Comparison of Parameter Estimator Efficiency Levels of Path Analysis with Bootstrap and Jack Knife (Delete-5) Resampling Methods on Simulation Data

In practice, the assumptions of normality are often not met, this causes the estimation of the resulting parameters to be less efficient. Problems with the assumption that normality is not met can be overcome by resampling. The use of resampling allows data to be applied free from distribution assumptions. In this study, a simulation study was carried out by applying bootstrap resampling and jackknife resampling (delete-5) on path analysis assuming that the normality of the alignment was not met and the resampling amount set at 1000 with the degree of closeness between variables consisting of low closeness, medium closeness, high closeness and closeness level representing the level low to high closeness. Based on the simulation results, the resampling 1000 magnitude is able to overcome the problem of the assumptions of unmet normality. In addition, a comparison between bootstrap and jackknife resampling for conditions of side normality assumptions is not fulfilled and the closeness of the relationship between low, medium, high and closeness variables representing low to high closeness levels, the estimation results of path analysis parameters obtained by resampling jackknife are more efficient than resampling bootstrap.


INTRODUCTION
Path analysis is an analysis that can facilitate complex forms of relationships and the presence of intervening endogenous variables. Path analysis is the development of regression analysis. Path analysis has several assumptions, one of which is the residual normality or, in other words, the residual normal distribution. But, sometimes, the assumption of normality is not fulfilled, causing the resulting hypothesis testing less efficient. The problem of unfulfilled normality assumption can be overcome by resampling. The use of resampling allows data to be free from distribution assumptions or not to require normality assumption [11].
Based on previous studies, the Jackknife resampling method is more appropriate for estimating parameter values and parameter confidence interval values in the regression analysis with a narrow range of confidence intervals [1]. In algorithm studies of bootstrap and jackknife (delete-one) resampling methods for parameter estimation in the regression analysis, it is known that the parameter bias, standard errors, and confidence interval of Jackknife confidence are 354 Adji Achmad Rinaldo Fernandes greater than the Bootstrap [8]. Therefore, in this research, we are interested in comparing the bootstrap and jackknife resampling methods in path analysis.
The data used in this research resulted from a simulation study. Based on the simulation study, Bootstrap resampling method can be used as an alternative method that can produce a very close regression parameter estimation to population parameters and fairly narrow confidence interval [12].
In contrast to previous studies, this research applied resampling methods in path analysis. Besides, this research was also intended to determine which resampling method is more efficient (bootstrap or jackknife (delete-5)) with a simulation study. Efficiency is relatively used to compare the variance of predictors between bootstrap and jackknife resampling methods.

LITERATURE REVIEW
Path analysis is an analysis developed from a linear regression analysis. According to [6], path analysis is an analysis used to evaluate direct and indirect effects through another cause. [9] stated that path analysis is a regression analysis with standardized variables. Another understanding suggests that path analysis determines the strength of the path shown in a path diagram [5]. Based on some of these notions, path analysis is an analysis used to study the direct and indirect effects of certain variables with standardized variables and it employs path diagrams to illustrate the relationships between variables.
According to [7], there are variable terms in path analysis. Exogenous variables are variables that are not determined by other variables in the model while endogenous variables are variables that are partly determined by other variables in the model.

Types of Effect on Path Analysis
According [9], there are three types of effect on path analysis. The three types of effect on path analysis are illustrated with the following path diagrams.

Direct Effect
Direct effects refer to the direct impact of exogenous variables on endogenous variables without intermediaries of other variables. Here is the path diagram illustrating direct effects.

Indirect Effect
Indirect effects refer to the impact of exogenous variables on endogenous variables that occurs through the intermediaries of other variables. , it can be seen that path diagrams illustrate the causal relationships of the four variables. Variable X is an endogenous variable while Variable Y 1 , Y 2 , and Y 3 are endogenous variables. Variable Y 3 is a pure endogenous variable while Variable Y 1 and Y 2 are intervening endogenous variables.

Path Analysis Model
The path analysis in this research is following the path diagram as shown in Figure 2.3. (c). The path analysis model formed according to the diagram can be written as follows.
In Equation (2.1) to 2.3, i = 1, 2, ..., n, and n is the number of observations. In path analysis, the variables to be used are standardized first. The purpose of standardization is to equalize the averages and variances so that the path coefficient can be compared with other path coefficients [11]. Data standardization is done by standardizing the average to 0 and the variance to 1 with the following formula [6].
In the form of a matrix, Equation (2.5) can be written like Equation (2.6).
The above matrix form can be written into the following equation.
In the standardization process, the coefficient  is equivalent to the correlation coefficient.
Furthermore, estimation of path coefficients can be done based on path analysis model.

Path Coefficient Estimation
Estimation is done to obtain the path coefficient on the model. Path coefficients show the extent of influence between variables. One estimation of path analysis parameters is the Ordinary Least Square (OLS) method. The OLS method can only be used if linearity assumptions are met. Parameter estimation using the OLS method is done by minimizing the number of remaining squares.
The equation of remaining squares above is derived to  and equates to zero as follows.

Path Analysis Asumption
According to [4] and [10], assumptions underlying path analysis are as follows:

Linearity Assumptions
The path analysis model assumes that the relationship between variables is linear. According to [4], linearity assumptions can be tested using the Ramsey's Regression Specification Error (RESET) test. Here are the steps for Ramsey's RESET. a. For example, the first equation is a linear r ssion model parameter estimation from the equation is conducted. After that, the determination coefficient is calculated with the following formula (2.10).
For example, the second equation is a quadratic model, Similar to the previous stage, parameter estimation is then also made in a quadratic model. Moreover, the determination coefficient is calculated with the following formula (2.11).

Central Limit Theorem
The central limit theorem has an important role in the sampling distribution. One of its roles is to estimate the parameters of the average and variance of a population. According to Mandenhall (1981) If n U (n-sized random variable) with n > 30, it can be approximated by the standard normal distribution.

Resampling
Resampling is the process of drawing repeated samples from existing or original samples so that a new sample is obtained. The new sample is obtained from original sized samples taken at random either with or without returns. The re-sampling method can be applied as an alternative if the number of observations does not meet the research need which can cause the parameter estimation incorrect. Besides, the implementation of the resampling method allows the validity of data that is free from assumptions or, in other words, does not need the normality assumption [11]. Adji Achmad Rinaldo Fernandes  : the size of resampling

Jackknife
Jackknife is a resampling method introduced by Quenouille in 1949 for estimation of bias. Then, in 1958, Tukey introduced jackknife to estimate standard deviations. The principle of the jackknife method is to eliminate five observations from n-sized samples and take other observations without returns. In the next stage, the deleted sample is returned and so are the other five observations until all observations from the population have a chance to be deleted. Based on the process, the Jackknife resampling process generally can be seen as shown in the following   : Average parameter estimator on the i-Jackknife process

Relative Efficiency
Comparison of the resampling methods is measured based on the relative efficiency value. According to [13], relative efficiency is calculated by comparing the variance between the twoparameter estimators. The relative efficiency of the two estimators can be written as follows.

METHODOLOGY
The data used in this research are simulation study data with one exogenous variable, two intervening endogenous variables, and one pure endogenous variable. The exogenous variable was raised and standardized while the endogenous variables were determined based on the values of exogenous variables, path coefficients, and residuals generated following the Weibull distribution. The path coefficients in this research were raised at several levels of closeness with a

RESULT AND DISCUSSION
The relative efficiency results for the low level of closeness can be seen in Table 4.1. From Table 4.1., it can be seen that the relative efficiency values of all path analysis coefficients were > 1, indicating that the Jackknife method had a smaller variance than the Bootstrap method. Therefore, it can be concluded that path analysis with the Jacknife resampling method is three times more efficient than that with the bootstrap resampling method.
The relative efficiency results for the medium level of closeness can be seen in Table 4 Table 4.3., it can be seen that the relative efficiency values of all path analysis coefficients were > 1, indicating that the Jackknife method had a smaller variance than the Bootstrap method. Therefore, it can be concluded that path analysis with the Jacknife resampling method is three times more efficient than that with the bootstrap resampling method.
The relative efficiency results for the low-to-high level of closeness can be seen in Table 4.4.  Table 4.4., it can be seen that the relative efficiency values of all path analysis coefficients were > 1, indicating that the Jackknife method had a smaller variance than the Bootstrap method. Therefore, it can be concluded that path analysis with the Jacknife resampling method is three times more efficient than that with the bootstrap resampling method. In contrast to [8] study, the Jacknife resampling in this research was done by removing one observation only so that the resulting Jacknife samples were less than the bootstrap samples. This caused Bootstrap resampling to be better than Jacknife resampling.

CONCLUSION
Based on the simulation study that has been done, the use of Bootstrap and Jackknife resampling methods on data with normality assumption is not fulfilled, indicating that both bootstrap and jackknife resampling methods can be applied and able to overcome normality assumption. The calculated relative efficiency results in various closeness levels of relationship show that the Jacknife resampling method (delete-5) is three times more efficient than the Bootstrap resampling method.