CONTACT:  

Siti Hadijah Hasanah          sitihadijah@ecampus.ut.ac.id         Statistics Department, Faculty of Science and 

Technology, Universitas Terbuka, Tangerang Selatan 15437, Indonesia  

 

       The article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.51-58    

 

Multivariate Adaptive Regression Splines (MARS)  

for Modeling The Student Status at Universitas Terbuka 
 

 

Siti Hadijah Hasanah 

Department of Statistics, Universitas Terbuka, Indonesia 
 

 
 

Article history: 

Received Dec 23, 2020 

Revised May 10, 2021 

Accepted May 30, 2021 

 

 

Kata Kunci:  

Fungsi basis, GCV, 

Multivariate, Recursive, 

Splines 

 

Abstrak. Multivariate Adaptive Regression Splines (MARS) 

digunakan untuk memodelkan status mahasiswa aktif program studi 

Statistika Universitas Terbuka serta untuk mengetahui faktor-faktor 

yang mempengaruhi variabel respon. Penelitian ini terdiri dari 9 

variabel yaitu jenis kelamin, umur, pendidikan, status pernikahan, 

pekerjaan, tahun registrasi awal, jumlah registrasi, SKS, dan IPK, 

tetapi setelah dilakukan pemodelan menggunakan metode MARS 

maka variabel penjelas yang dapat mempengaruhi variabel respon 

adalah tahun registrasi awal, jumlah registrasi, IPK, dan SKS. 

Berdasarkan hasil output R dan menggunakan selang kepercayaan 

95%, maka masing-masing fungsi basis 1 sampai 10 adalah signifikan 

secara parsial dengan nilai-p dari fungsi basis 1-10 lebih kecil dari 

0,05 dan secara simultan dengan nilai nilai-p lebih kecil dari 0,05, 

sehingga model di atas memiliki pengaruh yang signifikan secara 

parsial maupun simultan terhadap variabel respon. Dari hasil tersebut 

maka disimpulkan bahwa model MARS layak digunakan untuk 

menentukan faktor-faktor yang mempengaruhi status aktif siswa. 

 

Keywords:  

Basis function, GCV, 

Multivariate, Recursive, 

Splines 

 

 

Abstract. Multivariate Adaptive Regression Splines (MARS) used to 

model the active student’s status in the Department of Statistics at 

Universitas Terbuka and determine the factors that influence the 

response variable. This study consists of 9 variables, namely gender, 

age, education, marital status, job, initial registration year, number of 

registrations, credits, and GPA, but after modeling using the MARS 

method, the explanatory variable can affect the response variable is 

the initial registration year. Several registrations, GPA, and credits. 

Based on the results of the R output and using a 95% confidence 

interval, each base 1 to 10 function is partially significant with the p-

value of the base 1-10 function being smaller than 0.05 and 

simultaneously with a smaller p-value. of 0.05, so that the above 

model has a significant effect partially or simultaneously on the 

response variable. From these results, it is concluded that the MARS 

model is suitable for determining the factors that affect the active 

status of students. 

  

 

How to cite: 

S. H. Hasanah, “Multivariate Adaptive Regression Splines (MARS) For Modeling The Student 

Status at Universitas Terbuka”, J. Mat. Mantik, vol. 7, no. 1, pp. 51-58, May 2021. 

 

 

  

Jurnal Matematika MANTIK 
Vol. 7, No. 1, May 2021, pp. 51-58 
 
ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:sitihadijah@ecampus.ut.ac.id
https://doi.org/10.15642/mantik.2021.7.1.51-58
https://doi.org/10.15642/mantik.2021.7.1.51-58


Jurnal Matematika MANTIK 

Vol. 7, No. 1, May 2021, pp. 51-58 
 

52 

 

1. Introduction 
 

Universitas Terbuka has 39 service offices spread from Sabang to Merauke. The 

advantages of Universitas Terbuka are that there are no restrictions on the period of 

completion of the study, no implementation of the dropout system, no restrictions on the 

year of graduation or age, and registration time throughout the year [1]. Based on the 

advantages and ease of studying at Universitas Terbuka there is a problem that must be 

faced one of them is the active status of UT students, students who have been registered at 

Universitas Terbuka are given the facility to register at any time so many students with 

inactive status in a semester and do not know when status as an active student again. This 

study was conducted by analyzing the factors that affect the status of student active, 

especially in the Department of Statistics based on several indicators, namely age, gender, 

education, marital status, employment status, year of initial registration, number of 

registrations, credits, and GPA. 

Several statistical methods used to determine the effect of explanatory variables on 

categorical response variables include the Binary Logistic Regression method and the 

Multivariate Adaptive Regression Spline (MARS) [2]. Both methods are included in 

regression analysis, which is a statistical method that studies the mathematical relationship 

pattern between one or more explanatory variables and the response variable [3]. Logistic 

regression has the advantage that it does not need to fulfill assumptions [4], can be used on 

non-linear data, and is easy to interpret. However, logistic regression has a weakness, 

namely that there is multicollinearity between the explanatory variables which is one of the 

problems that make the parameter estimation results unstable [5]. The MARS method is 

able to accommodate interactions between variables even for high-dimensional data [6], 

[7]. 

MARS is a relatively flexible classification method used to determine the pattern of 

relationships between explanatory variables and response variables without using initial 

assumptions about the form of functional relationships. This method is a complex 

combination of spline and recursive partitioning and involves a high data dimension, 

namely the number of observations and a large number of variables [8]. In addition, MARS 

can effectively explore the hidden non-linear relationship between response variables and 

predictor variables as well as the interaction effects on complex data structures [9]. MARS 

can solve the problem of high and non-continuous dimensions in nodes. This method can 

analyze 50-1000 amounts of data with 3-20 explanatory variables [10]. Dynamic non-linear 

patterns and interactions can be explained by this method [11]. MARS can also be applied 

in credit assessment [12], [13], transportation [9], and software engineering [14]. So the 

method we use to determine the active status of Universitas Terbuka students in this study 

is Multivariate Adaptive Regression Splines (MARS). This research is an alternative study 

program to determine the best step in overcoming the many inactive student statuses, 

especially in the Statistics Study Program. 

 

2. Multivariate Adaptive Regression Splines (MARS) 
 

Suppose 𝑦  a single response variable depends on 𝑛 predictor variables 𝑥, where            
𝑥′ = (𝑥1,𝑥2,… ,𝑥𝑛) then the regression model is : 

𝑦 = 𝑓(𝑥1,𝑥2,…,𝑥𝑛)+ 𝜀 
   (1) 

Assume 𝑓 is a linear combination of the base functions of 𝐵𝑚(𝑥),  with the base of 
each subregion  𝑚 = 1,2,…,𝑀. 

𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑚𝐵𝑚(𝑥)

𝑀

𝑚=1

 
(    (2) 

 



Siti Hadijah Hasanah 

Multivariate Adaptive Regression Splines (MARS) For Modeling The Student Status at Universitas Terbuka 

 

53 

 

𝑎0,𝑎1,𝑎2,… ,𝑎𝑀 =  regression coefficient 
𝐵𝑚(𝑥) =  m - base function 
𝑀 =  maximum base function 

Each base function is a truncated power splines function. Univariate truncated power 

basis can be described as an indicator function. The functions of the base of univariate 

splines base from right and left are as follows : 

 

𝑏𝑞
+(𝑥,𝑐) = [+(𝑥 −𝑐)]+

𝑞
,   𝑏𝑞

−(𝑥,𝑐) = [−(𝑥 − 𝑐)]+
𝑞

 

A single base function can be specified: 

𝐵𝑚(𝑥) = ∏ [𝑠ℎ𝑚(𝑥𝑖(ℎ,𝑚) − 𝑐ℎ𝑚)]
𝐻𝑚

ℎ=1 +

𝑞

 
(3) 

Hm =  the number of interactions on the M based function 

shm =  value ±1, if the knot is located to the right or left of the subregion 

xi(h,m) =  predictor variables (1,2, …, n), h-interactions (1,2, …, M) and base of each m 

subregion (1,2,…, M) 

chm =  knot point position 

q =  order of splines 

 

 The MARS model can be stated as follows : 

𝑓(𝑥) = 𝑎0 +∑ 𝑎𝑚
𝑀

𝑚=1

∏ [𝑠ℎ𝑚(𝑥𝑖(ℎ,𝑚) − 𝑐ℎ𝑚)]
𝐻𝑚

ℎ=1

 
     (4) 

Location selection and the number of knots on MARS using forward stepwise and 

backward stepwise steps: 

1. Forward Stepwise 
This stage is to determine the location of the knot and the maximum base function 

based on the data by minimizing the average sum of square residual (ASR) [15]. The 

addition of the base function is continued until it reaches the maximum base function.  

2. Backward  Stepwise 
This stage is to determine the size of the appropriate base function. At this stage, 

the removal of the base function that contributes to the estimated value of the small 

response until a balance between bias and variety and a suitable model is obtained, that 

is, by minimizing the value of generalized cross-validation (GCV) [16]. 

The minimum GCV functions are as follows : 

𝐺𝐶𝑉(𝑀) =
𝐴𝑆𝑅

[1−
𝐶(�̂�)
𝑛
]

2 =

1
𝑛
∑ [𝑦𝑖 − 𝑓𝑀(𝑥𝑖)]

2𝑛
𝑖=1

[1 −
𝐶(�̂�)
𝑛
]

2  (5) 

yi  =  response variable 

𝑓𝑀(𝑥𝑖)  =  the value of the response variable estimates on the M  base function 
N =  many observations 

𝐶(�̂�)   = 𝐶(𝑀)+𝑑𝑀 
𝐶(𝑀) = Trace [𝐵(𝐵𝑀

𝑇𝐵𝑀)
−1𝐵𝑀

𝑇 ] +1  
d =  value when each base function achieves optimization ( 2 ≤  𝑑 ≤  4 ) 

 
2.1 Decomposition ANOVA 

 Interpretation of the MARS model through decomposition ANOVA [17] : 



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54 

 

𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑚
𝑀

𝑚=1

[𝑠1𝑚(𝑥𝑖(1,𝑚) − 𝑐1𝑚)]

+ ∑ 𝑎𝑚
𝑀

𝑚=1

[𝑠1𝑚(𝑥𝑖(1,𝑚) − 𝑐1𝑚)][𝑠2𝑚(𝑥𝑖(2,𝑚) − 𝑐2𝑚)]

+ ∑ 𝑎𝑚
𝑀

𝑚=1

[𝑠1𝑚(𝑥𝑖(1,𝑚) − 𝑐1𝑚)][𝑠2𝑚(𝑥𝑖(2,𝑚) − 𝑐2𝑚)][𝑠3𝑚(𝑥𝑖(3,𝑚)

− 𝑐3𝑚)]+ ⋯ 

(6) 

Can be written as follows: In general, it can be written as follows: 

𝑓(𝑥) = 𝑎0 + ∑ 𝑓𝑖(𝑥𝑖)
𝐻𝑚=1

+∑ 𝑓
𝑖𝑗
(𝑥
𝑖
,𝑥𝑗

𝐻𝑚=2

) +∑ 𝑓
𝑖𝑗𝑘
(𝑥
𝑖
,𝑥𝑗,𝑥𝑘

𝐻𝑚=3

) + ⋯ 
(7) 

 

2.2 Estimation of MARS Model 

 Parameters Suppose the base function 𝐵𝑚(𝑥), 𝑚 = 0,1,2,…,𝑀, to estimate the 
regression coefficient 𝑎𝑚 using the smallest square method. 

�̂�𝑀𝐾𝑇 = (𝐵
𝑇𝐵)−1𝐵𝑇 𝑌, (8) 

𝑎 = (𝑎0,𝑎1,… ,𝑎𝑀)
𝑇

 (9) 

𝑌 = (𝑦1,𝑦2,… ,𝑦𝑛)
𝑇

 (10) 

𝐵 =

(

 
 
 
 
 
1  ∏ 𝑠1𝑚(𝑥1(1,𝑚) − 𝑐1𝑚)…

𝐻1

ℎ=1

∏ 𝑠𝑀𝑚(𝑥1(𝑀,𝑚) − 𝑐𝑀𝑚)
𝐻𝑀

ℎ=1

1  ∏ 𝑠1𝑚(𝑥2(1,𝑚) − 𝑐1𝑚)…
𝐻1

ℎ=1

∏ 𝑠𝑀𝑚(𝑥2(𝑀,𝑚) − 𝑐𝑀𝑚)
𝐻𝑀

ℎ=1
  …

  …

1  ∏ 𝑠1𝑚(𝑥𝑛(1,𝑚) − 𝑐1𝑚)…
𝐻1

ℎ=1

∏ 𝑠𝑀𝑚(𝑥𝑛(𝑀,𝑚) − 𝑐𝑀𝑚)
𝐻𝑀

ℎ=1 )

 
 
 
 
 

 
(11) 

 

2.3 MARS Model Significance Test 

The significance test of the MARS Model is on the base function which includes 

simultaneous and partial tests, Simultaneous tests with the following hypotheses: 

 

𝐻0: 𝑎1 = 𝑎2 = ⋯ = 𝑎𝑚 = 0 
𝐻1: there is at least one 𝑎𝑚 ≠ 0;𝑚 = 1,2,…,𝑀 
F test : 

𝐹 =
𝑆𝑆𝑅

𝑆𝑆𝐸
~𝐹(𝑀,𝑁−𝑀−1) 

(12) 

 

Partial test with the following hypotheses: 

𝐻0:𝑎𝑚 = 0 

𝐻1:𝑎𝑚 ≠ 0 ;𝑚 = 1,2,…,𝑀 

t test : 

𝑡 =
�̂�𝑚

𝑠𝑒(�̂�𝑚)
~𝑡

(
𝑎
2
 ,𝑁−𝑀−1)

 
(13) 

 

 

 



Siti Hadijah Hasanah 

Multivariate Adaptive Regression Splines (MARS) For Modeling The Student Status at Universitas Terbuka 

 

55 

 

3. Data and Methods 
 

The data used in this study is secondary data, which is the data of the student 

characteristics of the Department of Statistics, Universitas Terbuka from year 2009.1 to 

2019.2. The data is divided into two parts, namely training data 1046 and testing data 447 

which consists of 9 explanatory variables and one response variable : 

 
Table 1. Characteristics Students at Department of Statistics, Universitas Terbuka 

Variable Information Scale Category 

X1 Gender Nominal 1 = Man 

   2 = Woman 

X2 Age Interval  

X3 Education Ordinal 1 = Senior High School 

   2 = Associate Degree 

   3 = Bachelor Degree 

   4 = Master Degree 

   5 = Doctoral Degree 

X4 Marital Status Nominal 1 = Single 

   2 = Married 

X5 Job Nominal 1 = Unemployment 

   2 = Private Employees  

   3 = Entrepreneur 

   4 = Civil Servants 

   5 = Army/Police 

X6 
Initial Registration 

Year 
Interval  

X7 
Number of 

Registrations 
Interval  

X8 Credits Interval  

X9 GPA Interval  

Y Student Status Nominal 0  = Not Active 

      1   = Active 

 

 The steps for applying the MARS method are as follows: 

1. Determining the number of base functions 
Base function limit between 2 - 4 times the number of explanatory variables. 

2. Determination of maximum interaction (MI) 
The MI used are 1, 2, and 3. 

3. Determination of minimum observations in each knot (MO) 
4. Perform a trial and error process 

Combine base, MI, and MO functions to get minimum GCV value. 

5. Test the significance of the regression coefficient 
Partial test statistics with t and F test statistics to test the significance of the regression 

coefficient simultaneously. 

 

4. Result 
 

By using the R software and simulating the number of basis functions, MI, MO, and 

GCV, the MARS model is obtained as follows: 

 

𝑌 = −0,003𝐵𝐹1 − 0,082𝐵𝐹2 + 0,029𝐵𝐹3 −0,027𝐵𝐹4 − 0,233𝐵𝐹5 −0,272𝐵𝐹6   
+0,254𝐵𝐹7 −0,004𝐵𝐹8 − 0,008𝐵𝐹9 +0,006𝐵𝐹10 

 (14) 



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56 

 

𝐵𝐹1   =  ℎ(20152 −𝑌𝑒𝑎𝑟_ 𝐸𝑎𝑟𝑙𝑦_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛)  
𝐵𝐹2   =  ℎ(10 −𝑇𝑜𝑡𝑎𝑙_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛)  
𝐵𝐹3   =  ℎ(𝑌𝑒𝑎𝑟_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛_𝐸𝑎𝑟𝑙𝑦 − 20121)    
𝐵𝐹4   =  ℎ(𝑌𝑒𝑎𝑟_𝐸𝑎𝑟𝑙𝑦_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛 −20142)   
𝐵𝐹5   =  ℎ(𝐺𝑃𝐴−0,33) 
𝐵𝐹6   =  ℎ(0,33 − 𝐺𝑃𝐴)                     
𝐵𝐹7   =  ℎ(𝐺𝑃𝐴−1,63)     
𝐵𝐹8   =  ℎ(𝐶𝑟𝑒𝑑𝑖𝑡𝑠 − 21)  
𝐵𝐹9   =  ℎ(21 −𝐶𝑟𝑒𝑑𝑖𝑡𝑠)  
𝐵𝐹10 =  ℎ(𝐶𝑟𝑒𝑑𝑖𝑡𝑠 −70) 

 
Table 2.  MARS Model Interpretation  

Basis Function 

(BF) 
Interpretation 

1 
Base 1 function contributes to the model of -0.003, when students 

register early in 2015 semester 2 then the student status is active.  

2 
Base 2 function contributes to the model by -0.082, if the student 

register 10 times then the student status is active.  

3 
Base 3 function contributes to the model of 0.029, when the student 

registers early in 2012 semester 1 then the student status is active. 

4 
Base 4 function contributes to the model of -0.027, if the student 

registers early in 2014 semester 2 then the student status is active. 

5 
Base 5 function contributes to the model by -0.233, if the student has 

a GPA of 0.33 then the student status is active. 

6 
Base 6 function contributes to the model by -0.272, if the student has 

a GPA of 0.33 then the student status is active.  

7 
Base 7 function contributes to the 0,254 models. if the student has a 

GPA of 1.63 then the student status is active.  

8 
The base 8 functions contribute to the model of -0.004, when a 

student takes 21 credits then the student status is active.  

9 
The base 9 functions contribute to the model of -0.008, when a 

student takes 21 credits then the student status is active.  

10 
The base 10 function contributes to the model of 0.006, when a 

student takes 70 credits then the student status is active. 

 

The significance test is then performed on each of the above Basis functions as 

follows: 

 
Table 3. Significance Test 

Basis 

Function 
T P-value F P-value 

𝐵𝐹1 -2,252 0,024537 297,7 < 2,2e-16 
𝐵𝐹2 -16,277 < 2e-16 

  

𝐵𝐹3 12,425 < 2e-16 
  

𝐵𝐹4 -10,58 < 2e-16 
  

𝐵𝐹5 -7,577 7,86e-14 
  

𝐵𝐹6 -2,327 0,02014 
  

𝐵𝐹7 4,945 8,90e-07 
  

𝐵𝐹8 -4,684 3,18e-06 
  

𝐵𝐹9 -3,873 0,000114 
  

  𝐵𝐹10 4,976 7,60e-07     

 

Previously, based on research data, there were 9 explanatory variables used to 

determine the effect of student activeness status, but after modeling using the MARS 



Siti Hadijah Hasanah 

Multivariate Adaptive Regression Splines (MARS) For Modeling The Student Status at Universitas Terbuka 

 

57 

 

method. These explanatory variables could affect the active status of students majoring in 

Statistics were initial registration year, number of registrations, GPA, and credits. Based 

on the output of the R software in table 3 and using a 95% confidence interval, each of the 

base 1 to 10 functions is partially significant (t-statistic) with the p-value of the base 

function 1-10 being smaller than 0,05 and automatically simultaneous (f-statistic) with p-

value less than 0,05 so that the above model has a partially and simultaneously significant 

influence on the response variable. From these results, it is concluded that the MARS model 

is suitable for determining the factors. 

 

5. Conclusions 
 

There are nine explanatory variables used to determine the effect of student active 

status, including gender, age, education, marital status, job, initial registration year, number 

of registrations, credits, GPA, but after modeling using the MARS method, the explanatory 

variables can affect Statistics student active status is the initial registration year, number of 

registrations, GPA, and credits. Based on the results of the R application output and using 

a 95% confidence interval, each of the base 1 to 10 functions is partially significant                          

(t-statistic) with the p-value of the base function 1-10 less than 0,05 and simultaneously             

(f-statistic) with a p-value less than 0,05, so that the above model has a significant effect 

partially or simultaneously on the response variable. From these results, it is concluded that 

the MARS model is suitable for determining the factors that affect student active status. 

The results of this study are as a way to find solutions for study programs in overcoming 

the many inactive student statuses, especially in the Statistics Study Program, namely by 

paying attention to these four factors, the study program will provide treatment in further 

research so that students can graduate on time so that can reduce the number of inactive 

students in each semester. 

 

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