Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure (GEM) in East Java


CAUCHY – Jurnal Matematika Murni dan Aplikasi 
Volume 7(1) (2021), Pages 105-117 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: July 25, 2021 Reviewed: October 10, 2021 Accepted: November 05, 2021 
DOI: https://doi.org/10.18860/ca.v7i1.12993     

 Spline Nonparametric Regression to Identify Factors Affecting 
Gender Empowerment Measure (GEM) in East Java 

Luluk Mahfiroh1, Yuniar Farida2* 

1,2 Department of Mathematics, Faculty of Science and Technology, Sunan Ampel State 
Islamic University Surabaya, Indonesia 

 
Email: mahfirohluluk@gmail.com, yuniar_farida@uinsby.ac.id* 

*Corresponding Author 

ABSTRACT 

Gender is a multidimensional issue that's not limited to gender discrimination, but also includes 
the economic, educational, and health aspects, which then become the focus of almost all the 
Sustainable Development Goals (SDGs). Evaluation of the development devoted to the perspective 
of the gender using several indicators, Gender Development Index (GDI) and Gender 
Empowerment Measure (GEM). GEM describes the role of women in the economic sphere and is 
measured by equality in political participation. GEM of East Java for 5 consecutive years (2014 – 
2018) is lower than the average national GEM. This study aims to identify factors affecting GEM in 
East Java using nonparametric regression spline quadratic. The result of the regression model 
shows that all variables selected are affecting GEM in East Java, they are the Labor Force 
Participation Rate (LFPR) population of women (𝑥1), School Participation Rate (SPR) high school 
population of women (𝑥2), Percentage of Population Female that Working in the formal sector (𝑥3), 
sex ratio (𝑥4), Percentage of Population Female that Working as members of People’s 
Representative Council (𝑥5), Percentage of Population Female that working as Civil Servants (𝑥6), 
and rate of women's income donations (𝑥7). The model generates 𝑅

2 value of 93.74% and MAPE 
of 3.22%. This research contributes to the implementation of non-parametric spline regression in 
identifying various factors that influence social phenomena. 

Keywords: Gender Empowerment Measure (GEM); Gender Development Index (GDI); 
Nonparametric Regression; Spline; Generalized Cross-Validation (GCV) 
 

INTRODUCTION 

Discussion about gender is indeed inseparable from the concept of gender equality 
and justice. The goal is to kill the patriarchal culture that overrides the role of women. At 
the beginning of the world's development, history records that very few female figures 
played important events. Information about women's roles in social, economic, and 
political movements is also minimal [1]. The male domination of women influences it. Such 
dominance is inseparable from the patriarchal ideology adopted by most of the world's 
people since a long time ago [2].  

Gender is a multidimensional issue that's not limited to gender discrimination, but 
also includes the economic, educational, and health aspects, which then become the focus 
of almost all the Sustainable Development Goals (SDGs) [3]. In Indonesia, evaluation of 

https://doi.org/10.18860/ca.v7i1.12993
mailto:mahfirohluluk@gmail.com
mailto:yuniar_farida@uinsby.ac.id*


Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 106 

development outcomes that are devoted to a gender perspective uses several indicators, 
namely the Gender Development Index (GDI) and the Gender Empowerment Measure 
(GEM). GDI explains the gap in human development between men and women. Meanwhile, 
GEM describes women's role in the economic sphere and is measured by equality in 
political participation [4]. GEM is the Percentage of women who work as professionals, 
managers, technicians, and all forms of leadership [5]. GEM East Java's achievements in 
2017 and 2018 were 69.37 and 69.71. It is lower than the national GEM average of 71.74 
in 2017 and 72.1 in 2018 [6]. 

GEM's calculations indicate the proportion of managers, administrative staff, 
professional workers, and technicians; the balance of representation in parliament; and 
non-farm workers' wages. In addition to these indicators, other factors that may influence 
it, including the Labor Force Participation Rate (LFPR), School Participation Rate (SPR), 
the Percentage of the population working in the formal sector, the gender ratio, the 
Percentage of income contributions, the Percentage of the population female that working 
as members of People’s Representative Council and civil servants. To find out the 
relationship of these factors to GEM values can be done by mathematical analysis. One of 
the analytical methods used to solve the problem is mathematical modeling [7]. 
Mathematical modeling can be done using regression analysis to determine the causal 
relationship with other variables.  [8]. Regression analysis is grouped into 3, namely 
parametric, nonparametric, and semiparametric [9].   

Research conducted by Adawiyah [10], it is obtained that the spline estimator is 
strongly influenced by the number of knots, the number of orders, the location of knot 
points, and the best spline regression model using two-knot points. Conducted by 
Fadhilah's research [11], the best spline model relies heavily on determining the optimal 
knot point with a minimum Generalized Cross-Validation (GCV) value. The best-truncated 
spline regression model lies in order 2 [12]. The regression function estimation with a 
nonparametric approach is done with the spline technique to adjust effectively to data 
[13]. 

This research used modeling spline nonparametric regression to determine the 
factors that affect the Gender Empowerment Measure in East Java. Nonparametric 
regression is used because the data is not in a particular pattern or the regression curve's 
shape is limited. 

 

METHODS  

Data and Data Processing 

The data used in this study is secondary data in 2013-2018 sourced from BPS (Badan 
Pusat Statistik) of East Java Province both through the official website of www.bps.go.id 
and publication books available at the Service Center. In this study, there are eight (8) 
variables, which consist of one dependent variable (𝑦) dan seven independent variables 
(𝑥). They are Gender Empowerment Measure (GEM) (𝑦), Labor Force Participation Rate 
(LFPR) of the female population (𝑥1), School Participation Rate (SPR) of Female 
Population in the High School Level (𝑥2), Percentage of Female Population in the Formal 
Sector (𝑥3), Sex Ratio (𝑥4), Percentage of Female that working as members of People’s 
Representative Council (𝑥5), Percentage of Female Population that working as Civil 
Servants (𝑥6), and Percentage of Women's Income Donations (𝑥7). 

The data obtained in this study were analyzed using the nonparametric spline 
regression method for longitudinal data with the RStudio program. The steps to achieve 
the objectives of this research are as follows: 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 107 

 
a) Create a descriptive statistical analysis and scatter plot for each variable 

A Scatter plot is used to detect relationship patterns between the y variable with each 
x variable, which is predicted to be a factor in its influence. Scatter plots provide 
information on the pattern of regression curve shapes used in modeling [14]. 

b) Create a Model of GEM with nonparametric spline regression. 
Nonparametric regression models, in general, can be presented as follows: 

 
 𝑦𝑖 = 𝑓(𝑥𝑖) +  𝜀𝑖 ;            𝑖 = 1, 2, 3, … , 𝑛 (1) 

with 𝑦𝑖 is the response variable;  𝑥𝑖  is predictor variable; 𝑓(𝑥𝑖) is a regression function 
that doesn't follow a particular pattern; 𝜀𝑖 =  (𝜀1, 𝜀2, … , 𝜀𝑛)

𝑇 is a free mutual error 
vector with zero alignments and diversity 𝜎2 [15]. 

Estimation of function 𝑓(𝑥𝑖) in nonparametric regression is performed with the spline 
estimator [16]. If a regression curve is 𝑓 is an additive model and approached with the 
spline function, the regression model is: 

 
𝑦𝑖𝑗 =  ∑ 𝛽ℎ𝑖𝑥𝑖𝑗𝑝

ℎ + ∑ 𝛼𝑙𝑖(𝑥𝑖𝑗𝑝 −  𝐾𝑙𝑖)+
𝑞

+  𝜀𝑖𝑗

𝑚

𝑙=1

𝑞

ℎ=0

 

 

 
(2) 

with 𝑦𝑖𝑗 is response variables in the i-subject and j-time observations; 𝑥𝑖𝑗𝑝 is p-

predictor variables on the i-subject and j-time observations; 𝜀𝑖𝑗 is a random error on i-

subject and j-time observation; 𝑞 is polynomial degrees. In this study, the value of 𝑖 is 
1, 2, ..., 38, that represent the subject/region in East java (38 cities/regencies); 𝑗  is the 
number of observations (2013-2018) with a value of 1, 2, ..., 6; 𝐾𝑙𝑖 is the number of 

knots. Then, the function of (𝑥𝑖𝑗𝑝 −  𝐾𝑙𝑖)+
𝑞

 is given by [17]:  

(𝑥𝑖𝑗𝑝 −  𝐾𝑙𝑖)+
𝑞

=  {
(𝑥 − 𝐾𝑙𝑖)

𝑞 ;     𝑥𝑖𝑗𝑝  ≥  𝐾𝑙𝑖
0 ;     𝑥𝑖𝑗𝑝  <  𝐾𝑙𝑖

 

This study used a nonparametric regression spline with a second-degree polynomial 
curve or quadratic for longitudinal data. The quadratic curves are selected because 
they can give a smaller error than linear curves and are suitable for data with more 
complex patterns. 
 
In this study, the ordinary least square (OLS) method is used to estimate the parameter 
value of 𝛽 in equation [2]. By using the spline regression model as a smooth curve 
estimation 𝑓(𝑥), The estimation equation is as follows [18]: 

 
 �̂� = (𝑋𝑇𝑋)−1𝑋𝑇𝑦  (3) 

with matrix X as follows: 

𝑋 =  

[
 
 
 
 
 1 𝑥111 … 𝑥111

𝑞
(𝑥111 −  𝐾11)+

𝑞
… (𝑥111 −  𝐾𝑚1)+

𝑞
… 𝑥11𝑝 … 𝑥11𝑝

𝑞
(𝑥11𝑝 −  𝐾𝑚11)+

𝑞
… (𝑥11𝑝 −  𝐾𝑚𝑝1)

+

𝑞

1 𝑥211 … 𝑥211
𝑞

(𝑥211 −  𝐾22)+
𝑞

… (𝑥211 −  𝐾𝑚2)+
𝑞 …

⋮⋱⋮ 𝑥21𝑝 … 𝑥21𝑝
𝑞

(𝑥21𝑝 −  𝐾𝑚12)+
𝑞

… (𝑥21𝑝 −  𝐾𝑚𝑝2)
+

𝑞

1 𝑥𝑛𝑡1 … 𝑥𝑛𝑡1
𝑞

(𝑥𝑛𝑡1 −  𝐾𝑚𝑛)+
𝑞

… (𝑥𝑛𝑡1 −  𝐾𝑚𝑛)+
𝑞

… 𝑥𝑛𝑡𝑝 … 𝑥𝑛𝑡𝑝
𝑞

(𝑥𝑛𝑡𝑝 −  𝐾𝑚1𝑛)+
𝑞

… (𝑥𝑛𝑡𝑝 −  𝐾𝑚𝑝𝑛)
+

𝑞

]
 
 
 
 
 

 

 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 108 

 
c) Select the optimal knot point using the GCV method. 

The selection of optimal knot points is critical in nonparametric regression models. 
Knot points are joint fusion points that have behavior changes in the data. One 
appropriate method for selecting the optimal knot points is Generalized Cross-
Validation (GCV) [19]. It is said to be the optimal knot point when obtained the lowest 
or minimum GCV value. The GCV function, according to Eubank  [20], is as follows: 

 
𝐺𝐶𝑉(𝐾𝑙𝑖) =  

𝑀𝑆𝐸(𝐾𝑙𝑖)

(𝑛−1𝑡𝑟𝑎𝑐𝑒[𝐼 −  𝐻𝐾𝑙𝑖])
2

 
 

(4) 

with 𝑛 is the amount of data; I is an identity matrix, and MSE is Mean Square Error. 
Then,  𝐻 = 𝑋(𝑋𝑇𝑋)−1𝑋𝑇 and 𝑀𝑆𝐸(𝐾𝑙𝑖) is defined as: 

𝑀𝑆𝐸(𝐾𝑙𝑖) =  𝑛
−1 ∑ ∑(𝑦𝑖𝑗 − 𝑓(𝑥𝑖𝑗𝑝))

2
𝑡

𝑗=1

𝑛

𝑖=1

 

d) Modeling GEM using spline with optimal knot points. 
e) Interpret the model and conclude.  

RESULTS AND DISCUSSION  

Characteristics of GEM in East Java Year 2013-2018 

An overview of the data can be seen in the following table. 

Table 1. The Descriptive Statistics of GEM Variable (y) and Various Variable Affecting the GEM  
                  (𝑥1 ….. 𝑥7) 

Variable Average Variance Maximum Value Minimum Value 
𝑦 65.82 75.82 83.29 42.09 
𝑥1 55.19 33.55 72.80 43.56 
𝑥2 71.55 190.07 100 25.30 
𝑥3 31.87 269.49 67.61 5.98 
𝑥4 97.13 6.40 101.64 90.64 
𝑥5 17.13 87.80 51.52 0 
𝑥6 48.11 33.96 83.88 29.93 
𝑥7 32.73 18.88 40.34 22.81 

 
Table 1 shows that the average GEM in East Java in 2013-2018 was 65.82 with a 

variance of 75.82. High variance data shows the data used is spread far from the average 
caused by the outlier value. This happens because of differences in women's participation 
in areas with city and district status. In districts, the majority of women are only 
housewives, while women's participation in the public sector is minimal. 

The maximum value is 83.29, which is the GEM value of Surabaya city in 2018, while 
the minimum value is 42.09, which is the GEM value of the Sampang Regency in 2013. 
Some variables that have an average score below 50% indicate a gender gap where 
women's involvement in the variable is lower than that of men. 

Characteristics of variable x5 or Percentage of Women as Members of Parliament in 
East Java in 2013-2018 obtained the minimum value is 0 this is because in Bangkalan 
Regency in the period 2014 – 2018 there are no members of the DPRD of the female sex. 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 109 

Relationship Patterns Between 𝒚 (GEM) Variable with 𝒙 Variables 

Relationship patterns between 𝑦 variables with 𝑥 variables do not form a specific 
pattern for each year. It can be seen from the random plot spread. So that all variables can 
be used as nonparametric components. The Scatter plot of their relationship is shown in 
Figures 1 to 7 as follows:  

2016 2017 2018 

   

Figure 1. Scatter Plots Between 𝑦 Variable and𝑥1 (Labor Force Participation Rate (LFPR) of the 
female population) Variable 

 

2016 2017 2018 

   

Figure 2. Scatter Plots Between 𝑦 Variable and 𝑥2 (School Participation Rate (SPR) High School 
Level Female Population) Variable 

 

2016 2017 2018 

   

Figure 3. Scatter Plots Between 𝑦 Variable and 𝑥3 (Percentage of Female Population in the 
Formal Sector) Variable 

 

 

 

 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 110 

2016 2017 2018 

   

Figure 4. Scatter Plots Between 𝑦 Variables and 𝑥4 (Sex Ratio) Variable 

 

2016 2017 2018 

   

Figure 5. Scatter Plots Between 𝑦 Variables and 𝑥5 (Percentage of Females that working as 
members of People’s Representative Council) Variable 

 

2016 2017 2018 

   

Figure 6. Scatter Plots Between 𝑦 Variables and 𝑥6 (Percentage of Female Population that 
working as Civil Servants) Variable 

 

2016 2017 2018 

   

Figure 7. Scatter Plots Between 𝑦 Variables and 𝑥7 (Percentage of Women's Income Donations) 
Variables 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 111 

Scatter plot between variable y (GEM) and all variables x (𝑥1,  𝑥2,  𝑥3,  𝑥4,  𝑥5,  𝑥6, 𝑥7) are 
random spread and do not follow a specific pattern. So those variables are included in 
nonparametric components. 
 
Spline Quadratic Nonparametric Regression Modeling with One-Knot Points 

The nonparametric regression model of a quadratic spline with the one-knot point in 
the GEM data of East Java Province is as follows:   

𝑦𝑖𝑗 =  𝛽0𝑖 +  𝛽1𝑖𝑥1𝑖𝑗 +  𝛽1𝑖(𝑥1𝑖𝑗)
2
+  𝛼1𝑖(𝑥1𝑖𝑗 −  𝐾1𝑖)

2
+  𝛽2𝑖𝑥2𝑖𝑗 +  𝛽2𝑖(𝑥2𝑖𝑗)

2
 

       + 𝛼1𝑖(𝑥2𝑖𝑗 −  𝐾1𝑖)
2
+ ⋯ +  𝛽7𝑖𝑥7𝑖𝑗 +  𝛽7𝑖(𝑥7𝑖𝑗)

2
+  𝛼1𝑖(𝑥7𝑖𝑗 −  𝐾1𝑖)

2
+  𝜀𝑖𝑗 

Here is the GCV value for a one-knot point. 

Table 2. Smallest GCV Value with One-Knot Point 
Order Regency/City 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 𝒙𝟓 𝒙𝟔 𝒙𝟕 GCV 

23 

Pacitan 70.89 65.98 13.3 95.37 16.31 45.61 38.8 

1.07E-26 

Ponorogo 59.57 74.24 18.08 99.84 12.47 45.67 34.65 

Trenggalek 62.56 64.25 14.78 98.71 15.28 47.22 36.75 

Tulungagung 57.63 74.18 28.49 95.12 6.69 50.7 37.7 

Blitar 53.75 64.25 27.71 100.24 15.98 53.95 40.09 

Kediri 52.1 78.59 34.61 100.56 29.23 52.29 30.65 

Malang 51.15 58.56 31.93 101.01 17.04 50.46 36.5 

Lumajang 46.99 55.7 22.74 95.22 14.55 45.91 23.16 

⋮ ⋮ 

Malang City 51.57 80.62 60.92 97.35 23.44 50.3 34.22 

Probolinggo City 50.17 75.5 54.51 97.15 24.47 49.55 30.83 

Pasuruan City 53.74 72.88 50.9 97.98 6.98 50.45 31.02 

Mojokerto City 56.47 81.99 50.69 96.52 28.46 50.7 36.47 

Madiun City 54.1 81.2 48.4 93.71 34.27 54.13 38.42 

Surabaya City  52.06 71.02 64.72 97.59 39.04 68.19 35.1 

Batu City 55.93 83.4 41.47 101.35 25.63 53.31 29.97 

 
The table above shows that the minimum GCV is 1.07E-26 in the 23rd order segment 

with optimal knot points on each 𝑥  variable; it only has a one-knot point. 

Spline Quadratic Nonparametric Regression Modeling with Two-Knot Points 

The next step is to create quadratic spline regression with two-knot points. The 
nonparametric regression model of the quadratic spline with two-knot points in GEM data 
of East Java Province is as follows:  

𝑦𝑖𝑗 =  𝛽0𝑖 +  𝛽1𝑖𝑥1𝑖𝑗 +  𝛽1𝑖(𝑥1𝑖𝑗)
2
+ 𝛼1𝑖(𝑥1𝑖𝑗 −  𝐾1𝑖)

2
+ 𝛼2𝑖(𝑥1𝑖𝑗 −  𝐾2𝑖)

2
  

 + 𝛽2𝑖𝑥2𝑖𝑗 +  𝛽2𝑖(𝑥2𝑖𝑗)
2
+  𝛼1𝑖(𝑥2𝑖𝑗 −  𝐾1𝑖)

2
+  𝛼2𝑖(𝑥2𝑖𝑗 −  𝐾2𝑖)

2
 + ⋯ 

+ 𝛽7𝑖𝑥7𝑖𝑗 +  𝛽7𝑖(𝑥7𝑖𝑗)
2
+ 𝛼1𝑖(𝑥7𝑖𝑗 −  𝐾1𝑖)

2
+ 𝛼2𝑖(𝑥7𝑖𝑗 −  𝐾2𝑖)

2
 +  𝜀𝑖𝑗 

Here are the resulting GCV values. 

(5) 

(6) 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 112 

Table 3. Smallest GCV Value with Two-Knot Points 

Order Regency/City 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 𝒙𝟓 𝒙𝟔 𝒙𝟕 GCV 

32 

Pacitan 
43.79 53.46 10.44 95.32 15.06 69.87 38.36 

2.83E-28 

51.91 82.78 19.65 94.84 6.8 57.55 36.63 

Ponorogo 
46.44 71.67 14.59 95.39 16.89 71.36 38.99 

54.06 94.26 23.4 94.91 11.13 66.47 38.05 

Trenggalek 
44.47 64.44 14.36 99.75 11.17 57.8 34.05 

47.04 71.52 14.08 95.04 20.17 48.66 30.65 

Tulungagung 
46.22 78.7 19.77 99.88 13.06 60.38 34.92 

50.69 80.54 18.43 95.47 25.77 59.11 31.54 

Blitar 
45.52 50.21 10.05 98.67 11.29 58.95 35.92 

44.61 56.7 14.03 97.08 12.12 47.62 24.97 

Kediri 
47.99 70.63 16.92 98.73 17.1 64.2 37.13 

47.28 74.39 24.3 97.57 16.04 53.8 25.84 

⋮ ⋮ 

Malang City 
52.8 74.94 39.9 99.98 29.54 56.49 35.2 

51.94 85.71 53.21 96.66 35.61 58 36.69 

Probolinggo City 
48.65 66.19 26.26 98.9 8.41 46.53 26.58 

52.56 67.42 45.15 93.57 27 52.35 37.84 

Pasuruan City 
51.72 80.73 32.06 99.17 21.61 51.94 27.22 

54.85 87.47 49.88 93.78 37.57 54.89 38.68 

Mojokerto City 
49.55 61.96 19.55 98.69 16.33 44.11 24 

54.91 60.59 62.27 97.54 28.48 50.53 34.7 

Madiun City 
51.86 79.87 24.99 98.82 27.01 51.1 25.1 

74.22 75.76 65.83 97.61 43.84 52.76 35.29 

Surabaya City 
50.23 78.87 20.1 97.33 11.2 48.99 28.95 

49.82 76.73 35.53 101.1 20.24 53.59 29.45 

Batu City 
53.02 86.7 24.82 97.42 14.11 54.47 30.02 

54.9 86.42 44.16 101.46 28.08 57 30.21 

  
Optimal Knot Point Selection 

The selection of the best model is based on the selection of optimal knot points with 
the minimum GCV grades. GCV with one-knot point produces 48 alternate knot points with 
each GCV value and obtained a minimum value of 1.07E-26, while GCV with two-knot 
points also has 48 alternate knot points GCV value obtained a minimum value of 2.83E-28. 
So, selected nonparametric regression modeling spline quadratic using 2 knot points with 
the GEM model in each city or regency in East Java is: 

𝑦𝑖𝑗 =  𝛽01 +  𝛽11𝑥1𝑖𝑗 +  𝛽12(𝑥1𝑖𝑗)
2
+  𝛼11𝑗(𝑥1𝑖𝑗 − 𝐾11𝑖)

2
+  𝛼21𝑖(𝑥1𝑖𝑗 −  𝐾21𝑖)

2
  

+ 𝛽21𝑥2𝑖𝑗 + 𝛽22(𝑥2𝑖𝑗)
2
+  𝛼12𝑖(𝑥2𝑖𝑗 − 𝐾12𝑖)

2
+  𝛼22𝑖(𝑥2𝑖𝑗 −  𝐾22𝑖)

2
  

+ 𝛽31𝑥3𝑖𝑗 + 𝛽32(𝑥3𝑖𝑗)
2
+  𝛼13𝑖(𝑥3𝑖𝑗 − 𝐾13𝑖)

2
+  𝛼23𝑖(𝑥3𝑖𝑗 −  𝐾23𝑖)

2
 

 + 𝛽41𝑥4𝑖𝑗 +  𝛽42(𝑥4𝑖𝑗)
2
+  𝛼14𝑖(𝑥4𝑖𝑗 −  𝐾14𝑖)

2
+  𝛼24𝑖(𝑥4𝑖𝑗 −  𝐾24𝑖)

2
 

+ 𝛽51𝑥5𝑖𝑗 + 𝛽52(𝑥5𝑖𝑗)
2
+  𝛼15𝑖(𝑥5𝑖𝑗 − 𝐾15𝑖)

2
+  𝛼25𝑖(𝑥5𝑖𝑗 −  𝐾25𝑖)

2
 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 113 

+ 𝛽61𝑥6𝑖𝑗 + 𝛽62(𝑥6𝑖𝑗)
2
+  𝛼16𝑖(𝑥6𝑖𝑗 − 𝐾16𝑖)

2
+  𝛼26𝑖(𝑥6𝑖𝑗 −  𝐾26𝑖)

2
 

+ 𝛽71𝑥7𝑖𝑗 + 𝛽72(𝑥7𝑖𝑗)
2
+  𝛼17𝑖(𝑥7𝑖𝑗 − 𝐾17𝑖)

2
+ 𝛼27𝑖(𝑥7𝑖𝑗 −  𝐾27𝑖)

2
  

Overall GEM model in each city or regency in East Java obtains 𝑅2 value is 93.74%, which 
means that the model can explain the diversity of GEM variable values of East Java 
Province by 93.74%; while the residual (6,26%) is explained by other variables that are 
not in the regression model. This model has a Mean Absolute Percentage Error (MAPE) 
value of 3.22%.  

Parameter Testing of Quadratic Spline Nonparametric Regression Model 

The test results are displayed in the following ANOVA Table 4. 

Table 4. ANOVA Results on Model 

Source df SS MS Fcount p-value Decision 

Regresi 6 16134.355 2689.089 512.053 1.96E-120 Failed to Reject H0 

Error 205 1076.755 5.2516 - - - 

Total 227 17211.111 - - - - 

 
The decision obtained is Reject 𝐻0, This means there is at least one significant 

parameter in the quadratic spline nonparametric regression model. The considerable 
parameter is an entire estimator 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 in each year from 2013-2018. 

Interpretation of Results of The Quadratic Spline Nonparametric Regression Model 

For example, the implementation of the quadratic spline nonparametric regression 
model in longitudinal data is used in Surabaya. The estimated coefficient of parameters 
and knot points obtained is substituted in Equation (7), so it obtained a model of 
nonparametric regression of quadratic spline with longitudinal data for the city of 
Surabaya as follows. In GEM data, Surabaya City is 37th out of a total of 38 cities/regencies, 
hence the value 𝑖 = 37 

𝑦37𝑗 =  0.0008 −  0.0134𝑥1.37.𝑗 −  0.0363(𝑥1.37.𝑗)
2
+  0.0049(𝑥1.37.𝑗 −  50.32)

2

+  0.3059(𝑥1.37.𝑗 −  49.82)
2
−  0.0191𝑥2.37.𝑗 −  0.0391(𝑥2.37.𝑗)

2

+  0.0007(𝑥2.37.𝑗 −  78.87)
2
−  0.0132(𝑥2.37.𝑗 −  76.73)

2
−  0.0302𝑥3.37.𝑗

+  0.0048(𝑥3.37.𝑗)
2
+  0.2975(𝑥3.37.𝑗 −  20.10)

2
+  0.002(𝑥3.37.𝑗 −  35.53)

2

+  0.0457𝑥437𝑗 +  0.0458(𝑥437𝑗)
2
+  0.1919(𝑥437𝑗 −  97.33)

2

−  0.0056(𝑥4.37.𝑗 −  101.1)
2
+  0.111𝑥5.37.𝑗 +  0.0843(𝑥5.37.𝑗)

2

+  0.4336(𝑥5.37.𝑗 −  11.2)
2
−  0.0274(𝑥5.37.𝑗 −  20.24)

2
+  0.0028𝑥6.37.𝑗

+  0.0011(𝑥6.37.𝑗)
2
−  0.0042(𝑥6.37.𝑗 −  48.99)

2
+  0.0255(𝑥6.37.𝑗 −  53.59)

2

+  0.009𝑥7.37.𝑗 +  0.3078(𝑥7.37.𝑗)
2
−  0.0122(𝑥7.37.𝑗 −  28.95)

2

− 0.0009(𝑥7.37.𝑗 −  29.45)
2

 

From the model above, if the 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 assumed to be constant, the 
influence of LFPR Female Population (𝑥1) on GEM is: 

(7) 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 114 

�̂� =  − 0.0134𝑥1 −  0.0363(𝑥1)
2 + 0.0049(𝑥1 − 50.32)

2 + 0.3059(𝑥1 − 49.82)
2 

If the LFPR Female Population variable (𝑥1) is below 50.32 and then there is an 
increase of 1 unit, then the GEM value tends to decrease in value by 0.0363. At the time of 
LFPR Female Population (𝑥1)  Between the moderate intervals of 49.82 and 50.32 and 
there is an increase of 1 unit, the GEM value will decrease by 0.0314. Lastly, if the LFPR 
Female Population (𝑥1) is greater than or equal to 49.82 then there is an increase of 1 unit, 
then GEM has an increase of 0.2745. Overall, the higher the LFPR Female Population score 
(𝑥1)  Then the GEM value will also be higher, and vice versa. This is because the LFPR is a 
representation of the large number of people participating in the economy, in this case is 
counted by the participation of working women. 

The model's interpretation about SPR High School Level Female Population variable 
(𝑥2) is if the 𝑥1, 𝑥3, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 assumed to be constant, the influence of SPR on GEM 
is: 

�̂� =  − 0.0191𝑥2 −  0.0391(𝑥2)
2 + 0.0007(𝑥2 − 78.87)

2 − 0.0132(𝑥2 − 76.73)
2 

If the SPR High School Level Female Population (𝑥2)  is below 78.87 and then there is 
an increase of 1 unit, then the GEM value will decrease by 0.0391 units. When the APS is 
between the moderate intervals of 76.73 and 78.87 and there is an increase of 1 unit, the 
GEM value will tend to decrease by 0.0384. Lastly, if the SPR High School Level Female 
Population (𝑥2)  is greater than or equal to 76.73 then there is an increase of 1 unit, then 
GEM also decreases by about 0.0516 units. Overall, the higher the SPR High School Female 
Population (𝑥2)  score, the higher the GEM score, and vice versa. This is because the higher 
the level of education that a person finishes, the more it will encourage his participation 
in the job market. A person's chances of getting a job also tend to be in line with their level 
of education, especially the share of the current job market usually increases their 
education. 

The model's interpretation about Percentage of Female Population Working Age 
Working in the Formal Sector variable (𝑥3) is if the 𝑥1, 𝑥2, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 assumed to be 
constant, the influence of Formal Sector on GEM is: 

�̂� =  − 0.0302𝑥3 +  0.0048(𝑥3)
2 + 0.2975(𝑥3 − 20.10)

2 + 0.0020(𝑥3 − 35.53)
2 

If the Percentage of Female Population Working Age Working in the Formal Sector (𝑥3)  
is below 20.10 and then there is an increase of 1 unit, then the GEM value will increase by 
0.0048 units. When the Percentage of Female Population Working Age Working in the 
Formal Sector (𝑥3) was between the moderate interval segments of 20.10 and 35.53 and 
there was an increase of 1 unit, the GEM value also increased by 0.3023. Lastly, if the 
percentage of the female population working age working in the formal sector (𝑥3) is 
greater than or equal to 35.53, there is an increase of 1 unit, the GEM also increases by 
about 0.3043 units. So overall, if the percentage of the female population working age 
working in the formal sector (𝑥3) increases, GEM value will also increase, and vice versa. 
This is because, the role of women in the economic field as measured in GEM is women 
who work as professional workers, leadership, technicians, and technical or skilled 
workers. 

 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 115 

The model's interpretation about the Sex Ratio variable is if the 𝑥1, 𝑥2, 𝑥3, 𝑥5, 𝑥6 and 𝑥7 
assumed to be constant, the influence of Sex Ratio on GEM is: 

�̂� =  0.0457𝑥4 +  0.0458(𝑥4)
2 + 0.1919(𝑥4 − 97.33)

2 − 0.0056(𝑥4 − 101.1)
2 

If the Sex Ratio (𝑥4)is below the segment of 97.33 then if the value increases by one 
unit, GEM will increase by 0.0458. Whereas if the value of the Sex Ratio (𝑥4)is located 
between the moderate intervals of 97.33 and 101.1 then if the ratio value increases by one 
unit, GEM tends to increase by 0.2377. If the Sex Ratio (𝑥4) is worth more than or equal to 
101.1 then if there is an increase of one unit will affect the increase in GEM by 0.2321. So 
overall, if the value of the Sex Ratio (𝑥4) increases then the GEM value will also increase, 
and vice versa. This is because the sex ratio is a comparison of the number of male 
population per 100 female population in a given region and time. 

The model's interpretation about Percentage Female that working as members of 
People’s Representative Council variable (𝑥5) is if the 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥6 and 𝑥7 assumed to 
be constant, the influence of the People’s Representative Council on GEM is: 

�̂� =  0.111𝑥5 +  0.0843(𝑥5)
2 + 0.4336(𝑥5 − 11.20)

2 − 0.0274(𝑥5 − 20.24)
2 

If the Percentage of females that work as members of the People's Representative 
Council(𝑥5) is below the segment of 11.20 then if the value increases by one unit, GEM will 
experience an increase in value of 0.0843. Meanwhile, if the percentage value is located 
between the moderate intervals of 11.20 and 20.24 then if the ratio value increases by one 
unit, GEM also tends to increase by 0.5179 If the percentage value is more than or equal to 
20.24 then if there is an increase of one unit will affect the increase in GEM by 0.49. So 
overall, if the value of Percentage Female that working as members of the People's 
Representative Council (𝑥5) increases, then the GEM value will also increase, and vice 
versa. This is because, GEM also highlights the decision of women to participate in politics, 
so the role of women in the field of political decision-making is measured by the 
membership of the People’s Representative Council. 

The model's interpretation about Percentage of Population Female that working as 
Civil Servants variable (𝑥6) is if the 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5 and 𝑥7 assumed to be constant, the 
influence of Civil Servants on GEM is: 

�̂� =  0.0028𝑥6 +  0.0011(𝑥6)
2 − 0.0042(𝑥6 − 48.99)

2 + 0.0255(𝑥6 − 53.59)
2 

 
If the Percentage of Women Working as Civil Servants(𝑥6) is below the segment of 

48.99 then if the value increases by one unit, GEM will experience an increase in value of 
0.0011. Whereas if the percentage value is located between the moderate intervals of 
48.99 and 53.59 then if the percentage value increases by one unit, GEM also tends to 
decrease by 0.0031. If the Percentage of Women Working as Civil Servants(𝑥6)is worth 
more than or equal to 53.59 then if there is an increase of one unit will affect the increase 
in gem value by 0.0224. So overall, if the percentage of population female that working as 
Civil Servants (𝑥6) increases, GEM value will also increase, and vice versa. This is because, 
the employment status of civil servants is included in the status of formal sector 
employment but within the scope of statehood, the percentage of population female that 
working as Civil Servants is considered to affect GEM. 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 116 

The model's interpretation about the Percentage of Women's Income Donations 
variable (𝑥7) is if the 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5 and 𝑥6 assumed to be constant, the influence of 
Income on GEM is: 

 
�̂� =  0.0090𝑥7 +  0.3078(𝑥7)

2 − 0.0122(𝑥7 − 28.95)
2 − 0.0009(𝑥7 − 29.45)

2 
 
If the percentage of Women's Income Donations (𝑥7) is below 28.95 and there is an 

increase of 1 unit, then the GEM value will increase by 0.3078 units. When the percentage 
value is between the medium interval segments of 28.95 and 29.45 and there is an increase 
of 1 unit, the GEM value also increases by 0.296. Lastly, if the percentage of women's 
income Donations (𝑥7) is greater than or equal to 29.45, there is an increase of 1 unit, then 
GEM also continues to increase by about 0.295 units. So overall, if the percentage value of 
Women's Income Donations (𝑥7) increases, then the value of GEM will also increase, and 
vice versa. This is because the income contribution is a contribution of the value of the 
proceeds received in return from members of households who work in this case women. 

CONCLUSIONS 

Identification of factors that affect GEM using spline nonparametric regression results in 
a highly optimized model with MAPE at 3.22%. All factors (variable x) studied have a 
significant influence on GEM value (y) so that the value  𝑅2  is 93.74% GEM (y). This value 
indicates that the resulting model is already excellent. All research variables showed 
significant results on the model so that the factors that influenced Gender Empowerment 
Measure (GEM) East Java province is Labor Force Participation Rate (LFPR) population of 
Women (𝑥1), School Participation Rate (SPR) High School Level Female Population (𝑥2), 
Percentage of Female Population that Working in the Formal Sector (𝑥3), Sex Ratio (𝑥4), 
Percentage of Female Population that Working as members of People’s Representative 
Council (𝑥5), Percentage of Female Population that Working as Civil Servants (𝑥6), and 
Percentage of Female's Income Donations (𝑥7). Areas below the interval are dominated by 
regions with district status, and Madura Islands have the most areas below the interval. 

REFERENCES 

[1] Gadis Arifia and Nur Iman Subono, “A Hundred Years of Feminism in Indonesia An 
Analysis of Actors, Debates, and Strategies,” Ctry. The study, pp. 1–28, 2017, [Online]. 
Available: www.fes-asia.org. 

[2] KPPA and BPS, Pembangunan Manusia Berbasis Gender 2016. Jakarta: CV. Lintas 
Khatulistiwa, 2016. 

[3] S. Akter et al., “Women’s empowerment and gender equity in agriculture: A different 
perspective from Southeast Asia,” Food Policy, vol. 69, pp. 270–279, 2017, DOI: 
10.1016/j.foodpol.2017.05.003. 

[4] KPPA and BPS, Pembangunan Manusia Berbasis Gender 2018. Jakarta: KPPA, 2018. 
[5] KPPA and BPS, Pembangunan Manusia Berbasis Gender 2013. Jakarta: CV. Lintas 

Khatulistiwa, 2013. 
[6] Badan Pusat Statistik, “Indeks Pemberdayaan Gender Menurut Provinsi, 2010-

2018.”  
[7] R. P. Rangel, M. de L. G. Magaña, R. U. Azpeitia, and E. Nesterova, “Mathematical 

Modeling in Problem Situations of Daily Life,” J. Educ. Hum. Dev., vol. 5, no. 1, pp. 62–
76, 2016, doi: 10.15640/jehd.v5n1a7. 



Spline Nonparametric Regression to Identify Factors Affecting Gender Empowerment Measure 
(GEM)  

Luluk Mahfiroh 117 

[8] R. Kurniawan and B. Yuniarto, Analisis Regresi Dasar dan Penerapannya dengan R. 
Jakarta: Kencana, 2016. 

[9] N. A. Erilli, “Non-Parametric Regressiın Estimation For Data With Equal Values,” Eur. 
Sci. J. Febr. 2014, vol. 10, no. 4, pp. 70–82, 2014. 

[10] R. Syam, W. Sanusi, and R. Adawiyah, “Model Regresi Nonparametrik dengan 
Pendekatan Spline ( Studi Kasus : Berat Badan Lahir Rendah di Rumah Sakit Ibu dan 
Anak Siti Fatimah Makassar ),” 2017. 

[11] K. N. Fadhilah, “PEMODELAN REGRESI SPLINE TRUNCATED UNTUK DATA 
LONGITUDINAL ( Studi Kasus : Harga Saham Bulanan pada Kelompok Saham 
Perbankan Periode Januari 2009 – Desember 2015 ),” vol. 5, pp. 447–454, 2016. 

[12] M. F. F. Mardianto, E. Tjahjono, and M. Rifada, “Statistical modeling for prediction of 
rice production in Indonesia using semiparametric regression based on three forms 
of fourier series estimator,” ARPN J. Eng. Appl. Sci., vol. 14, no. 15, pp. 2763–2770, 
2019. 

[13] F. N. Hidayah, Analisis Regresi Nonparametrik Spline Linear. Yogyakarta: Fakultas 
Saintek UIN Sunan Kalijaga, 2019. 

[14] Y. H. Chan, C. D. Correa, and K. L. Ma, “The generalized sensitivity scatterplot,” IEEE 
Trans. Vis. Comput. Graph., vol. 19, no. 10, pp. 1768–1781, 2013, doi: 
10.1109/TVCG.2013.20. 

[15] P. Taylan, G. W. Weber, L. Liu, and F. Yerlikaya-Özkurt, “On the foundations of 
parameter estimation for generalized partial linear models with B-splines and 
continuous optimization,” Comput. Math. with Appl., vol. 60, no. 1, pp. 134–143, 
2010, doi: 10.1016/j.camwa.2010.04.040. 

[16] X. Ni, H. H. Zhang, and D. Zhang, “Automatic model selection for partially linear 
models,” J. Multivar. Anal., vol. 100, no. 9, pp. 2100–2111, 2009, doi: 
10.1016/j.jmva.2009.06.009. 

[17] A. Pawar et al., “Adaptive FEM-based nonrigid image registration using truncated 
hierarchical B-splines,” Comput. Math. with Appl., vol. 72, no. 8, pp. 2028–2040, 
2016, doi: 10.1016/j.camwa.2016.05.020. 

[18] H. A. Farahani, A. Rahiminezhad, L. Same, and K. Immannezhad, “A comparison of 
Partial Least Squares (PLS) and Ordinary Least Squares (OLS) regressions in 
predicting of couples mental health based on their communicational patterns,” 
Procedia - Soc. Behav. Sci., vol. 5, pp. 1459–1463, 2010, doi: 
10.1016/j.sbspro.2010.07.308. 

[19] A. Prahutama, Suparti, and T. W. Utami, “Modelling fourier regression for time series 
data - A case study: Modelling inflation in foods sector in Indonesia,” J. Phys. Conf. 
Ser., vol. 974, no. 1, 2018, doi: 10.1088/1742-6596/974/1/012067. 

[20] R. L. Eubank, Nonparametric Regression and Spline Smoothing, 2nd Editio. New York: 
Mercel Dekker, 1988.