Pemodelan Data Rata-Rata Ujian Nasional Pada Sekolah Menengah Negeri Di Jawa Barat.
ON MODELLING THE AVERAGE SCORES OF
NATIONAL EXAMINATION FOR PUBLIC
SENIOR HIGH SCHOOL IN WEST JAVA
KARIN AMELIA SAFITRI
GRADUATE SCHOOL
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2015
STATEMENT OF ORIGINALITY
The author hereby states that this thesis which is entitled On Modelling the
Average Scores of National Examination for Public Senior High School in West
Java is based on author’s own research with from supervisors comission and has
not been published before. Information source which was either derived or cited
from either published or unpublished work from another writer has been
mentioned in the text and listed in the bibliography which is the final part of this
thesis.
The writer hereby bestows this copyright from this research to Bogor
Agricultural University.
Bogor, September 2015
Karin Amelia Safitri
NIM G152130511
Copyright transfer of joint research collaboration with external agency of IPB shall
meet the MoU between parties
RINGKASAN
KARIN AMELIA SAFITRI. Pemodelan Data Rata-Rata Ujian Nasional pada
Sekolah Menengah Negeri di Jawa Barat. Dibawah bimbingan oleh ANANG
KURNIA dan KHAIRIL ANWAR NOTODIPUTRO.
Pendidikan formal di Indonesia terdiri dari pedidikan pra-sekolah, Sekolah
Dasar (SD), Sekolah Menengah (SMP-SMA), dan perguruan tinggi. Pemerintah
Indonesia sudah berusaha keras untuk meningkatkan kualitas pendidikan di
Indonesia. Peningkatan kualitas pendidikan dapat direpresentasikan dalam
bentuk Ujian Nasional (UN) yang diujikan pada siswa.
Penelitian ini dilakukan dengan tujuan untuk melihat faktor apa saja yang
mempengaruhi UN dan mendapatkan model yang terbaik. Data yang digunakan
pada penelitian ini adalah data panel yang berupa data UN siswa SMA Negeri
dari tahun 2011-2014 dan peubah bebas yang digunakan adalah Indeks
Pmbangunan Manusia (IPM), Pendapatan Dometik Regional Bruto (PDRB),
bahan pembelajaran, proses pembelajaran, tenaga pendidik, fasilitas sekolah,
total akreditasi dan rata-rata Ujian Sekolah (US).
Skor rata-rata UN untuk kasus ini tidak bebas dan sangat bervariasi untuk
setiap SMA Negeri dan setiap tahun sehingga SMA Negeri dianggap unit
individu dan tahun bisa diasumsikan pengaruh tetap dan acak. Sehingga, data
pada penelitian ini dimodelkan dengan Linear Mixed Model (LMM),
Generalized Estimating Equation (GEE), dan model tetap. Dengan
membandingkan nilai R-square, regresi dengan menambahkan pengaruh tetap
sekolah dan pengaruh tetap waktu menghasilkan model terbaik dalam
menjelaskan keragaman data UN dibandingkan jika dengan GEE dan LMM.
Model dengan efek tetap sekolah dan waktu dengan menambahkan peubah
dummy dapat menangkap keragaman dari setiap subjek baik sekolah maupun
waktu. Jika sekolah dan waktu diasumsikan sebagai pengaruh acak tidak
menghasilkan model yang baik. Begitupula jika tahun mengikuti autoregressive
atau AR(1) model yang dihasilkan menghasilkan R-square yang cukup tinggi
tetap ada kemungkinan bahwa tahun tidak mengikuti AR(1) tetapi lebih tepatnya
sekolah dan waktu diasumsikan sebagai pengaruh tetap. Semua peubah ebas
memberikan kontribusinya masing-masing dalam menaikkan skor UN dimana
peubah bebas tersebut adalah IPM, PDRB, bahan pembelajaran, proses
pembelajaran, tenaga pendidik, fasilitas sekolah, total akreditasi dan rata-rata US.
Kata kunci: pengaruh tetap, GEE, linear mixed model, pengaruh random, ujian
nasional.
SUMMARY
KARIN AMELIA SAFITRI. On Modelling the Average Scores of National
Examination for Public Senior High School in West Java. Supervised by ANANG
KURNIA and KHAIRIL ANWAR NOTODIPUTRO.
Formal education in Indonesia is commonly divided into stages such as
preschool, primary school (SD), Secondary School (SMP-SMA), and
universities/colleges. Indonesian government has been taking serious efforts on
how to improve the quality of education in Indonesia. The roadmap for continous
improvement of education quality can be designed based on the results of
National Examination (UN) taken regularly by high school students.
This research was aimed at exploring informations on how the scores of
UN can be linked with other explanatory variables and to find the best model. A
panel data which consists of average scores of UN for all public senior high
schools (SMA Negeri) in West Java Provinces during 2011-2014 and other
related variables such as total scores of accreditation, regional domestic product,
human development index, scores of school’s facilities and its infrastructure,
scores of school’s educators, average scores of final school exams, were used in
this research.
The average scores of UN in this case were dependent on variations
between high schools and time periods as well as other explanatory variables in
which the effects were either fixed or random. The data of this research was
modelled with linear mixed models, the Generalized Estimating Equation (GEE),
two fixed effects approach. This research showed that by comparing the
coefficient of determination, the regression with adding two fixed effects of time
and school provided a model of better performance than the GEE and linear
mixed model in explaining the variability of the response variable which was the
average scores of UN.
A model with fixed effect which used dummy variables had an ability to
catch the variability of the subjects in this case the subjects were schools and time,
had the highest R-squared value was because time effect is suitable if being
assumed to be fixed, not fitted if assumed to be either AR(1) or random effect. All
explanatory variables gave its own conribution at increasing the average scores of
UN which is human development index, regional domestic product, the average
scores of US and five components of accreditation (the scores of educators,
facilities, content, process, and total accreditation.
Key words: fixed effects, GEE, linear mixed model, national examination,
random effects.
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Prohibited to quote some or all of this paper without including or
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Prohibited to announce and reproduce part or all of this paper in any form
without permission IPB
ON MODELLING THE AVERAGE SCORES OF
NATIONAL EXAMINATION FOR PUBLIC
SENIOR HIGH SCHOOL IN WEST JAVA
KARIN AMELIA SAFITRI
A Thesis submitted to the Graduate School of
Bogor Agricultural University
as a partial fullfilment of the requirement for the degree of
Master of Science
at
Department of Statistics
GRADUATE SCHOOL
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2015
External Examiner : Dr Ir Indahwati, MSi
ACKNOWLEDGEMENT
All praises and gratitude is only to Allah Subhanahu WaTa’ala for the grace
and the blessings He never stops giving so as the writer can accomplish the
writing of this thesis which has a title “On Modelling the Average Scores of
National Examination for Public Senior High School in West Java”.
The writer also gives best thank to the most awesome advisors, Dr. Anang
Kurnia, M.Si and Prof. Dr. Khairil Anwar Notodiputro, M.S who have been
helping, critisizing, teaching the writer about everything related to her thesis. Big
thank you to the best parents in the world for the prayers, support, encouragement
and everything they have been giving to the writer. Thank you too for all the staffs
of Applied Statistics Study Program, all of her friends in Statistics for their
helping hands and advices.
The writer realizes that the thesis still need critics and opinion for the
implementation of the next research in the future. The writer hopes this research
will be useful of the scientific work.
Bogor, September 2015
Karin Amelia Safitri
CONTENTS
LIST OF TABLES
vi
LIST OF FIGURES
vi
LIST OF APPENDICES
vi
1 INTRODUCTION
Background
Research Questions
Objectives
Research Benefit
1
1
2
3
3
2 LITERATURE REVIEWS
Panel Data
Fixed Effect Model
Linear Mixed Model
Estimation Method
Generalized Estimation Equation
Sampling Technique
3
3
4
4
6
7
7
3 MATERIALS AND METHODS
Data
Variables
Methodology
9
9
9
10
4 RESULT AND DISCUSSION
Principal Component Analysis
The Estimated Model
Overview of the Best Model
Prelimary Analysis of Required Sampling Technique
12
14
15
17
20
5 CONCLUSION AND RECOMMENDATION
Conclusion
Recommendation
22
22
23
REFERENCES
23
APPENDICES
25
BIOGRAPHY
31
LIST OF TABLES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Structures of working correlation matrix
The descriptive statistics of natinal examination
Coefficient correlation between response and explanatory variables
for natural science
Coefficient correlation between response and explanatory variables
for social science
Variance inflation factor of explanatory variables
Eigen analysis of the correlation matrix
Model comparison
Plots of the actual and predicted values in each model
Parameter estimates for the first five components of natural science
Parameter estimates for the three selected components of natural
science
Parameter estimates for the first five components of social science
Parameter estimates for the two selected components of natural
science
Number of sampled cluster with different bound of errors
Variance of average of UN
7
13
13
13
14
14
15
16
17
17
18
18
21
22
LIST OF FIGURES
1.
2.
3.
4.
Scatterplot of the average score of UN versus years for a) natural
science b) social science in West Java
Normality plots for a) natural science b) social science
Residual versus predicted value a) natural science b) social science
Sampling scheme
12
19
20
20
LIST OF APPENDICES
1
2
3
4
5
6
7
8
9
Analysis of variance for the five principal component of natural
science
Analysis of variance for the three selected principal component of
natural science
Analysis of variance of the five principal component of social science
Analysis of variance of the two selected principal component of
social science
Eigen vectors for natural science
Eigen vectors for social science
Statistics descriptive of natural science
Statistics descriptive of social science
Statistics descriptive of each municipality/regency for natural science
in 2014
25
25
25
25
26
26
26
26
27
10 Statistics descriptive of each municipality/regency for social science
in 2014
11 The formula of Estimated mean of three stage sampling ( stratified
sampling – cluster sampling – simple random sampling)
12 The formula of Estimated variance of three stage sampling ( stratified
sampling – cluster sampling – simple random sampling)
28
29
30
INTRODUCTION
Background
Education plays important roles for the national development. Quality
of human resources can be improved through better quality of education
continuously. In Indonesia, there are several types of education namely
formal, non-formal and informal education which are complemented each
other. The level of formal education comprises of elementary education,
high school or intermediate education, and higher education. A good quality
school must provide quality teaching, curriculum, management, and
facilities. The quality of a school can be reflected by the national
examination scores and the school examination scores, in Indonesia namely
as UN and US, respectively.
UN is a standard evaluation system of primary and secondary
education in Indonesia. In order to control the quality of education
nationwide, UN is conducted once in a year which not only consideration
for students to continue to next level study and for government to deliver
helps in order to develop education system of education units, but also
represent the quality of education mapping itself (Permendikbud No.5). UN
case has been urgently required for this research due to the importance of
respresenting the education in Indonesia. Some previous research related to
UN was carried out by Sinaga (2011) analyzed correlation between the
scores of UN, US and school grades using canonical analysis. Mongi (2014)
also did research about relation between scores of UN and accreditation
standards in West Java.
Senior High School which is commonly abbreviated as SMA in
Indonesia acknowledged its role in preparing students to continue to the
college-level studies. Therefore, This research has analyzed the data of UN
scores all public SMAs in West Java. These SMAs have been referred as
observation units. This kind of data is panel data, since the data is a
combination of cross sectional data and time series data. Data was taken in
four years sequently from 2011-2014.
A linear mixed model is an appropriate approach which has beenused
in this research since response variable is normally distributed. Linear
mixed model is also well suited for the analysis of panel data, where each
time series constitues an individual curve as a cluster. The model
accommodates both fixed effects and random effect. The linear mixed
model is an extention of linear model for data that are collected in groups.
linear mixed model was useful in the analysis of panel data because it has
accommodated the correlation of repeated measurements which are made on
the same unit, the heterogeneity among subjects, and inference toward
certain subjects.
The panel data also could be modelled by fixed effect model with
various intercept and constant slope by adding dummy variables. This
model assumed that the different intercepts were from cross-section and
time unit.
2
Another approach that is used for analysis in this research is
Generalized Estimating Equation (GEE). GEE is used to model the
correlated data from repeated measure studies (panel data) and basically, it
uses quasi-score likelihood for estimating the model parameters. GEE
covers the extentions of Generalized Linear Model (GLM) to panel data.
Specifically, GEE is a population averaging method that models panel data
in which the response is a member of the exponential family of distribution,
for example continous, binary, grouped and count data.
It is important to mention this research considered that UN scores can
be affected by internal factors as well as external factors. The external
factors are the factors within school, education system, and the effect of
school policies. Economics condition of each regency/municipality in West
Java where the schools are located may affect the quality of education.
Hence, it will affect the UN scores of the schools. This economic condition
can be measured by the gross regional income per capita (commonly
mentioned as PDRB in Indonesia) just like what Ahmad (2011) has stated
that the government funds allocated for education sector can affect
economic growth or PDRB and vice versa. According to Statistics Indonesia
(2008), the growth rate of PDRB also influences human development index
which is abbreviated as IPM in Indonesia through household expenditures
for daily primary needs including foods, medicines, and school stuffs. The
growth rate of PDRB may also influence IPM through domestic expenditure
policies made by government including priority funds for social aspects. In
contrast, IPM may influence PDRB through qualified human resources in
terms of good health and education services.
Another reason why UN is highlighted in this research because of the
students pass rate is no longer measured by UN since 2015, due to the
change of education system. This is not wise way to spend much money for
performing UN every year for every school. On the other hand UN is solely
used for measuring things that do not maximally improve the education in
Indonesia. Therefore, UN is better to be conducted for some schools as a
focused sample to mindmap the education instead of using the population
of all schools in Indonesia as measured target. The critical analysis is
required as preliminary task to do due to evaluate how the sampling can be
applied for UN implementation used for the consideration.
Research Questions
1.
2.
What is the best model in modelling between average scores of UN all
public SMAs for both natural science major and social science major
and other explanatory variables such as IPM, PDRB, average scores of
US, total accreditation scores, educators scores, school facilities scores,
teaching content scores, and teaching process scores?
Of all explanatory variables mentioned above, how much do it
contribute at affecting the average scores of UN?
3
Objectives
The main objectives of this study are
1. To develop the best model for modelling between average scores of UN
all public SMAs for both natural science major and social science major
and another explanatory variables mentioned above.
2. To evaluate how much do the explanatory variables contribute at
affecting the average scores of UN.
Apart from the main objectives mentioned above, this study also has a
secondary objective which is to preliminary analyze the possibility of
designing samples for evaluating student achievement.
Research Benefit
The best models which is obtained in this research will lead to what
influence UN the most. As the first preliminary step to evaluate the process
of conducting the UN since there has been provision that UN is no longer
used to measure student pass rate.
LITERATURE REVIEWS
Panel Data
A panel data set is one that follows a given sample of individuals over
time, and thus provides multiple observations on each individual on the
sample (Hsiao 2003). Observing a panel data set is basically observing a
broad section of subjects over time, and thus allows us to study dynamic, as
well as cross sectional, aspects of a problem (Frees 2004).
Baltagi (2005) mentioned the main advantages of panel data analysis
such as.
1. Solves individual heterogeneity.
2. Provides more informative dataset, handles collinearity of variables,
controls some types of ommited variables without observing them,
obtains degree of freedom in a bigger amount.
3. Studies comprehensively the dynamics of adjustments.
4. Identifies and estimates effects which cannot be taken into traditional
technique (e.g. time series analysis and usual linear regression).
5. Reduces bias when estimating because of plenty number observations.
Define following general panel data model for observation
section data i at time t
of cross
(1)
4
where is a vector of regression coefficient with K x 1 where K is number
of explanatory variables,
is a response variable observed for individual
unit i at time t,
is a vector of explanatory variables individual unit i at
time t,
is the error term.
Fixed Effect Model
Fixed effect model is a statistical model that represents the observed
quantities in terms of explanatory variables that are treated as if the
quantities were non random. Fixed and random effects to respectively refer
to the population average and subject specific effects. Fixed effect model
which has form as.
(2)
where denoted subject dimension which equals 1, 2, ... N. denoted time
dimension which equals 1, 2, ..., T and with specification of residual
components is
. Model 2 also can be written as.
(3)
where
,
[
]
[
]
[
]
where is the number of explanatory variables.
, with
is an
identity matrix,
is a
vector. is
matrix of dummies which is included in the regression to estimate α if the
parameters are assumed to be fixed (Baltagi,2008).
Linear Mixed Model
According to Jiang (2007), linear mixed model takes general form that
can be written as
y = Xβ + Zα+ ε
(4)
where y is an observation vector of dimension ni, X is a known covariate
matrix, β is a regression coefficient vector (fixed effect), Z is a design
matrix that contains only 1 and 0, α is a normally distributed random effect
vector, ε is an error vector.
5
The Basic assumption in linear mixed model is the random effect and
error have mean zero and certain varians, e.g. var(α) = G and var(ε)= R. It is
also assumed that α and ε must be uncorrelated. If the random effect and
error are assumed to be normally distributed, then the assumptions can be
written as
α~N(0, G)
ε~N(0, R)
Furthermore, the distribution of response variable is y~N(Xβ,V) where V =
Var (y) = ZGZ’+R. Otherwise, if the random effect and error are not
assumed to be normally distributed, then model (4) is non Gaussian linear
mixed model.
Jiang (2007) also stated that estimation in linear mixed model consists
of estimation toward both the fixed effect and random effect that can be
shown below. Estimation of fixed effect is carried out using Best Linear
Unbiased Estimator (BLUE)
̂
(5)
and the random effect prediction which is Best Linear Unbiased Prediction
(BLUP) can be written as
̂
̃
(
(6)
In fact, variance component V is unknown therefore, in estimating fixed
effect and random effect, V is replaced by the estimator ̂ and the estimation
result is empirical best linear unbiased predictor (EBLUP)
̂
̂
̂
(7)
̂
̂
where and are estimator using Maximum Likelihood Estimation (MLE)
or Restricted Maximum Likelihood (REML).
Jiang (2007) mentioned that the estimation result using REML for
variance componen is a consistent estimator without assuming normality
from random variable and error. REML is also an estimation method that is
more robust and practical.
Henderson (1959) introduced the solutions which can produce
simultaneously the GLS estimator of and BLUP of .
and
the joint density of and is
{
[
The log of
{
}
}
]
is
[
The partial derivatives of log of
with respect to
.
and
are
].
.
6
Setting these results to zero will obtain the following equations
{
.
These can be written as
]
[
[
].
(8)
Estimating The Model Parameters
Some approaches used to estimate the model parameters in linear
mixed model are the traditional approach, Maximum Likelihood Estimation
(MLE), and Restricted Maximum Likelihood (REML). One of those
approaches used for this study is REML.
REML can produce the unbiased or nearly estimator for variance and
it does not require more complex computation just like the MLE estimators.
Searle (1992) stated that the REML approach is now more preferable to
estimate the variance parameters in mixed model.
y is a response vector that normally distributed which is transformed
for z = A’y which has distribution z ~ N(0, A’VA). Define a non-zero n x
(n-p) matrix A for A’X = 0 so then A A’ = Q where
̂
and AA’ = I then define z = A’y = (
. The
expected value of z is 0 and covariance matrix is A’VA. The density of z is
{
{
(
}
̂
According to Rao (1973) that
̂
̂
̂
(
(
(
then we come to the final expression of the desity of z
̂
{
}
(
(
(
̂
̂
̂
}
(
.
(9)
the final density of z leads to the restricted or residual log likelihood which
is written below.
.
(10)
Just like estimating using MLE, log likelihood REML is maximized
also requires numerical technique for example Newton Raphson with minor
7
adaptations (Longford 2003). By subtituting the estimator ̂
into GLS formula
̂
̂
( ̂
̂
eventhough y is normal distribution
is not identical ̂
̂
(11)
Generalized Estimating Equation
Zeger and Liang (1986) introduced an alternative approach to
estimate the parameter in maximum likelihood case. This approach is
usually known as generalized estimating equation (GEE). Basically, GEE
uses quasi score likelihood theory.
GEE approach in
for estimating by using quasiscore likelihood equation which is shown as follow
∑
[ ]
with
[ ]
⁄
dan
⁄
⁄
is diagonal matrix with element var( ),
is working
correlation matrix for , and is constant (Dobson 2002). Some general
forms of structure of working correlation matrix presented in the Table 1 as
follow
Stuctures
Name
Independence
Exchange
-able
Table 1. Structures of working correlation matrix
Structures
Stuctures
Structures
Name
Autoregressive
Unstructured
)
)
)
Sampling Technique
Stratified Random Sampling
Scheaffer et al. (2006) stated that a stratified random sample is
obtained by separating the population elements into nonoverlapping groups,
called strata, then selecting a simple random sample from each stratum. The
principal reasons for using stratified random sampling areas follows:
8
1. Stratification may produce a small bound on the error of estimation.
This result is particularly true if measurements within strata are
homogenous
2. The cost per observation in the survey may be reduced by
stratification of the population elements into convenient groupings
3. Estimates of population parameters may be desired for subgroups of
the population. These subgroups should then be identifiable strata.
The first step in the selection of a stratified random sample is to
clearly specify the strata then each sampling unit of the population is placed
into its appropriate stratum. After the sampling units have been divided into
strata, select a simple random sample from each stratum.
Estimator of the population mean is
̅
∑
̅
where
is number of sampling units in stratum i, i = 1, 2, .. L, ̅ denotes
the sample mean for the simple random sample selected from stratum i.
Estimator variance of ̅ is
̂ ̅
[
∑
̂ ̅
(
(
̂ ̅
(
̂ ̅ ]
Two Stage Cluster Sampling
A two stage cluster sample is obtained by first selecting a probability
sample of clusters and then selecting a probability sample of elements from
each sampled cluster (Scheaffer et al. 2006). Cluster must be as
heterogeneous (different) as possible within, and one cluster should look
very much like another in order for the economic advantages of cluster
sampling to pay off. If a population being divided into nonoverlapping
groups of elements. These groups are considered to be clusters, then a
simple random sample of groups is selected, and the sampled groups are
then subsampled.
The advantages of two stage cluster sampling are
1. A frame listing of all elements in the population may be impossible
or costly to obtain, whereas obtaining a list of all clusters may be
easy
2. The cost of obtaining the data may be inflated by travel costs if the
sampled elements are spread over a large geographic area. Thus,
sampling clusters of elements that are physically close together is
often economical.
By assuming simple random sampling at each stage, the unbiased
estimator of the population mean with is the number of cluster in the
9
population, is the number of clusters selected in a simple random sample,
is the number of elements in cluster .
is the number of elements
selected in a simple random sample from cluster
equals ∑
̅ is
∑
and
is the th
the sample mean for the ith cluster
observation in the sample from the th cluster, is
∑
∑
̅
̂ (
̅
The estimated variance of
where
and
̂
̂
(
(
̅
̅
is
̅
∑
̅
∑
(
∑
̅
(
̅
is simply the sample variance among the terms
the sample variance for the sample selected from cluster .
̅ and that
is
MATERIALS AND METHODS
Data
1. The average scores of UN and US for both natural science diciplines and
social science diciplines of all SMANs in West Java 2011-2014. Data was
accessed from Educational Research Center, Ministry of Education and
Culture with the lowest score is zero and the highest one is 10 then it is
multipled with 10 therefore the score range becomes between 10 and
100.
2. Data of total accreditation scores, educators scores, school facilities
scores, teaching content scores, teaching process scores, school
management scores, school finance scores, and graduates competence
scores were accessed in National Accreditation Institution.
3. Data of human development index and gross regional product in West
Java were accessed through Statistics Indonesia website.
Variables
Variables which are used in this research are
10
Y : The average scores of UN each SMAN for both natural science and
social science
X1 : IPM scores each regency/municipality in West Java
X2 : PDRB each regency/municipality in West Java
X3 : Teaching content scores each SMAN in West Java
X4 : Teaching process scores each SMAN in West Java
X5 : Educators scores each SMAN in West Java
X6 : School facilities scores each SMAN in West Java
X7 : Total accreditation scores each SMAN in West Java
X8 : The average scores of US each SMAN in West Java
Research Methodology
This research is divided into two to reach the goals, first is modelling,
and the second is sampling technique. Some steps in data analysis process in
this research are
Modelling The Data
1.
2.
3.
Exploratory data analysis
In this step, the variability and mean of response variable in
each regency/municipality in West Java was investigated variable
to visualize the information was provided.
Identification of the multicollinearity among explanatory variables
The multicollinearity was identified by calculating the
coefficient of correlation among the explanatory variables. The
data was then analyzed using principal component analysis to
remove the multicollinearity.
Modelling the data with the linear mixed model (LMM)
Models were fitted by incorporating fixed and random effects
which accomodated that every school i at time t had their own
intercept. The model which contained mixed effect took form as:
̂ ̂
̂ , where ̂ is a vector of estimated
̂
parameters with K x 1 where K is the number of principal
components, ̂ is an estimated response variable observed for
individual unit i at time t,
is a vector of principal
components, ̂ is the estimated school effect, ̂ is the estimated
time effect.
In this modelling, the school effect is assumed to be fixed
because schools as sample in West Java is determined from the
beginning. In contrast, the time effect
is assumed to be random
because year can not be limited and there will always be next year
therefore the data includes 2011, 2012, 2013 and 2014.
The school effect is assumed to be fixed then those variables
are transformed into dummy variables. Because there are 421
schools in West Java that have natural science and 419 schools that
have social science program then it means there 420 and 418
11
dummy variables respectively. The model parameters was
estimated using restricted maximum likelihood (REML).
4. Modelling the data with generalized estimating equation (GEE).
̂ ̂ , where ̂ is the estimated school effect
̂
approached to be fixed and this model was estimated using GEE
which accomodates the correlated response variable using the
working correlation matrix : AR(1) in condition of the time series
data which was taken solely covers 4 years. This model is
considered because it is able to accomodate the heterogeneity
among school effect using dummy variables. GEE model with
response variable is normally distributed using identity link
function. The working correlation matrix has the following form as
shown below
)
Modelling with GEE by adding school effect which was
assumed to be fixed into the equation took form as
̂
The model parameters were estimated using quasi-score likelihood
where the variance of as a function of the mean, contained the
working correlation matrix autoregressive.
5. Modelling the data with fixed effects
This model was built with consideration that the numbers of
both of schools and time were defined from the beginning. Because
of the effects of schools and time were decided to be fixed, it has
been added dummy variables into the model which could be written
as
̂ ̂
̂
̂
where ̂ denoted the estimated school effect and ̂ is the estimated
time effect. The model parameters were estimated by using analysis
of variance.
6. Evaluation of the models by comparing the R-squared value and
mean square error of each model. The model with the largest value
is assumed as the best model.
7. Interpretation of the data analysis results.
Sampling Technique
1. Divided population in this case covers West Java into two strata
which are city and regency.
2. West Java has 9 cities and 19 regencies in which is considered as
population clusterings.
3. Defined the number ofsampled clusters of both city and regency
blocks by calculating approximate n cluster with different bounds
12
4.
5.
6.
7.
B on error of estimation which are in this research 0.05, 0.07, and
0.09, respectively.
n was defined, chose element of sampled cluster by using simple
random sampling.
Making an experiment by taking schools in each cluster with
simple random sampling in terms of the characteristics how many
schools each municipality/regency has.
Calculated the variance of the average scores of UN for each school
in each municipality/regency in which there had 6 variances found
out with three stages sampling formula which include stratified
sampling, cluster sampling, and simple random sampling.
Compared every variance and the most minimum variance of
sample with three stages sampling method can be representative
toward the population.
13
RESULT AND DISCUSSION
As it has been commonly acknowledged in Indonesia, there are two
programs available in many senior high schools namely natural science and
social science. This research has both utilized and analyzed the data of those
two different programs separately due to not every senior high schools in
West Java has both natural science and social science and it then can not be
concluded that students learning ability in natural science is as equal as in
social science.
There were two initial models established. Those two models were
built based on the data of natural science and social science. Evaluating the
model and selecting the best model was done by comparing the R-squared
value and mean square error of each model.
The averages scores of UN every year have been presented in Figure
1.
90
a) The Average Score of UN 20112014
90
70
70
50
50
30
2010
2011
2012
2013
2014
b) The Average Score of UN 20112014
30
2010
2011
2012
2013
2014
Figure 1. Scatterplot of the average score of UN versus years for a) natural
science b) social science in West Java
The Figure 1 showed that there was such a decline on the slope between
year 2012 and 2013. The most current years which are 2013 and 2014, the
average scores of UN for natural science has decreased. Sharp slope also
happened between year 2012 and 2013 on the plot of social science major.
Another exploration obtained was the mean and variability of the
average scores of UN in the very curren year, 2014 for each
municipality/regency were also provided in Appendix 9 and Appendix 10.
Both of natural and social science, Bandung City has the higest mean of the
average scores of UN. Bandung City is one of big cities in Indonesia in
which its goverment pays much attention for education there.
The descriptive statistics of the average scores of UN of all public
senior high schools in West Java also shown in the Table 2.
14
Table 2. The descriptive statistics of national examination
Term
2011
2012
2013
2014
Natural
Maximum
96.07
97.67
79.40
95.75
Science
Minimum
64.90
24.08
41.83
43.58
Mean
82.56
82.17
65.53
58.50
Variance
19.20
18.23
17.47
33.41
Social
Maximum
89.95
94.83
91.33
81.45
Science
Minimum
58.30
13.67
7.33
43.18
Mean
79.75
76.27
61.95
53.05
Variance
25.81
23.91
11.16
24.11
Based on Table 2, started from 2013 both in natural and social
science program, there was a newer system that has been implemented for
accomplishing UN that caused the average score of UN fell down. In that
system of UN, enforced that each task paper has been made exactly different
for each student which was implemented since 2013 in order to dismiss
every kind of cheating students possibly could do in UN performance.
Another descriptive statistics measured the relationship between
response variable and explanatory variable has been statistically illustrated
by coefficient correlation shown in the following table.
Table 3. Coefficient correlation between response and explanatory variables
for natural science
Year
X1
X2
X3
X4
X5
X6
X7
X8
2011 Y 0.079 -0.017 0.282 0.219 0.359 0.468 0.449 0.127
2012 Y 0.086 -0.004 0.073 0.002 0.131 0.163 0.141 0.127
2013 Y 0.460 0.176 0.148 0.171 0.309 0.287 0.292 0.068
2014 Y 0.565 0.229 0.271 0.215 0.462 0.423 0.450 0.187
Table 4. Coefficient correlation between response and explanatory variables
for social science
Year
X1
X2
X3
X4
X5
X6
X7
X8
2011 Ya 0.175 -0.015 0.234 0.181 0.355 0.421 0.406 0.103
2012 Y 0.344 0.148 0.196 0.132 0.309 0.315 0.321 0.064
2013 Y 0.336 0.102 0.167 0.209 0.283 0.267 0.301 -0.013
2014 Y 0.460 0.104 0.239 0.215 0.455 0.393 0.427 0.220
a
Variables used in this research; Y : The average score of UN each SMAN in West Java, X1
: IPM scores each regency/municipality in West Java, X2 : PDRB each
regency/municipality in West Java, X3 : Teaching content scores each SMAN in West Java
X4
: Teaching process scores each SMAN in West Java, X5 : Educators scores each
SMAN in West Java, X6 : School facilities scores each SMAN in West Java, X7 : Total
accreditation scores each SMAN in West Java, X8 : The average scores of US each SMAN in
West Java
From Table 3 and Table 4, known that for the natural science the average
scores of Y is correlated with all X variables observed for 4 years. However
social science has a slight difference toward natural science and both for
15
natural and social science the correlation between Y and X is not very
strong.
Another exploration to make sure by calculating variance inflation
factor (VIF) to quantify the severity of multicollinearity in an ordinary least
squares regression analysis. The following table showed VIF of all X.
Table 5. Variance inflation factor of explanatory variables
Predictor
VIF
X1
1.25
X2
1.16
X3
2.34
X4
2.10
X5
2.79
X6
5.25
X7
11.80
X8
1.07
Table 5 showed that X6 and X7 has the highest VIF. It surely means that
there was multicollinearity which could cause a serious problem which
make the standard error get higher if it was not removed and would lead to
the wrong conclusion for this study.
Principal Component Analysis
From the result shown Table 5, it is clear that multicollinearity
among explanatory variables has been existed. Principal component analysis
has been widely used to eliminate the correlation between explanatory
variables through transformation which has been presented in Table 5. It
then can be ensured that all explanatory variables is independent. Many
methods to solve multicollinearity yet principal component analysis has
been chosen in order to include all informations in X without eliminating an
X that has high VIF value. Table 6 of eigen analysis for both natural and
social science is shown below and eigen vectors which was generated
shown by Appendix 5 and Appendix 6.
eigenvalue
proportion
cumulative
eigenvalue
proportion
cumulative
Table 6. Eigen analysis of the correlation matrix
Natural Science
PC1 PC2
PC3 PC4
PC5
PC6
3.51 1.41
0.98
0.78
0.58
0.38
0.44 0.18
0.12
0.09
0.07
0.05
0.44 0.62
0.74
0.83 0.91
0.99
Social Science
3.54 1.39
0.96
0.79
0.59
0.38
0.44 0.17
0.12
0.09
0.07
0.05
0.44 0.62
0.74
0.83
0.91
0.99
PC7
0.31
0.04
0.99
PC8
0.06
0.01
1.00
0.30
0.04
0.99
0.06
0.01
1.00
16
Table 6 showed that the cumulative proportions of variance explained by the
first five components have shown 0.91 both for natural and social science
which means that more than 90% information of eigen explanatory
variables could be explained by five components. Moreover, these five
components have been orthogonal each other. Hence, the problem of
multicollinearity has been removed.
The Estimated Model
The response (the average scores of UN) is continous variable which
was assumed normally distributed.This assumption was justified by the
central limit theorem. Here the response variable (the average scores of UN)
has been modelled with those five components for each program
respectively. Every modelling which has been done in order to find best
model gave different results by comparing the value of R-square, R-square
adjusted, and mean square error. The Table 7 shows the result of 3 different
modellings in the simple version.
Table 7. Model comparison
Model1
Model2
Diciplines
School Eff
Fixed
Fixed
Time Effect
Random
AR(1)
Natural
R-Square
5.12
77.30
R-square Adj
3.28
76.87
Science
MSE
175.95
42.11
214.78
Social
R-Square
2.60
71.95
Science
R-square Adj
0.70
71.40
MSE
137.11
39.64
163.21
-
Model3
Fixed
Fixed
91.89
91.74
15.03
89.18
88.97
15.29
-
Table 7 showed that it was directly identified that the best model was model
3 in which both school effect and time effect were assumed to be fixed with
the highest R-squared value, 91,89 for natural science and 89,23 for social
science, and with the lowest MSE 15,03.
The data also has been modelled with linear mixed model in Model 1.
This model contained two effects, fixed and random effects outright. The
response (average scores of UN) is a normally distributed outcome which is
allowed to use the linear mixed model. School effect was assumed to be
fixed effect and time effect was assumed to be random effect. This model
was not the best model because time should not be random effect. In fact,
time was fixed effect because the sequence of year of observed data was
defined by the researcher from the beginning.
Model 2 which was built with Generalized Estimating Equation with
autoregressive working correlation matrix AR(1). This model was not
concluded to be the best model because of the possiblity of the imprecision
of utilizating working correlation matrix AR(1). This data had a short
17
sequence of time which only included 4 years observed data. Therefore it
could mean that possibly data of 2014 was not affected by the previous year,
data of 2013. It also occured in data of 2013 was not affected by data of
2012 and so forth.
Another plot to see that the best model was a model which was good
at predicting. The accuracy at predicting can be presented in a plot between
the average scores of UN and its predicted values shown in the following
table.
Table 8. Plots of the actual and predicted values in each model
Natural Science
Social Science
Modelling with Generalized Estimating Equation
Modelling with Linear Mixed Model
Modelling with Fixed School Effect and Fixed Time Effect
Table 8 showed that the model with both school and time effect which
were assumed to be fixed was good at predicting marked by the dots spread
18
around the diagonal line. Both in model 3 for natural science and social
science, standardized residual was normally distributed.
Overview of Model 3 (Fixed Effect Model)
̂ ̂
̂ where ̂ is the
Model 3 which had form as ̂
estimated model parameters, ̂ is the estimated response variable observed
for school i at time t,
is the vector of principal components for school i
at time t, ̂ is the estimated schools effect, ̂ is the estimated time effect
which were assumed to be fixed was ensured because of the numbers of
schools and time (the observed years) were both defined and chosen from
the beginning by researcher.
Model 3 is a model with fixed effect which used dummy variables had
an ability to catch the variability of the subjects in this case the subjects
were schools and time, had the highest R-squared value was because time
effect more fits if it was assumed to be fixed rather than to be random or
AR(1). Beside that, the variance of time ( 2011 – 2014 ) was quite high
because there was always modification of system of implementation of UN
globally every year. The model parameters were resulted in the Table 9.
Table 9. Parameter estimates for the first five components of natural science
Term
Coef
SE coef
T-Value
P-Value
w1
1.82
0.72
2.52
0.01
w2
0.60
1.52
0.40
0.69
w3
2.95
0.44
6.68
0.00
w4
0.22
1.14
0.19
0.85
w5
-7.56
1.83
-4.13
0.00
Table 9 showed that the p-value for
and
(0.01,
NATIONAL EXAMINATION FOR PUBLIC
SENIOR HIGH SCHOOL IN WEST JAVA
KARIN AMELIA SAFITRI
GRADUATE SCHOOL
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2015
STATEMENT OF ORIGINALITY
The author hereby states that this thesis which is entitled On Modelling the
Average Scores of National Examination for Public Senior High School in West
Java is based on author’s own research with from supervisors comission and has
not been published before. Information source which was either derived or cited
from either published or unpublished work from another writer has been
mentioned in the text and listed in the bibliography which is the final part of this
thesis.
The writer hereby bestows this copyright from this research to Bogor
Agricultural University.
Bogor, September 2015
Karin Amelia Safitri
NIM G152130511
Copyright transfer of joint research collaboration with external agency of IPB shall
meet the MoU between parties
RINGKASAN
KARIN AMELIA SAFITRI. Pemodelan Data Rata-Rata Ujian Nasional pada
Sekolah Menengah Negeri di Jawa Barat. Dibawah bimbingan oleh ANANG
KURNIA dan KHAIRIL ANWAR NOTODIPUTRO.
Pendidikan formal di Indonesia terdiri dari pedidikan pra-sekolah, Sekolah
Dasar (SD), Sekolah Menengah (SMP-SMA), dan perguruan tinggi. Pemerintah
Indonesia sudah berusaha keras untuk meningkatkan kualitas pendidikan di
Indonesia. Peningkatan kualitas pendidikan dapat direpresentasikan dalam
bentuk Ujian Nasional (UN) yang diujikan pada siswa.
Penelitian ini dilakukan dengan tujuan untuk melihat faktor apa saja yang
mempengaruhi UN dan mendapatkan model yang terbaik. Data yang digunakan
pada penelitian ini adalah data panel yang berupa data UN siswa SMA Negeri
dari tahun 2011-2014 dan peubah bebas yang digunakan adalah Indeks
Pmbangunan Manusia (IPM), Pendapatan Dometik Regional Bruto (PDRB),
bahan pembelajaran, proses pembelajaran, tenaga pendidik, fasilitas sekolah,
total akreditasi dan rata-rata Ujian Sekolah (US).
Skor rata-rata UN untuk kasus ini tidak bebas dan sangat bervariasi untuk
setiap SMA Negeri dan setiap tahun sehingga SMA Negeri dianggap unit
individu dan tahun bisa diasumsikan pengaruh tetap dan acak. Sehingga, data
pada penelitian ini dimodelkan dengan Linear Mixed Model (LMM),
Generalized Estimating Equation (GEE), dan model tetap. Dengan
membandingkan nilai R-square, regresi dengan menambahkan pengaruh tetap
sekolah dan pengaruh tetap waktu menghasilkan model terbaik dalam
menjelaskan keragaman data UN dibandingkan jika dengan GEE dan LMM.
Model dengan efek tetap sekolah dan waktu dengan menambahkan peubah
dummy dapat menangkap keragaman dari setiap subjek baik sekolah maupun
waktu. Jika sekolah dan waktu diasumsikan sebagai pengaruh acak tidak
menghasilkan model yang baik. Begitupula jika tahun mengikuti autoregressive
atau AR(1) model yang dihasilkan menghasilkan R-square yang cukup tinggi
tetap ada kemungkinan bahwa tahun tidak mengikuti AR(1) tetapi lebih tepatnya
sekolah dan waktu diasumsikan sebagai pengaruh tetap. Semua peubah ebas
memberikan kontribusinya masing-masing dalam menaikkan skor UN dimana
peubah bebas tersebut adalah IPM, PDRB, bahan pembelajaran, proses
pembelajaran, tenaga pendidik, fasilitas sekolah, total akreditasi dan rata-rata US.
Kata kunci: pengaruh tetap, GEE, linear mixed model, pengaruh random, ujian
nasional.
SUMMARY
KARIN AMELIA SAFITRI. On Modelling the Average Scores of National
Examination for Public Senior High School in West Java. Supervised by ANANG
KURNIA and KHAIRIL ANWAR NOTODIPUTRO.
Formal education in Indonesia is commonly divided into stages such as
preschool, primary school (SD), Secondary School (SMP-SMA), and
universities/colleges. Indonesian government has been taking serious efforts on
how to improve the quality of education in Indonesia. The roadmap for continous
improvement of education quality can be designed based on the results of
National Examination (UN) taken regularly by high school students.
This research was aimed at exploring informations on how the scores of
UN can be linked with other explanatory variables and to find the best model. A
panel data which consists of average scores of UN for all public senior high
schools (SMA Negeri) in West Java Provinces during 2011-2014 and other
related variables such as total scores of accreditation, regional domestic product,
human development index, scores of school’s facilities and its infrastructure,
scores of school’s educators, average scores of final school exams, were used in
this research.
The average scores of UN in this case were dependent on variations
between high schools and time periods as well as other explanatory variables in
which the effects were either fixed or random. The data of this research was
modelled with linear mixed models, the Generalized Estimating Equation (GEE),
two fixed effects approach. This research showed that by comparing the
coefficient of determination, the regression with adding two fixed effects of time
and school provided a model of better performance than the GEE and linear
mixed model in explaining the variability of the response variable which was the
average scores of UN.
A model with fixed effect which used dummy variables had an ability to
catch the variability of the subjects in this case the subjects were schools and time,
had the highest R-squared value was because time effect is suitable if being
assumed to be fixed, not fitted if assumed to be either AR(1) or random effect. All
explanatory variables gave its own conribution at increasing the average scores of
UN which is human development index, regional domestic product, the average
scores of US and five components of accreditation (the scores of educators,
facilities, content, process, and total accreditation.
Key words: fixed effects, GEE, linear mixed model, national examination,
random effects.
© IPB Copyright, 2015
Copyright Reserved
Prohibited to quote some or all of this paper without including or
mentioning the source. Quoting is only for educational purposes, research,
scientific thesis, report writing, criticism, or reviewing an issue; and citations are
not detrimental to the interests of IPB
Prohibited to announce and reproduce part or all of this paper in any form
without permission IPB
ON MODELLING THE AVERAGE SCORES OF
NATIONAL EXAMINATION FOR PUBLIC
SENIOR HIGH SCHOOL IN WEST JAVA
KARIN AMELIA SAFITRI
A Thesis submitted to the Graduate School of
Bogor Agricultural University
as a partial fullfilment of the requirement for the degree of
Master of Science
at
Department of Statistics
GRADUATE SCHOOL
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2015
External Examiner : Dr Ir Indahwati, MSi
ACKNOWLEDGEMENT
All praises and gratitude is only to Allah Subhanahu WaTa’ala for the grace
and the blessings He never stops giving so as the writer can accomplish the
writing of this thesis which has a title “On Modelling the Average Scores of
National Examination for Public Senior High School in West Java”.
The writer also gives best thank to the most awesome advisors, Dr. Anang
Kurnia, M.Si and Prof. Dr. Khairil Anwar Notodiputro, M.S who have been
helping, critisizing, teaching the writer about everything related to her thesis. Big
thank you to the best parents in the world for the prayers, support, encouragement
and everything they have been giving to the writer. Thank you too for all the staffs
of Applied Statistics Study Program, all of her friends in Statistics for their
helping hands and advices.
The writer realizes that the thesis still need critics and opinion for the
implementation of the next research in the future. The writer hopes this research
will be useful of the scientific work.
Bogor, September 2015
Karin Amelia Safitri
CONTENTS
LIST OF TABLES
vi
LIST OF FIGURES
vi
LIST OF APPENDICES
vi
1 INTRODUCTION
Background
Research Questions
Objectives
Research Benefit
1
1
2
3
3
2 LITERATURE REVIEWS
Panel Data
Fixed Effect Model
Linear Mixed Model
Estimation Method
Generalized Estimation Equation
Sampling Technique
3
3
4
4
6
7
7
3 MATERIALS AND METHODS
Data
Variables
Methodology
9
9
9
10
4 RESULT AND DISCUSSION
Principal Component Analysis
The Estimated Model
Overview of the Best Model
Prelimary Analysis of Required Sampling Technique
12
14
15
17
20
5 CONCLUSION AND RECOMMENDATION
Conclusion
Recommendation
22
22
23
REFERENCES
23
APPENDICES
25
BIOGRAPHY
31
LIST OF TABLES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Structures of working correlation matrix
The descriptive statistics of natinal examination
Coefficient correlation between response and explanatory variables
for natural science
Coefficient correlation between response and explanatory variables
for social science
Variance inflation factor of explanatory variables
Eigen analysis of the correlation matrix
Model comparison
Plots of the actual and predicted values in each model
Parameter estimates for the first five components of natural science
Parameter estimates for the three selected components of natural
science
Parameter estimates for the first five components of social science
Parameter estimates for the two selected components of natural
science
Number of sampled cluster with different bound of errors
Variance of average of UN
7
13
13
13
14
14
15
16
17
17
18
18
21
22
LIST OF FIGURES
1.
2.
3.
4.
Scatterplot of the average score of UN versus years for a) natural
science b) social science in West Java
Normality plots for a) natural science b) social science
Residual versus predicted value a) natural science b) social science
Sampling scheme
12
19
20
20
LIST OF APPENDICES
1
2
3
4
5
6
7
8
9
Analysis of variance for the five principal component of natural
science
Analysis of variance for the three selected principal component of
natural science
Analysis of variance of the five principal component of social science
Analysis of variance of the two selected principal component of
social science
Eigen vectors for natural science
Eigen vectors for social science
Statistics descriptive of natural science
Statistics descriptive of social science
Statistics descriptive of each municipality/regency for natural science
in 2014
25
25
25
25
26
26
26
26
27
10 Statistics descriptive of each municipality/regency for social science
in 2014
11 The formula of Estimated mean of three stage sampling ( stratified
sampling – cluster sampling – simple random sampling)
12 The formula of Estimated variance of three stage sampling ( stratified
sampling – cluster sampling – simple random sampling)
28
29
30
INTRODUCTION
Background
Education plays important roles for the national development. Quality
of human resources can be improved through better quality of education
continuously. In Indonesia, there are several types of education namely
formal, non-formal and informal education which are complemented each
other. The level of formal education comprises of elementary education,
high school or intermediate education, and higher education. A good quality
school must provide quality teaching, curriculum, management, and
facilities. The quality of a school can be reflected by the national
examination scores and the school examination scores, in Indonesia namely
as UN and US, respectively.
UN is a standard evaluation system of primary and secondary
education in Indonesia. In order to control the quality of education
nationwide, UN is conducted once in a year which not only consideration
for students to continue to next level study and for government to deliver
helps in order to develop education system of education units, but also
represent the quality of education mapping itself (Permendikbud No.5). UN
case has been urgently required for this research due to the importance of
respresenting the education in Indonesia. Some previous research related to
UN was carried out by Sinaga (2011) analyzed correlation between the
scores of UN, US and school grades using canonical analysis. Mongi (2014)
also did research about relation between scores of UN and accreditation
standards in West Java.
Senior High School which is commonly abbreviated as SMA in
Indonesia acknowledged its role in preparing students to continue to the
college-level studies. Therefore, This research has analyzed the data of UN
scores all public SMAs in West Java. These SMAs have been referred as
observation units. This kind of data is panel data, since the data is a
combination of cross sectional data and time series data. Data was taken in
four years sequently from 2011-2014.
A linear mixed model is an appropriate approach which has beenused
in this research since response variable is normally distributed. Linear
mixed model is also well suited for the analysis of panel data, where each
time series constitues an individual curve as a cluster. The model
accommodates both fixed effects and random effect. The linear mixed
model is an extention of linear model for data that are collected in groups.
linear mixed model was useful in the analysis of panel data because it has
accommodated the correlation of repeated measurements which are made on
the same unit, the heterogeneity among subjects, and inference toward
certain subjects.
The panel data also could be modelled by fixed effect model with
various intercept and constant slope by adding dummy variables. This
model assumed that the different intercepts were from cross-section and
time unit.
2
Another approach that is used for analysis in this research is
Generalized Estimating Equation (GEE). GEE is used to model the
correlated data from repeated measure studies (panel data) and basically, it
uses quasi-score likelihood for estimating the model parameters. GEE
covers the extentions of Generalized Linear Model (GLM) to panel data.
Specifically, GEE is a population averaging method that models panel data
in which the response is a member of the exponential family of distribution,
for example continous, binary, grouped and count data.
It is important to mention this research considered that UN scores can
be affected by internal factors as well as external factors. The external
factors are the factors within school, education system, and the effect of
school policies. Economics condition of each regency/municipality in West
Java where the schools are located may affect the quality of education.
Hence, it will affect the UN scores of the schools. This economic condition
can be measured by the gross regional income per capita (commonly
mentioned as PDRB in Indonesia) just like what Ahmad (2011) has stated
that the government funds allocated for education sector can affect
economic growth or PDRB and vice versa. According to Statistics Indonesia
(2008), the growth rate of PDRB also influences human development index
which is abbreviated as IPM in Indonesia through household expenditures
for daily primary needs including foods, medicines, and school stuffs. The
growth rate of PDRB may also influence IPM through domestic expenditure
policies made by government including priority funds for social aspects. In
contrast, IPM may influence PDRB through qualified human resources in
terms of good health and education services.
Another reason why UN is highlighted in this research because of the
students pass rate is no longer measured by UN since 2015, due to the
change of education system. This is not wise way to spend much money for
performing UN every year for every school. On the other hand UN is solely
used for measuring things that do not maximally improve the education in
Indonesia. Therefore, UN is better to be conducted for some schools as a
focused sample to mindmap the education instead of using the population
of all schools in Indonesia as measured target. The critical analysis is
required as preliminary task to do due to evaluate how the sampling can be
applied for UN implementation used for the consideration.
Research Questions
1.
2.
What is the best model in modelling between average scores of UN all
public SMAs for both natural science major and social science major
and other explanatory variables such as IPM, PDRB, average scores of
US, total accreditation scores, educators scores, school facilities scores,
teaching content scores, and teaching process scores?
Of all explanatory variables mentioned above, how much do it
contribute at affecting the average scores of UN?
3
Objectives
The main objectives of this study are
1. To develop the best model for modelling between average scores of UN
all public SMAs for both natural science major and social science major
and another explanatory variables mentioned above.
2. To evaluate how much do the explanatory variables contribute at
affecting the average scores of UN.
Apart from the main objectives mentioned above, this study also has a
secondary objective which is to preliminary analyze the possibility of
designing samples for evaluating student achievement.
Research Benefit
The best models which is obtained in this research will lead to what
influence UN the most. As the first preliminary step to evaluate the process
of conducting the UN since there has been provision that UN is no longer
used to measure student pass rate.
LITERATURE REVIEWS
Panel Data
A panel data set is one that follows a given sample of individuals over
time, and thus provides multiple observations on each individual on the
sample (Hsiao 2003). Observing a panel data set is basically observing a
broad section of subjects over time, and thus allows us to study dynamic, as
well as cross sectional, aspects of a problem (Frees 2004).
Baltagi (2005) mentioned the main advantages of panel data analysis
such as.
1. Solves individual heterogeneity.
2. Provides more informative dataset, handles collinearity of variables,
controls some types of ommited variables without observing them,
obtains degree of freedom in a bigger amount.
3. Studies comprehensively the dynamics of adjustments.
4. Identifies and estimates effects which cannot be taken into traditional
technique (e.g. time series analysis and usual linear regression).
5. Reduces bias when estimating because of plenty number observations.
Define following general panel data model for observation
section data i at time t
of cross
(1)
4
where is a vector of regression coefficient with K x 1 where K is number
of explanatory variables,
is a response variable observed for individual
unit i at time t,
is a vector of explanatory variables individual unit i at
time t,
is the error term.
Fixed Effect Model
Fixed effect model is a statistical model that represents the observed
quantities in terms of explanatory variables that are treated as if the
quantities were non random. Fixed and random effects to respectively refer
to the population average and subject specific effects. Fixed effect model
which has form as.
(2)
where denoted subject dimension which equals 1, 2, ... N. denoted time
dimension which equals 1, 2, ..., T and with specification of residual
components is
. Model 2 also can be written as.
(3)
where
,
[
]
[
]
[
]
where is the number of explanatory variables.
, with
is an
identity matrix,
is a
vector. is
matrix of dummies which is included in the regression to estimate α if the
parameters are assumed to be fixed (Baltagi,2008).
Linear Mixed Model
According to Jiang (2007), linear mixed model takes general form that
can be written as
y = Xβ + Zα+ ε
(4)
where y is an observation vector of dimension ni, X is a known covariate
matrix, β is a regression coefficient vector (fixed effect), Z is a design
matrix that contains only 1 and 0, α is a normally distributed random effect
vector, ε is an error vector.
5
The Basic assumption in linear mixed model is the random effect and
error have mean zero and certain varians, e.g. var(α) = G and var(ε)= R. It is
also assumed that α and ε must be uncorrelated. If the random effect and
error are assumed to be normally distributed, then the assumptions can be
written as
α~N(0, G)
ε~N(0, R)
Furthermore, the distribution of response variable is y~N(Xβ,V) where V =
Var (y) = ZGZ’+R. Otherwise, if the random effect and error are not
assumed to be normally distributed, then model (4) is non Gaussian linear
mixed model.
Jiang (2007) also stated that estimation in linear mixed model consists
of estimation toward both the fixed effect and random effect that can be
shown below. Estimation of fixed effect is carried out using Best Linear
Unbiased Estimator (BLUE)
̂
(5)
and the random effect prediction which is Best Linear Unbiased Prediction
(BLUP) can be written as
̂
̃
(
(6)
In fact, variance component V is unknown therefore, in estimating fixed
effect and random effect, V is replaced by the estimator ̂ and the estimation
result is empirical best linear unbiased predictor (EBLUP)
̂
̂
̂
(7)
̂
̂
where and are estimator using Maximum Likelihood Estimation (MLE)
or Restricted Maximum Likelihood (REML).
Jiang (2007) mentioned that the estimation result using REML for
variance componen is a consistent estimator without assuming normality
from random variable and error. REML is also an estimation method that is
more robust and practical.
Henderson (1959) introduced the solutions which can produce
simultaneously the GLS estimator of and BLUP of .
and
the joint density of and is
{
[
The log of
{
}
}
]
is
[
The partial derivatives of log of
with respect to
.
and
are
].
.
6
Setting these results to zero will obtain the following equations
{
.
These can be written as
]
[
[
].
(8)
Estimating The Model Parameters
Some approaches used to estimate the model parameters in linear
mixed model are the traditional approach, Maximum Likelihood Estimation
(MLE), and Restricted Maximum Likelihood (REML). One of those
approaches used for this study is REML.
REML can produce the unbiased or nearly estimator for variance and
it does not require more complex computation just like the MLE estimators.
Searle (1992) stated that the REML approach is now more preferable to
estimate the variance parameters in mixed model.
y is a response vector that normally distributed which is transformed
for z = A’y which has distribution z ~ N(0, A’VA). Define a non-zero n x
(n-p) matrix A for A’X = 0 so then A A’ = Q where
̂
and AA’ = I then define z = A’y = (
. The
expected value of z is 0 and covariance matrix is A’VA. The density of z is
{
{
(
}
̂
According to Rao (1973) that
̂
̂
̂
(
(
(
then we come to the final expression of the desity of z
̂
{
}
(
(
(
̂
̂
̂
}
(
.
(9)
the final density of z leads to the restricted or residual log likelihood which
is written below.
.
(10)
Just like estimating using MLE, log likelihood REML is maximized
also requires numerical technique for example Newton Raphson with minor
7
adaptations (Longford 2003). By subtituting the estimator ̂
into GLS formula
̂
̂
( ̂
̂
eventhough y is normal distribution
is not identical ̂
̂
(11)
Generalized Estimating Equation
Zeger and Liang (1986) introduced an alternative approach to
estimate the parameter in maximum likelihood case. This approach is
usually known as generalized estimating equation (GEE). Basically, GEE
uses quasi score likelihood theory.
GEE approach in
for estimating by using quasiscore likelihood equation which is shown as follow
∑
[ ]
with
[ ]
⁄
dan
⁄
⁄
is diagonal matrix with element var( ),
is working
correlation matrix for , and is constant (Dobson 2002). Some general
forms of structure of working correlation matrix presented in the Table 1 as
follow
Stuctures
Name
Independence
Exchange
-able
Table 1. Structures of working correlation matrix
Structures
Stuctures
Structures
Name
Autoregressive
Unstructured
)
)
)
Sampling Technique
Stratified Random Sampling
Scheaffer et al. (2006) stated that a stratified random sample is
obtained by separating the population elements into nonoverlapping groups,
called strata, then selecting a simple random sample from each stratum. The
principal reasons for using stratified random sampling areas follows:
8
1. Stratification may produce a small bound on the error of estimation.
This result is particularly true if measurements within strata are
homogenous
2. The cost per observation in the survey may be reduced by
stratification of the population elements into convenient groupings
3. Estimates of population parameters may be desired for subgroups of
the population. These subgroups should then be identifiable strata.
The first step in the selection of a stratified random sample is to
clearly specify the strata then each sampling unit of the population is placed
into its appropriate stratum. After the sampling units have been divided into
strata, select a simple random sample from each stratum.
Estimator of the population mean is
̅
∑
̅
where
is number of sampling units in stratum i, i = 1, 2, .. L, ̅ denotes
the sample mean for the simple random sample selected from stratum i.
Estimator variance of ̅ is
̂ ̅
[
∑
̂ ̅
(
(
̂ ̅
(
̂ ̅ ]
Two Stage Cluster Sampling
A two stage cluster sample is obtained by first selecting a probability
sample of clusters and then selecting a probability sample of elements from
each sampled cluster (Scheaffer et al. 2006). Cluster must be as
heterogeneous (different) as possible within, and one cluster should look
very much like another in order for the economic advantages of cluster
sampling to pay off. If a population being divided into nonoverlapping
groups of elements. These groups are considered to be clusters, then a
simple random sample of groups is selected, and the sampled groups are
then subsampled.
The advantages of two stage cluster sampling are
1. A frame listing of all elements in the population may be impossible
or costly to obtain, whereas obtaining a list of all clusters may be
easy
2. The cost of obtaining the data may be inflated by travel costs if the
sampled elements are spread over a large geographic area. Thus,
sampling clusters of elements that are physically close together is
often economical.
By assuming simple random sampling at each stage, the unbiased
estimator of the population mean with is the number of cluster in the
9
population, is the number of clusters selected in a simple random sample,
is the number of elements in cluster .
is the number of elements
selected in a simple random sample from cluster
equals ∑
̅ is
∑
and
is the th
the sample mean for the ith cluster
observation in the sample from the th cluster, is
∑
∑
̅
̂ (
̅
The estimated variance of
where
and
̂
̂
(
(
̅
̅
is
̅
∑
̅
∑
(
∑
̅
(
̅
is simply the sample variance among the terms
the sample variance for the sample selected from cluster .
̅ and that
is
MATERIALS AND METHODS
Data
1. The average scores of UN and US for both natural science diciplines and
social science diciplines of all SMANs in West Java 2011-2014. Data was
accessed from Educational Research Center, Ministry of Education and
Culture with the lowest score is zero and the highest one is 10 then it is
multipled with 10 therefore the score range becomes between 10 and
100.
2. Data of total accreditation scores, educators scores, school facilities
scores, teaching content scores, teaching process scores, school
management scores, school finance scores, and graduates competence
scores were accessed in National Accreditation Institution.
3. Data of human development index and gross regional product in West
Java were accessed through Statistics Indonesia website.
Variables
Variables which are used in this research are
10
Y : The average scores of UN each SMAN for both natural science and
social science
X1 : IPM scores each regency/municipality in West Java
X2 : PDRB each regency/municipality in West Java
X3 : Teaching content scores each SMAN in West Java
X4 : Teaching process scores each SMAN in West Java
X5 : Educators scores each SMAN in West Java
X6 : School facilities scores each SMAN in West Java
X7 : Total accreditation scores each SMAN in West Java
X8 : The average scores of US each SMAN in West Java
Research Methodology
This research is divided into two to reach the goals, first is modelling,
and the second is sampling technique. Some steps in data analysis process in
this research are
Modelling The Data
1.
2.
3.
Exploratory data analysis
In this step, the variability and mean of response variable in
each regency/municipality in West Java was investigated variable
to visualize the information was provided.
Identification of the multicollinearity among explanatory variables
The multicollinearity was identified by calculating the
coefficient of correlation among the explanatory variables. The
data was then analyzed using principal component analysis to
remove the multicollinearity.
Modelling the data with the linear mixed model (LMM)
Models were fitted by incorporating fixed and random effects
which accomodated that every school i at time t had their own
intercept. The model which contained mixed effect took form as:
̂ ̂
̂ , where ̂ is a vector of estimated
̂
parameters with K x 1 where K is the number of principal
components, ̂ is an estimated response variable observed for
individual unit i at time t,
is a vector of principal
components, ̂ is the estimated school effect, ̂ is the estimated
time effect.
In this modelling, the school effect is assumed to be fixed
because schools as sample in West Java is determined from the
beginning. In contrast, the time effect
is assumed to be random
because year can not be limited and there will always be next year
therefore the data includes 2011, 2012, 2013 and 2014.
The school effect is assumed to be fixed then those variables
are transformed into dummy variables. Because there are 421
schools in West Java that have natural science and 419 schools that
have social science program then it means there 420 and 418
11
dummy variables respectively. The model parameters was
estimated using restricted maximum likelihood (REML).
4. Modelling the data with generalized estimating equation (GEE).
̂ ̂ , where ̂ is the estimated school effect
̂
approached to be fixed and this model was estimated using GEE
which accomodates the correlated response variable using the
working correlation matrix : AR(1) in condition of the time series
data which was taken solely covers 4 years. This model is
considered because it is able to accomodate the heterogeneity
among school effect using dummy variables. GEE model with
response variable is normally distributed using identity link
function. The working correlation matrix has the following form as
shown below
)
Modelling with GEE by adding school effect which was
assumed to be fixed into the equation took form as
̂
The model parameters were estimated using quasi-score likelihood
where the variance of as a function of the mean, contained the
working correlation matrix autoregressive.
5. Modelling the data with fixed effects
This model was built with consideration that the numbers of
both of schools and time were defined from the beginning. Because
of the effects of schools and time were decided to be fixed, it has
been added dummy variables into the model which could be written
as
̂ ̂
̂
̂
where ̂ denoted the estimated school effect and ̂ is the estimated
time effect. The model parameters were estimated by using analysis
of variance.
6. Evaluation of the models by comparing the R-squared value and
mean square error of each model. The model with the largest value
is assumed as the best model.
7. Interpretation of the data analysis results.
Sampling Technique
1. Divided population in this case covers West Java into two strata
which are city and regency.
2. West Java has 9 cities and 19 regencies in which is considered as
population clusterings.
3. Defined the number ofsampled clusters of both city and regency
blocks by calculating approximate n cluster with different bounds
12
4.
5.
6.
7.
B on error of estimation which are in this research 0.05, 0.07, and
0.09, respectively.
n was defined, chose element of sampled cluster by using simple
random sampling.
Making an experiment by taking schools in each cluster with
simple random sampling in terms of the characteristics how many
schools each municipality/regency has.
Calculated the variance of the average scores of UN for each school
in each municipality/regency in which there had 6 variances found
out with three stages sampling formula which include stratified
sampling, cluster sampling, and simple random sampling.
Compared every variance and the most minimum variance of
sample with three stages sampling method can be representative
toward the population.
13
RESULT AND DISCUSSION
As it has been commonly acknowledged in Indonesia, there are two
programs available in many senior high schools namely natural science and
social science. This research has both utilized and analyzed the data of those
two different programs separately due to not every senior high schools in
West Java has both natural science and social science and it then can not be
concluded that students learning ability in natural science is as equal as in
social science.
There were two initial models established. Those two models were
built based on the data of natural science and social science. Evaluating the
model and selecting the best model was done by comparing the R-squared
value and mean square error of each model.
The averages scores of UN every year have been presented in Figure
1.
90
a) The Average Score of UN 20112014
90
70
70
50
50
30
2010
2011
2012
2013
2014
b) The Average Score of UN 20112014
30
2010
2011
2012
2013
2014
Figure 1. Scatterplot of the average score of UN versus years for a) natural
science b) social science in West Java
The Figure 1 showed that there was such a decline on the slope between
year 2012 and 2013. The most current years which are 2013 and 2014, the
average scores of UN for natural science has decreased. Sharp slope also
happened between year 2012 and 2013 on the plot of social science major.
Another exploration obtained was the mean and variability of the
average scores of UN in the very curren year, 2014 for each
municipality/regency were also provided in Appendix 9 and Appendix 10.
Both of natural and social science, Bandung City has the higest mean of the
average scores of UN. Bandung City is one of big cities in Indonesia in
which its goverment pays much attention for education there.
The descriptive statistics of the average scores of UN of all public
senior high schools in West Java also shown in the Table 2.
14
Table 2. The descriptive statistics of national examination
Term
2011
2012
2013
2014
Natural
Maximum
96.07
97.67
79.40
95.75
Science
Minimum
64.90
24.08
41.83
43.58
Mean
82.56
82.17
65.53
58.50
Variance
19.20
18.23
17.47
33.41
Social
Maximum
89.95
94.83
91.33
81.45
Science
Minimum
58.30
13.67
7.33
43.18
Mean
79.75
76.27
61.95
53.05
Variance
25.81
23.91
11.16
24.11
Based on Table 2, started from 2013 both in natural and social
science program, there was a newer system that has been implemented for
accomplishing UN that caused the average score of UN fell down. In that
system of UN, enforced that each task paper has been made exactly different
for each student which was implemented since 2013 in order to dismiss
every kind of cheating students possibly could do in UN performance.
Another descriptive statistics measured the relationship between
response variable and explanatory variable has been statistically illustrated
by coefficient correlation shown in the following table.
Table 3. Coefficient correlation between response and explanatory variables
for natural science
Year
X1
X2
X3
X4
X5
X6
X7
X8
2011 Y 0.079 -0.017 0.282 0.219 0.359 0.468 0.449 0.127
2012 Y 0.086 -0.004 0.073 0.002 0.131 0.163 0.141 0.127
2013 Y 0.460 0.176 0.148 0.171 0.309 0.287 0.292 0.068
2014 Y 0.565 0.229 0.271 0.215 0.462 0.423 0.450 0.187
Table 4. Coefficient correlation between response and explanatory variables
for social science
Year
X1
X2
X3
X4
X5
X6
X7
X8
2011 Ya 0.175 -0.015 0.234 0.181 0.355 0.421 0.406 0.103
2012 Y 0.344 0.148 0.196 0.132 0.309 0.315 0.321 0.064
2013 Y 0.336 0.102 0.167 0.209 0.283 0.267 0.301 -0.013
2014 Y 0.460 0.104 0.239 0.215 0.455 0.393 0.427 0.220
a
Variables used in this research; Y : The average score of UN each SMAN in West Java, X1
: IPM scores each regency/municipality in West Java, X2 : PDRB each
regency/municipality in West Java, X3 : Teaching content scores each SMAN in West Java
X4
: Teaching process scores each SMAN in West Java, X5 : Educators scores each
SMAN in West Java, X6 : School facilities scores each SMAN in West Java, X7 : Total
accreditation scores each SMAN in West Java, X8 : The average scores of US each SMAN in
West Java
From Table 3 and Table 4, known that for the natural science the average
scores of Y is correlated with all X variables observed for 4 years. However
social science has a slight difference toward natural science and both for
15
natural and social science the correlation between Y and X is not very
strong.
Another exploration to make sure by calculating variance inflation
factor (VIF) to quantify the severity of multicollinearity in an ordinary least
squares regression analysis. The following table showed VIF of all X.
Table 5. Variance inflation factor of explanatory variables
Predictor
VIF
X1
1.25
X2
1.16
X3
2.34
X4
2.10
X5
2.79
X6
5.25
X7
11.80
X8
1.07
Table 5 showed that X6 and X7 has the highest VIF. It surely means that
there was multicollinearity which could cause a serious problem which
make the standard error get higher if it was not removed and would lead to
the wrong conclusion for this study.
Principal Component Analysis
From the result shown Table 5, it is clear that multicollinearity
among explanatory variables has been existed. Principal component analysis
has been widely used to eliminate the correlation between explanatory
variables through transformation which has been presented in Table 5. It
then can be ensured that all explanatory variables is independent. Many
methods to solve multicollinearity yet principal component analysis has
been chosen in order to include all informations in X without eliminating an
X that has high VIF value. Table 6 of eigen analysis for both natural and
social science is shown below and eigen vectors which was generated
shown by Appendix 5 and Appendix 6.
eigenvalue
proportion
cumulative
eigenvalue
proportion
cumulative
Table 6. Eigen analysis of the correlation matrix
Natural Science
PC1 PC2
PC3 PC4
PC5
PC6
3.51 1.41
0.98
0.78
0.58
0.38
0.44 0.18
0.12
0.09
0.07
0.05
0.44 0.62
0.74
0.83 0.91
0.99
Social Science
3.54 1.39
0.96
0.79
0.59
0.38
0.44 0.17
0.12
0.09
0.07
0.05
0.44 0.62
0.74
0.83
0.91
0.99
PC7
0.31
0.04
0.99
PC8
0.06
0.01
1.00
0.30
0.04
0.99
0.06
0.01
1.00
16
Table 6 showed that the cumulative proportions of variance explained by the
first five components have shown 0.91 both for natural and social science
which means that more than 90% information of eigen explanatory
variables could be explained by five components. Moreover, these five
components have been orthogonal each other. Hence, the problem of
multicollinearity has been removed.
The Estimated Model
The response (the average scores of UN) is continous variable which
was assumed normally distributed.This assumption was justified by the
central limit theorem. Here the response variable (the average scores of UN)
has been modelled with those five components for each program
respectively. Every modelling which has been done in order to find best
model gave different results by comparing the value of R-square, R-square
adjusted, and mean square error. The Table 7 shows the result of 3 different
modellings in the simple version.
Table 7. Model comparison
Model1
Model2
Diciplines
School Eff
Fixed
Fixed
Time Effect
Random
AR(1)
Natural
R-Square
5.12
77.30
R-square Adj
3.28
76.87
Science
MSE
175.95
42.11
214.78
Social
R-Square
2.60
71.95
Science
R-square Adj
0.70
71.40
MSE
137.11
39.64
163.21
-
Model3
Fixed
Fixed
91.89
91.74
15.03
89.18
88.97
15.29
-
Table 7 showed that it was directly identified that the best model was model
3 in which both school effect and time effect were assumed to be fixed with
the highest R-squared value, 91,89 for natural science and 89,23 for social
science, and with the lowest MSE 15,03.
The data also has been modelled with linear mixed model in Model 1.
This model contained two effects, fixed and random effects outright. The
response (average scores of UN) is a normally distributed outcome which is
allowed to use the linear mixed model. School effect was assumed to be
fixed effect and time effect was assumed to be random effect. This model
was not the best model because time should not be random effect. In fact,
time was fixed effect because the sequence of year of observed data was
defined by the researcher from the beginning.
Model 2 which was built with Generalized Estimating Equation with
autoregressive working correlation matrix AR(1). This model was not
concluded to be the best model because of the possiblity of the imprecision
of utilizating working correlation matrix AR(1). This data had a short
17
sequence of time which only included 4 years observed data. Therefore it
could mean that possibly data of 2014 was not affected by the previous year,
data of 2013. It also occured in data of 2013 was not affected by data of
2012 and so forth.
Another plot to see that the best model was a model which was good
at predicting. The accuracy at predicting can be presented in a plot between
the average scores of UN and its predicted values shown in the following
table.
Table 8. Plots of the actual and predicted values in each model
Natural Science
Social Science
Modelling with Generalized Estimating Equation
Modelling with Linear Mixed Model
Modelling with Fixed School Effect and Fixed Time Effect
Table 8 showed that the model with both school and time effect which
were assumed to be fixed was good at predicting marked by the dots spread
18
around the diagonal line. Both in model 3 for natural science and social
science, standardized residual was normally distributed.
Overview of Model 3 (Fixed Effect Model)
̂ ̂
̂ where ̂ is the
Model 3 which had form as ̂
estimated model parameters, ̂ is the estimated response variable observed
for school i at time t,
is the vector of principal components for school i
at time t, ̂ is the estimated schools effect, ̂ is the estimated time effect
which were assumed to be fixed was ensured because of the numbers of
schools and time (the observed years) were both defined and chosen from
the beginning by researcher.
Model 3 is a model with fixed effect which used dummy variables had
an ability to catch the variability of the subjects in this case the subjects
were schools and time, had the highest R-squared value was because time
effect more fits if it was assumed to be fixed rather than to be random or
AR(1). Beside that, the variance of time ( 2011 – 2014 ) was quite high
because there was always modification of system of implementation of UN
globally every year. The model parameters were resulted in the Table 9.
Table 9. Parameter estimates for the first five components of natural science
Term
Coef
SE coef
T-Value
P-Value
w1
1.82
0.72
2.52
0.01
w2
0.60
1.52
0.40
0.69
w3
2.95
0.44
6.68
0.00
w4
0.22
1.14
0.19
0.85
w5
-7.56
1.83
-4.13
0.00
Table 9 showed that the p-value for
and
(0.01,