Nonparametric Path Analysis on Consumer Satisfaction and Consumer

Engagement in PT Pertamina

ADJI ACHMAD RINALDO FERNANDES, SOLIMUN, LAILIL MUFLIKHAH, AISYAH ALIFA,

ENDANG KRISNAWATI, NI MADE AYU ASTARI BADUNG,

ERLINDA CITRA LUCKI EFENDI

Brawijaya University

Jl. Veteran, Malang 65145 East Java

INDONESIA

Abstract: - The purpose of this research is to apply nonparametric path analysis on consumer satisfaction and

consumer engagement of PT Pertamina. The results of the analysis are expected to be able to provide an

estimate of the function in determining consumer satisfaction and consumer engagement of PT Pertamina. This

study uses primary data involving five variables, namely Digitalization (X1), Consumer Needs (X2), Consumer

Service (X3), Consumer Satisfaction (Y1), Consumer Engagement (Y3). Variable measurement technique is

done by calculating the average score on the items. Sampling in this study used a purposive sampling technique

with the respondent's criteria being company leaders. The result of this research is the estimation of

nonparametric Path function using MARS approach on various interactions. The best estimate of the function

of obedient behavior in paying credit is when it involves 3 variables, namely the digitization variable (X1),

Consumer Needs (X2), Consumer Service (X3) with a value ofgeneralized cross-validation The smallest (GCV)

obtained is 0.2833. The originality of this research is that the variables used are the results of DNA analysis

(Discourse Network Analysis), where the analysis extracts information from cyberspace which is then formed

as the main issue and becomes a variable. In addition, there is no previous research that examines

nonparametric path analysis on PT Pertamina's consumer satisfaction and engagement.

Key-Words: - Non-Paremetric Path, Consumer Satisfaction, Consumer Engagement, PT Pertamina, Function

Estimation, Flexible Model

Received: March 18, 2021. Revised: November 14, 2021. Accepted: December 19, 2021. Published: January 9, 2022.

1 Introduction

Path analysis is an extension of multiple linear

regression analysis that has more than one equation

in the form of a system. In path analysis, the terms

exogenous and endogenous variables are used. So

that the model obtained is unbiased and can be used

accurately, there are several assumptions that must

be met. However, often the data found in the field

do not meet the assumptions in the analysis to be

used, namely the assumptions of normality and

heteroscedasticity. Another thing that needs to be

considered before starting the analysis is the form of

the relationship between variables. There are times

when the relationship between variables is linear,

quadratic and even more than cubic. Therefore,

Path analysis based on nonparametric regression

is a path/regression approach that is suitable for the

pattern of the relationship between exogenous and

endogenous which is not/not yet known [1]. The

unknown pattern of the relationship between

endogenous and exogenous can be estimated using

the Spline function approach [2]. This study

developed a nonparametric regression-based path

analysis using a spline path estimator. To get the

regression curve estimation, Penalized Least Square

(PLS) optimization or Penalized Weighted Least

Square (PWLS) optimization is used [3].

Statistics has an important role in various fields,

one of which is in the field of BUMN. PT Pertamina

is a State-Owned Enterprise (BUMN) in charge of

managing oil and gas mining in Indonesia. In

general, PT Pertamina's market is divided into two,

namely retail (Business to Consumer - B2C) and

industrial (Business to Business - B2B). Until now,

PT Pertamina continues to innovate to improve

customer satisfaction, both B2C and B2B. The

Industrial and Marine Business at PT Pertamina, for

example, is currently developing a single point of

contact with the concept of "Pertamina One

Solution". This concept is expected to facilitate

service to consumers, especially in the industrial

market so that PT Pertamina can provide all the

needs and desires of consumers.

Customer satisfaction is a response from

consumers on the performance that has been given

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.3

Adji Achmad Rinaldo Fernandes,

Solimun, Lailil Muflikhah, Aisyah Alifa,

Endang Krisnawati, Ni Made Ayu Astari Badung,

Erlinda Citra Lucki Efendi

E-ISSN: 2224-2880

Volume 21, 2022

in accordance with customer expectations.

According to Wang in [4] customer satisfaction is a

level where the needs, desires and expectations of

customers can be met which will result in repeat

purchases or continued loyalty. Customer

satisfaction is the company's benchmark for how

things are going in the future or even there are some

things that must be changed because customers feel

dissatisfied or disadvantaged which will affect

customer engagement. Someone who returns to buy,

and will tell others about his good experience with

the product, it can be said that the customer is

satisfied [5].

Based on the description that has been

explained, it is very important to do research on

Nonparametric Path analysis with a spline approach

on PT Pertamina's consumer satisfaction and

engagement. This research is very important

because there are still weaknesses in the parametric

path which requires the relationship between

variables to be linear and the data pattern that must

be known. So it is necessary to develop a

nonparametric Path model that is able to estimate

the function when the variable relationship is not

linear and the data pattern is not known.

2 Literature Review

2.1 Nonparametric Path Analysis

Nonparametric path is an appropriate approach for

the pattern of relationships between exogenous and

endogenous, as well as fellow endogens whose form

is unknown, or there is no complete past

information about the pattern of relationships [1]. In

the Nonparametric Path approach, the estimation

form of the relationship pattern model is determined

based on the existing data patterns. The unknown

pattern of the relationship between endogenous and

endogenous can be estimated using the Spline

function approach or the Fourier Series [6]. The

spline approach has high flexibility and is able to

handle patterns of data relationships whose behavior

changes at certain sub-intervals [1][7]. This has also

been shown by [8] who compared the smoothing

spline function with the kernel numerically, and [9]

who compared the spline smoothing function with a

numerically better kernel spline function. If given

paired data following a simple Nonparametric

Path model i.e:

1 1.1 1 1

()

i i i

y f x





2 1.2 1 2.2 2 1

( ) ( ) ; 1,2,...,

i i i i

y f x f y i n



   

2.2 Spline Regression

Spline regression is a nonparametric regression

method that aims to minimize diversity and estimate

the behavior of data that tends to differ. The spline

approach has the ability to overcome data patterns

that show a sharp rise or fall with the help of knot

points, and the resulting curve is relatively smooth.

Knot points are joint points that indicate changes in

data behavior patterns [1]. Spline is a piecewise

polynomial of order q and has a continuous

derivative with knots of order (q-1) [10][11]. In a

univariate spline with K knots, the base function is

as follows:

In general, the spline approach states the

relationship between p predictors and a single

response with the model shown as follows:

   

 

1 1 2 2 ; 1,2, ,

i i i p pi

y f x f x f x i n



     

The equation can be written in the following form:

 

, 1, 2, ,

i i ji i

y f x i n





  



approached with a trucated spline function

which has

order of polynomials and

knots then

the equation can be written as:

 

1 1 1

;

1, 2, ,

pq rq

i jl jl jh ji jh i

j i h

y x y x t

  

  



    







  

   

qji jh ji jh

ji jh

x t x t

xt xt













Information:

: response to observation to-

: predictor to-

on the observation



: intercept



: polynomial coefficient on predictor th

and

the order of

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DOI: 10.37394/23206.2022.21.3

Adji Achmad Rinaldo Fernandes,

Solimun, Lailil Muflikhah, Aisyah Alifa,

Endang Krisnawati, Ni Made Ayu Astari Badung,

Erlinda Citra Lucki Efendi

E-ISSN: 2224-2880

Volume 21, 2022

: truncated coefficient on the th predictor

and knots to-

: value of knots on predictor

and knots to-

: number of knots

: order of polynomials truncated spline

: number of predictors

: many observations



: random error on the th observation

2.3 Multivariate Adaptive Regression Spline

(MARS)

MARS is an approach to non-parametric regression.

MARS was first introduced by [10]. The MARS

model is focused on overcoming the problem of

high dimensions and discontinuities in the data.

MARS is able to estimate the contribution of the

basis function to the response variable, by capturing

not only the adaptive effect but also the interaction

effect between predictors. MARS is an extension of

the approachRecursive Partition Regression (RPR)

which produces a discontinuous model at knots.

According to [10] the MARS method can be used

for as many predictor variables.

According to [12] things that need to be considered

in building the MARS model are:

a. Knot, is the value of the predictor variable

when slope a regression line undergoes a

change that can be defined as the end of one

segment as well as the beginning of another

segment. At each knot point, it is expected

that there will be continuity of the basis

function between oneregion with region

other. The maximum observations between

knots (MO) are 0, 1,2 and 3 observations

[13].

b. The basis function (B) is the interval between

successive knots. In general, the selected

basis function is a polynomial with a

continuous derivative at every knot point. The

maximum allowable basis function is 2 to 4

times as many predictor variables. There is a

limitation of the base function used so that the

resulting model is not too complex.

c. Interaction is the result of cross multiplication

between correlated variables. The maximum

number of interactions (MI) allowed is 1, 2 or

3. If MI>3 the resulting model is increasingly

complex and difficult to interpret.

The MARS model is generally expressed in the

following equation:

 

0 1 ( , )

MKm

i m k km v k m km i

y a a S x t







    





The estimation of the MARS model is written in the

following equation:

 

0 1 ( , )

MKm

i m k km v k m km

f x a a S x t





   





with,

 

: base function

: coefficient of the base function



: coefficient of the th function basis

: number of base functions

: number of interactions on the th basis

function

: is +1 if the knot is to the right of the

subregion, and -1 if the knot is to the left of

the subregion

( , )v k m

: predictor variable to-

, select to

and

subregion to

: knot value of predictor variable

2.4 MARS Best Model Selection

According to [10], in determining the best MARS

model, it is by looking at the smallest generalized

cross-validation (GCV) value. The following is the

GCV formula:

3 Methodology

This study uses primary data involving five

variables, namely Digitalization (X1), Consumer

Needs (X2), Consumer Service (X3), Consumer

Satisfaction (Y1), Consumer Engagement (Y3). The

research instrument used in primary data is a

questionnaire with a Likert scale. Variable

measurement technique is done by calculating the

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.3

Adji Achmad Rinaldo Fernandes,

Solimun, Lailil Muflikhah, Aisyah Alifa,

Endang Krisnawati, Ni Made Ayu Astari Badung,

Erlinda Citra Lucki Efendi

E-ISSN: 2224-2880

Volume 21, 2022

average score on the items. Sampling in this study

used a purposive sampling technique with the

respondent's criteria being company leaders. The

total respondents in this study were 200

respondents. This study uses nonparametric path

analysis with the MARS method approach. The

combination of the base functions used in this study

is 2 to 4 times the exogenous variable with a

minimum observation (MO) of 0.1,2,3 and a

maximum interaction of 1,2,3.

4 Result

4.1 Linearity Check

Information about the pattern of relationships

between variables is very necessary in statistical

modeling because the information is used to

determine the method used is a parametric

regression approach or a nonparametric regression

approach. In nonparametric regression modeling.

This study uses data exploration using a scatter plot

to check linearity.

Fig. 1: Scatter Plot Exogenous variable with

variable Y1

Figure 1 shows that the exogenous variable to the

Y1 variable has unknown patterns of relationships.

This indicates that the use of nonparametric paths

with the MARS approach for model estimation is

appropriate.

Fig. 2: Scatter Plot Exogenous variable with

variable Y1

Figure 2 also shows that the exogenous variable

to the Y2 variable has unknown patterns of

relationships. This indicates that the use of

nonparametric paths with the MARS approach for

model estimation is appropriate.

4.2 Nonparametric Path Function Estimation

with MARS Approach

Based on linearity examination, it was found that

the relationship between variables was unknown as

well as random data patterns so that the

nonparametric path approach using the MARS

method was appropriate. In the estimation of the

Nonparametric Path function with the MARS

approach, it is formed by a combination of the Basis

Function, Maximum Interaction (MI) and Maximum

Observation (MO) which is shown in Table 4.2. The

basis function used is 2 to 4 times the number of

predictor variables. There is a limitation on the basis

function used, so that the resulting model is not

complex. Because this study uses 3 exogenous

variables, the basis functions used are 6, 9, 12. The

maximum interactions used are 1,2 and 3. The

maximum observations used are 0.1,2 and 3. Table

Table 1. MARS Nonparametric Path GCV Values

Inter

action

Function

Base

Function

GCV

Total

GCV

Function

0.0963

0.2889

Function

0.1925

Function

0.1064

0.2835

Function

0.1771

Function

0.1010

0.2833

Function

0.1823

Based on Table 1, the best estimated function with 3

interactions with the smallest GCV value in the first

function is 0.1010 and the second function has the

smallest GCV value of 0.1823. The best estimates of

the first and second functions are obtained when

involving three exogenous variables,

namelyDigitization (X1), Consumer Needs (X2),

Consumer Services (X3). Best first function

estimation in model development Path

nonparametric with MARS approach is shown in

equations (1) and (2).

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Adji Achmad Rinaldo Fernandes,

Solimun, Lailil Muflikhah, Aisyah Alifa,

Endang Krisnawati, Ni Made Ayu Astari Badung,

Erlinda Citra Lucki Efendi

E-ISSN: 2224-2880

Volume 21, 2022

(1)

(2)

In nonparametric path function estimation

with the MARS approach, there are several

statistics used to measure the magnitude of the

error from the obtained function estimation.

Figure 4.1 shows the magnitude of the

generalization R2 (GRSq) and R2 (RSq) when a

number of exogenous variables are used. The

greater the value of generalized R2, the better the

estimated function obtained.

Fig. 3: MARS Nonparametric Path Model Selection

Graph with 3 interactions

Based on Figure 3, it can be seen that the

selected model is a model that uses 3 exogenous

variables, namely the generalizationR2 is 0.3426

and R2 is 0.4531. Figure 3 shows if the predictor

variables used are more than three, then there is a

decrease in generalizatioR2 and if the predictor

variables used are less than three then the values

of generalizatioR2 and R2 are getting smaller.

The optimal basis functions of Path MARS

equations (1) and (2) are taken using a backward

process based on the minimum GCV value for

each response partially. In selecting this optimal

basis function, the minimum GCV value for

response 1 is 0.1010 and for response 2 is

0.1823. The selection of the optimal basis

function is important, because when it involves

too many basic functions, the model obtained

will be more complex (not parsimony).

5 Conclusion

Based on the results and discussions conducted in

the study, it can be concluded that:

1. The estimation of the nonparametric path

model on the compliance behavior data is

when it involves 3 interactions. The smallest

GCV value for the estimation of the first

function is 0.1010 while the smallest GCV

value for the second estimation function is

0.1823. In addition, the generalized value

obtained R2 as big as 0.3426 and R2 of 0.4531

with a total GCV value of 0.2833. Model

estimation with minimum GCV value is

obtained when involving three exogenous

variables for the first and second functions.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.3

Adji Achmad Rinaldo Fernandes,

Solimun, Lailil Muflikhah, Aisyah Alifa,

Endang Krisnawati, Ni Made Ayu Astari Badung,

Erlinda Citra Lucki Efendi

E-ISSN: 2224-2880

Volume 21, 2022

2. Variables that have a dominant influence on

consumer satisfaction and engagement are the

digitization variable (X1), Consumer Needs

(X2), and Consumer Service (X3). This can

be used as information for PT Pertamina in

increasing consumer satisfaction and

engagement. Consumer satisfaction and

engagement will increase if Digitization,

Consumer Needs, and Consumer Service.

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DOI: 10.37394/23206.2022.21.3

Adji Achmad Rinaldo Fernandes,

Solimun, Lailil Muflikhah, Aisyah Alifa,

Endang Krisnawati, Ni Made Ayu Astari Badung,

Erlinda Citra Lucki Efendi

E-ISSN: 2224-2880

Volume 21, 2022