Statistical Analysis for Non-Normal and Correlated Outcome in Panel Data
STATISTICAL ANALYSIS FOR NON-NORMAL
AND CORRELATED OUTCOME IN PANEL DATA
ANNISA GHINA NAFSI RUSDI
DEPARTMENT OF STATISTICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
INSTITUT PERTANIAN BOGOR
BOGOR
2014
THE BACHELOR THESIS STATEMENT AND SOURCES OF
INFORMATION AND COPYRIGHT DEVOLUTION
I hereby declare that the bachelor thesis entitled Statistical Analysis for
Non-Normal and Correlated Outcome in Panel Data is my work under the
guidance of the supervisory committee and it has not been submitted in any form
to any college. Resources derived or quoted from works published and
unpublished from other writers are mentioned in the text and listed in the
Bibliography at the end of this bachelor thesis.
I hereby bestow the copyright of my papers to the Bogor Agricultural
University.
Bogor, October 2014
Annisa Ghina Nafsi Rusdi
NIM G14100069
ABSTRACT
ANNISA GHINA NAFSI RUSDI. Statistical Analysis for Non-Normal and
Correlated Outcome in Panel Data. Supervised by ASEP SAEFUDDIN and
ANANG KURNIA.
There are many cases that cannot fulfill some assumptions in statistical
analysis, such as normality and independence. In many practical problems, the
normality as well independent assumption is not reasonable. For example, data
that repeated over time tend to be correlated. If analysis ignores the non
independent outcome the Standard Error (SE) on the parameter estimates tends to
be too small. Generalized Linear Model (GLM), Generalized Estimating Equation
(GEE), and Generalized Linear Mixed Model (GLMM) can be used for nonnormal data using the link functions. GEE includes working correlation matrix to
accommodate the correlation in the data. GLMM may overcome the repeated
observation and allows individual have different baseline/intercept. The study is
aim at comparing result based on those approaches. The data that are used in this
study are from BPS and the outcome that are used is poverty proportion. Based on
the study shows that GEE approach is better than GLM for marginal model,
and GLMM approach is better than GEE with dummy variable.
Keywords: correlation outcome, GEE, GLM, GLMM, non-normal
STATISTICAL ANALYSIS FOR NON-NORMAL
AND CORRELATED OUTCOME IN PANEL DATA
ANNISA GHINA NAFSI RUSDI
A Bachelor Thesis
in partial fulfillment of the requirements for the degree of
bachelor of statistics
in
Department of Statistics
DEPARTMENT OF STATISTICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
INSTITUT PERTANIAN BOGOR
BOGOR
2014
Title
Name
NIM
: Statistical Analysis for Non-Normal and Correlated Outcome in
Panel Data
: Annisa Ghina Nafsi Rusdi
: G14100069
Approved by
Prof Dr Ir Asep Saefuddin, MSc
Advisor I
Acknowledged by
Dr Anang Kurnia, Msi
Head of Department
Graduation date:
Dr Anang Kurnia, MSi
Advisor II
PREFACE
All best praise and regards to llah Subhanahu Wata’ala. Because of Allah,
this research and bachelor thesis could be done. This research theme is about nonnormal and correlated outcome with the title is Statistical Analysis for NonNormal and Correlated Outcome in Panel Data and it is conducted from February
2014 until September 2014.
Thank you Prof Asep Saefuddin and Dr Anang Kurnia as the advisors of
this research. Also big thanks for both of my dearest parents (ayah ibu, I love
you), my dearest sister (ade, you are the best partner in my life), my roommate Ka
Husnul Puncul, statistika 47, all of the staff in Department of Statistics, Sakinah
Gang (you are the best), AgCN (ma pal, you all guys are so cool), and all of my
friends, your support has always been my spirit and encouragement. I hope this
research will give benefit for academics.
Bogor, October 2014
Annisa Ghina Nafsi Rusdi
TABLE OF CONTENTS
TABLE LIST
vi
TABLE OF FIGURE
vi
TABLE OF APPENDIX
vi
INTRODUCTION
1
Background
1
Objective of The Study
2
DATA AND METHODOLOGY
2
Data Source
2
Methodology
2
RESULT
6
Exploration of the Data
6
GLM Approach
6
GEE Approach
7
GLMM Approach
8
GEE Approach with Dummy Variable
9
Model Comparison
9
CONCLUSION
10
REFERENCES
10
APPENDIX
11
BIOGRAPHY
14
TABLE LIST
1.
2.
3.
4.
Example of the link function
Common working correlation matrix
Parameter estimates and standard errors (GLM approach)
Parameter estimates and standard errors (GEE approach)
3
4
7
8
TABLE OF FIGURE
1.
Line graph for poverty proportion of 33 provinces in Indonesia
6
TABLE OF APPENDIX
1.
2.
3.
Parameter estimates and standard errors (GLMM approach)
Parameter estimates and standard errors (GEE approach with dummy
variable)
Scatter plot between prediction values and residual
11
12
13
INTRODUCTION
Background
Methods of statistical analysis depend on the measurement scales of the
response and explanatory variables. In many practical problems, the normality
assumption is not reasonable. In some cases the response variable can be
transformed to improve linearity and homogeneity of variance. But this approach
has some drawbacks such as response variable has changed (not original) and
transformation must simultaneously improve linearity and homogeneity of
variance.
Panel analysis combines the time series and cross-sectional data. Models
that usually used in panel analysis are Pooled Model, Fixed Effects Model, and
Random Effects Model. The outcome that are used in this study is not normal and
correlated.
Data that repeated over time tend to be correlated or dependent each other.
Individuals tend to be more similar to themselves over time than the other
independent individuals. Group of individuals may have dependent outcome. The
Standard Error (SE) of the estimate is very small in the case of ignore dependent
outcome.
If the assumptions are violated, such as response variable is not normally
distributed and correlated, the classic model like Panel Regression may produce
missleading conclusion. There are some approaches for this kind of data (nonnormal and correlated outcome). Pooled Model is approached by Generalized
Linear Model (GLM) and Generalized Estimating Equation (GEE), Fixed Effects
Model is approached by Generalized Estimating Equation with dummy variable,
and Random Effects Model is approached by Generalized Linear Mixed Model
(GLMM).
GLM can solve the problem of non-normal data distributed. GLM
introduced by Nelder and Wedderburn (1972) are standard method used to fit
regression model for univariate data that are presumed to follow an exponential
family distribution (Horton and Lipsitz 1999). It is possible to fit models of data
from normal, inverse Gaussian, gamma, Poisson, binomial, geometric, and
negative binomial by suitable choice of the link function g(.).
GEE can solve the problem of non-normal and correlated data. Liang and
Zeger (1986) introduced GEE to take into account correlation between
observations in GLM. GEE takes into account the dependency of observation by
specifying a “working correlation matrix”. The working correlation matrix is not
usually known and must be estimated.
GLMM is a little bit different than GLM and GEE. It has random effect
where you can have different intercept for every observation if you assume that
the intercept of each observation is different.
GLM and GEE are marginal model, the parameter estimates the marginal
population mean. In contrast, GLMM is a mixed effect modeling approach.
Parameters in GLMM have a subject-specific interpretation.
2
Objectives of The Study
The objective of the study is to determine the most appropriate model for
non-normal and correlated outcome.
DATA AND METHODOLOGY
Data Source
The data that was used in this study are poverty proportion by province
(POV) that was obtained from BPS (2008-2012) for all provinces in Indonesia.
The explanatory variable that were used are Human Development Index by
province (1-100) (HDI), Unemployment Rate by province (%) (UR), Gross
Domestic Regional Product per Capita by province (Rp10.000) (GDRPC).
Methodology
1.
The procedures of data analysis implemented are:
Exploration of the data
Make a plot for the outcome (poverty) to obtaine information on province
with the highest and lowest poverty.
Data structure (Dobson 2002):
i
y=
,
i
i
,
so y has length
i
ni , where N is the amount of the
ini
object and ni is the observation in the object (n).
xi is a p x vector of explanatory variable,
xi
xi
xip
so xi
,
xi
xip
β
and β is the p x
vector of regression parameters β
the ith column of the design matrix X
xT
X=
x
T
=
x
x
x
x
p
.
p
Panel regression that are usually used (Gujarati 2004):
Pooled model: yit xi β it
βp
. The vector xi is
3
Fixed Effects Model: y xi β i
Random Effects Model: y xi β
2.
it
i
; where i
it ; where
.
i
Applying Generalized Linear Model (GLM)
GLM model for independent data are characterized by:
g
g
i
xi β
i
where i
i , g is a link function. Some examples of the link function can
be seen in Table 1.
Table 1 Example of link function
Outcome
Distribution Link
Function
(Y)
continuous Normal
identity g( i ) = i
g( i ) = logit( i )
proportion Binomial
logit
= log
count
Poisson
V( i ) = 1
V( i ) = i (1- i )
i
- i
g( i ) = log( i )
log
Var
V( i ) =
i
Response variable which is assumed to share the same distribution from
exponential family. The values of the β coefficients are obtained by
maximum likelihood estimation. The maximum likelihood estimator of the
p x parameter vector β is obtained by solving estimating equation for β.
m
i
β
i
vi
yi
i
β
with variance-covariance matrix (V) is:
V
V
V
V
Vi is vector of variance, and assuming that responses for different subjects are
independent, where O denotes a matrix of zero, (Dobson 2002).
Hence the GLM model for binomial distribution using logit link function was
expressed as the following:
log
3.
i
- i
β
β
β U
β
.
Applying Generalized Estimating Equation (GEE)
GEE is a method of estimation of regression model parameters when dealing
with dependent data and extension of Generalized Linear Model (GLM) for
4
longitudinal data analysis using quasi-likelihood estimation. When data are
collected on the same units across in time, these repeated observations are
dependent over times. If this dependent is not taken into account then the
Standard Error (SE) of the parameter estimates will not be valid and
hypothesis testing result will be non-replicable.
GEE model for correlated outcome is the same as GLM:
g
i
g
i
xi β
GEE accommodated correlated outcome using Working Correlation Matrix.
GEE includes a working correlation matrix in the SE calculation.
Table 2 Common working correlation matrix
Structure
Example
Explanation
No correlation
between repeated
observation
Independence
Correlation between
repeated observation
is the same accross all
individuals
Exchangeable
Correlation varies
with each group of
observations
n
n
Unstructured
n
n
n
n
AutoRegressive
n
Correlation is a
function of time
between
measurements
(Horton and Lipsitz 1999)
Working variance-covariance matrix for
equals:
where is n x n diagonal matrix with elements var(yik ), i is n x n “working”
correlation matrix for yi , is a constant to allow for over dispersion.
G estimator of β is the solution of:
5
i
i
β
i
β
i
GEE model for binomial distribution using logit link function for this
study was:
log
4.
β
i
- i
β
β U
β G
.
Applying Generalized Linear Mixed Model (GLMM)
Subject-specific models assume that every region has its own intercept. To
avoid correlation away regions, each province is assumed to have its own
model. The GLMM of subject-specific models rope with this condition.
GLMM model for binomial distribution using logit link function with random
intercept was expressed as the following:
g
g
i
i
xi β vi
where vi is the random effect (one of each subject). These random effect
represent the influence of subject i on repeated subjects that is not captured by
observed covariates.
The mixed model equations are (Henderson 1984)
y
y
β
v
GLMM model for this study is:
log
i
β
β U
β
vi
i
5.
where:
β is fixed and population mean of POV
β is fixed effect of the hdi on the mean of POV
β is fixed effect of the ur on the mean of POV
β is fixed effect of the gdrpc on the mean of POV
vi is random effect for the ith individual
Random effects assume that vi
v and independence of the
fixed effects and the error terms .
Comparing the models by seeing the goodnest of fit with marginal R2 (
)
An extension of the R2 measure is calculating using:
n
i
n
i
ni
t
ni
t
it
it
it
6
This measure is interpreted as the proportion of variance in the outcome that
is explained by the model (Hardin and Hilbe 2003).
RESULT
Exploration of the Data
The poverty proportion of 33 provinces in Indonesia can be seen in Figure 1,
shows that the poverty proportion of 33 provinces are random. As we can see in
Figure 1, the poverty proportions for most of provinces are decreasing every year.
As we can see in Figure 1, the provinces that have high poverty proportion
are Papua, Papua Barat, Maluku, Nusa Tenggara Timur (NTT), Nusa Tenggara
Barat (NTB), Nangro Aceh Darussalan (NAD), Lampung, Sulawesi Tengah,
Bengkulu, and Sulawesi Tenggara. The provinces that have low poverty
proportion are Jakarta, Bali, Kalimantan Selatan, Banten, Kalimantan Tengah,
Kepulauan Bangka Belitung, Kalimatan Timur, Kepulauan Riau, Jambi, Sulawesi
Utara. The rest provinces are in the middle class.
The provinces that have the low poverty proportion are commonly from
West Indonesia, such as Java Island, Sumatera Island, and Kalimantan Island. It is
very opposite with East Indonesia where most of the provinces that have high
poverty proportion come from, such as Papua Island, Maluku Island, NTT and
NTB.
0.40
poverty proportion
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
2008
2009
2010
2011
2012
year
Figure 1 Line graph for poverty proportion of 33 provinces in Indonesia
GLM Approach
The constructions of GLM are by deciding on response and explanatory
variables of the data and choosing and appropriate link function and response
7
probability distribution. Because the response variable in the data is proportion so
the probability distribution for the response variable is binomial with log
i
- i
as
the link function.
The parameters estimate and Standard Errors (SEs) for GLM approach can
be seen in Table 3.
Table 3 Parameters estimate and standard errors (GLM approach)
Parameter
Estimate
Intercept
HDI
UR
GDRPC
Standard Error
3.7367
-0.0708
-0.0394
-0.0025
0.0030
0.0000
0.0000
0.0000
31.1569%
The model estimation for GLM approach in this study is:
log
i
.
.
.
U
.
i
and
for GLM approach is 31.1569%. It is interpreted as the 31.1569%
proportion of variance in the outcome that is explained by the model.
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale are still large, that are from -0.12 until 0.23. It is bacause the
approach that used is not enough to explain the data and it is in accordance with
the
from GLM approach that is only 31.1569%.
GEE Approach
The construction of GEE is by deciding the working correlation matrix that
we will use. In this study, working correlation matrix that was used is
autoregressive (AR(1)). The reason for using AR working correlation matrix is
measurement closer in time are likely to be more correlated than measurement
further apart in time.
Working correlation matrix (AR(1)) for this approach is:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The parameters estimate and Standard Errors (SEs) for GEE approach can
be seen in Table 4.
8
Table 4 Parameters estimate and standard errors (GEE approach)
Parameter
Estimate
Intercept
HDI
UR
GDRPC
9.3656
-0.1616
0.0055
0.0008
Standard Error
1.3063
0.0194
0.0083
0.0009
37.8220%
The model estimation for GEE approach (using autoregressive working
correlation matrix) in this study is:
i
log
.
.
.
U
.
i
and
for GEE approach is 37.8220%. It is interpreted as the 37.8220%
proportion of variance in the outcome that is explained by the model.
for
GEE is higher than GLM.
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale are still large, that are from -0.14 until 0.16. It is in accordance
with the
from GEE approach that is 37.8220%. GEE approach has better
than GLM even though it is not significant difference.
GLMM Approach
In the previous approaches, GLM and GEE are marginal model where you
estimate model for population. As we can see in Figure 1, the province variance is
very high. It can give missleading output if you assume that every province has
same mean in poverty proportin (intercept). To solve that problem we can apply
GLMM approach so that every province can have different intercept by assuming
that intercept is random effect.
The parameters estimate and Standard Errors (SEs) for GLMM approach
can be seen in Appendix 1.
for GLMM approach is 99.2349%. It is
interpreted as the 99.2349% proportion of variance in the outcome that is
explained by the model.
As we can see in Appendix 1, it is assumed that the intercept is random
effect and different for every province. The variance of the random intercepts on
the logit scale is estimated as
.
The model estimation for GLMM approach in this study is:
log
i
.
i
.
.
U
.
G
vi
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale is from -0.03 until 0.03. That is much better that the previous
9
approaches. It is in accordance with the
99.2349%.
from GLMM approach that is
GEE Approach with Dummy Variable
G
approach doesn’t have coefficient or variable that shows the subject
specifict/effect as in GLMM. It is kind of not suitable to compare between GEE
and GLMM. The model that uses GEE approach in this study take subject
cepecifit, in this study is provinces, as dummy variable. Dummy variable put into
the model in GEE approach. Dummy variable that used for every province can be
seen in Appendix 2. This method had been done by Anwar (2012) in his bachelore
thesis.
The difference between GLMM and GEE with dummy variable is, in
GLMM we assume that there are random effects but in GEE with dummy variable
we assume that all variable is fixed.
The working correlation matrix that used is Auto-Regressive. Working
correlation matrix (AR(1)) for this approach is:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The parameters estimate and Standard Errors (SEs) for GLMM approach
can be seen in Appendix 2.
for GLMM approach is 99.2227%. It is
interpreted as the 99.2227% proportion of variance in the outcome that is
explained by the model.
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale is from -0.03 until 0.03. It is similar with GLMM approach.
The
from GEE approach with dummy variable is 99.2227% and it is also
similar with GLMM approach.
Model Comparison
The SEs of parameter estimate for GLM approach are very small which are
0.000 (under estimate), in contrast GEE, GLMM, and GEE with dummy variable
approach have bigger SEs of parameter estimate. That problem occured because
GLM ignore the corrrelation between outcomes.
The
for every approach in this study are GLM approach is 31.1569%,
GEE is 37.8220%, GLMM is 99.2366%, and GEE with dummy variable is
99.2227%. For marginal model, GEE is better than GLM in
because GEE
can overcomes the correlation problem. For subject-specifict model, GLMM has
than GEE with dummy variable. GLMM approach has the best
better
which is means that it is better in explaine the proportion of variance in the
outcome by the model.
Scatter plot between prediction values and residual from GLM and GEE
approaches still haven’t fulfil the exppected result as in the theory. Hence, it is
10
possible to execute model emendation with another approach such as quasi
likelihood. For GLMM and GEE with dummy variable approaches, the scatter
plots between prediction values and residual are much better than the previous
approaches.
CONCLUSION
For population mean model, GEE approach is better than GLM to overcome
non-normal and correlated outcome in the case of this study which
for GEE
is 37.8220%. For subject-specific model, GLMM approach is better than GEE
with dummy variable to overcomes non-normal and correlated outcome in the
case of poverty proportion in this study where the variance of subjects is very big.
The
for GLMM approach in this study is 99.2366%.
REFERENCES
Anwar, N. 2012. Pemodelan Tingkat Pengangguran di Lima Negara Anggota
ASEAN dengan Regresi Data Panel dan Generalized Estimating Equation.
[bachelor thesis]. Bogor: Faculty of Mathematics and Natural Sciences,
Institut Pertanian Bogor.
Dobson, A.J. 2002. Introduction to Generalized Linear Models. Second Edition.
New York: CRC Press.
Gujarati D. N. 2004. Basic Econometrics, Fourth Edition. The McGraw-Hill
Companies.
Hardin, J.W. and Hilbe, J.M. 2003. Generalized Estimating Equations. Florida:
Chapman & Hall.
Henderson, C. R. 1984. Best Linear Unbiased Prediction of Performance and
Breeding Value. University of Guelph.
Horton N.J. and Lipsitz S.R. 1999. Review of Software to Fit Generalized
Estimating Equation Regression Models. The American Statiscian, 53, 160169.
McCulloch, C.E. and Searle, S.R. 2001. Generalized, Linear, and Mixed Model,
New York : Wiley.
Nelder, J.A. and Wedderburn, R. W. M. 1989. Generalized Linear Models,
Journal of Royal Statistical Society, 135(3), 370-84.
Zeger, S. L., Liang, K. Y. 1986. Longitudinal Data Analysis Using Generalized
Linear Models. Biometrics, 73, 13–22.
11
Appendix 1 Parameter estimates and standard errors (GLMM approach)
Parameter
Intercept
Random effects
HDI
UR
GDRPC
Estimate Standard Error
province 1
province 2
province 3
province 4
province 5
province 6
province 7
province 8
province 9
province 10
province 11
province 12
province 13
province 14
province 15
province 16
province 17
province 18
province 19
province 20
province 21
province 22
province 23
province 24
province 25
province 26
province 27
province 28
province 29
province 30
province 31
province 32
province 33
6.8584
-0.2940
-0.7291
-0.9828
-0.8720
-1.1839
-1.1333
-0.5083
-1.4401
-0.2370
-0.4231
-1.8808
-0.9600
-1.6976
-0.4281
-0.0561
-0.6538
-1.8445
-0.9199
-0.5994
-1.4980
-1.1841
-2.0085
-1.2635
-0.7809
-0.3263
-0.5243
-1.0000
-0.7086
0.0092
-1.4221
-0.3158
-0.8956
0.0000
-0.1584
0.0053
0.0008
0.0531
0.0024
0.0039
0.0038
0.0053
0.0053
0.0034
0.0031
0.0035
0.0036
0.0026
0.0087
0.0028
0.0022
0.0030
0.0051
0.0023
0.0030
0.0035
0.0030
0.0019
0.0044
0.0019
0.0063
0.0054
0.0027
0.0024
0.0025
0.0022
0.0030
0.0028
0.0036
0.0026
0.0531
0.0003
0.0002
0.0000
99.2349%
12
Appendix 2 Parameter estimates and standard errors (GEE approach with dummy
variable)
Parameter
Intercept
HDI
UR
GDRPC
Prov 1
Prov 2
Prov 3
Prov 4
Prov 5
Prov 6
Prov 7
Prov 8
Prov 9
Prov 10
Prov 11
Prov 12
Prov 13
Prov 14
Prov 15
Prov 16
Prov 17
Prov 18
Prov 19
Prov 20
Prov 21
Prov 22
Prov 23
Prov 24
Prov 25
Prov 26
Prov 27
Prov 28
Prov 29
Prov 30
Prov 31
Prov 32
Estimate
10.0993
-0.1588
0.0051
0.0008
-0.1766
-0.4860
-0.7605
-0.5233
-0.8644
-0.9563
-0.3213
-1.2500
-0.0515
-0.3211
-1.4056
-0.8188
-1.6325
-0.2694
0.2584
-0.5294
-1.6913
-1.0987
-0.6920
-1.4869
-0.9102
-1.9612
-0.9072
-0.4576
-0.2844
-0.4282
-0.8906
-0.6702
0.1013
-1.4316
-0.5136
-0.8705
Standard Error
0.8494
0.0122
0.0078
0.0007
0.0601
0.0777
0.0784
0.0696
0.0699
0.0701
0.0632
0.0497
0.0744
0.0641
0.1684
0.0798
0.0733
0.0720
0.1002
0.0378
0.0515
0.0613
0.0712
0.0399
0.0682
0.0316
0.1119
0.1191
0.0711
0.0503
0.0645
0.0533
0.0901
0.0640
0.0694
0.0602
99.2227%
13
0.21
0.21
0.16
0.16
0.11
0.11
p - p hat
p - p hat
Appendix 3 Scatter plot between prediction values and residual
0.06
0.01
-0.04 0
0.2
0.4
0.06
0.01
-0.04 0
-0.09
-0.09
-0.14
-0.14
�
(a) GLM Approach
0.1
0.2
0.3
0.4
�
(c) GLMM Approach
0.21
0.21
0.16
0.16
0.11
p - p hat
p - p hat
0.11
0.06
0.01
-0.04 0
0.2
0.01
-0.04
0
0.2
-0.09
-0.09
-0.14
-0.14
(b) GEE Approach
0.4
0.06
�
�
(d) GEE with dummy variable
approach
0.4
14
2
BIOGRAPHY
The author was born in Indramayu, West Java on November 7, 1992 as the
first child of Rusdi and Ratna Dewi couple. The author began her education in AlIrsyad Al-Islamiyah kindergarden in 1997, then continue to Muhammadiyah 1
Elementary School in 1998. The author continue her education at SMPN 1
Cirebon. Graduated from SMPN 1 Cirebon in 2007, the author continued her
education in SMAN 2 Cirebon. After graduating from SMAN 2 Cirebon in 2010,
author accepted in the Department of Statistics, Institut Pertanian Bogor via
USMI.
The author was active in Gamma Sigma Beta (GSB) as the member of Division
of Data Base Center in 2012/2013 and also a committee of Statistika Ria 8 in 2012.
The autor was an apprentice in Ministry of Education and Culture in August until
September 2013.
AND CORRELATED OUTCOME IN PANEL DATA
ANNISA GHINA NAFSI RUSDI
DEPARTMENT OF STATISTICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
INSTITUT PERTANIAN BOGOR
BOGOR
2014
THE BACHELOR THESIS STATEMENT AND SOURCES OF
INFORMATION AND COPYRIGHT DEVOLUTION
I hereby declare that the bachelor thesis entitled Statistical Analysis for
Non-Normal and Correlated Outcome in Panel Data is my work under the
guidance of the supervisory committee and it has not been submitted in any form
to any college. Resources derived or quoted from works published and
unpublished from other writers are mentioned in the text and listed in the
Bibliography at the end of this bachelor thesis.
I hereby bestow the copyright of my papers to the Bogor Agricultural
University.
Bogor, October 2014
Annisa Ghina Nafsi Rusdi
NIM G14100069
ABSTRACT
ANNISA GHINA NAFSI RUSDI. Statistical Analysis for Non-Normal and
Correlated Outcome in Panel Data. Supervised by ASEP SAEFUDDIN and
ANANG KURNIA.
There are many cases that cannot fulfill some assumptions in statistical
analysis, such as normality and independence. In many practical problems, the
normality as well independent assumption is not reasonable. For example, data
that repeated over time tend to be correlated. If analysis ignores the non
independent outcome the Standard Error (SE) on the parameter estimates tends to
be too small. Generalized Linear Model (GLM), Generalized Estimating Equation
(GEE), and Generalized Linear Mixed Model (GLMM) can be used for nonnormal data using the link functions. GEE includes working correlation matrix to
accommodate the correlation in the data. GLMM may overcome the repeated
observation and allows individual have different baseline/intercept. The study is
aim at comparing result based on those approaches. The data that are used in this
study are from BPS and the outcome that are used is poverty proportion. Based on
the study shows that GEE approach is better than GLM for marginal model,
and GLMM approach is better than GEE with dummy variable.
Keywords: correlation outcome, GEE, GLM, GLMM, non-normal
STATISTICAL ANALYSIS FOR NON-NORMAL
AND CORRELATED OUTCOME IN PANEL DATA
ANNISA GHINA NAFSI RUSDI
A Bachelor Thesis
in partial fulfillment of the requirements for the degree of
bachelor of statistics
in
Department of Statistics
DEPARTMENT OF STATISTICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
INSTITUT PERTANIAN BOGOR
BOGOR
2014
Title
Name
NIM
: Statistical Analysis for Non-Normal and Correlated Outcome in
Panel Data
: Annisa Ghina Nafsi Rusdi
: G14100069
Approved by
Prof Dr Ir Asep Saefuddin, MSc
Advisor I
Acknowledged by
Dr Anang Kurnia, Msi
Head of Department
Graduation date:
Dr Anang Kurnia, MSi
Advisor II
PREFACE
All best praise and regards to llah Subhanahu Wata’ala. Because of Allah,
this research and bachelor thesis could be done. This research theme is about nonnormal and correlated outcome with the title is Statistical Analysis for NonNormal and Correlated Outcome in Panel Data and it is conducted from February
2014 until September 2014.
Thank you Prof Asep Saefuddin and Dr Anang Kurnia as the advisors of
this research. Also big thanks for both of my dearest parents (ayah ibu, I love
you), my dearest sister (ade, you are the best partner in my life), my roommate Ka
Husnul Puncul, statistika 47, all of the staff in Department of Statistics, Sakinah
Gang (you are the best), AgCN (ma pal, you all guys are so cool), and all of my
friends, your support has always been my spirit and encouragement. I hope this
research will give benefit for academics.
Bogor, October 2014
Annisa Ghina Nafsi Rusdi
TABLE OF CONTENTS
TABLE LIST
vi
TABLE OF FIGURE
vi
TABLE OF APPENDIX
vi
INTRODUCTION
1
Background
1
Objective of The Study
2
DATA AND METHODOLOGY
2
Data Source
2
Methodology
2
RESULT
6
Exploration of the Data
6
GLM Approach
6
GEE Approach
7
GLMM Approach
8
GEE Approach with Dummy Variable
9
Model Comparison
9
CONCLUSION
10
REFERENCES
10
APPENDIX
11
BIOGRAPHY
14
TABLE LIST
1.
2.
3.
4.
Example of the link function
Common working correlation matrix
Parameter estimates and standard errors (GLM approach)
Parameter estimates and standard errors (GEE approach)
3
4
7
8
TABLE OF FIGURE
1.
Line graph for poverty proportion of 33 provinces in Indonesia
6
TABLE OF APPENDIX
1.
2.
3.
Parameter estimates and standard errors (GLMM approach)
Parameter estimates and standard errors (GEE approach with dummy
variable)
Scatter plot between prediction values and residual
11
12
13
INTRODUCTION
Background
Methods of statistical analysis depend on the measurement scales of the
response and explanatory variables. In many practical problems, the normality
assumption is not reasonable. In some cases the response variable can be
transformed to improve linearity and homogeneity of variance. But this approach
has some drawbacks such as response variable has changed (not original) and
transformation must simultaneously improve linearity and homogeneity of
variance.
Panel analysis combines the time series and cross-sectional data. Models
that usually used in panel analysis are Pooled Model, Fixed Effects Model, and
Random Effects Model. The outcome that are used in this study is not normal and
correlated.
Data that repeated over time tend to be correlated or dependent each other.
Individuals tend to be more similar to themselves over time than the other
independent individuals. Group of individuals may have dependent outcome. The
Standard Error (SE) of the estimate is very small in the case of ignore dependent
outcome.
If the assumptions are violated, such as response variable is not normally
distributed and correlated, the classic model like Panel Regression may produce
missleading conclusion. There are some approaches for this kind of data (nonnormal and correlated outcome). Pooled Model is approached by Generalized
Linear Model (GLM) and Generalized Estimating Equation (GEE), Fixed Effects
Model is approached by Generalized Estimating Equation with dummy variable,
and Random Effects Model is approached by Generalized Linear Mixed Model
(GLMM).
GLM can solve the problem of non-normal data distributed. GLM
introduced by Nelder and Wedderburn (1972) are standard method used to fit
regression model for univariate data that are presumed to follow an exponential
family distribution (Horton and Lipsitz 1999). It is possible to fit models of data
from normal, inverse Gaussian, gamma, Poisson, binomial, geometric, and
negative binomial by suitable choice of the link function g(.).
GEE can solve the problem of non-normal and correlated data. Liang and
Zeger (1986) introduced GEE to take into account correlation between
observations in GLM. GEE takes into account the dependency of observation by
specifying a “working correlation matrix”. The working correlation matrix is not
usually known and must be estimated.
GLMM is a little bit different than GLM and GEE. It has random effect
where you can have different intercept for every observation if you assume that
the intercept of each observation is different.
GLM and GEE are marginal model, the parameter estimates the marginal
population mean. In contrast, GLMM is a mixed effect modeling approach.
Parameters in GLMM have a subject-specific interpretation.
2
Objectives of The Study
The objective of the study is to determine the most appropriate model for
non-normal and correlated outcome.
DATA AND METHODOLOGY
Data Source
The data that was used in this study are poverty proportion by province
(POV) that was obtained from BPS (2008-2012) for all provinces in Indonesia.
The explanatory variable that were used are Human Development Index by
province (1-100) (HDI), Unemployment Rate by province (%) (UR), Gross
Domestic Regional Product per Capita by province (Rp10.000) (GDRPC).
Methodology
1.
The procedures of data analysis implemented are:
Exploration of the data
Make a plot for the outcome (poverty) to obtaine information on province
with the highest and lowest poverty.
Data structure (Dobson 2002):
i
y=
,
i
i
,
so y has length
i
ni , where N is the amount of the
ini
object and ni is the observation in the object (n).
xi is a p x vector of explanatory variable,
xi
xi
xip
so xi
,
xi
xip
β
and β is the p x
vector of regression parameters β
the ith column of the design matrix X
xT
X=
x
T
=
x
x
x
x
p
.
p
Panel regression that are usually used (Gujarati 2004):
Pooled model: yit xi β it
βp
. The vector xi is
3
Fixed Effects Model: y xi β i
Random Effects Model: y xi β
2.
it
i
; where i
it ; where
.
i
Applying Generalized Linear Model (GLM)
GLM model for independent data are characterized by:
g
g
i
xi β
i
where i
i , g is a link function. Some examples of the link function can
be seen in Table 1.
Table 1 Example of link function
Outcome
Distribution Link
Function
(Y)
continuous Normal
identity g( i ) = i
g( i ) = logit( i )
proportion Binomial
logit
= log
count
Poisson
V( i ) = 1
V( i ) = i (1- i )
i
- i
g( i ) = log( i )
log
Var
V( i ) =
i
Response variable which is assumed to share the same distribution from
exponential family. The values of the β coefficients are obtained by
maximum likelihood estimation. The maximum likelihood estimator of the
p x parameter vector β is obtained by solving estimating equation for β.
m
i
β
i
vi
yi
i
β
with variance-covariance matrix (V) is:
V
V
V
V
Vi is vector of variance, and assuming that responses for different subjects are
independent, where O denotes a matrix of zero, (Dobson 2002).
Hence the GLM model for binomial distribution using logit link function was
expressed as the following:
log
3.
i
- i
β
β
β U
β
.
Applying Generalized Estimating Equation (GEE)
GEE is a method of estimation of regression model parameters when dealing
with dependent data and extension of Generalized Linear Model (GLM) for
4
longitudinal data analysis using quasi-likelihood estimation. When data are
collected on the same units across in time, these repeated observations are
dependent over times. If this dependent is not taken into account then the
Standard Error (SE) of the parameter estimates will not be valid and
hypothesis testing result will be non-replicable.
GEE model for correlated outcome is the same as GLM:
g
i
g
i
xi β
GEE accommodated correlated outcome using Working Correlation Matrix.
GEE includes a working correlation matrix in the SE calculation.
Table 2 Common working correlation matrix
Structure
Example
Explanation
No correlation
between repeated
observation
Independence
Correlation between
repeated observation
is the same accross all
individuals
Exchangeable
Correlation varies
with each group of
observations
n
n
Unstructured
n
n
n
n
AutoRegressive
n
Correlation is a
function of time
between
measurements
(Horton and Lipsitz 1999)
Working variance-covariance matrix for
equals:
where is n x n diagonal matrix with elements var(yik ), i is n x n “working”
correlation matrix for yi , is a constant to allow for over dispersion.
G estimator of β is the solution of:
5
i
i
β
i
β
i
GEE model for binomial distribution using logit link function for this
study was:
log
4.
β
i
- i
β
β U
β G
.
Applying Generalized Linear Mixed Model (GLMM)
Subject-specific models assume that every region has its own intercept. To
avoid correlation away regions, each province is assumed to have its own
model. The GLMM of subject-specific models rope with this condition.
GLMM model for binomial distribution using logit link function with random
intercept was expressed as the following:
g
g
i
i
xi β vi
where vi is the random effect (one of each subject). These random effect
represent the influence of subject i on repeated subjects that is not captured by
observed covariates.
The mixed model equations are (Henderson 1984)
y
y
β
v
GLMM model for this study is:
log
i
β
β U
β
vi
i
5.
where:
β is fixed and population mean of POV
β is fixed effect of the hdi on the mean of POV
β is fixed effect of the ur on the mean of POV
β is fixed effect of the gdrpc on the mean of POV
vi is random effect for the ith individual
Random effects assume that vi
v and independence of the
fixed effects and the error terms .
Comparing the models by seeing the goodnest of fit with marginal R2 (
)
An extension of the R2 measure is calculating using:
n
i
n
i
ni
t
ni
t
it
it
it
6
This measure is interpreted as the proportion of variance in the outcome that
is explained by the model (Hardin and Hilbe 2003).
RESULT
Exploration of the Data
The poverty proportion of 33 provinces in Indonesia can be seen in Figure 1,
shows that the poverty proportion of 33 provinces are random. As we can see in
Figure 1, the poverty proportions for most of provinces are decreasing every year.
As we can see in Figure 1, the provinces that have high poverty proportion
are Papua, Papua Barat, Maluku, Nusa Tenggara Timur (NTT), Nusa Tenggara
Barat (NTB), Nangro Aceh Darussalan (NAD), Lampung, Sulawesi Tengah,
Bengkulu, and Sulawesi Tenggara. The provinces that have low poverty
proportion are Jakarta, Bali, Kalimantan Selatan, Banten, Kalimantan Tengah,
Kepulauan Bangka Belitung, Kalimatan Timur, Kepulauan Riau, Jambi, Sulawesi
Utara. The rest provinces are in the middle class.
The provinces that have the low poverty proportion are commonly from
West Indonesia, such as Java Island, Sumatera Island, and Kalimantan Island. It is
very opposite with East Indonesia where most of the provinces that have high
poverty proportion come from, such as Papua Island, Maluku Island, NTT and
NTB.
0.40
poverty proportion
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
2008
2009
2010
2011
2012
year
Figure 1 Line graph for poverty proportion of 33 provinces in Indonesia
GLM Approach
The constructions of GLM are by deciding on response and explanatory
variables of the data and choosing and appropriate link function and response
7
probability distribution. Because the response variable in the data is proportion so
the probability distribution for the response variable is binomial with log
i
- i
as
the link function.
The parameters estimate and Standard Errors (SEs) for GLM approach can
be seen in Table 3.
Table 3 Parameters estimate and standard errors (GLM approach)
Parameter
Estimate
Intercept
HDI
UR
GDRPC
Standard Error
3.7367
-0.0708
-0.0394
-0.0025
0.0030
0.0000
0.0000
0.0000
31.1569%
The model estimation for GLM approach in this study is:
log
i
.
.
.
U
.
i
and
for GLM approach is 31.1569%. It is interpreted as the 31.1569%
proportion of variance in the outcome that is explained by the model.
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale are still large, that are from -0.12 until 0.23. It is bacause the
approach that used is not enough to explain the data and it is in accordance with
the
from GLM approach that is only 31.1569%.
GEE Approach
The construction of GEE is by deciding the working correlation matrix that
we will use. In this study, working correlation matrix that was used is
autoregressive (AR(1)). The reason for using AR working correlation matrix is
measurement closer in time are likely to be more correlated than measurement
further apart in time.
Working correlation matrix (AR(1)) for this approach is:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The parameters estimate and Standard Errors (SEs) for GEE approach can
be seen in Table 4.
8
Table 4 Parameters estimate and standard errors (GEE approach)
Parameter
Estimate
Intercept
HDI
UR
GDRPC
9.3656
-0.1616
0.0055
0.0008
Standard Error
1.3063
0.0194
0.0083
0.0009
37.8220%
The model estimation for GEE approach (using autoregressive working
correlation matrix) in this study is:
i
log
.
.
.
U
.
i
and
for GEE approach is 37.8220%. It is interpreted as the 37.8220%
proportion of variance in the outcome that is explained by the model.
for
GEE is higher than GLM.
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale are still large, that are from -0.14 until 0.16. It is in accordance
with the
from GEE approach that is 37.8220%. GEE approach has better
than GLM even though it is not significant difference.
GLMM Approach
In the previous approaches, GLM and GEE are marginal model where you
estimate model for population. As we can see in Figure 1, the province variance is
very high. It can give missleading output if you assume that every province has
same mean in poverty proportin (intercept). To solve that problem we can apply
GLMM approach so that every province can have different intercept by assuming
that intercept is random effect.
The parameters estimate and Standard Errors (SEs) for GLMM approach
can be seen in Appendix 1.
for GLMM approach is 99.2349%. It is
interpreted as the 99.2349% proportion of variance in the outcome that is
explained by the model.
As we can see in Appendix 1, it is assumed that the intercept is random
effect and different for every province. The variance of the random intercepts on
the logit scale is estimated as
.
The model estimation for GLMM approach in this study is:
log
i
.
i
.
.
U
.
G
vi
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale is from -0.03 until 0.03. That is much better that the previous
9
approaches. It is in accordance with the
99.2349%.
from GLMM approach that is
GEE Approach with Dummy Variable
G
approach doesn’t have coefficient or variable that shows the subject
specifict/effect as in GLMM. It is kind of not suitable to compare between GEE
and GLMM. The model that uses GEE approach in this study take subject
cepecifit, in this study is provinces, as dummy variable. Dummy variable put into
the model in GEE approach. Dummy variable that used for every province can be
seen in Appendix 2. This method had been done by Anwar (2012) in his bachelore
thesis.
The difference between GLMM and GEE with dummy variable is, in
GLMM we assume that there are random effects but in GEE with dummy variable
we assume that all variable is fixed.
The working correlation matrix that used is Auto-Regressive. Working
correlation matrix (AR(1)) for this approach is:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The parameters estimate and Standard Errors (SEs) for GLMM approach
can be seen in Appendix 2.
for GLMM approach is 99.2227%. It is
interpreted as the 99.2227% proportion of variance in the outcome that is
explained by the model.
Scatter plot between prediction values and residual in Appendix 3 shows
that residual scale is from -0.03 until 0.03. It is similar with GLMM approach.
The
from GEE approach with dummy variable is 99.2227% and it is also
similar with GLMM approach.
Model Comparison
The SEs of parameter estimate for GLM approach are very small which are
0.000 (under estimate), in contrast GEE, GLMM, and GEE with dummy variable
approach have bigger SEs of parameter estimate. That problem occured because
GLM ignore the corrrelation between outcomes.
The
for every approach in this study are GLM approach is 31.1569%,
GEE is 37.8220%, GLMM is 99.2366%, and GEE with dummy variable is
99.2227%. For marginal model, GEE is better than GLM in
because GEE
can overcomes the correlation problem. For subject-specifict model, GLMM has
than GEE with dummy variable. GLMM approach has the best
better
which is means that it is better in explaine the proportion of variance in the
outcome by the model.
Scatter plot between prediction values and residual from GLM and GEE
approaches still haven’t fulfil the exppected result as in the theory. Hence, it is
10
possible to execute model emendation with another approach such as quasi
likelihood. For GLMM and GEE with dummy variable approaches, the scatter
plots between prediction values and residual are much better than the previous
approaches.
CONCLUSION
For population mean model, GEE approach is better than GLM to overcome
non-normal and correlated outcome in the case of this study which
for GEE
is 37.8220%. For subject-specific model, GLMM approach is better than GEE
with dummy variable to overcomes non-normal and correlated outcome in the
case of poverty proportion in this study where the variance of subjects is very big.
The
for GLMM approach in this study is 99.2366%.
REFERENCES
Anwar, N. 2012. Pemodelan Tingkat Pengangguran di Lima Negara Anggota
ASEAN dengan Regresi Data Panel dan Generalized Estimating Equation.
[bachelor thesis]. Bogor: Faculty of Mathematics and Natural Sciences,
Institut Pertanian Bogor.
Dobson, A.J. 2002. Introduction to Generalized Linear Models. Second Edition.
New York: CRC Press.
Gujarati D. N. 2004. Basic Econometrics, Fourth Edition. The McGraw-Hill
Companies.
Hardin, J.W. and Hilbe, J.M. 2003. Generalized Estimating Equations. Florida:
Chapman & Hall.
Henderson, C. R. 1984. Best Linear Unbiased Prediction of Performance and
Breeding Value. University of Guelph.
Horton N.J. and Lipsitz S.R. 1999. Review of Software to Fit Generalized
Estimating Equation Regression Models. The American Statiscian, 53, 160169.
McCulloch, C.E. and Searle, S.R. 2001. Generalized, Linear, and Mixed Model,
New York : Wiley.
Nelder, J.A. and Wedderburn, R. W. M. 1989. Generalized Linear Models,
Journal of Royal Statistical Society, 135(3), 370-84.
Zeger, S. L., Liang, K. Y. 1986. Longitudinal Data Analysis Using Generalized
Linear Models. Biometrics, 73, 13–22.
11
Appendix 1 Parameter estimates and standard errors (GLMM approach)
Parameter
Intercept
Random effects
HDI
UR
GDRPC
Estimate Standard Error
province 1
province 2
province 3
province 4
province 5
province 6
province 7
province 8
province 9
province 10
province 11
province 12
province 13
province 14
province 15
province 16
province 17
province 18
province 19
province 20
province 21
province 22
province 23
province 24
province 25
province 26
province 27
province 28
province 29
province 30
province 31
province 32
province 33
6.8584
-0.2940
-0.7291
-0.9828
-0.8720
-1.1839
-1.1333
-0.5083
-1.4401
-0.2370
-0.4231
-1.8808
-0.9600
-1.6976
-0.4281
-0.0561
-0.6538
-1.8445
-0.9199
-0.5994
-1.4980
-1.1841
-2.0085
-1.2635
-0.7809
-0.3263
-0.5243
-1.0000
-0.7086
0.0092
-1.4221
-0.3158
-0.8956
0.0000
-0.1584
0.0053
0.0008
0.0531
0.0024
0.0039
0.0038
0.0053
0.0053
0.0034
0.0031
0.0035
0.0036
0.0026
0.0087
0.0028
0.0022
0.0030
0.0051
0.0023
0.0030
0.0035
0.0030
0.0019
0.0044
0.0019
0.0063
0.0054
0.0027
0.0024
0.0025
0.0022
0.0030
0.0028
0.0036
0.0026
0.0531
0.0003
0.0002
0.0000
99.2349%
12
Appendix 2 Parameter estimates and standard errors (GEE approach with dummy
variable)
Parameter
Intercept
HDI
UR
GDRPC
Prov 1
Prov 2
Prov 3
Prov 4
Prov 5
Prov 6
Prov 7
Prov 8
Prov 9
Prov 10
Prov 11
Prov 12
Prov 13
Prov 14
Prov 15
Prov 16
Prov 17
Prov 18
Prov 19
Prov 20
Prov 21
Prov 22
Prov 23
Prov 24
Prov 25
Prov 26
Prov 27
Prov 28
Prov 29
Prov 30
Prov 31
Prov 32
Estimate
10.0993
-0.1588
0.0051
0.0008
-0.1766
-0.4860
-0.7605
-0.5233
-0.8644
-0.9563
-0.3213
-1.2500
-0.0515
-0.3211
-1.4056
-0.8188
-1.6325
-0.2694
0.2584
-0.5294
-1.6913
-1.0987
-0.6920
-1.4869
-0.9102
-1.9612
-0.9072
-0.4576
-0.2844
-0.4282
-0.8906
-0.6702
0.1013
-1.4316
-0.5136
-0.8705
Standard Error
0.8494
0.0122
0.0078
0.0007
0.0601
0.0777
0.0784
0.0696
0.0699
0.0701
0.0632
0.0497
0.0744
0.0641
0.1684
0.0798
0.0733
0.0720
0.1002
0.0378
0.0515
0.0613
0.0712
0.0399
0.0682
0.0316
0.1119
0.1191
0.0711
0.0503
0.0645
0.0533
0.0901
0.0640
0.0694
0.0602
99.2227%
13
0.21
0.21
0.16
0.16
0.11
0.11
p - p hat
p - p hat
Appendix 3 Scatter plot between prediction values and residual
0.06
0.01
-0.04 0
0.2
0.4
0.06
0.01
-0.04 0
-0.09
-0.09
-0.14
-0.14
�
(a) GLM Approach
0.1
0.2
0.3
0.4
�
(c) GLMM Approach
0.21
0.21
0.16
0.16
0.11
p - p hat
p - p hat
0.11
0.06
0.01
-0.04 0
0.2
0.01
-0.04
0
0.2
-0.09
-0.09
-0.14
-0.14
(b) GEE Approach
0.4
0.06
�
�
(d) GEE with dummy variable
approach
0.4
14
2
BIOGRAPHY
The author was born in Indramayu, West Java on November 7, 1992 as the
first child of Rusdi and Ratna Dewi couple. The author began her education in AlIrsyad Al-Islamiyah kindergarden in 1997, then continue to Muhammadiyah 1
Elementary School in 1998. The author continue her education at SMPN 1
Cirebon. Graduated from SMPN 1 Cirebon in 2007, the author continued her
education in SMAN 2 Cirebon. After graduating from SMAN 2 Cirebon in 2010,
author accepted in the Department of Statistics, Institut Pertanian Bogor via
USMI.
The author was active in Gamma Sigma Beta (GSB) as the member of Division
of Data Base Center in 2012/2013 and also a committee of Statistika Ria 8 in 2012.
The autor was an apprentice in Ministry of Education and Culture in August until
September 2013.