Maximum Likelihood Estimation Approach using the CB-SEM Method: Case Study of Service Quality

Abstrak. Penelitian ini


INTRODUCTION
The level of service, satisfaction and loyalty cannot be measured directly.Therefore, to analyze causal relationships in structural unobserved variables, analytical methods are needed that take into account the nature of these relationships (Rasoolimanesh et al., 2021;Zhang et al., 2021).One method that can be used to analyze causal relationships as discussed above is Structural Equation Modeling (SEM) (Bullock et al., 1994;Lowry & Gaskin, 2014).According to Legate et al. (2023) and Zyphur et al. (2023) One of the advantages of SEM is the ability to model constructs as latent variables or variables that are not measured directly, but are estimated in the model from the measured variables which are assumed to have a relationship with the latent variable.
Generally, there are two types of SEM that are widely known, namely Covariance Based-Structural Equation Modeling (CB-SEM) developed by Jöreskog et al. (2016) and Partial Least Square Structural Equation Modeling (PLS-SEM) often called variance or componentbased structural equation modeling developed by Hair Jr et al. (2017).CB-SEM aims to estimate structural models based on strong theoretical studies to test causal relationships between constructs as well as measure the feasibility of the model and confirm it according to empirical data (Daryono et al., 2023;Hidayat & Wulandari, 2022).CB-SEM demands a strong theoretical base, meets various parametric assumptions and meets model feasibility tests (goodness of fit).Seeing this phenomenon, researchers are interested in examining the influence of service levels on visitor satisfaction and loyalty in the Sariringgung Beach tourist area using the Maximum Likelihood estimation method with the CB-SEM approach to test the theory and obtain the truth of the test with a series of complex analyzes.

METHOD
The data needed in the research is primary data with an infinite population, namely visitors to the Sariringgung Beach tourist area in 2018, so a sample size of (n = 200) was taken by applying a simple random sampling technique, namely random sampling of visitors to Sariringgung Beach.Primary data is data collected directly by the researchers themselves by giving questionnaires to respondents who visited Sariringgung Beach as a case study.
The steps in this research method are as follows: 1. Model Specifications Designing structural models and measurement models used to carry out testing.This research consists of 3 latent variables, namely loyalty (η1) and satisfaction (η2) and service (ξ1) and 13 variables observed are X1, X2, X3, X4, X5, X6.

Construction of a path diagram
Constructing a path diagram means building relationships between latent variables, namely ξ1, η1, η2 and writing parameter symbols for each loading factor value.

Test the overall suitability of the model
Evaluate the results of the goodness-of-fit test to see the feasibility of the model using the maximum likelihood estimation method (Deva & Husein, 2017;Levene & Kononovicius, 2021).This comparison was carried out by looking at the Goodness Of Fit (GOF) values in the Chi-Square, NCP, GFI, RMSEA, AGFI, PNFI and NFI test statistics.

Parameter Estimation
This research uses the maximum likelihood method with the following steps: a. Forming a likelihood function derived from structural equations.b.Maximize the function obtained to obtain estimated parameters.c.Find the first derivative of the maximum likelihood function ln for the parameter to be estimated and equate it to zero. 5. Using Lisrel 8.80 software to obtain estimated values for parameters γ, β and λ.

Testing the significance of parameters in the measurement model
Evaluation is carried out by looking at the t value of factor loadings ≥1.96 and standard factor loadings ≥0.05. 7. View direct, indirect effects and calculate the total effect between latent variables.8. Evaluation of the CB-SEM model with Lisrel 8.80 software based on the coefficient of determination value in the model.

RESULT AND DISCUSSION
a.Estimated Parameters Suppose η and ξ are multinormal random variables of size n with   = (η1, η2, …, ηm) and   = (ξ1, ξ2, …, ξn), because they are assumed to be normal then η~N (βη-ξ; Ʃ).So the probability density function is: If x= β η-ξ then the joint density function for stochastically independent and identical random samples at x is as follows: with the likelihood function: Based on the derivative properties of the matrix f(x)= x΄Ax then   = (A+A΄)x, so that the derivative of β is obtained as follows: 0= -0 -0 -0 + ξ΄΄ξ)

𝜕𝜸
; with Based on the derivative properties of the matrix f(x)= x΄Ax then   = (A+A΄)x, so that the derivative of γ is obtained as follows: For example, Y is a random variable with   = (y1, y2, …, yp), because it is assumed to be normal then Y~N (yη; ).The probability density function is: If x = yη then the joint density function for random samples is stochastically independent and identical at x, as follows: With likelihood function: = (A+A΄)x, so that the derivative of   is obtained as follows: Let X be a random variable with   = (x1, x2, …, xq), because it is assumed to be normal then X~N (xξ; ).The probability density function is: ( −   ξ)΄( −   ξ)} If x =   ξ then the joint density function for stochastically independent and identical random samples at x is as follows: with the likelihood function: Estimation of parameter x = (A+A΄)x, so that the derivative of   is obtained as follows: In determining the estimated parameters γ, β and Γ using the maximum likelihood method, a closed form for the estimated parameters was not obtained so this was overcome using the help of Lisrel 8.80 software.

b. Maximum Likelihood Method Parameter Estimation with Lisrel 8.80
The results of the standardized solution and model estimation from Liserel 8.80 are as follows:  where the smaller the chi-square value, the better and the chi-square value obtained can be considered quite good.The RMSEA value obtained is 0.038, which means close fit (not good), while for the p-value obtained, it is 0.06691, which is said to be good because the pvalue is > 0.05, so it can be said that the data supports the desired model estimation.
The criteria for the significance test on the Standardized Loading Factor (SLF) and t-value are that the standard factor loading size is ≥ 0.5 and the t-value is ≥ 1.96, so it is said to be very significant or valid (Shrestha, 2021).To measure reliability, Construct Reliability (CR) ≥ 0.70 and Variance Extracted (VE) ≥ 0.50 are used (Elias et al., 2022;Fang et al., 2022).For more details, information regarding the validity and reliability of indicators in the measurement model can be seen in Table 1 below: This value also shows that service (ξ1) has a positive effect on satisfaction (η1) and loyalty (η2).
The indirect influence value is obtained by multiplying the path coefficients from service (ξ1) to satisfaction (η1) and from satisfaction (η1) to loyalty (η2).So the results obtained from multiplying the indirect influence path coefficient with the intermediate variable satisfaction (η1) are as follows: Indirect influence through satisfaction (η1) indirect influence = (ξ1 → η1) x (η1 → η2) = (0.77) x ( 0.50) = 0.385 Based on these results, it can be concluded that service (ξ1) significantly influences loyalty (η2) through the intermediary variable satisfaction (η1) of 0.385.The total influence of service (ξ1) on loyalty (η2) with the intermediary variable satisfaction (η1) is 0.735, which means that the influence of service on customer loyalty through satisfaction is 73.5%.

d. Evaluation of the CB-SEM model
After analysis using the maximum likelihood method with Lisrel 8.80, an R-Square value was obtained which can state that service variability (ξ1) explains 0.60 of customer satisfaction (η1), which means that satisfaction variability (η1) can be explained by 60% by service variability.(ξ1) and service variability (ξ1) explain 0.54 of the variability in customer loyalty (η2), which means that variability in customer loyalty (η2) can be explained by service variability (ξ1) of only 54%.

CONCLUSION
Test the suitability of the entire model using goodness of fit criteria each value shows a good match, namely χ 2 (79.45) with the smaller the value, the better, NCP (17.45) with the smaller the value, the better, GFI (0.94) with the criterion value > 0.9, RMSEA (0.038) with a value measure of 0.05 ≤ good fit ≤ 0.08, AGFI (0.92) with a criterion value measure > 0.9, PNFI (0.77) with a value measure that increases the higher the better and NFI (0 .97) with a criterion value > 0.9.So it can be concluded that the overall model is good.The results of the analysis show that the influence of total service (ξ1) on loyalty (η2) with the intermediary variable satisfaction (η1) is 0.735 which is This means that service has a big influence on customer loyalty through satisfaction namely 73.5%.ξ1 η1 η2 Based on the derivative properties of the matrix f(x)= x΄Ax then Based on the derivative properties of the matrix f(x)= x΄Ax then

Figure 2 .
Figure 2. Path diagram of estimation results.Based on Figures1 and 2, it shows that the results of the standardized solution and estimates have the same close relationship value for each latent variable, namely from the service variable (ξ) to the satisfaction variable (η1) of 0.77, the service variable (ξ) to loyalty (η2) is 0.35 and the satisfaction variable (η1) to the loyalty variable (η2) is 0.50.Then the magnitude of the influence value on each indicator variable has almost the same value or not much different.With a sample size of 200, a chi-square value of 80.15 was obtained, where the smaller the chi-square value, the better and the chi-square value obtained can be considered quite good.The RMSEA value obtained is 0.038, which means close fit (not good), while for the p-value obtained, it is 0.06691, which is said to be good because the pvalue is > 0.05, so it can be said that the data supports the desired model estimation.

Table 1 .
Measurement Model Significance Test

Table 1
(Radam et al., 2022)sults of calculating validity and reliability in the measurement model.The results show that all indicators on the latent variables have met the validity criteria with an SLF value for each indicator/construct of ≥ 0.5 with a t-value ≥ 1.96(Radam et al., 2022).For each latent variable reliability of service, satisfaction and loyalty has a value of CR ≥ 0.70 and VE ≥ 0.5 so it can be said that the reliability of the latent variable service, satisfaction and loyalty in the measurement model is good.Apart from that, reliability can also be shown by the black standard error value on the path diagram, so it can be said that the indicator meets the reliability criteria in the measurement model.

Table 2 ,
it shows that the service path coefficient (ξ1) on satisfaction (η1) is 0.77, service (ξ1) on loyalty (η2) 0.35.So it can be concluded that the direct influence between