Detection
The Ramsey RESET test is an effective instrument for identifying misspecification in regression models, particularly concerning nonlinearity. If the model is determined to be nonlinear, it is essential to modify it by incorporating additional variables or exploring a more suitable data transformation.
ggplot(data.frame(ksi1, eta), aes(x = ksi1, y = eta)) +
geom_point(shape = 21, colour = "blue",fill = "blue", size = 2.5) +
labs(title = expression(paste("Scatterplot of ", eta, " vs ", xi[1])),
x = expression(xi[1]),
y = expression(eta)) +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5,size = 16, face = "bold"))
ggplot(data.frame(ksi2, eta), aes(x = ksi2, y = eta)) +
geom_point(shape = 21, colour = "blue",fill = "blue", size = 2.5) +
labs(title = expression(paste("Scatterplot of ", eta, " vs ", xi[2])),
x = expression(xi[2]),
y = expression(eta)) +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5,size = 16, face = "bold"))
ggplot(data.frame(ksi3, eta), aes(x = ksi3, y = eta)) +
geom_point(shape = 21, colour = "blue",fill = "blue", size = 2.5) +
labs(title = expression(paste("Scatterplot of ", eta, " vs ", xi[3])),
x = expression(xi[3]),
y = expression(eta)) +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5,size = 16, face = "bold"))
ggplot(data.frame(ksi4, eta), aes(x = ksi4, y = eta)) +
geom_point(shape = 21, colour = "blue",fill = "blue", size = 2.5) +
labs(title = expression(paste("Scatterplot of ", eta, " vs ", xi[4])),
x = expression(xi[4]),
y = expression(eta)) +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5,size = 16, face = "bold"))
#Linearity Test
cat("\nLinearity Test of η vs ξ1\n")##
## Linearity Test of η vs ξ1reset(eta~ksi1)##
## RESET test
##
## data: eta ~ ksi1
## RESET = 6.1373, df1 = 2, df2 = 381, p-value = 0.00238cat("\nLinearity Test of η vs ξ2\n")##
## Linearity Test of η vs ξ2reset(eta~ksi2)##
## RESET test
##
## data: eta ~ ksi2
## RESET = 3.8642, df1 = 2, df2 = 381, p-value = 0.02181cat("\nLinearity Test of η vs ξ3\n")##
## Linearity Test of η vs ξ3reset(eta~ksi3)##
## RESET test
##
## data: eta ~ ksi3
## RESET = 6.5776, df1 = 2, df2 = 381, p-value = 0.001555cat("\nLinearity Test of η vs ξ4\n")##
## Linearity Test of η vs ξ4reset(eta~ksi4)##
## RESET test
##
## data: eta ~ ksi4
## RESET = 3.8637, df1 = 2, df2 = 381, p-value = 0.02182cat("\nLinearity Test of η vs ξ1,ξ2,ξ3,ξ4\n")##
## Linearity Test of η vs ξ1,ξ2,ξ3,ξ4reset(formula = eta ~ ksi1 + ksi2 + ksi3 + ksi4, power = 2, type = "fitted")##
## RESET test
##
## data: eta ~ ksi1 + ksi2 + ksi3 + ksi4
## RESET = 3.9548, df1 = 1, df2 = 379, p-value = 0.04746PLS SEM Model
PLS is an iterative algorithm that breaks down the measurement model blocks and, in the next step, estimates the path coefficients in the structural model.
reg1 = lm(eta ~ ksi1 + ksi2 + ksi3 + ksi4)
summary(reg1)##
## Call:
## lm(formula = eta ~ ksi1 + ksi2 + ksi3 + ksi4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.98399 -0.46419 0.08742 0.48245 2.79007
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.650e-16 3.776e-02 0.000 1.00000
## ksi1 1.008e-01 5.322e-02 1.894 0.05894 .
## ksi2 2.424e-01 5.812e-02 4.170 3.77e-05 ***
## ksi3 3.045e-01 4.229e-02 7.201 3.23e-12 ***
## ksi4 2.055e-01 5.317e-02 3.866 0.00013 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.741 on 380 degrees of freedom
## Multiple R-squared: 0.4567, Adjusted R-squared: 0.451
## F-statistic: 79.86 on 4 and 380 DF, p-value: < 2.2e-16MARSPLS
MARSPLS is an extension of the partial least squares structural equation model (PLS-SEM) designed for scenarios where the relationship between latent variable scores is nonlinear or indeterminate.
nilai_nk = c()
nilai_degree = c()
nilai_minspan = c()
nilai_gcv = c()
nilai_rsq = c()
for (i in c(9,13,17)){
for (j in 1:3){
for (k in 1:3){
a = earth(x=xi,y=eta, nk=i,
minspan=k,endspan=k, degree=j, penalty=-1, pmethod="none")
nilai_nk = c(nilai_nk,a$nk)
nilai_degree = c(nilai_degree,j)
nilai_minspan = c(nilai_minspan,k)
nilai_gcv = c(nilai_gcv,a$gcv)
nilai_rsq = c(nilai_rsq,a$rsq)
}
}
}
optimumGCVMars=cbind(nilai_nk, nilai_degree, nilai_minspan,
nilai_gcv,nilai_rsq)
#sorting the minimum GCV values
GCVmin1=optimumGCVMars[order(optimumGCVMars[,ncol(optimumGCVMars)],decreasing = TRUE),]
cat("\nOrder Minimum GCV:\n")##
## Order Minimum GCV:GCVmin1## nilai_nk nilai_degree nilai_minspan nilai_gcv nilai_rsq
## [1,] 17 3 1 0.4580398 0.5407673
## [2,] 17 3 2 0.4595267 0.5392766
## [3,] 17 2 2 0.4603609 0.5384402
## [4,] 17 2 1 0.4605128 0.5382879
## [5,] 17 3 3 0.4626696 0.5361256
## [6,] 17 2 3 0.4649667 0.5338224
## [7,] 13 3 2 0.4763623 0.5223972
## [8,] 13 3 1 0.4764795 0.5222796
## [9,] 13 3 3 0.4772823 0.5214748
## [10,] 13 2 2 0.4779725 0.5207828
## [11,] 13 2 1 0.4782670 0.5204876
## [12,] 13 2 3 0.4785700 0.5201837
## [13,] 17 1 2 0.4894849 0.5092404
## [14,] 17 1 3 0.4897877 0.5089368
## [15,] 17 1 1 0.4916520 0.5070677
## [16,] 9 2 2 0.4996054 0.4990935
## [17,] 9 3 2 0.4996054 0.4990935
## [18,] 9 2 1 0.4998353 0.4988630
## [19,] 9 3 1 0.4998353 0.4988630
## [20,] 9 2 3 0.5001736 0.4985238
## [21,] 9 3 3 0.5001736 0.4985238
## [22,] 13 1 1 0.5011985 0.4974963
## [23,] 13 1 2 0.5016551 0.4970385
## [24,] 13 1 3 0.5019854 0.4967073
## [25,] 9 1 1 0.5107132 0.4879569
## [26,] 9 1 2 0.5108536 0.4878161
## [27,] 9 1 3 0.5109255 0.4877440best = as.data.frame(t(GCVmin1[1,]))
model_best = earth(x=xi,y=eta, nk=best$nilai_nk,
minspan=best$nilai_minspan,endspan=best$nilai_minspan,
degree=best$nilai_degree, penalty=-1)
cat("\nSummary of Basis Function:\n")##
## Summary of Basis Function:summary(model_best, style = "bf")## Call: earth(x=xi, y=eta, degree=best$nilai_degree, nk=best$nilai_nk,
## minspan=best$nilai_minspan, endspan=best$nilai_minspan, penalty=-1)
##
## eta =
## -1.232224
## - 0.217952 * bf1
## + 0.3002871 * bf2
## - 0.6868161 * bf3
## + 0.2694033 * bf4
## - 1.137398 * bf5
## + 0.2076821 * bf6
## + 0.07866258 * bf7 * bf1
## + 4.760677 * bf8 * bf1
## + 0.2545884 * bf9 * bf3
## - 0.01166664 * bf10 * bf3
## - 7.249754 * bf11 * bf4
## + 0.08063738 * bf12 * bf4
## + 0.9861144 * bf10 * bf3 * bf13
## + 1.27991 * bf10 * bf3 * bf14
## - 1.263136 * bf10 * bf3 * bf15
##
## bf1 h(-0.577095-ξ2)
## bf2 h(ξ2--0.577095)
## bf3 h(-0.574144-ξ3)
## bf4 h(ξ3--0.574144)
## bf5 h(-3.10291-ξ4)
## bf6 h(ξ4--3.10291)
## bf7 h(0.453342-ξ1)
## bf8 h(ξ1-0.453342)
## bf9 h(-1.78984-ξ1)
## bf10 h(ξ1--1.78984)
## bf11 h(-2.33084-ξ2)
## bf12 h(ξ2--2.33084)
## bf13 h(ξ4--1.24754)
## bf14 h(-1.24754-ξ4)
## bf15 h(ξ4--0.461635)
##
## Selected 16 of 16 terms, and 4 of 4 predictors
## Termination condition: Reached nk 17
## Importance: ξ2, ξ3, ξ4, ξ1
## Number of terms at each degree of interaction: 1 6 6 3
## GCV 0.4580398 RSS 176.3453 GRSq 0.5407673 RSq 0.5407673cat("\nSummary The Best MARS Model:\n")##
## Summary The Best MARS Model:summary(model_best)## Call: earth(x=xi, y=eta, degree=best$nilai_degree, nk=best$nilai_nk,
## minspan=best$nilai_minspan, endspan=best$nilai_minspan, penalty=-1)
##
## coefficients
## (Intercept) -1.2322243
## h(-0.577095-ξ2) -0.2179520
## h(ξ2- -0.577095) 0.3002871
## h(-0.574144-ξ3) -0.6868161
## h(ξ3- -0.574144) 0.2694033
## h(-3.10291-ξ4) -1.1373977
## h(ξ4- -3.10291) 0.2076821
## h(0.453342-ξ1) * h(-0.577095-ξ2) 0.0786626
## h(ξ1-0.453342) * h(-0.577095-ξ2) 4.7606770
## h(-1.78984-ξ1) * h(-0.574144-ξ3) 0.2545884
## h(ξ1- -1.78984) * h(-0.574144-ξ3) -0.0116666
## h(-2.33084-ξ2) * h(ξ3- -0.574144) -7.2497538
## h(ξ2- -2.33084) * h(ξ3- -0.574144) 0.0806374
## h(ξ1- -1.78984) * h(-0.574144-ξ3) * h(ξ4- -1.24754) 0.9861144
## h(ξ1- -1.78984) * h(-0.574144-ξ3) * h(-1.24754-ξ4) 1.2799101
## h(ξ1- -1.78984) * h(-0.574144-ξ3) * h(ξ4- -0.461635) -1.2631365
##
## Selected 16 of 16 terms, and 4 of 4 predictors
## Termination condition: Reached nk 17
## Importance: ξ2, ξ3, ξ4, ξ1
## Number of terms at each degree of interaction: 1 6 6 3
## GCV 0.4580398 RSS 176.3453 GRSq 0.5407673 RSq 0.5407673cat("\nEstimate variable importances:\n")##
## Estimate variable importances:evimp(model_best)## nsubsets gcv rss
## ξ2 15 100.0 100.0
## ξ3 14 65.5 65.5
## ξ4 13 47.4 47.4
## ξ1 11 38.2 38.2# Prediction
eta_pred <- predict(model_best)
# Calculate MSE and RMSE
MSE <- mean((eta - eta_pred)^2)
RMSE <- sqrt(MSE)
# Calculate AIC, AICc, dan BIC
k_mars=length(coef(model_best)[coef(model_best)!=0])-1
R2_mars <- summary(model_best)$rsq # R-squared
R2_adj <- 1 - ((1 - R2_mars) * (n - 1) / (n - k_mars - 1))
RSS <- sum((eta - eta_pred)^2)
AIC <- n * log(RSS/n) + 2*k
AICc <- AIC + (2*k_mars*(k_mars+1))/(n-k_mars-1) # AIC correction for small samples
BIC <- n * log(RSS/n) + log(n) * k_mars
# Show Accuracy
Akurasi = as.matrix(c(MSE,RMSE,R2_mars,R2_adj,AIC,AICc,BIC),ncol=1)
colnames(Akurasi) = "Goodness of Fit"
rownames(Akurasi) = c("MSE","RMSE","R2_MARS","R2_Adj","AIC","AICc","BIC")
cat("\nAkurasi Model\n")##
## Akurasi Modelprint(Akurasi)## Goodness of Fit
## MSE 0.4580398
## RMSE 0.6767864
## R2_MARS 0.5407673
## R2_Adj 0.5220994
## AIC -294.6076582
## AICc -293.3068452
## BIC -211.3090082# wich = 1 untuk Model Selection dan wich = 4 untuk Residual
plot(model_best, which = 1)
Save Output of MARSPLS
Save the MARSPLS model output into several csv files to facilitate
further analysis.
# Save Output of MARSPLS Model
write.csv(model_best$bx,"Basis Function.csv", row.names = FALSE)
write.csv(model_best$coefficients,"Coefficients of MARSPLS.csv",
row.names = FALSE)
result = cbind(ksi1,ksi2,ksi3,ksi4,eta,model_best$bx,model_best$fitted.values,model_best$residuals)
colnames(result) = c("ξ1","ξ2","ξ3","ξ4","η",variable.names(model_best$bx),"fitted","residual")
write.csv(result,"Result MARSPLS.csv", row.names = FALSE)