İSTATİSTİK ARAŞTIRMA DERGİSİ Journal of Statistical Research İ l

Transkript

TÜRKİYE İSTATİSTİK KURUMU
İSTATİSTİK
ARAŞTIRMA DERGİSİ
Journal of Statistical
Research
İSTATİSTİK
Research
Cilt-Volume: 10 Sayı-Number: 02
Temmuz-July 2013
Temmuz-July 2013
ISSN 1303-6319
Turkish Statistical Institute
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Editör Notu
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Prof. Dr. Fetih YILDIRIM
Dergi Editörü
TÜİK, İstatistik Araştırma Dergisi, Temmuz 2013
TurkStat, Journal of Statistical Research, July 2013
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Editör Editor
Prof. Dr. Fetih YILDIRIM Prof. Dr. Fetih YILDIRIM
(GLW|U<DUGÕPFÕVÕ Assistant Editor
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Buket AKGÜN
Z.Nur EMRE
Nurdan ELVER
V
Contents
İçindekiler
VII
İçindekiler
VIII
Contents
Aim, Target, Principles
Amaç, Kapsam, İlkeler
IX
Amaç, Kapsam, İlkeler
Aim, Target, Principles
Referee List
Hakem Listesi
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REFEREES WHO PROVIDED SCIENTIFIC CONTRIBUTIONS FOR THIS
VOLUME OF THE JOURNAL
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XI
XI
Volume: 10 Number: 02 Category: 02 Page: 1-24 ISSN No: 1303-6319 Cilt: 10 Sayı: 02 Kategori: 02 Sayfa: 1-24
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Construction of a Life Table by Using
the Turkish General Death Data
Türkiye Geneli Ölüm Verileri Kullanılarak
Yaşam Tablosunun Oluşturulması
ülkenin kendi demografik özellikOHULQL \DQVÕWDQ \DúDP WDEORVXQD GX\GX÷X LKWL\Do
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Kintner, 2004; Keyfitz ve Caswell, 2005; Bell ve Miller, 2005 vb). Bir ülkenin belirli
ELU]DPDQDUDOÕ÷ÕQGDNLPHYFXWGHPRJUDILN \DSÕVÕQÕ \DQVÕWDQYHEXDUDOÕNWDNL \DúD|]el
|OPKÕ]ODUÕED]DOÕQDUDNROXúWXUXODQ\DúDPWDEORVXQDG|QHP\DúDPWDEORVXGHQLU (Pfaff
vd, 2012) %X E|OPGH YH \ÕOODUÕ LoLQ |OP YH \DúDP YHULOHUL
NXOODQÕODUDN 7UNL\H g]HWOHQPLú '|QHP <DúDP 7DEORVX FLQVL\HW YH \Dú JUXSODUÕ
ED]ÕQGD HOGH HGLOPLúWLU %X oDOÕúPDQÕQ \|QWHPL DúD÷ÕGD DQODWÕOPÕúWÕU gQFHOLNOH
GHPRJUDIL ELOLPLQGH KÕ] KHUKDQJL ELU ROD\ÕQ |OP GR÷XP J|o YE JHUoHNOHúPH
VD\ÕVÕQÕQ EX ULVNH PDUX] NDODQ NLúL \ÕO VD\ÕVÕQD E|OP\OH HOGH HGLOLU 7DEORQXQ
ROXúWXUXOPDVÕ LoLQ WDQÕPODQDFDN LON GH÷LúNHQ \DúD |]HO |OP KÕ]Õ RODUDN DGODQGÕUÕODQ
n M x , belirli bir zamaQ DUDOÕ÷ÕQGD x ve x n \DúODUÕ DUDVÕQGD |OHQ NLúL VD\ÕVÕQÕQ yine
D\QÕ\DúDUDOÕ÷ÕQGD\DúDQÕODQ NLúL\ÕOVD\ÕVÕQDE|OQPHVL\OHHlde edilir (Arias, 2010).
n
Mx
Dx
| n mx
n Nx
1
n
ax LOH J|VWHULOHQ RUWDODPD \DúDQDQ NLúL \ÕO VD\ÕVÕ G|UW IDUNOÕ \|QWHPH J|UH
hesaplanabilir 3UHVWRQ YG %X oDOÕúmada gerçek gözlemler kullaQÕOPÕú \DQL
RUWDODPD \DúDQDQ NLúL \ÕO VD\ÕVÕ |OP VD\ÕODUÕQÕQ KHU ELU \Dú JUXEX LoLQ D÷ÕUOÕNOÕ
RUWDODPDODUÕDOÕQDUDNHOGHHGLOPLúWLU
n
n
ax
d x . 0 d x 1. 1 d x 2 . 2 d x 3 . 3 d x 4 . 4 n dx
,
n
5
2
<DúD |]HO |OP RODVÕOÕ÷Õ n qx , x \DúÕQGa bir kimsenin n \ÕO LoLQGHNL |OP olaVÕOÕ÷ÕQÕ
LIDGH HGHU YH \DúD |]HO ölüm KÕ]Õ YH \DúDQDQ RUWDODPD NLúL \ÕO VD\ÕVÕ GH÷LúNHQOHUL
NXOODQÕODUDN HOGH HGLOLU %X G|QúP JHUoHN NXúDN GDYUDQÕúODUÕQÕ ED] DOÕU YH DúD÷ÕGDNL
gibi elde edilir.
dx
lx
n. n mx
1 1 n ax n mx
n
Lx
n . lx n n ax . n d x o n qx
n
Lx
: %LUNXúDNWD[YH[Q\ÕOODUÕDUDVÕQGD\DúDQDQNLúL\ÕOVD\ÕVÕNXúDNGDYUDQÕúODUÕQÕ
ED]DOÕU
: %LUNXúDNWD\DúD\DQNLúLOHUden EXDUDOÕNWD\DúD\DQNLúL\ÕOVD\ÕVÕ
: <DúDQDQRUWDODPDNLúL\ÕO VD\ÕVÕ
: %LUNXúDNQIXVXQGDQEXDUDOÕNWD|OHQNLúLOHULQVD\ÕVÕ
n . lxn
n ax
n
dx
n
<DúD\DQNXúDNNLúL\ÕOVD\ÕVÕDúD÷ÕGDNLJLEL\HQLGHQG]HQOHQLU
n
Lx
n. lx
lx
n lx n d x n ax . n d x
n
Lx n . n d x n ax . n d x
1
ª n Lx n n ax . n d x º¼
n¬
.XúDN GDYUDQÕúODUÕ NXOODQÕODUDN HOGH HGLOHQ lx yani x \DúÕQGD \DúD\DQ ELUH\OHULQ VD\ÕVÕ
n q x IRUPOQGH\HULQH\D]ÕOÕU
n. n d x
n dx
n qx
lx
n Lx n n a x . n d x
Pay ve payda n Lx ’e bölünür:
d
n. n x
n Lx
n qx
L
d
n x
n n ax . n x
n Lx
n Lx
n . n mx
1 n n ax . n d x
'DKDVRQUD\DúD|]HO|OPRODVÕOÕ÷ÕDúD÷ÕGDNLJLELHOGHHGLOLU
n
qx
n. n mx
1 1 n ax n mx
and
qf
1
3
%XG|QúPGreYLOOHDQG&KLDQJWDUDIÕQGDQWDQÕPODQPÕúWÕUYHVDGHFHELU
parametre gerektirir ve o parametre ise n ax ’tir. 2UWDODPD\DúDQDQNLúL\ÕOVD\ÕVÕG|QHP
\DúDP WDEORODUÕQÕQ ROXúWXUXOPDVÕQGD |QHPOL ELU \HUH VDKLSWLU YH n mx o n qx G|QúP
LOH \DúD |]HO |OP RODVÕOÕNODUÕQÕQ J|]OHQHQ YHULOHUGHQ HOGH HGLOPHVLQH RODUDN VD÷ODU
(Preston vd., 2001). 'DKD VRQUD VÕUDVÕ\OD \DúDP WDEORVX IRQNVL\RQODUÕ RODQ \DúDPD
RODVÕOÕNODUÕ n px \DúDQDQ NLúL \ÕO VD\ÕVÕ n Lx WRSODP NLúL \ÕO VD\ÕVÕ Tx HOGH HGLOPLúWLU
(Selvin, 2008) <DúDP WDEORVX DQDOL]LQGHQ ID\GDODQÕODUDN x \DúÕQGD ELU NLPVHQLQ
EHNOHQHQ\DúDP|PUDúD÷ÕGDNLJLELKHVDSODQÕU
0
ex
Tx
lx
'R÷XúWDEHNOHQHQ\DúDP|PULVHDúD÷ÕGDNLJLELHOGHHGLOLU
0
e0
T0
l0
3. UYGULAMA
%X oDOÕúPDGD 7hø. WDUDIÕQGDQ \D\ÕPODQDQ YH \ÕOÕ 7UNL\H JHQHOL
QIXV |OP YH GR÷XP YHULOHUL NXOODQÕODUDN YH \ÕOODUÕ LoLQ FLQVL\HW
ED]ÕQGD g]HWOHQPLú '|QHP <DúDP 7DEORODUÕ ROXúWXUXOPXúWXU $GUHVH 'D\DOÕ 1IXV
DQFDN\ÕOÕQGDELU|QFHNL\ÕODJ|UHEXDUDOÕNWDHUNHNOHUGHµOLNELUDUWÕúROGX÷X
J|]OHPOHQPLúWLU - \Dú DUDOÕ÷ÕQGD |OP RODVÕOÕNODUÕQGDNL D]DOÕú HUNHNOHUGH NDGÕQODUGD LVH ROGX÷X YH \ÕOÕQGD HUNHNOHUGH |OP RODVÕOÕ÷ÕQÕQ DUWWÕ÷Õ
J|]OHPOHQPLúWLU - \Dú DUDOÕ÷ÕQGD |OP RODVÕOÕNODUÕQGD NL D]DOÕú HUNeklerde %1.6,
NDGÕQODUGD LVH ROGX÷X YH \ÕOÕQGD D\QÕ WUHQGLQ HUNHNOHUGH D]DODUDN GHYDP
HWWL÷L J|]OHPOHQPLúWLU *|UOG÷ JLEL EHEHN YH oRFXN |OPOHULQGH YH \Dú
VRQUDVÕQGD HUNHNOHUGH \ÕOlarD J|UH D]DODQ |OP RODVÕOÕ÷Õ NDGÕQODUD J|UH GDKD GúNWU
%XQXQ WHUVLQH RUWD \Dú DUDOÕ÷ÕQGD GDKD \NVHNWLU $QFDN \ÕOÕQGDQ LWLEDUHQ \DúÕQGDQ VRQUD HUNHNOHULQ \DúD |]HO |OP RODVÕOÕNODUÕQGD ELU DUWÕú WUHQGLQLQ EDúODGÕ÷Õ
J|UOPúWU
<DúDP WDEORVX DQDOL]L NXOODQÕODUDN HOGH HGLOHQ EHNOHQHQ \DúDP VUHOHUL NDGÕQODUÕQ
HUNHNOHUGHQ GDKD X]XQ \DúDGÕ÷ÕQÕ J|VWHUPHNWHGLU 7DEOR ¶GH EHOLUOL \DúODU LoLQ HOGH
HGLOHQEHNOHQHQ|PUOHUJ|VWHULOPLúWLU
Tablo 2. 2009-2011 yÕOODUÕ cLQVL\HW ED]ÕQGD belirli yDúODU LoLQ ortalama yDúDP süresi (beklenen
yDúDPsüresi) (“ “=2009, “*”=2010, “**”=2011)
<Dú
Tüm
Tüm* Tüm**
.DGÕQ .DGÕQ .DGÕQ
Erkek Erkek* Erkek**
0
75.8
76.5
76.8
78.5
79.1
79,4
73.1
73.9
74,1
20
57.6
58.1
58.3
60.2
60.7
60.9
55.0
55.6
55.7
40
38.3
38.8
39.0
40.7
41.1
41.4
35.9
36.4
36.5
60
20.5
20.9
21.0
22.2
22.6
22.8
18.6
19.0
19.0
80
7.4
7.6
7.7
7.9
8.1
8.3
6.6
6.8
6.8
Tablo 2’de elde edilen sonuçlara göre 7UNL\H¶QLQGR÷XúWDEHNOHQHQ\DúDPVUHVL
\ÕOÕQGD LNHQ \ÕOÕQGD \ÕOÕQGD LVH ¶H \NVHOPLúWLU 7UNL\H¶QLQ
GR÷XúWDEHNOHQHQ \DúDPVUHVL¶OLNELUDUWÕúJ|VWHUPLúWLU%HNOHQHQ\DúDPVUHVL
NDGÕQODUGD\ÕOÕQGD\ÕOÕQGDLNHQ\ÕOÕQGD¶HHUNHNOHUGHLVH
\ÕOÕQGD\ÕOÕQGDLNHQ\ÕOÕQGD¶H\NVHOPLúWLU\ÕOÕQGD
\ÕOÕQD J|UH7UNL\HQIXVXQXQEHNOHQHQ \DúDPVUHVL \ÕOX]DPÕúWÕU.DGÕQODUÕQ
GR÷XúWDEHNOHQHQ\DúDPVUHVL D\X]DUNHQHUNHNOHULQ|PU\ÕOX]DPÕúWÕU
Tablo 3. 2011 yÕOÕQGD2009 yÕOÕQDJöre beklenen yDúDPsüresinde meydana gelen aUWÕúoUDQODUÕ
<Dú
Tüm .DGÕQ Erkek
0
1.32%
1.15%
1.37%
20
1.22%
1.16%
1.27%
40
1.83%
1.72%
1.67%
60
2.44%
1.80%
2.15%
80
4.05%
2.53%
3.03%
7DEOR¶WHHOGHHGLOHQVRQXoODUDJ|UHEHNOHQHQ\DúDPVUHVLQGHNLDUWÕúRUDQÕHUNHNOHUGH
ROXUNHQ NDGÕQODUGD ROPXúWXU +HU ELU \Dú JUXEX LoLQ RUDQODU
LQFHOHQGL÷LQGH HUNHNOHULQ EHNOHQHQ \DúDP VUHVLQLQ \Dú JUXEX KDULo NDGÕQODUGDQ
GDKD KÕ]OÕ X]DGÕ÷Õ J|UOPúWU %X da erkekOHULQ EHNOHQHQ \DúDP VUHVLQLQ zamanla
NDGÕQODUÕQNLQH \DNODúDFD÷ÕQÕQELU J|VWHUJHVLGLU7DEOR¶WH J|UOHQELUEDúNDVRQXoLVH
EHNOHQHQ\DúDPVUHVLQGHNLDUWÕúRUDQÕQÕQ\DúOÕQIXVDGR÷UX\NVHOPHVLGLU*|UOG÷
]HUH \DúÕQGD RUWDODPD |PUGHNL DUWÕú RUDQÕ ¶WLU %X GXUXP \DúOÕ QIXVXQ
|PUQQ\ÕOODULWLEDUL\OHDUWÕúJ|VWHUGL÷LQLDoÕNODPDNWDGÕU
4. %8/*8/$59(7$57,ù0$
%XoDOÕúPDGDJHUoHNQIXVYHGHPRJUDILNYHULOHUNXOODQÕODUDN7UNL\HLoLQg]HWOHQPLú
'|QHP <DúDP 7DEORVX ROXúWXUXOPXúWXU <DúDP WDEORVX DQDOL]L YH oHúLWOL GHPRJUDILN
\DNODúÕPODUNXOODQÕODUDN|OPGDYUDQÕúODUÕLQFHOHQPLúYH\DúJUXEXYHFLQVL\HWED]ÕQGD
bekOHQHQ \DúDP VUHOHUL KHVDSODQPÕúWÕU %XQD J|UH \DúD |]HO |OP RODVÕOÕ÷ÕQÕQ
EHEHNOHUGH YH JHQo QIXVWD Gúú J|VWHUGL÷L YH EX RODVÕOÕ÷ÕQ \DúOÕ QIXVD GR÷UX DUWÕú
J|VWHUGL÷LJ|UOPúWU%XQXQELUVRQXFXRODUDN7UNL\H¶QLQ\ÕOODULWLEDUL\OHGDKDX]XQ
\DúDGÕ÷ÕJ|UOPúWU.DGÕQODUÕQHUNHNOHUGHQGDKDX]XQ\DúDGÕ÷ÕDQFDNEHNOHQHQ\DúDP
VUHVLQGHNL DUWÕúÕQ HUNHNOHUGH GDKD ID]OD ROGX÷X |QHPOL ELU EXOJX RODUDN NDUúÕPÕ]D
oÕNPÕúWÕU %XQXQ DQODPÕ LVH HUNHNOHULQ LOHUOH\HQ \ÕOODUGD NDGÕQODUÕQ RUWDODPD \DúDP
sürelerine \DNODúDFDNODUÕGÕU
gOPGDYUDQÕúODUÕLO-LOoHYH7UNL\HJHQHOLED]ÕQGD\D\ÕPODQDQYHULLOHNDUúÕODúWÕUÕOPÕú
YH JHQHOGDYUDQÕúÕQD\QÕROGX÷XDQFDN \ÕOÕ LWLEDUL\OHYHULND\ÕWVLVWHPLQHN|\ YH
EXFDNODUÕQÕQ GDKLO HGLOPHVL\OH ELUOLNWH EHOOL \Dú JUXSODUÕ LoLQ IDUNOÕOÕN J|VWHUGL÷L
J|UOPúWU%XGD7UNL\H¶QLQJHUoHNGHPRJUDILN\DSÕVÕQÕ\DQVÕWPDGÕ÷ÕLoLQ\ÕOÕ
|QFHVLQGH\D\ÕPODQDQ|OPYHULOHULQLQNXOODQÕOPDVÕQÕQ\DQOÕúVRQXoODUGR÷XUDELOHFH÷LQL
J|VWHUPLúWLU $\UÕFD \ÕOÕ LWLEDUL\OH |OP YHULOHULQLQ güncellenmesiyle birlikte
PHYFXWQIXVXQGHPRJUDILN\DSÕVÕQÕ\DQVÕWDQ'|QHP<DúDP7DEORODUÕQÕQROXúWXUXOPDVÕ
PPNQNÕOÕQPÕúWÕU
ùHNLOYHùHNLO¶GHEXoDOÕúPDGDHOGHHGLOHQ\DúD|]HO|OPRODVÕOÕNODUÕoD\UÕ\DúDP
tablosu ile FLQVL\HW ED]ÕQGD NDUúÕODúWÕUÕOPÕúWÕU %X WDEORODU VÕUDVÕ\OD ONHPL]GH KDOHQ
NXOODQÕODQ &62 <DúDP 7DEORVX The Commissioners 1980 Standard Ordinary
0RUWDOLW\ 7DEOH EX oDOÕúPDGD ROXúWXUXODQ \DúDP WDEORVX 7XUNL\H YH Hayat
6LJRUWDODUÕ %LOJL 0HUNH]L +$<0(5 için \DSÕODQ projede 7UNL\H LoLQ ROXúWXUXODQ
\DúDPWDEORVXGXUTRH-2010).
ùHNLOLQFHOHQGL÷LQGHHOGHHGLOHQ\DúD|]HO|OPRODVÕOÕNODUÕQÕQIDUNOÕ\DúDUDOÕNODUÕLoLQ
GDYUDQÕúODUÕ\RUXPODQPÕúWÕU%XQDJ|UH-\DúDUDOÕ÷ÕLoLQONHPL]LoLQROXúWXUXODQLNL
tabloda CSO’ya J|UH D\QÕ GDYUDQÕúODUÕ J|VWHUPLúWLU $QFDN EX oDOÕúPDGD HOGH HGLOHQ
tabloya göre bebek ölümleri CSO ve TRH-2010 tablosuna göre daha yüksek, 1-\Dú
DUDOÕ÷ÕQGDLVH|OPRODVÕOÕNODUÕ75+-WDEORVXQDJ|UH&62¶\DGDKD\DNÕQoÕNPÕúWÕU
%HQ]HUúHNLOGHD\QÕdurum 65-\DúJUXEXQDNDGDUGHYDPHWPLúWLU\DúVRQUDVÕQGD
LVH &62¶\D J|UH |OP RODVÕOÕNODUÕ \NVHOPLúWLU EX GD ONH QIXVXQXQ \DúODQGÕ÷ÕQÕ
J|VWHUPHNWHGLU$QFDNEXDUDOÕNWD|OPRODVÕOÕNODUÕ75+-WDEORVXQGDEXoDOÕúPD\D
J|UH &62¶\D GDKD \DNÕQ oÕNPÕúWÕU <DQL EX oDOÕúPDGD EHNOHQHQ |PUGHNL DUWÕúÕQ
ND\QD÷Õ \DúODQDQ QIXV ROXUNHQ 75+- WDEORVXQGD DUWÕúÕQ QHGHQL GDKD oRN JHQo
QIXVWDNL|OPRODVÕOÕNODUÕQÕQGúNO÷ROPXúWXU
10
11
EHEHN YH oRFXN |OPOHULQGHNL GúúWHQ GDKD oRN \DúODQDQ ELU 7UNL\H QIXVXQXQ
ROGX÷XQXJ|VWHUPLúWLU
Tablo 4. Belirli yDúODULoLQcinsiyet bD]ÕQGDbeklenen yDúDPsüresi
.DGÕQ
Erkek
<Dú
CSO
Türkiye
Haymer
CSO
Türkiye
Haymer
0
75.8
79.1
78.0
70.8
73.9
71.9
20
57.0
60.7
58.9
52.4
55.6
54.0
40
38.4
41.1
39.2
34.1
36.4
34.9
60
21.2
22.6
20.8
17.5
19.0
17.6
80
7.5
8.1
7.0
6.2
6.8
6.0
7DEOR¶WHNDGÕQYHHUNHNOHULoLQHOGHHGLOHQEHNOHQHQ\DúDPVUHOHULNDUúÕODúWÕUÕOPÕúWÕU
%XQDJ|UHNDGÕQODUGDEHNOHQHQ\DúDPVUHVL75+-¶DJ|UH\ÕOHUNHNOHUGHLVH
\ÕO GDKD X]XQGXU <DúOÕ QIXV LoLQ \Dú LQFHOHQGL÷LQGH EX oDOÕúPDGD KHVDSODQDQ
EHNOHQHQ \DúDP VUHVL TRH-201’a göre NDGÕQODUGD \ÕO HUNHNOHUGH \ÕO GDKD
uzundur.
%X oDOÕúPDGDQ HOGH HGLOHQ EXOJXODU J|VWHUPLúWLU NL ONHPL]GH KDOHQ DNWLI RODUDN
NXOODQÕODQ EDúND ONHOHULQ WDEORODUÕ ONHPL]LQ JHUoHN GHPRJUDILN \DSÕVÕQÕ
\DQVÕWPDPDNWDGÕU %X GXUXP EDúWD VRV\DO JYHQOLN VLVWHPL YH VLJRUWD VHNW|U LoLQ
tehlike arz etmektedir. Bu tehlikelerin önlenmesi için Türkiye kendi demografik
\DSÕVÕQÕ\DQVÕWDQ\DúDPWDEORVXQXROXúWXUPDOÕYHDNWLIRODUDNNXOODQPDOÕGÕU
7hø.WDUDIÕQGDQ\D\ÕPODQDQ|OPYHULOHULELOLQPH\HQ\DúDDLWYHULOHUL de içermektedir
DQFDNEXoDOÕúPDGDELOLQPH\HQ\DúDDLWRODQ|OPVD\ÕODUÕYHUL\HGDKLOHGLOPHPLúWLU%X
oDOÕúPD LoLQ ELU HNVLNOLNWLU YH oHúLWOL LVWDWLVWLNVHO \|QWHPOHU NXOODQÕODUDN EX HNVLNOLN
giderilebilir.
$\UÕFD EX oDOÕúPDGD g]HWOHQPLú <DúDP 7DEORVX YH \ÕOODUÕ LoLQ D\UÕ
D\UÕROXúWXUXOPXúWXU$QFDNG|QHP\DúDPWDEORVXKHU\ÕOD\UÕD\UÕROXúWXUXODELOGL÷LJLEL
\ÕOOÕN \D GD EHQ]HUL G|QHPOHU LoLQGH ROXúWXUXODELOLU %X oDOÕúPDGD |UQHN ROPDVÕ LoLn
WHN \ÕO YH \Dú JUXSODUÕ LoLQ ROXúWXUXOPXúWXU øOHUOH\HQ oDOÕúPDODUGD WHN \DúODU LoLQ
ROXúWXUXOPDVÕKHGHIOHQPHNWHGLU
6RQ RODUDN JHOHFHN \ÕOODU LWLEDUL\OH HNOHQHFHN \HQL |OP YHULOHUL KHU \ÕO ONHPL] LoLQ
G|QHP \DúDP WDEORVXQXQ \HQLOHQPHVLQH YH X\JXQ \ÕO VD\ÕVÕQD XODúÕOGÕ÷ÕQGD |OP
RODVÕOÕNODUÕQÕQ YH EHNOHQHQ \DúDP VUHOHULQLQ JHOHFHN LoLQ NHVWULPLQH RODQDN
VD÷OD\DFDNWÕU
5. KAYNAKLAR
Alpay A., 1969. Abridged Life Tables for Selected Regions and Cities of Turkey,
Turkish Demography: Proceedings of a Conference. H.Ü. Nüfus Etütleri Enstitüsü, 83108.
Arias E., 2010. United States Life Tables by Hispanic Origin, National Center for
Health Statistics. Vital Health Stat 2(152).
12
Australian Bureau of Statistics (ABS). 2008. Life Tables. Retrieved 2012,
from http://www.abs.gov.au/AUSSTATS/[email protected]/webpages/statistics?opendocument.
Bell, F., C., Miller, M., L., 2005. Life Tables for the United States Social Security Area
1900-2100, Social Security Administration. Actuarial Study, 120.
Chiang, C. L., 1968. Introduction to Stochastic Processes in Biostatistics, New York:
John-wiley and Sons.
Chiang, C. L., 1972. On Constructing Current Life Tables, Journal of the American
Statistical Association 67, 538-541.
&Rúkun, Y., 2002. Estimation of Adult Mortality by Using the Orphanhood Method
from the 1993 and 1998 Turkish Demographic and Health Surveys, H. Ü. Nüfus
Etütleri$QNDUDøngilizce).
Demirbüken, D., 2001. An Evaluation of Burial Records of Ankara City Cemeteries, H.
Ü. Nüfus Etütleri Enstitüsü, Ankara. (øQJLOL]FH).
Demirci, M., 1987. 7UNL\H¶QLQgOPOON<Dú <DSÕVÕQD0RGHO<DúDP 7DEORODUÕQGDQ
(Q8\JXQ.DOÕEÕQ6HoLPL, H. Ü. Fen Bilimleri Enstütüsü, Ankara. (Türkçe).
Duransoy, M. L., 1993. Türk Mortalite Tablosu (1980-2000), Mimar Sinan Üniversitesi,
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14
15
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46.446.199
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0,9636
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33.402
4.499.262
22.649.110
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18.149.848
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50
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0,0668
0,9332
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59.107
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0,1160
0,8840
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95.682
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11.671.643
14,1
75
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0,2020
0,7980
729.403
147.308
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7.829.807
10,7
80
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1,9568
0,3251
0,6749
582.095
189.246
2.334.554
4.609.912
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0,4638
0,5362
392.848
182.213
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35
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0,9787
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20.169
4.670.359
27.862.224
29,5
55
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0,0328
0,9672
925.230
30.374
4.534.114
23.191.865
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1,9683
0,0539
0,9461
894.856
48.225
4.328.074
18.657.752
20,9
65
0,0184
2,0616
0,0874
0,9126
846.631
73.983
4.015.761
14.329.678
16,9
70
0,0312
2,0057
0,1425
0,8575
772.648
110.111
3.533.532
10.313.917
13,3
75
0,0522
2,1130
0,2267
0,7733
662.537
150.216
2.879.011
6.780.385
10,2
80
0,0881
1,9427
0,3471
0,6529
512.321
177.850
2.017.868
3.901.374
7,6
85
0,1416
1,6205
0,4788
0,5212
334.472
160.149
1.131.134
1.883.507
5,6
90+
0,2317
3,2246
1,0000
0,0000
174.323
174.323
752.373
752.373
4,3
19
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0,9965
987.147
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3.938.675
72.916.407
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5
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0,9977
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4.910.890
68.977.732
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64.066.843
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20
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54.278.259
55,6
25
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0,9961
972.555
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4.851.704
49.408.328
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30
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0,9955
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4.830.628
44.556.624
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35
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0,9941
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4.805.909
39.725.996
41,2
40
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2,1784
0,0093
0,9907
958.751
8.946
4.768.513
34.920.087
36,4
45
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2,0870
0,0165
0,9835
949.805
15.697
4.703.302
30.151.574
31,7
50
0,0060
2,2004
0,0293
0,9707
934.109
27.385
4.593.876
25.448.272
27,2
55
0,0093
1,9758
0,0453
0,9547
906.724
41.046
4.409.488
20.854.396
23,0
60
0,0153
1,9468
0,0730
0,9270
865.678
63.224
4.135.353
16.444.908
19,0
65
0,0243
2,0427
0,1135
0,8865
802.454
91.058
3.742.986
12.309.555
15,3
70
0,0396
1,9676
0,1767
0,8233
711.396
125.703
3.175.797
8.566.569
12,0
75
0,0628
2,0501
0,2650
0,7350
585.693
155.233
2.470.543
5.390.772
9,2
80
0,1048
1,8875
0,3951
0,6049
430.460
170.069
1.622.967
2.920.229
6,8
85
0,1626
1,5819
0,5226
0,4774
260.391
136.078
836.825
1.297.263
5,0
90+
0,2700
2,9682
1,0000
0,0000
124.313
124.313
460.438
460.438
3,7
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78.132.955
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5
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0,9977
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2.290
4.919.251
74.187.814
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10
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1.502
4.911.002
69.268.563
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15
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64.357.561
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20
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0,0016
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25
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0,9981
978.229
1.904
4.885.805
54.560.565
55,8
30
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1,9708
0,0024
0,9976
976.326
2.337
4.874.550
49.674.759
50,9
35
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0,9966
973.989
3.353
4.860.183
44.800.210
46,0
40
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2,1562
0,0055
0,9945
970.636
5.316
4.838.061
39.940.026
41,1
45
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2,0061
0,0082
0,9918
965.320
7.945
4.802.811
35.101.965
36,4
50
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2,1722
0,0131
0,9869
957.375
12.554
4.751.372
30.299.154
31,6
55
0,0041
1,9569
0,0204
0,9796
944.820
19.272
4.665.452
25.547.782
27,0
60
0,0073
2,0090
0,0360
0,9640
925.548
33.275
4.528.213
20.882.329
22,6
65
0,0132
2,0920
0,0636
0,9364
892.273
56.743
4.296.355
16.354.117
18,3
70
0,0244
2,0550
0,1138
0,8862
835.529
95.052
3.897.724
12.057.761
14,4
75
0,0434
2,1875
0,1932
0,8068
740.478
143.074
3.299.996
8.160.037
11,0
80
0,0781
1,9866
0,3160
0,6840
597.404
188.807
2.418.071
4.860.041
8,1
85
0,1305
1,6450
0,4539
0,5461
408.597
185.449
1.420.801
2.441.970
6,0
90+
0,2185
3,3281
1,0000
0,0000
223.148
223.148
1.021.170
1.021.170
4,6
21
Ek 7. 7UNL\Hg]HWOHQPLú'|QHP<DúDP7DEORVX
ŐĞ
22
Ŷ ŵ ǆ
Ŷ a ǆ
Ŷ Ě ǆ
d ǆ Ğ ǆ 0
Ŷ Ƌ ǆ
Ŷ p ǆ
ů ǆ
ŶLǆ
0,0117
0,9883
1.000.000
11.720
988.280
76.755.744
76,8
0
0,0117
1
0,0008
1,1269
0,0032
0,9968
988.280
3.152
3.944.064
75.767.464
76,7
5
0,0004
1,6902
0,0021
0,9979
985.128
2.064
4.918.809
71.823.400
72,9
10
0,0003
2,0768
0,0016
0,9984
983.064
1.525
4.910.863
66.904.591
68,1
15
0,0005
2,1151
0,0027
0,9973
981.539
2.629
4.900.111
61.993.728
63,2
20
0,0005
2,0568
0,0027
0,9973
978.910
2.657
4.886.730
57.093.618
58,3
25
0,0006
2,0218
0,0029
0,9971
976.253
2.839
4.872.811
52.206.888
53,5
30
0,0007
1,9993
0,0033
0,9967
973.414
3.259
4.857.291
47.334.077
48,6
35
0,0009
2,2168
0,0047
0,9953
970.155
4.555
4.838.097
42.476.786
43,8
40
0,0014
2,0620
0,0071
0,9929
965.600
6.852
4.807.868
37.638.689
39,0
45
0,0024
2,0649
0,0119
0,9881
958.748
11.419
4.760.223
32.830.821
34,2
50
0,0040
2,0285
0,0198
0,9802
947.329
18.798
4.680.783
28.070.598
29,6
55
0,0067
1,9424
0,0327
0,9673
928.530
30.386
4.549.744
23.389.815
25,2
60
0,0111
2,0380
0,0536
0,9464
898.144
48.182
4.348.009
18.840.071
21,0
65
0,0182
2,0949
0,0864
0,9136
849.963
73.415
4.036.535
14.492.063
17,1
70
0,0308
2,0790
0,1413
0,8587
776.547
109.716
3.562.258
10.455.527
13,5
75
0,0524
2,2384
0,2288
0,7712
666.831
152.593
2.912.749
6.893.269
10,3
80
0,0861
1,9454
0,3408
0,6592
514.238
175.254
2.035.866
3.980.520
7,7
85
0,1412
1,5985
0,4769
0,5231
338.984
161.664
1.145.025
1.944.654
5,7
90+
0,2218
3,0890
1,0000
0,0000
177.321
177.321
799.629
799.629
4,5
ŐĞ
Ŷ ŵ ǆ
Ŷ a ǆ
d ǆ Ğ ǆ 0
Ŷ Ƌ ǆ
Ŷ p ǆ
ů ǆ
Ŷ Ě ǆ
Ŷ L ǆ
0,0123
0,9877
1.000.000
12.284
987.716
74.091.496
74,1
0
0,0123
1
0,0008
1,1333
0,0033
0,9967
987.716
3.232
3.941.598
73.103.780
74,0
5
0,0004
1,7252
0,0021
0,9979
984.484
2.099
4.915.544
69.162.182
70,3
10
0,0003
2,1797
0,0017
0,9983
982.385
1.714
4.907.090
64.246.638
65,4
15
0,0007
2,1384
0,0035
0,9965
980.671
3.461
4.893.451
59.339.548
60,5
20
0,0007
2,0744
0,0036
0,9964
977.210
3.555
4.875.650
54.446.097
55,7
25
0,0008
1,9811
0,0038
0,9962
973.655
3.653
4.857.246
49.570.447
50,9
30
0,0009
1,9931
0,0043
0,9957
970.002
4.163
4.837.491
44.713.201
46,1
35
0,0012
2,2068
0,0058
0,9942
965.839
5.558
4.813.667
39.875.710
41,3
40
0,0018
2,0954
0,0091
0,9909
960.280
8.766
4.775.940
35.062.043
36,5
45
0,0031
2,0968
0,0156
0,9844
951.515
14.845
4.714.473
30.286.103
31,8
50
0,0055
2,0392
0,0270
0,9730
936.669
25.303
4.608.429
25.571.630
27,3
55
0,0092
1,9501
0,0449
0,9551
911.366
40.887
4.432.129
20.963.201
23,0
60
0,0153
2,0138
0,0732
0,9268
870.479
63.743
4.162.044
16.531.072
19,0
65
0,0244
2,0591
0,1138
0,8862
806.736
91.792
3.763.728
12.369.029
15,3
70
0,0397
2,0351
0,1775
0,8225
714.944
126.885
3.198.522
8.605.300
12,0
75
0,0639
2,2307
0,2713
0,7287
588.059
159.546
2.498.466
5.406.778
9,2
80
0,1044
1,8739
0,3935
0,6065
428.512
168.618
1.615.445
2.908.312
6,8
85
0,1664
1,5586
0,5290
0,4710
259.894
137.496
826.284
1.292.867
5,0
90+
0,2623
2,7725
1,0000
0,0000
122.397
122.397
466.583
466.583
3,8
23
(N7UNL\Hg]HWOHQPLú'|QHP<DúDP7DEORVX.DGÕQ
ŐĞ
24
Ŷ ŵ ǆ
Ŷ a ǆ
Ŷ Ě ǆ
ŶLǆ
d ǆ Ğ ǆ 0
Ŷ Ƌ ǆ
Ŷ p ǆ
ů ǆ
0,0111
0,9889
1.000.000
11.124
988.876
79.433.927
79,4
0
0,0111
1
0,0008
1,1197
0,0031
0,9969
988.876
3.067
3.946.668
78.445.052
79,3
5
0,0004
1,6520
0,0021
0,9979
985.808
2.027
4.922.254
74.498.384
75,6
10
0,0003
1,9364
0,0013
0,9987
983.781
1.326
4.914.844
69.576.130
70,7
15
0,0004
2,0667
0,0018
0,9982
982.455
1.752
4.907.139
64.661.287
65,8
20
0,0004
2,0187
0,0018
0,9982
980.704
1.717
4.898.401
59.754.148
60,9
25
0,0004
2,0998
0,0020
0,9980
978.987
1.989
4.889.167
54.855.747
56,0
30
0,0005
2,0107
0,0024
0,9976
976.998
2.325
4.878.040
49.966.579
51,1
35
0,0007
2,2329
0,0036
0,9964
974.673
3.529
4.863.601
45.088.539
46,3
40
0,0010
1,9983
0,0050
0,9950
971.144
4.833
4.841.213
40.224.938
41,4
45
0,0016
2,0034
0,0082
0,9918
966.311
7.895
4.807.895
35.383.726
36,6
50
0,0025
2,0047
0,0125
0,9875
958.416
11.968
4.756.230
30.575.831
31,9
55
0,0042
1,9259
0,0206
0,9794
946.448
19.506
4.672.273
25.819.600
27,3
60
0,0072
2,0855
0,0352
0,9648
926.941
32.650
4.539.550
21.147.327
22,8
65
0,0127
2,1558
0,0613
0,9387
894.292
54.817
4.315.545
16.607.778
18,6
70
0,0236
2,1384
0,1107
0,8893
839.475
92.943
3.931.414
12.292.233
14,6
75
0,0431
2,2475
0,1927
0,8073
746.532
143.891
3.336.601
8.360.819
11,2
80
0,0751
2,0048
0,3067
0,6933
602.642
184.836
2.459.590
5.024.218
8,3
85
0,1285
1,6246
0,4480
0,5520
417.806
187.196
1.457.176
2.564.628
6,1
90+
0,2082
3,2219
1,0000
0,0000
230.610
230.610
1.107.452
1.107.452
4,8
25
Mutlu KAYA, Emel ÇANKAYA
dHúLWOL ND\QDNODUGDQ HOGH HGLOHQ |Q ELOJLQLQ WDKPLQ VUHFLQH GDKLO HGLOPHVLQH RODQDN
VD÷OD\DQ%D\HVFL\DNODúÕPGDGDPRGHOVHoLPLSUREOHPLSRSOHUELUoDOÕúPDDODQÕRODUDN
RUWD\D oÕNPDNWDGÕU %D\HVFL PRGHOOHPH \DSDQ DUDúWÕUPDFÕODU EX DPDo LoLQ VÕNOÕNOa
%D\HVIDNW|UQNXOODQPD\ÕWHUFLKHWPLúOHUGLU.DVVYH5DIWHU\øNLUDNLSPRGHO
NÕ\DVODQPDN LVWHQGL÷LQGH NXOODQÕODQ %D\HV IDNW|U PRGHOOHU KDNNÕQGDNL |Q ELOJLGHQ
VRQELOJL\HJHoLúWHPRGHOOHUGHQELULOHKLQHRGGVODUGDQHNDGDUOÕNELUGH÷LúLPROGX÷Xnun
|OoVGU 0RGHOOHU KDNNÕQGD ELU |Q ELOJLQLQ ROPDPDVÕ GXUXPXQGD GDKL PRGHOOHULQ
GR÷UXOX÷XQD GDLU ELU RODVÕOÕN KHVDSODQPDVÕQD RODQDN WDQÕPDVÕ YH EXQODUÕQ
RUDQODQPDVÕQGDQ HOGH HGLOPHVL \RUXPX NROD\ WXWDUOÕ YH RWRPDWLN RODUDN PRGHOGH
“basitlik prensibinH´ VDKLS ROPDVÕ DoÕVÕQGDQ WHUFLK HGLOLU ELU \|QWHP ROPXúWXU $\UÕFD
IRUPO\OHoRNOXPRGHONDUúÕODúWÕUPDVÕGD\DSÕODELOPHNWHGLU0DUWLQR
2007).
%D\HV IDNW|UQQ KLSRWH] WHVWOHULQGHNL NXOODQÕPÕ -HIIUH\V WDUDIÕQGDQ PRGHO
VHoLPLQGHNLNXOODQÕPÕLVHLONRODUDN6FKZDU]YH5DIWHU\WDUDIÕQGDQRUWD\D
NRQXOPXúWXU0FFXOORJKYH5RVVLHúOHQLN|QVHOOHUL.DVVYH:DVVHUPDQ
isH UHIHUDQV |QVHOOHUL NXOODQDUDN PRGHO NDUúÕODúWÕUPDVÕQGD %D\HV IDNW|U X\JXODPDODUÕ
VXQPXúODUGÕU +DQ YH &DUOLQ %D\HV IDNW|U KHVDEÕ LoLQ Markov Zinciri Monte
Carlo (MCMC) metodunu, Sinharay ve Stern (2002), Bayes faktörünün önsel
GD÷ÕOÕPODUD GX\DUOÕOÕ÷ÕQÕ LQFHOHPLú ROXS %D\HV IDNW|U YH DOWHUQDWLI oHúLWOHULQLQ
NXOODQÕPÕ $UDXMR YH 3HUHLUD ¶QÕQ oDOÕúPDVÕQGD ED]Õ VLPODV\RQ X\JXODPDODUÕ\OD
J|VWHULOPLúWLU
0DUMLQDO RODELOLUOLNOHU RUDQÕ RODQ %D\HV IDNW|UQQ DQDOLWLN RODUDN KHVDSODQPDVÕQÕQ
mümkQ ROPDGÕ÷Õ GXUXPODUGD /DSODFH \DNODúÕP \|QWHPL |QHP örneklemesi, Gauss
NDUHOHúWLUPH ve 0&0& VLPODV\RQX HQ oRN NXOODQÕODQ \|QWHPOHU DUDVÕQGDGÕU
(Rosenkranz ve Raftery, 1994). %LOHúLN PRGHO-SDUDPHWUH X]D\Õ WDUDPD \|QWHPOHULQGHQ
Carlin ve Chib (1995) yöntHPLLVHPRGHOEHOLUVL]OL÷LQLELUPRGHOJ|VWHUJHVL\DUGÕPÕ\OD
EHOLUOH\LS 0&0& \|QWHPLQL NXOODQDUDN PRGHOOHULQ NDUúÕODúWÕUÕOPDVÕQGD %D\HV
IDNW|UQQ HOGH HGLOPHVLQL J|VWHUPLúWLU. $\UÕFD EHOOL úDUWODU DOWÕQGD, Schwarz (1978)
WDUDIÕQGDQ JHOLúWLULOHQ YH PRGHO SDUDPHWUHOHULQH |QVHO GD÷ÕOÕP EHOLUOHPH\L
JHUHNWLUPH\HQ%,&6FKZDU]|OoW \DUGÕPÕ\OD%D\HVIDNW|UQQNDEDFDELUWDKPLQL
elde edilebilmektedir (Kass ve Wasserman, 1995).
%D\HV IDNW|U LOH PRGHO NDUúÕODúWÕUPDVÕQÕQ PRGHO SDUDPHWUH VD\ÕVÕQÕ ELOPH\L
JHUHNWLUPHVLSDUDPHWUHVD\ÕVÕQÕQJ|]OHPOHUGHQVD\ÕFDID]ODROGX÷XNDUPDúÕNKL\HUDUúLN
PRGHOOHUGH KHVDSODQPDVÕQÕQ PPNQ ROPDPDVÕ JLEL SUREOHPOHU *HOIDQG YH 'H\
1994; Kass ve Raftery, 1995), AIC’ye benzer prHQVLSWHoDOÕúDQ\HQLELU|OoWQ6DSPD
Bilgi Ölçütü (',& DGÕ\OD JHOLúWLULOPHVLQH QHGHQ ROPXúWXU 6SLHJHOKDOWHU vd., 2002).
.XOODQÕPÕ VRQ G|QHPOHUGH ROGXNoD DUWDQ YH KDOHQ DNWLI DUDúWÕUPD NRQXVX RODQ EX
ölçütün özelliklerLYHX\JXODPDODUÕ(Da Silva vd., 2004), Mislevy (2006), Spiegelhalter
(2006a), Martino (2007), Asseburg (2007) ve Wilberg ve Bence (2008)’de bulunabilir.
%XoDOÕúPDGD\DUÕSDUDPHWULNPRGHOOHPHQLQ|]HOELUIRUPXRODQNXDQWDOPRGHOOHPHQLQ
OLWHUDWUGHNL ELU X\JXODPDVÕ VRQXFX RUWD\D oÕNDQ DOWHUQDWLI LNL PRGHOLQ NÕ\DVODPDVÕ
%D\HV IDNW|U YH ',& KHVDSODQDUDN \DSÕODFDNWÕU %D\HV IDNW|U KHVDEÕ NXOODQÕP
NROD\OÕ÷Õ DoÕVÕQGDQ &DUOLQ YH &KLE \|QWHPL LOH \DSÕOPÕúWÕU øNL \|QWHPLQ NXOODQÕP
amaçlarÕQGDNL IDUNOÕOÕNWDQ GROD\Õ Gelman vd., 2004; Spiegelhalter vd., 2002),
VRQXoODUÕQ NÕ\DVODQDELOLUOL÷LQL VD÷ODPDN DoÕVÕQGDQ PRGHOOHU LoLQ KHVDSODQDQ %,&
26
An Application of the Bayesian Model Selection by Using Bayes Factor,
Bayesian Information Criterion and Deviance Information Criterion
Bayes Faktörü, Bayesci Bilgi Ölçütü ve Sapma Bilgi Ölçütü
Kullanımıyla Bayesci Model Seçiminin Bir Uygulaması
GH÷HUOHUL GH oDOÕúPDGD VXQXOPXúWXU 7P LúOHPOHU LoLQ :LQEXJV SDNHW SURJUDPÕ
NXOODQÕOPÕúWÕU
2. YÖNTEM
2.1 Bayes Faktörü
6RQOXVD\ÕGDPRGHOOHUDUDVÕQGDQ³HQL\L´PRGHOLVHoPHSUREOHPL\OHLOJLOHQHOLP5DNLS
modeller kümesi olmak üzere gözlenen veri setinin,
(
) PRGHOLWDUDIÕQGDQ
vektörü ile
UHWLOGL÷L YDUVD\ÕOVÕQ +HU PRGHOLQ IDUNOÕ ELOLQPH\HQ SDUDPHWUHOHUL
gösterilsin.
Model
için,
, PRGHOLQ |QVHO RODVÕOÕ÷Õ
SDUDPHWUH YHNW|UQQ |QVHO GD÷ÕOÕPÕ YH
WDQÕPODQÕUVDELOHúLNGD÷ÕOÕP
, model
, olabilirlik fonksiyonu olarak
olur.
%LOHúLNGD÷ÕOÕPÕQ parametre vektörüne göre marjinallenmesi ve gözlenen veri üzerine
NRúXOODQGÕUÕOPDVÕVRQXFXKHUELUPRGHOHLOLúNLQVRQVDOPRGHORODVÕOÕNODUÕ,
ile hesaplanabilir (Martino, 2007). Burada
marjinal olabilirlik olup
úHNOLQGHKHVDSODQÕU
$OWHUQDWLI PRGHO VD\ÕVÕQÕQ LNL ROGX÷X GXUXPGD NÕ\DVODPD LúOHPL VRQVDO PRGHO
RODVÕOÕNODUÕQÕQRUDQODQPDVÕ\ROX\OD \DSÕOÕU%LUELULQHUDNLS
ve
modelleri için bu
oran;
GÕU%XUDGDPDUMLQDORODELOLUOLNOHURUDQÕ%D\HVIDNW|UGU%XHúLWOLN sözel olarak,
[Sonsal Odds] = [Önsel Odds]
[Bayes faktörü]
ile ifade edilirse,
27
úHNOLQGH RGGV RUDQÕ RODUDN WDQÕPODQDQ
RUDQÕQD ¶\H NÕ\DVOD
lehine Bayes
IDNW|U DGÕ YHULOLU 5DIWHU\ %LU EDúND GH\LúOH
|QVHOGHQ VRQVDOD JHoLúWH
odds’larda
OHKLQH QH NDGDUOÕN ELU GH÷LúLP ROGX÷XQXQ ELU |OoVGU Modellerin
WHUFLK HGLOHELOLUOL÷L NRQXVXQGD ELU |Q ELOJLQLQ ROPDPDVÕ GXUXPXQGD |QVHO RODVÕOÕNODU
RODUDN VHoLOLU YH GROD\ÕVÕ\OD %D\HV IDNW|U VRQVDO RGGV¶D HúLW
ROXU6RQVDOPRGHORODVÕOÕNODUÕQÕQWRSODPÕELUHHúLWROGX÷XQGDQformül 4,
úeklinde yeniden ifade edilebilir. Hesaplanan Bayes faktörünün yorumu, Tablo 1
NXOODQÕODUDN\DSÕOÕU
7DEOR%D\HVIDNW|UGH÷HULQLQPRGHOVHoLPLQGHNL\RUXPX-HIIUH\V
%D\HVIDNW|UGH÷HUL(
Yorum
Ok yönünde artan derecede
BFij 0.1
M j OHKLQHJoONDQÕW
0.1 BFij 0.3
M j lehine makul NDQÕW
0.3 BFij 1
M j OHKLQH]D\ÕI NDQÕW
1 BFij 3
M i OHKLQH]D\ÕI NDQÕW
3 BFij 10
M i OHKLQHPDNXONDQÕW M i tercih edilir
BFij ! 10
*
)
M j tercih edilir
M i OHKLQHJoONDQÕW
BFij : j. PRGHOHNÕ\DVODi. PRGHOOHKLQHKHVDSODQDQ%D\HVIDNW|UGH÷HUL
%D\HV IDNW|UQQ DQDOLWLN RODUDN KHVDSODQDELOPHVL HúLWOLN ¶GH YHULOHQ LQWHJUDO
LúOHPOHULQLQ \DSÕOPDVÕQÕ JHUHNWLULU 0RGHOOHUGH \HU DODQ ELOLQPH\HQ SDUDPHWUH VD\ÕVÕ
DUWWÕ÷ÕQGDEXLúOHPOHULJHUoHNOHúWLUPHNROGXNoDJoOHúLUKDWWDED]ÕGXUXPODUGDPPNQ
GH÷ildir. .DUPDúÕN LQWHJUDV\RQ WHNQLNOHUL X\JXODPDN \HULQH NRúXOOX GD÷ÕOÕPODUGDQ
|UQHNOHP oHNPHN \ROX\OD SDUDPHWUH WDKPLQOHULQLQ HOGH HGLOPHVLQL VD÷OD\DQ 0&0&
\|QWHPLPRGHOVHoLPLSUREOHPOHULQGHGHHWNLQELUúHNLOGHNXOODQÕOPDNWDGÕU*LONVYG
1996). Bu yaNODúÕPDVDKLS&DUOLQYH&KLE\|QWHPLELUVRQUDNLE|OPGHD\UÕQWÕOÕolarak
DoÕNODQPÕúWÕU
.Õ\DVODQPDN LVWHQHQ PRGHOOHULQ |Q RODVÕOÕNODUÕQÕQ HúLW ROGX÷X GXUXPODUGD %D\HV
IDNW|UQQ \DNODúÕN KHVDEÕ %,& \DUGÕPÕ\OD DúD÷ÕGDNL formül NXOODQÕODUDN GD
\DSÕODELOPHNWHGLU (Kass ve Wasserman, 1995):
28
ya da
Burada Q |UQHNJHQLúOL÷LS PRGHOGHNLSDUDPHWUHVD\ÕVÕYH Tˆ = parametre vektörünün
en çok olabilirlik tahmin edicisi olmak üzere, BIC i ve BIC j ;
formülü NXOODQÕODUDN, i. ve j. modeller için D\UÕD\UÕhesaplanPÕúGH÷HUOHUGLU Modeldeki
SDUDPHWUH VD\ÕVÕQÕQ DUWPDVÕ EX |OoWQ GH÷HULQLQ ORJQ RUDQÕQGD E\PHVLQH QHGHQ
RODFD÷ÕQGDQ %,& GH EDVLW \D GD ER\XWX NoN PRGHOOHUL WHUFLK HWPH H÷LOLPLQGHGLU
'R÷UXPRGHOLQROGX÷XYDUVD\ÕPÕ\ODHQNoN%,&GH÷HUli modelin en iyi model oOGX÷X
\|QQGH\RUXPODQPDVÕ|QHULOPHNWHGLUBurnham, 2004).
2.2 Carlin ve Chib Yöntemi
%D\HV IDNW|UQQ HOGH HGLOPHVLQGH JHUHNOL RODQ VRQVDO PRGHO RODVÕOÕNODUÕQÕQ
KHVDSODQPDVÕQD RODQDN VD÷OD\DQ \DNODúÕPODUGDQ ELUL 0&0& SUHQVLSOHULQL NXOODQDQ
CDUOLQ YH &KLE \|QWHPLGLU %X \DNODúÕPGD PRGHO EHOLUVL]OL÷L ELU SDUDPHWUH
RODUDN WDQÕPODQÕU YH DOGÕ÷Õ GH÷HUOHU ELU J|VWHUJH GH÷LúNHQL LOH EHOLUOHQLU %X SDUDPHWUH
\DUGÕPÕ\OD ELUELULQH UDNLS PRGHOLQ |UQHNOHP X]D\ODUÕ ELUELULQH ED÷ODQÕU YH E|\OHFH
0&0& |UQHNOHPHVLQLQ EX ELOHúLN PRGHO X]D\ÕQÕQ ELU IRUPXQGDQ |UQHNOHPOHU DOPDVÕ
VD÷ODQÕU
0RGHOSDUDPHWUHVLWDPVD\ÕGH÷HUOL ile ve tüm modellere ait YHNW|UOHULQLQELUOHúLPL
ROGX÷XQGD rakip model ioLQELOHúLNGD÷ÕOÕP
ise ile gösterilsin.
ROXU*|VWHUJHGH÷LúNHQ , hangi vektörünün LOHLOLúNLOLROGX÷XQXEHOLUOHGL÷LQGHQ
YHULOGL÷LQGH GL÷HU PRGHOOHULQ
YHNW|UQGHQ ED÷ÕPVÕ]GÕU $\UÕFD
modellere ait YHNW|UOHULQLQGHELUELULQGHQED÷ÕPVÕ]ROGX÷XYDUVD\ÕOÕU%XGXUXPGD
PRGHO DOWÕQGD
LoLQ NXOODQÕODQ GD÷ÕOÕPVDO IRUP %D\HVFL PRGHO
WDQÕPODPDVÕQGD |QHPOL ROPDGÕ÷ÕQGDQ EXQlar için “sözde (pseudo) önsel” seçimi
\DSÕODELOLU
%XUDGD V|]GH |QVHO JHUoHN ELU |QVHO ROPD\ÕS ELOHúLN PRGHO-SDUDPHWUH X]D\ÕQÕ
WDQÕPOD\DELOPHN LoLQ X\JXQ úHNLOGH VHoLOPLú ELU ED÷ODQWÕ GD÷ÕOÕPÕGÕU *LEEV
örneklemesini yürütmek için gerekli ’nin tam koúXOOXGD÷ÕOÕPÕ
RODUDNWDQÕPODQÕU%XUDGD
ROGX÷XQGDWPNRúXOOXRODVÕOÕNODUJHoHUOLolan model
’den,
ROGX÷XQGD ise sözde önselinden üretilir.
29
Sonuçta Gibbs örneklemesinin
RUDQWÕODQDUDNVRQVDOPRGHO
PRGHOL ]L\DUHW VD\ÕVÕ WP |UQHNOHPOHU VD\ÕVÕQD
RODVÕOÕNODUÕQÕQ tahminleri,
úHNOLQGHHOGHHGLOHELOLU%XUDGD
*LEEVLWHUDV\RQVD\ÕVÕGÕU&DUOLQYH&KLE
%X WDKPLQ GH÷HUOHULQGHQ HOGH HGLOHQ VRQVDO RGGV |QVHO RGGV LOH HúLWOLN 4’de ifade
HGLOGL÷L JLEL ELUOHúWLULOLUVH PRGHOOHULQ LNLúHUOL NÕ\DVODPDODUÕ DPDoOÕ %D\HV IDNW|U
hesaplanabilir.
2.3 Sapma Bilgi Ölçütü (DIC)
.ODVLN ELU PRGHOOHPH oDOÕúPDVÕQGD PRGHO NÕ\DVODPDVÕ \DSÕOPDN LVWHQGL÷LQGH YHULQLQ
PRGHOH X\XPXQX |OoHQ ELU VDSPD LVWDWLVWL÷L LOH PRGHOGHNL SDUDPHWUH VD\ÕVÕQÕQ
EHOLUOHGL÷L PRGHO NDUPDúÕNOÕ÷Õ DUDVÕQGD GHQJH NXUPD\D GD\DOÕ |OoWOHU NXOODQÕOÕU
%XQODUGDQ ED]ÕODUÕQÕQ PRGHOGHNL SDUDPHWUH VD\ÕVÕQÕ EHOLUOHPH\L JHUHNWLUPHVL YH
ED]ÕODUÕQÕQ LVH SDUDPHWUHOHUL J|]OHPOHUGHQ VD\ÕFD VWQ RODQ NDUPDúÕN KL\HUDUúLN
modellerde do÷rudDQ X\JXODQDPDPDVÕ *HOIDQG YG VRQ ]DPDQODUGD NXOODQÕPÕ
\D\JÕQ RODQ DOWHUQDWLI ELU %D\HVFL PRGHO VHoLP \öntemi Sapma Bilgi Ölçütünün
JHOLúWLULOPHVLQHVHEHSROPXúWXU
DICLNLELOHúHQGHQROXúPDNWDGÕU
DIC
8\XPL\LOL÷L
+
.DUPDúÕNOÕN
ĻĻ
Kuramsal bir model ile
Artan model parametre
|UQHNOHPYHULVLDUDVÕQGDNLVD\ÕVÕLoLQELUVÕQÕUODPDWHULPL
uyumun bir ölçüsü
=
DIC¶QLQ ELULQFL ELOHúHQL PRGHO \HWHUOLOL÷LQLQ ELU WDKPLQL ROXS 'HPSVWHU WDUDIÕQGDQNODVLNVDSPDLIDGHVLQHGD\DOÕGÕU
Burada
; model parametre vektörüQH GD\DOÕ RODELOLUOLN IRQNVL\RQX ROXS
ise
úHNOLQGHVDGHFHYHULQLQIRQNVL\RQXRODQVDELWELUWHULPGLU
%X WHULP PRGHO NDUúÕODúWÕUPDVÕQGD WP PRGHOOHU LoLQ DOÕQDUDN DúD÷ÕGD YHULOHQ
formüllerin WPQGHLKPDOHGLOPHVLVD÷ODQÕU&HOHX[YG
6DSPDQÕQVRQVDOGD÷ÕOÕPÕQÕQRUWDODPDVÕX\XPL\LOL÷i ölçüsü RODUDNNXOODQÕOÕU%XLIDGH
úHNOLQGHDGODQGÕUÕOÕUYH
ile gösterilir.
30
Burada C, burn-LQ \DNÕQVDPD SHUL\RGX oÕNDUÕOPÕú 0&0& VLPODV\RQODUÕQÕQ VD\ÕVÕ
olup
ise olabLOLUOLNIRQNVL\RQXQXQGR÷DOORJDULWPDVÕGÕU6SLHJHOKDOWHUYG
%X HúLWOLNWHQ J|UOG÷ JLEL
, Gibbs örneklemesinin bir iterasyonunun
VRQXQGDKHVDSODQPÕúORJ-RODELOLUOLNOHULQRUWDODPDVÕGÕU
NXOODQÕODUDN
DIC HúLWOL÷LQGH \HU DODQ LNLQFL ELOHúHQ LVH ¶QÕQ VRQVDO RUWDODPDVÕ
KHVDSODQDQVDSPD\DGD\DOÕGÕU
GL\HDGODQGÕUÕODQYH
ile gösterilen bu ifade,
úHNOLQGHWDQÕPOÕGÕU%LUEDúNDGH\LúOH ¶QÕQVRQVDORUWDODPDVÕQÕNXOODQDUDNKHVDSODQPÕú
log-olabilirliktir.
9HUL\OH HQ L\L X\XPX VD÷OD\DQ ELU PRGHOGH \HU DOPDVÕ JHUHNHQ HWNLQ SDUDPHWUH VD\ÕVÕ
RODUDN|OoOHQPRGHONDUPDúÕNOÕ÷Õ
ile gösterilir ve
HúLWOL÷L LOH HOGH HGLOLU %XUDGD
PRGHOGHNL SDUDPHWUH VD\ÕVÕ LoLQ ELU VÕQÕUODPD
WHULPLGLUYHPRGHOSDUDPHWUHVD\ÕVÕQÕQ\DNODúÕNGH÷HULQLYHULU
%XWDQÕPODUÕNXOODQDUDNDIC ölçütü,
úHNOLQGHLIDGHHGLOLU (Spiegelhalter vd., 2002).
/LWHUDWUGHPRGHOGHNLSDUDPHWUHVD\ÕVÕD\QÕNDOPDNNRúXOX\ODPRGHOSDUDPHWUHOHULQLQ
yeniden ifadelendirilmesi sonucu
¶QLQ GH÷HULQGH E\N GH÷LúLNOLNOHU RODELOHFH÷L
NRQXVXQGD\DSÕODQX\DUÕODUQHGHQL\OH6SLHJHOKDOWHUYH%XOO*HOPDQYG
\HULQH VDSPDQÕQ VRQVDO YDU\DQVÕQÕQ \DUÕVÕ úHNOLQGH WDQÕPOÕ
’nin
tarDIÕQGDQ
NXOODQÕPÕ|QHULOPLúWLU
Matematiksel ifadeyle,
olarak gösterilir.
9HUL\LUHWWL÷LYDUVD\ÕODQL\LELUPRGHOLQRODELOLUOL÷LQLQE\NGH÷HUOLROPDVÕ
¶QÕQ
GDKD NoN GH÷HUOHUH XODúPDVÕQÕ YH GROD\ÕVÕ\OD DIC¶QLQ NoN GH÷HUOL ROPDVÕQÕ
VD÷OD\DFDNWÕU 6RQXo RODUDN HQ NoN DIC GH÷HULQH VDKLS PRGHO UDkip modeller
DUDVÕQGDQHQL\LPRGHORODUDNVHoLOHELOLU
DIC GH÷HULGDLPDSR]LWLIROPD\ÕSQHJDWLIGH÷HUOHUGHYHUHELOPHNWHGLU6WDQGDUWVDSPDVÕ
küçük olan bir
RODELOLUOL÷LQLQVHEHSROGX÷XE|\OHVLGXUXPODUGDQHJDWLIGH÷HUOHU
GHGLNNDWHDOÕQDUDNHQNüçük DIC¶\HVDKLSPRGHOOHKLQGHVHoLP\DSÕOÕU6SLHJHOKDOWHU
2006b).
31
%D\HV IDNW|UQQ DNVLQH ',&¶QLQ PXWODN E\NO÷QQ PRGHO NDUúÕODúWÕUPDVÕQGD ELU
|QHPL \RNWXU gQHPOL RODQ VDGHFH ',& GH÷HUOHUL DUDVÕQGDNL IDUNÕQ PXWODN
E\NO÷GU (Spiegelhalter vd., PLQLPXP GH÷HUOL ',& GH÷HUL LOH DUDVÕQGD
¶GHQ GDKD D] IDUN RODQ PRGHOOHULQ HúLW GHUHFHGH L\L PRGHO RODUDN GLNNDWH DOÕQPDVÕ
JHUHNWL÷LQL í DUDVÕQGD IDUN GH÷HUOL PRGHOOHULQ LVH GDKD D] GHVWH÷H VDKLS PRGHOOHU
RODUDNGH÷HUOHQGLULOPHVLQL|QHUPLúWLU
3. BULGULAR
3.1 Kuantum Modelleme
øVWDWLVWLNVHOPRGHOOHPHQLQ|]HOELUIRUPXNXDQWDOPRGHOOHPHDGÕDOWÕQGDSHN oRNIDUNOÕ
GLVLSOLQGH NDUúÕPÕ]D oÕNPDNWDGÕU %LU \DUÕ-SDUDPHWULN PRGHOOHPH |UQH÷L RODQ NXDQWDO
modellemede; verilen herhangi bir örneklem verisinin bir temel birim olan kuantumun
WDPVD\Õ NDWODUÕ rastJHOH KDWD RODUDN LIDGH HGLOLS HGLOHPH\HFH÷L VRUXVXQD cevap
DUDQPDNWDGÕU %X VRUJXGDNL HQ |QHPOL QRNWD E|\OH ELU NXDQWXP GH÷HULQLQ YDU ROXS
ROPDGÕ÷ÕQÕQ |QFHGHQ ELOLQPHPHVLGLU $QFDN E|\OH ELU GH÷HU YDUVD YHULGHQ WDKPLQ
HGLOPHN\ROX\ODKHVDSODQPDOÕGÕU$FDU
örneklem verisi olmak üzere matematiksel olarak basit kuantum modeli,
olarak ifade edilir. Burada
= kuantum,
round (
úHNOLQGHKHVDSODQDQWDPVD\Õ
GH÷HUOHU URXQG E|OPQ HQ \DNÕQ WDPVD\Õ\D \XYDUODQPDVÕ
FLYDUÕQGDYH µ\HNÕ\DVODNoNGH÷HUOHUDODFDNWÕU
KDWD GH÷HUOHUL VÕIÕU
Kuantum modelleme ilk olaUDNøQJLOWHUHøVNRo\DYH*DOOHU¶GHSHNoRNVD\ÕGDEXOXQDQ
6WRQHKHQJH LVLPOL GDLUHVHO IRUPGDNL WDULKL WDú GLNLWOHULQ \DUÕoDSODUÕQÕQ (Thom, 1955)
WDUDIÕQGDQ |OoOPHVL VRQXFXQGD RUWD\D oÕNPÕúWÕU 0HJDOLWLN G|QHPL LQVDQODUÕQÕQ EX
\DSÕWODUÕ ROXúWXUXUNHQ temel bir ölçü birimi ve onun katlarÕQÕ NXOODQPÕú RODELOHFHNOHUL
VDYÕRUWD\DDWÕOPÕúYHNODVLNLVWDWLVWLNVHO\|QWHPOHUOHDQDOL]HGLOPHVLVRQXFXQGD = 5.44
LQo GH÷HULQLQ NXDQWXP \D GD OLWHUDWUGHNL QO DGÕ\OD ³0HJDOLWLN <DUG´ RODELOHFH÷LQH
GDLU|QHPOLEXOJXODUHOGHHGLOPLúWLU.HQGDOO).
%XSUREOHPLQ%D\HVFL\DNODúÕPODLONDQDOL]LLVH)UHHPDQWDUDIÕQGDQ\DSÕOPÕúYH
NXDQWXP LoLQ \DQÕ VÕUD YH GH÷HUOHULQLQ GH DOWHUQDWLI RODELOHFH÷L
oÕNDUVDPDVÕQGD EXOXQXOPXúWXU 0RGHUQ %D\HVFL \|QWHPOHUGHQ 0&0& NXOODQÕODUDN
problemin yeniden analizi (Acar, 2000) sonucunda elde edilen iki alternatif kuantum
WDKPLQGH÷HUL = 5.439 ve = 7.480, Freeman (¶ÕQVRQXoODUÕLOHWXWDUOÕGÕU
%X oDOÕúPDGD \XNDUÕGD |]HWOHGL÷LPL] oDOÕúPDODU VRQXFX HOGH HGLOPLú IDUNOÕ NXDQWXP
WDKPLQ GH÷HUOHULQLQ ROXúWXUGX÷X LNL EDVLW NXDQWXP PRGHOLQLQ NÕ\DVODQPDVÕ %D\HV
faktörü, BIC YH ',& NXOODQÕODUDN \DSÕOPÕúWÕU /LWHUDWUGH NXDQWDO ROPD |]HOOL÷LQH HQ
ID]OD VDKLS ROGX÷X VÕNOÕNOD DWIHGLOHQ JRRG ULQJV Q YHUL VHWL 7KRP EX
oDOÕúPDX\JXODPDVÕLoLQWHUFLKHGLOPLúWLU
32
Modeller matematiksel olarak,
Model 1 :
Model 2 :
úeklinde WDQÕPODQÕUBurada
,
,
,
,
,
olup
= 7.480 ve
= 5.439 ‘dur.
3.2 Kuantum Modellerinin Carlin vH&KLE<|QWHPL\OH.Õ\DVODQPDVÕ
)
Model göstergesi *LEEV|UQHNOHPHVLER\XQFD]LQFLULQUHWWL÷L GH÷HUOHULQL
NXOODQDUDN ELOHúLN PRGHOOHPH\L JHUoHNOHúWLUHQ ELU SDUDPHWUH ROXS NXDQWXP
PRGHOOHULQGHQ PRGHO YH PRGHO ¶\H LOLúNLQ ELOLQPH\HQ SDUDPHWUHOHU LVH VÕUDVÕ\OD
=(
ve =(
‘dir.
%DVLW NXDQWXP PRGHOOHPH oDOÕúPDODUÕQGD, VÕIÕUD \DNÕQ oRN NoN GH÷erlerin gerçek
NXDQWXPRODPD\DFD÷Õ YHD\UÕFDmaximum gözlemGHQGDKDE\NGH÷HUOLELU tahminin
PPNQ ROPDGÕ÷Õ YXUJXODQPÕú “Megalitik Yard” |UQH÷L verisinin klasik yöntemlerle
analizinde [q min =2, q max @ DUDOÕ÷ÕQGD WDUDPD \DSÕOarak bir tahmin sonucuna
XODúÕOPÕúWÕU (Kendal, 1974). Ancak GDKD JQFHO ELU oDOÕúPDGD (Acar, 2000), t=1/q
GH÷LúNHQLQLQ 8QLIRUP GD÷ÕOÕPD VDKLS ROGX÷X LVSDWODQPÕú YH WDUDPDQÕQ >W min =1/q max ,
t max =1/q min ] DUDOÕ÷ÕQGD \DSÕOPDVÕ önerilPLúWLU $\QÕ YHULQLQ NXOODQÕOGÕ÷Õ EX oDOÕúPDGD
bu parametre için |QVHO GD÷ÕOÕP RODUDN YH SDUDPHWUHOL 8QLIRUP GD÷ÕOÕP
VHoLOPLúWLU
Herhangi bir verinin kuantal ROGX÷XQGDQbahsedilebilmek için; model hata teriminin (H i )
eúitlik LOHWDQÕPOÕ GH÷HUOHULnin ±
µ\HNÕ\DVODNoNYH³´FLYDUÕQGD\R÷XQODúPDVÕ
ELU EDúND GH\LúOH KDWD YDU\DQVÕQÕQ (V2), WHVW HGLOHQ NXDQWXP GH÷HULQH NÕ\DVOD NoN
ROPDVÕ JHUHNPHNWHGLU. %X oDOÕúPDGD NXOODQÕODQ YHULnin, )UHHPDQ WDUDIÕQGDQ
%D\HVFL \DNODúÕPOD \DSÕODQ LON o|]POHQPHVLQGH KDWD YDU\DQVÕQÕQ ³´ den büyük
olmDVÕ GXUXPXQGD YHULGH YDU RODELOHFHN ELU NXDQWDO \DSÕQÕQ RUWD\D
2
oÕNDUÕODPD\DFD÷ÕQGDQ EDKVHGLOPLú ve bu parametre için F (2)
önVHO GD÷ÕOÕPÕ
kullanÕOPÕúWÕU'ROD\ÕVÕ\OD çaOÕúPDQÕQilerleyen bölümlerinde, V2 için bu önsel ile benzer
GH÷HUOHU UHWHELOHFHN úHNLOGH SDUDPHWUL]H HGLOPLú ELU *DPPD |QVHOL Winbugs model
formülasyonunda W = V parametresi için NXOODQÕOPÕúWÕU
Bu ELOJLOHU ÕúÕ÷ÕQGD, iki kuantal PRGHO NÕ\DVODPDVÕ LoLQ JHUHNOL |QVHO GD÷ÕOÕPODU
;
úHNOLQGHbelirlenPLúWLU Burada
ve
’dir.
33
%LOHúLN PRGHO WDQÕPODPDVÕQÕ VD÷ODPDN için gerekli sözde önseller D\QÕ YHUL VHWLnin
0&0& \DNODúÕPÕ LOH o|]POHQPHVLnden elde edilen (Acar, 2000) ve Tablo 2’de
sunulan parametre tahminlerinin %D\HVFLJYHQDUDOÕNODUÕQGDQ ID\GDODQÕOarak elde
HGLOPLúWLU.
Tablo 2. <DNÕQVDNOÕN WHVWOHULQL JHoHQ SDUDPHWUHOHULQ IDUNOÕ EDúODQJÕo GH÷HUOHUL LoLQ sonsal
tahminleri
=LQFLU3DUDPHWUH%DúODQJÕo'H÷HUL Sonsal Ortalama
1
2
%95 Güven $UDOÕ÷Õ
t
q
0.13414
--0.33128
---
0.134
7.480
1.371
0.586
[0.1323, 0.1352]
[7.395, 7.559]
[1.034, 1.707]
[0.3437, 0.9346]
t
q
0.16003
--0.67310
---
0.184
5.439
2.190
0.709
[0.183, 0.185]
[5.410, 5.470]
[0.949, 3.870]
[0.509, 1.030]
W
V
W
V
Not: *LEEV |UQHNOHPHVL EDúODQJÕo GH÷HUOHUL DWDPDVÕ sadece |QVHO GD÷ÕOÕP WDQÕPODQDQ PRGHO
parametreleri t ve W için gereklidir. Tahmin edilmek istenen q ve V SDUDPHWUHOHUL LoLQ ]LQFLU GH÷HUOHUL
t=1/q ve W= V–2 deterministik LOLúNLOHUNXOODQÕODUDNROXúWXUXOPDNWDGÕU
Bu durumda pseudo önselleri
Model 1 için (j=1),
Model 2 için (j=2),
,
(0.1323 , 0.1352) ,
(0.1829 , 0.1849) ,
(1 , 0.5)
(1 , 0.5)
RODUDNDOÕQPÕúWÕU
Formül 19’da verilen iki kuantum modelini, model göstergesi ( \DUGÕPÕ\OD
NÕ\DVODPDNYHELOHúLNPRGHOOHPH\LEHOLUOHPHNDPDFÕ\ODPRGHOOHULQ:LQEXJVSURJUDPÕ
içindeki grafiksel gösterimi ùHNLO ¶GH J|UOG÷ gibidir. Model parametresi M’nin
DODELOHFH÷L³1” ve “2” GH÷HUOHULQLUHWPHNDPDFÕ\OD:LQEXJVSURJUDPÕQGDWDQÕPOÕdcat
NDWHJRULNGD÷ÕOÕPÕNXOODQÕOPÕúWÕr. 'D÷ÕOÕP parametresi
; = 1, 2 ise modellerin önsel
RODVÕOÕNODUÕQDNDUúÕOÕNJHOip,
VD÷ODPDNWDGÕU ùHNLOGHNLkesikli çizgili oklar ise
GHWHUPLQLVWLNLOLúNLOHULJ|VWHUPHNDPDFÕ\ODNXOODQÕOPDNWDGÕU
34
M
dcat ( p[ ] )
t1
U (0.1 , 0.5)
q1
W1
W2
q2
1
Gamma (1 , 0.5)
Gamma (1 , 0.5)
1
t1
Oi1
t2
t2
U (0.1 , 0.5)
Oi 2
m1i
§Y ·
round ¨ i ¸
© q1 ¹
Model 1 ĂůƚŦŶĚĂ
t2 ~ U (0.1829, 0.1849)
W 2 ~ Ga (1, 0.5)
m2i
m1i q1
m2i q2
yi
N (Oij , W j
V j 2 )
i 1, 2,...,16
j 1, 2
§Y ·
round ¨ i ¸
© q2 ¹
Model 2 ĂůƚŦŶĚĂ
t1 ~ U (0.1323, 0.1352)
W 1 ~ Ga (1, 0.5)
ùHNLO&DUOLQYH&KLEyöntemi ile iki Kuantum modelinin kÕ\DVODQPDVÕQÕQgrafiksel gösterimi
(OGH HGLOHQ EX ELOJLOHU NXOODQÕODUDN &DUOLQ YH &KLE \|QWHPL LOH \D]ÕODQ SURJUDPÕQ
:LQEXJV3DNHW3URJUDPÕ¶QGDoDOÕúWÕUÕOPDVÕVRQXFXNXDQWXPPRGHOLQLQVRQVDOPRGHO
RODVÕOÕ÷ÕQÕQWDKPLQL
= 0.5005 olarak bulunur. (MC hata = 0.01259)
Bu durumda model ¶HNÕ\DVODPRGHOOHKLQH%D\HVIDNW|Uformül 4’den,
RODUDNKHVDSODQÕU%XKHVDSODPDGDPRGHOOHULQ|QVHORODVÕOÕNODUÕ]LQFLULQ GH÷HULQLHúLW
sD\ÕGD UHWPHVLQL VD÷ODPDN DPDFÕ\OD
= 0.999995 ve
= 0.000005 olarak
seçilPLúWLU.
Elde edilen
GH÷HULQLQ 7DEOR ¶H J|UH ¶GDQ ROGXNoD E\N ELU GH÷HU ROPDVÕ
VHEHEL\OHYHUL\LUHWWL÷LYDUVD\ÕODQHQL\LNXDQWXPPRGHOLQLQ
modeli; GROD\ÕVÕ\ODHQ
RODVÕNXDQWXPGH÷HULQLQROPDVÕ\|QQGHJoOELUNDQÕWÕQEXOXQGX÷XV|\OHQHELOLU
35
3.3 Kuantum Modellerinin BIC ve ',&.XOODQÕODUDN.Õ\DVODQPDVÕ
DIC KHVDEÕ (formül 16) için gerekli olan sapma;
(formül 12),
(formül 13),
(formül 14) ve
(formül NXOODQÕODUDN :LQEXJV SDNHW SURJUDPÕQGD \D]ÕODQ
SURJUDPODUÕQoDOÕúWÕUÕOPDVÕVRQXFXQGDKHULNLNXDQWXPPRGHOLLoLQD\UÕD\UÕKHVDSODQDQ
',& GH÷HUOHUL 7DEOR ¶WH J|UOG÷ JLEL HOGH HGLOPLúWLU Bilgi ölçütü temelinde
NÕ\DVODPD\DRODQDNWDQÕPDVÕDoÕVÕQGDQformül 7 LOHWDQÕPODQDQ%,&hesap GH÷HUOHULGH
D\QÕWDEORGDVXQXOPXúWXU
Tablo 3.
Model
ve
PRGHOOHULQLQ%,&YH',&KHVDSGH÷HUOHUL
Dbar
Dhat
p
pD
pV
DIC
BIC
57.662
55.706
2
1.957
1.930
3.8612
59.619
60.058
33.613
31.648
2
1.965
1.919
3.8377
35.578
36.018
Sonuçlar yorumlanGÕ÷ÕQGD HQNoN%,&YH',&GH÷HUOHULQLQD\QÕPRGHOLLúDUHWHWWL÷L,
PRGHOL ROGX÷X
ELU EDúND GH\LúOH veri setini en iyi temsil eden kuantum modelinin
görülmektedir.
0RGHO¶HNDUúÕPRGHOOHKLQH %D\HVIDNW|UGH÷HULLVHformül 6 NXOODQÕODUDN
úHNOLQGH EXOXQXU Bu GH÷HULQ, formül 22’de Carlin ve Chib yöntemiyle tam olarak
hesaplanan Bayes faktörünün kaba bir tahmini ROGX÷XQX KDWÕUODWDUDN
ROPDVÕVHEHEL\OH
PRGHOLQHNDUúÕ
modeli lehine NHVLQNDQÕWROGX÷X
VRQXFXQDELUNHUHGDKDXODúÕODELOLU
7$57,ù0$YH6218d
*QP] LVWDWLVWLNVHO PRGHOOHPH oDOÕúPDODUÕQGD QH NDGDU NRPSOHNV ROXUVD ROVXQ
hemen hemen her problemin PRGHOOHQPHVLQLQPPNQROGX÷XJ|UOPHNWHGLU0HYFXW
PRGHOOHU DUDVÕQGDQ HQ X\JXQ PRGHOL VHoPHN LoLQ ELU DUDo JHUHNVLQLPL JLWJLGH
DUWWÕ÷ÕQGDQKDOHQSHNoRNPHWRWJHOLúWLULOPHNWHGLU0HWRWODUÕQoHúLWOLOL÷LYHVD\ÕFDID]OD
ROPDVÕ SUREOHP oHúLWOLOL÷LQGHQ JHUHNOL DQDOLWLN LúOHPOHULQ ]RUOXN GHUHFHVLQGHQ \D GD
PRGHOLQROXúWXUXOPDDPDFÕQGDQND\QDNOÕGÕU
%X oDOÕúPDGD PRGHO VHoLPL SUREOHPLQH %D\HVFL \DNODúÕPODUGDQ %D\HV IDNW|U YH
6DSPD %LOJL gOoW ',& LONLQLQ OLWHUDWUGH VÕNOÕNOD NXOODQÕOPDVÕ YH GL÷HULQLQ VRQ
G|QHPOHUGH JHOLúWLULOPLú ROPDVÕ VHEHEL\OH WDQÕWÕOPÕú YH ELU X\JXODPDVÕ J|VWHULOPLúWLU
%D\HV IDNW|UQGHQ WDPDPHQ IDUNOÕ \DSÕGD YH SUHQVLSWH oDOÕúDQ ',&¶QLQ
NÕ\DVODQDELOLUOL÷LQL VD÷ODPDN DPDFÕ\OD %D\HVFL %LOJL gOoW %,& GH oDOÕúPD\D GDKLO
HGLOPLúWLU %D\HV IDNW|UQQ DQDOLWLN LúOHPOHUL LoLQ 0&0& VLPODV\RQXQD GD\DOÕ
36
&DUOLQ YH &KLE \|QWHPL X\JXODPD NROD\OÕ÷Õ DoÕVÕQGDQ WHUFLK HGLOPLúWLU %D\HV
IDNW|UQQ%,&\DUGÕPÕ\OD\DNODúÕNRODUDNKHVDEÕGDNÕVDFDDoÕNODQPÕúWÕU
øNLPRGHONDUúÕODúWÕUPDVÕQGDYH|]HOOLNOHKLSRWH]WHVWOHULQGHVÕNOÕNODWHUFLKHGLOHQ%D\HV
IDNW|UROGXNoDNXOODQÕúOÕROPDVÕQDUD÷PHQ|QVHOGD÷ÕOÕPVHoLPLQHYHPRGHOLQ\HQLGHQ
SDUDPHWUL]HHGLOPHVLQHNDUúÕGX\DUVÕ]GH÷LOGLUg]HOLNOHNÕ\DVODQDFDNPRGHOOHULQIDUNOÕ
ER\XWODUGDROPDVÕX\JXQVX] \DGDPX÷ODN|QVHOVHoLPL%D\HVIDNW|UQQNXOODQÕPÕQÕ
JHoHUVL]NÕOPDNWDGÕU+DOO$\UÕFDSDUDPHWUHVD\ÕVÕJ|]OHPVD\ÕVÕQGDQID]ODRODQ
KL\HUDUúLN PRGHOOHULQ NÕ\DVODQPDVÕQGD %D\HV IDNW|U X\JXODQDPDPDNWDGÕU %X
GH]DYDQWDMODUGDQ ED]ÕODUÕQÕQ VWHVLQGHQ JHOPHN LoLQ %D\HV IDNW|UQQ YHUVL\RQODUÕ
VRQVDO NÕVPL LoVHO YH NHVLUOL %D\HV IDNW|U JHOLúWLULOPLú ROPDVÕQD UD÷PHQ EXQODUÕQ
NXOODQÕPNRúXOODUÕGDKDOHQWDUWÕúPDNRQXVXGXU$UDXMRYH3HUHLUD
%,& IDUNOÕ VD\ÕGD SDUDPHWUH LoHUHQ IDUNOÕ PRGHOOHU DUDVÕQGDQ VHoLP \DSPDN LoLQ
NXOODQÕOPDVÕ |QHULOHQ |OoWOHUGHQ ELULGLU %XUQKDP $UWDQ GH÷LúNHQ VD\ÕVÕQÕQ
X\XPGD VD÷ODGÕ÷Õ L\LOHúPH\L DUWDQ SDUDPHWUH VD\ÕVÕ LOH ORJQ RUDQÕQGD
FH]DODQGÕUGÕ÷ÕQGDQ PRGHOGH EDVLWOLN SUHQVLELQH GD\DQÕU gUQHNOHP E\NO÷
DUWWÕ÷ÕQGDWXWDUOÕELUPHWRWWXU%LUEDúNDGH\LúOHH÷HUDOWHUQDWLIPRGHOOHUDUDVÕQGDGR÷UX
PRGHOYDULVHRQXEXOXU3DUDPHWUHOHULoLQ|QVHOGD÷ÕOÕPEHOLUWPH\LJHUHNWLUPHGL÷LQGHQ
|QELOJL\HWHUVL]OL÷LQGHWHUFLKHGLOHELOPHNWHGLU
Son dönemlerde, özellikle kompOHNV KL\HUDUúLN PRGHOOHPHGH NDUúÕODúÕODQ SUREOHPOHUL
JLGHUPHN DPDFÕ\OD JHOLúWLULOHQ ',& |OoW %D\HV IDNW|UQGHQ WDPDPHQ IDUNOÕ ELU
\DSÕGD ROXS $,& YH %,& LOH EHQ]HU SUHQVLSWH oDOÕúPDNWDGÕU %X \DNODúÕP YHUL\H
X\JXQOX÷XWHVW HGLOHQPRGHOGH \HUDOPDVÕ JHUHNHQHWNLQSDUDPHWUHVD\ÕVÕQÕQWDKPLQLQL
VD÷ODPDVÕ 0&0& \|QWHPL LOH NROD\OÕNOD KHVDSODQDELOPHVL |QVHO VHoLPLQH GX\DUOÕ
ROPDPDVÕ JLEL DYDQWDMODUD VDKLS ROPDVÕQD UD÷PHQ KHU PRGHOOHPH WUQH
X\JXODQDPDPDVÕ YH E\N |UQHNOHPOHUGH JHUHNVL] VD\ÕGD SDUDPHWUH\H VDKLS E\N
modelleri tercih etmesi gibi dezavantajlara da sahiptir.
%LU PRGHOOHPH oDOÕúPDVÕQGD KDQJL PHWRW \D GD PHWRWODUOD PRGHO VHoLPL \DSÕODFD÷Õ
PRGHOOHULQ ROXúWXUXOPD DPDFÕQD J|UH EHOLUOHQPHOLGLU (÷HU DPDo |QJ|U \DSPDN LVH
Bayes faktörü ve DIC (ya da BIC) NXOODQÕPÕ\OD PRGHO NÕ\DVODPDVÕQÕQ D\QÕ VRQXFD
LúDUHW HWPHVL EHNOHQPHPHOLGLU dQN LNLVL GH IDUNOÕ DPDFD KL]PHW HWPHNWHGLU %D\HV
IDNW|UVDGHFHELUPRGHOLQGR÷UXROGX÷XYDUVD\ÕPÕ\ODDOWHUQDWLIPRGHOOHUDUDVÕQGDQEX
GR÷UX PRGHOLQ VHoLOPHVL DPDFÕ\OD NXOODQÕOÕU ',& NXOODQÕPÕQGD LVH DPDo GL÷HU ELOJL
ölçütleri (AIC, BIC, AIC c YV NXOODQÕPODUÕQGD ROGX÷X JLEL GR÷UX PRGHO ROPDVD ELOH
YHUL\OH HQ X\XPOX PRGHOL VHoPHNWLU g]HOOLNOH KL\HUDUúLN PRGHOOHPHGH |QJ|UQQ
KL\HUDUúLQLQKDQJLVHYL\HVLLoLQ\DSÕODFD÷ÕQDED÷OÕRODUDNPRGHOVHoLP|OoWWHUFLKLQLQ
\DSÕOPDVÕJHUHNWL÷L|QHPOHYXUJXODQPÕúWÕU6SLHJHOKDOWHU6b).
øVWDWLVWLNVHO PRGHOOHPH DODQÕQGD IDUNOÕ ELU \HUH VDKLS NXDQWDO PRGHOOHPHGH LVH DPDo
J|]OHPOHQHQ YHULGH ROPDVÕ PXKWHPHO ELU \DSÕQÕQ NXDQWDO ROPD RUWD\D oÕNDUÕOPDVÕQD
\|QHOLN WDQÕPOD\ÕFÕ ELU PRGHO ROXúWXUPDNWÕU /LWHUDWUGH ³0HJDOLWLN <DUG´ DGÕ\OD
37
ELOLQHQ ELU |UQH÷LQGH DQWLN G|QHP LQVDQODUÕQÕQ \DSÕ LQúDDWODUÕQGD JQP] PHWULN
VLVWHPLQGHNLJLEL\HUOHúLNELUWHPHOX]XQOXN|OoVNXDQWXPNXOODQÕSNXOODQPDGÕNODUÕ
VRUXVXQD FHYDS DUDQPDNWDGÕU %XUDGD PRGHO SDUDPHWUHVL NXDQWXPXQ ELUGHQ ID]OD
WDKPLQL PPNQ ROGX÷XQGDQ SDUDPHWUHQLQ KDQJL GH÷HULQLQ YHULQLQ TXDQWDO \DSÕVÕQÕ
WDQÕPOD\DQ HQ L\LHQ GR÷UX GH÷HU ROGX÷X WHVSLWL %ayes faktörü, DIC ve BIC ile
\DSÕODELOLU %X oDOÕúPDGD ³0HJDOLWLN <DUG´ SUREOHPLQLQ ELU YHUL VHWLQH o \|QWHPLQ
X\JXODQPDVÕ VRQXFX 0HJDOLW \DSÕ LQúDDWÕQGD NXOODQÕOPÕú RODELOHFHN WHPHO |Oo ELULPL
için ¶HNÕ\DVOD LQoGH÷HULQLGHVWHNOHU\|QGHRUWDNEXOJX\DXODúÕOPÕúWÕU
%X oDOÕúPD LoLQ VHoLOHQ \|QWHPOHUGHQ D\QÕ GR÷UXOWXGD VRQXo HOGH HGLOPLúWLU $QFDN
PRGHO VHoLP |OoWOHULQLQ IDUNOÕ VRQXoODU YHUGL÷L ELU EDúND X\JXODPD oDOÕúPDVÕQGD
PRGHOOHPH DODQÕna YH DPDFÕQD |UQHNOHP JHQLúOL÷LQH PRGHOGHNL SDUDPHWUH VD\ÕVÕQD
PRGHOOHULQ LoLoH JHoPLú ROXS ROPDGÕ÷ÕQD YV EDNÕODUDN VRQXoODU \RUXPODQPDOÕGÕU
gUQH÷LQ E\N |UQHNOHPOHU LoLQ PRGHOLQ JHUHNVL] GHUHFHGH E\N ROPD LKWLPDOLQH
|QOHP RODUDN %,& WHUFLK HGLOLUNHQ NoN |UQHNOHPOHUGH $,& SUHQVLELQGH oDOÕúDQ ',&
daha iyi sonuçlar vermektedir. Bayes Faktörü, BIC, AIC ve DIC model seçim
\|QWHPOHULQLQ VLPODV\RQD GD\DOÕ ELU J|]GHQ JHoLUPH oDOÕúPDVÕ YH GDKD D\UÕQWÕOÕ
NÕ\DVODPDODUÕ:DUG¶GDEXOXQDELOLU
5. KAYNAKLAR
Acar, E., 2000. Extensions of Quantal Problems. PhD. Thesis. University of Sheffield
Department of Probability and Statistics, UK. (in English).
Araujo, M. I., Pereira, B.B., 2007. A Comparison of Bayes factors for Separated
Models: Some Simulation Results. Communications in Statistics, Simulation and
Computation, 36, 297-309.
Asseburg, C., 2007. An Introduction to Using WinBUGS for Cost-Effectiveness
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R.L.Launer & G. N. Wilkinson, (Eds.) Robustness in Statistics New York: Academic
Press, 201-236.
Burnham, K. P. 2004. Multimodel Inference: Understanding AIC and BIC in Model
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Carlin, B., Chib, S., 1995. Bayesian Model Choice via Markov Chain Monte Carlo
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Celeux, G., Forbes, F., Robert, C. P., Titterington, D. M., 2006. Deviance Information
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Bayesian Approach. Biostatistics, 1-17.
38
39
Raftery, A. E., 1995. Bayesian Model Selection in Social Research. Sociological
Methodology, Marsden, P. V. Cambridge, Mass., Blackwells, 111-196.
Rosenkranz, S. L., Raftery, A. E., 1994. Covariate Selection in Hierarchical Models of
Hospital Admission Counts: A Bayes Factor Approach No.268 Department of Statistics,
GN 22 University of Washington Seattle, Washington, 98195 USA.
Schwarz, G., 1978. Estimating the Dimension of a Model. Annals of Statistics, 6(2),
461-464.
Sinharay, S., Stern, H. S., 2002. On the Sensitivity of Bayes Factors to the Prior
Distributions. The American Statistician, 56(3), 196-201.
Spiegelhalter, D. J., Bull, K., 1997. Tutorial in Biostatistics Survival Analysis in
Observational Studies. Statistics in Medicine, 16(9), 1041-1074.
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., Van der Linde, A., 2002. Bayesian
Measures of Model Complexity and Fit. Journal of the Royal Statistical Society, Series
B, Methodological, 64(4), 583-616.
Spiegelhalter, D. J., 2006a. Two Brief Topics on Modelling with WinBUGS. MRC
Biostatistics Unit, Cambridge.
Spiegelhalter, D. J., 2006b. Some DIC Slides. MRC Biostatistics Unit, Cambridge.
Thom, A., 1955. A Statistical Examination of the Megalithic Sites in Britain. J. R.
Statist. Soc. A., 118, 275-295.
Thom, A., 1967. Megalithic Sites in Britain. Clarendon Press, Oxford.
8FDO 0 ù (NRQRPHWULN 0RGHO 6HoLP .ULWHUOHUL h]HULQH .ÕVD %LU øQFHOHPH
&høNWLVDGLYHøGDUL%LOLPOHU'HUJLVL-57.
Ward, E. J., 2008. A Review and Comparison of Four Commonly Used Bayesian and
Maximum Likelihood Selection Tools. Ecological Modelling, 211, 1-10.
Wilberg, M. J., Bence, J. R., 2008. Performance of Deviance Information Criterion
Model Selection in Statistical Catch-at-age Analysis. Fisheries Research, 93, 212-221.
40
AN APPLICATION OF THE BAYESIAN MODEL SELECTION BY
USING BAYES FACTOR, BAYESIAN INFORMATION
CRITERION AND DEVIANCE INFORMATION CRITERION
ABSTRACT
In statistical modelling studies, due to the advanced technology and
methodological developments, it is possible to construct alternative models
assumed to generate the data. Therefore, the process of choosing “the best
model” among available competing models appears to be one of the crucial
steps that has to be included in the modelling process. In this study, Bayes
factor, which is a preferred Bayesian approach to the solution of statistical
model selection problem, is introduced. For the cases when analytical
computation of Bayes factor is not possible, in addition to Bayesian
Information Criterion (BIC), Carlin and Chib method based on Markov
Chain Monte Carlo (MCMC) simulation is explained. Besides, a frequently
used criteria in the recent years of model selection applications, namely
Deviance Information Criterion (DIC), which has a completely different
working principle than Bayes factor, is described in detail. Two models
appeared in the literature as a result of an application of quantal modelling,
which is an example of a semi-parametric modelling, are compared by
means of Bayes factor, BIC and DIC.
Keywords: Bayes factor, Carlin and Chib method, DIC, MCMC, BIC.
41
Cilt: 10 Sayı: 02 Kategori: 02 Sayfa: 42-55 ISSN No: 1303-6319 Volume: 10 Number: 02 Category: 02 Page: 42-55
42
Investigation of Factors Effective on Credit Card Ownership of
Marmara University Students by Bayesian Logistic Regression
Marmara Üniversitesi Öğrencilerinin Kredi Kartı Sahibi Olmalarını
Etkileyen Faktörlerin Bayesci Lojistik Regresyon Yardımıyla İncelenmesi
DUDVÕQGDNLVHEHS-VRQXoLOLúNLVLQLLQFHOHUNHQLojistik UHJUHV\RQDQDOL]LNXOODQÕOÕU(Oktay
vd., 2009). Lojistik regresyon modeli ilk olarak \ÕOÕQGD %HUNVRQ WDUDIÕQGDQ
NXOODQÕOPÕúWÕU &R[ EX PRGHOL J|]GHQ JHoLUHUHN oHúLWOL X\JXODPDODUÕQÕ \DSPÕú,
|]HWJHOLúPHOHULVHLON$QGHUson (1979, 1983) WDUDIÕQGDQYHULOPLúWLU3UHJLERQLNL
grup Lojistik modelde etkin (influential) D\NÕUÕ RXWOLHU J|]OHPOHUL YH EHOLUOHPH
ölçütlerini, Lesaffre (1986), Lesaffre ve Albert (1989) ise çoklu grup Lojistik
PRGHOOHUGHHWNLQYHD\NÕUÕJ|]OHPOHUOHEHOLUOHPH|OoOHULQLLQFHOHPLúOHUGLU
Lojistik UHJUHV\RQ PRGHOOHULQLQ \D\JÕQ ELU ELoLPGH NXOODQÕOÕU KDOH JHOPHVL NDWVD\Õ
WDKPLQ \|QWHPOHULQLQ JHOLúWLULOPHVL YH Lojistik UHJUHV\RQ PRGHOOHULQLQ GDKD D\UÕQWÕOÕ
LQFHOHQPHVLQHVHEHSROPXúWXU&RUQILHOGLojistik UHJUHV\RQGDNLNDWVD\ÕWDKPLQ
LúOHPlerinde diskriminanW IRQNVL\RQX \DNODúÕPÕQÕ LON NH] NXOODQDUDN SRSOHU hale
JHWLUPLúWLU 5REHUW YG. (1987) Lojistik regresyonda standart Ki-kare, olabilirlik oran
(G2), “pseudo” en çok olabilirlik tahminleri, uyum mükemPHOOL÷L YH KLSRWH] WHVWOHUL
]HULQH DUDúWÕUPDODU \DSPÕúODUGÕU 'XII\ Lojistik regresyonda hata terimlerinin
GD÷ÕOÕúÕ YH SDUDPHWUH GH÷HUOHULQLQ JHUoHN GH÷HUOHUH \DNODúPDVÕQÕ LQFHOHPLúWLU %DúDUÕU
NOLQLN YHULOHUGH oRN GH÷LúNHQOL Lojistik regres\RQ DQDOL]L YH D\UÕPVDPD VRUXQX
]HULQHoDOÕúPÕúWÕU
6RQ\ÕOODUGDNODVLN\|QWHPOHUGHNLNÕVÕWODPDODU%D\HVFL\DNODúÕPDRODQLOJL\LDUWWÕUPÕúWÕU
%D\HVFL \DNODúÕPVXEMHNWLIGúQFHQLQWHPHOWDúÕRODUDNNDEXOHGLOHQ%D\HVWHRUHPLQH
GD\DQDUDNJHOLúWLULOPLúWLUYH\DNODúÕPDJ|UHSDUDPHWUHOHUNODVLN\DNODúÕPGDNLJLELVDELW
RODUDNGH÷LORODVÕOÕ÷DED÷OÕRODUDNWDQÕPODQÕU'ROD\ÕVÕLOHKHUELUSDUDPHWUH\HLOLúNLQELU
GD÷ÕOÕP V|] NRQXVXGXU %X RODVÕOÕNODU ³NDQDDW GHUHFHVL GHJUHHV RI EHOLHI´ RODUDN
WDQÕPODPDNWDGÕU %LU EDúND GH\LúOH SDUDPHWUHOHU UDVJHOH GH÷LúNHQ RODUDN HOH
DOÕQPDNWDGÕU ³%D\HVFL´ NHOLPHVL GH SDUDPHWUH WDKPLQOHUL LoLQ \DSÕODQ oÕNDUVDPDODUGD
7KRPDV%D\HV¶LQWHRUHPLQGHQID\GDODQÕOPDVÕQGDQGR÷PXúWXU(Ibrahim vd., 2001).
%D\HVFL \DNODúÕP NDUPDúÕN YHUL\L PRGHOOHPHGH |QVHO SULRU ELOJL\H EDúYXUPD
HVQHNOL÷L QHGHQL\OH NODVLN \|QWHPOHUH J|UH ROGXNoD DYDQWDMOÕGÕU (Ibrahim vd., 2001;
Wong vdgQVHOELOJLQLQHOGHHGLOPHVL%D\HVFLoÕNDUVDPDGD|QHPOLUROR\QDU
gQVHO GD÷ÕOÕPÕQ VRQXoODUÕ QH NDGDU GH÷LúWLUHFH÷L PRGHO VHoLP NULWHUOHUL LOH WHVSLW
HGLOPHOLGLU gQVHO GD÷ÕOÕPODU DoÕNOD\ÕFÕ RODQ LQIRUPDWLYH YH DoÕNOD\ÕFÕ ROPD\DQ
QRQLQIRUPDWLYH ROPDN ]HUH LNL WHPHO JUXED D\UÕOÕU $oÕNOD\ÕFÕ |QVHO ELOJLOHU, daha
|QFH \DSÕOPÕú oDOÕúPDODUGDQ HOGH HGLOHQ ELOJLOHU JHoPLú GHQH\LPOHU olarak
EHOLUOHQLUNHQ DoÕNOD\ÕFÕ ROPD\DQ |QVHO ELOJLOHU LVH SDUDPHWUHQLQ WDQÕPOÕ ROGX÷X DUDOÕN
ELOJLVL GÕúÕQGD KHUKDQJL ELU ELOJLQLQ ROPDPDVÕ RODUDN WDQÕPODQPDNWDGÕU %D\HVFL
\DNODúÕP |QVHO ELOJLOHU ÕúÕ÷ÕQGD J|]OHQHQ YHULQLQ VXEMHNWLI \RUXPXQD GD\DQÕU YH
EXUDGDQ KDUHNHWOH HOGH HGLOHQ \HQL ELOJLQLQ ELOHúLPL RODQ VRQVDO GD÷ÕOÕPOD DoÕNODQÕU
(Congdon, 2006; Wong vd., 2005). Hsu ve Leonard (1995) Lojistik regresyon
IRQNVL\RQODUÕQGD %D\HV WDKPLQOHULQLQ HOGH HGLOPHVL LúOHPOHUL ]HULQH oDOÕúPÕúODU YH
Bayesci Lojistik UHJUHV\RQGD 0RQWH &DUOR G|QúPQQ NXOODQÕODELOHFH÷LQL
J|VWHUPLúOHUGLU
dDOÕúPDQÕQ DPDFÕ %D\HV WDQÕPÕQGDQ \ROD oÕNDUDN YDU olan bilginin yeni bilgi ile
güncellenmesini göstermektir. %X DPDoOD \ÕOÕQGD *D]LRVPDQSDúD YH øQönü
QLYHUVLWH |÷UHQFLOHULQLQ NUHGL NDUWÕ VDKLSOL÷LQL HWNLOH\HQ IDNW|UOHUL EHOLUOHPH
oDOÕúPDVÕQGD X\JXODQDQ VRUXOXN DQNHW, Marmara üniversitesinde okuyan 200
|÷UHQFL\H X\JXODQPÕúWÕU 0DUPDUDQLYHUVLWHVL|÷UHQFLOHULQGHQGHUOHQHQ YHULOHU önsel
bilgi olDUDN NXOODQÕODQ *D]LRVPDQSDúD YH øQ|Q QLYHUVLWH |÷UHQFLOHULQLQ YHULOHUL LOH
43
Esin AVCI
ELUOHúWLULOHUHN%D\HVFLLojistik UHJUHV\RQX\JXODQPÕúYHNUHGLNDUWÕ VDKLSOL÷LQHHWNLHGHQ
IDNW|UOHUEHOLUOHQPLúWLU
2. YÖNTEM
2.1 Lojistik regresyon
øVWDWLVWLNVHO PRGHOOHULQ NXOODQÕOGÕ÷Õ ELUoRN ELOLPVHO DUDúWÕUPDGD VRQXoODUÕQ DQDOL]
edilmesinde en çok lineer olmayan PRGHOOHU NXOODQÕOPDNWDGÕU. Lojistik regresyon
modeli lineer-olmayan modellerden en önemlilerindendir. Lojistik regresyon modeli
ED÷ÕPOÕ GH÷LúNHQLQ NHVLNOL ED÷ÕPVÕ] GH÷LúNHQOHULQ LVH NHVLNOL YH\D VUHNOL ROPDVÕ
GXUXPXQGD ED÷ÕPOÕ GH÷LúNHQOH ED÷ÕPVÕ] GH÷LúNHQOHU DUVÕQGDNL VHEHS-VRQXo LOLúNLVLQLQ
RUWD\DNRQXOPDVÕQGDNXOODQÕOPDNWDGÕU (Eskandari ve Meshkani, 2006).
Lojistik regresyon analizi, diskriminant analizi ve çRNOX UHJUHV\RQ DQDOL]LQGHQ IDUNOÕ
RODUDN ED÷ÕPVÕ] GH÷LúNHQOHULQ GD÷ÕOÕPÕQD LOLúNLQ DUDúWÕUPDFÕODUFD NDUúÕODQPDVÕ JHUHNHQ
YDUVD\ÕPODU JHUHNWLUPH] 7DEDFKQLFN YH )LGHOO %LU EDúND GH\LúOH ED÷ÕPVÕ]
GH÷LúNHQOHULQQRUPDOGD÷ÕOPDVÕGR÷UXVDOOÕNYHYDU\DQV-NRYDU\DQVPDWULVOHULQLQHúLWOL÷L
gibi YDUVD\ÕPODUÕQ NDUúÕODQPDVÕJHUHNPH]'ROD\ÕVÕ\ODGDLojistik UHJUHV\RQXQGL÷HULNL
WHNQLNWHQ oRN GDKD HVQHN ROGX÷X LIDGH HGLOHELOLU Lojistik UHJUHV\RQXQ \DQVÕ] YH
VDSPDVÕ] LVWDWLVWLNOHU RUWD\D NR\PDVÕ LoLQ E\N |UQHNOHPOHU JHUHNWLUGL÷L
ELOGLULOPHNWHGLU g]HOOLNOH ED÷ÕPOÕ GH÷LúNHQLQ LNLGHQ ID]OD NDWHJRULVLQLQ ROGX÷X
GXUXPODUGD JHoHUOL ELU KLSRWH] WHVWL LoLQ KHU ED÷ÕPVÕ] GH÷LúNHQGH HQ D] NLúLOLN ELU
JUXS E\NO÷QH LKWL\Do YDUGÕU %D]Õ ND\QDNODUGD EX VD\ÕQÕQ KHU ED÷ÕPVÕ] GH÷LúNHQ
LoLQ PLQLPXP WRSODPGD PLQLPXP ROPDVÕ JHUHNWL÷L YXUJXODQPDNWDGÕU 'L÷HU
\DQGDQ |UQHNOHP E\NONOHULQLQ D\QÕ ROPDVÕ GXUXPXQGD ED÷ÕPOÕ GH÷LúNHQLQ KHU ELU
NDWHJRULVLQGH ED÷ÕPVÕ] GH÷LúNHQOHULQ oRN GH÷LúNHQOL QRUPDOOL÷H VDKLS ROPDVÕ Ker bir
NDWHJRUL LoLQ YDU\DQV YH NRYDU\DQVODUÕQ HúLWOL÷L YDUVD\ÕPODUÕQÕQ NDUúÕODQPDVÕ
GXUXPXQGD GDKD |QFH GH GH÷LQLOGL÷L JLEL GLVNULPLQDQW DQDOL]L Lojistik regresyon
analizine tercih edilmelidir. Bununla birlikte, Lojistik UHJUHV\RQ DQDOL]L LOH \DSÕODQ
çö]POHPHGHQ HOGH HGLOHQ PDWHPDWLNVHO PRGHOLQ \RUXPODQPDVÕQÕQ GDKD NROD\
ROGX÷XQX EHOLUWPHNWH \DUDU YDUGÕU $NNXú YH dHOLN *ULPP YH <DUQROG .DOD\FÕLeech vd., 2005; Poulsen ve French, 2008; Tabachnick ve Fidell, 1996;
7DWOÕGLO
Lojistik UHJUHV\RQ DQDOL]L DGÕQÕ ED÷ÕPOÕ GH÷LúNHQH X\JXODQDQ ORJLW G|QúPQden
ORJLW WUDQVIRUPDWLRQ DOPDNWDGÕU %X GXUXP D\QÕ ]DPDQGD KHP NHVWLULP KHP GH
\RUXPODPDVUHFLQGHED]ÕIDUNOÕOÕNODUDQHGHQROXU+DLU vd., 2006). Lojistik regresyon
analizi, bD÷ÕPOÕ GH÷LúNHQLQ |OoOG÷ |OoHN WUQH YH ED÷ÕPOÕ GH÷LúNHQLQ VHoHQHN
VD\ÕVÕQD J|UH oH D\UÕOPDNWDGÕU (÷HU ED÷ÕPOÕ GH÷LúNHQ LNL VHoHQHNOL ELU NDWHJRULN
GH÷LúNHQ LVH ³øNLOL Lojistik Regresyon Analizi (Binary Logistic Regression Analysis)”
DGÕQÕDOÕUgUQH÷LQELUDNDGHPLNSURJUDPÕELWLUPHGXUXPXQD J|UH|÷UHQFLOHULQEDúDUÕOÕ
YHEDúDUÕVÕ]RODUDNQLWHOHQGLULOPHVLGXUXPXQGDLNLOLLojistik UHJUHV\RQX\JXODQÕU(÷HU
ED÷ÕPOÕ GH÷LúNHQ LNLGHQ oRN NDWHJRULOL G]H\OL VÕQÕIODPDOÕ ELU GH÷LúNHQ LVH ³dRN
Kategorili/Düze\OL øVLPVHO Lojistik Regresyon Analizi (Multinominal Logistic
5HJUHVVLRQ $QDO\VLV´ DGÕQÕ DOÕU gUQH÷LQ o IDUNOÕ DNDGHPLN SURJUDPGD |÷UHQLP
J|UPHNWH RODQ |÷UHQFLOHUGHQ ROXúDQ ELU ED÷ÕPOÕ GH÷LúNHQLQ ROPDVÕ GXUXPXQGD oRN
düzeyli isimsel Lojistik regresyon X\JXODQÕU(÷HUED÷ÕPOÕGH÷LúNHQVÕUDODPD|OoH÷L\OH
HOGH HGLOPLú LVH EX GXUXPGD GD ³6ÕUDOÕ Lojistik Regresyon Analizi (Ordinal Logistic
5HJUHVVLRQ $QDO\VLV´ NXOODQÕOÕU gUQH÷LQ |÷UHQFLOHULQ |÷UHQLP J|UGNOHUL DNDGHPLN
44
SURJUDPGDNL EDúDUÕODUÕQÕQ ³GúN´ ³RUWD´ YH ³\NVHN´ RODUDN JUXSODQGÕ÷Õ GXUXPGD
VÕUDOÕ Lojistik UHJUHV\RQ X\JXODQÕU Lojistik UHJUHV\RQ ³WHN GH÷LúNHQOL Lojistik
UHJUHV\RQ ED÷ÕPVÕ] GH÷LúNHQLQ WHN ROGX÷X GXUXP´ YH ³oRN GH÷LúNHQOL Lojistik
UHJUHV\RQ ED÷ÕPVÕ] GH÷LúNHQLQ LNL YH\D GDKD ID]OD olGX÷X GXUXP´ RODUDN
VÕQÕIODQGÕUPD\DSÕOPDNWDGÕU (Stephenson, 2008).
'R÷UXVDO UHJUHV\RQ DQDOL]LQGH ED÷ÕPOÕ GH÷LúNHQLQ GH÷HUL NHVWLULOLUNHQ Lojistik
UHJUHV\RQ DQDOL]LQGH ED÷ÕPOÕ GH÷LúNHQLQ DODFD÷Õ GH÷HUOHUGHQ ELULQLQ JHUoHNOHúPH
RODVÕOÕ÷ÕNHVWLULOLU%XRODVÕOÕNGH÷HULDúD÷ÕGDNLPRGHONXOODQÕODUDNHOGHHGLOLU
S ( x)
exp( E 0 E 1 x1 ... E p x p )
(1)
1 exp( E 0 E 1 x1 ... E p x p )
Burada x ( x1 , x 2 ,..., x p ) ED÷ÕPVÕ] GH÷LúNHQOHU YHktörü ve
ED÷ÕPVÕ]GH÷LúNHQOHUHDLWSDUDPHWUHYHNW|UQJ|VWHUPHNWHGLU
E
( E 1 , E 2 ,..., E p )
økili de÷HUDODQED÷ÕPOÕGH÷LúNHQ y(0,1) için EúLWOLN¶GHYHULOHQifade, verilen x GH÷HUL
için y ¶QLQ¶HHúLWROPDNRúXOOXRODVÕOÕ÷ÕQÕYHUPHNWHGLU%XRODVÕOÕN
S ( x)
P( y
1 x)
(2)
HúLWOL÷LLOHVD÷ODQÕU%HQ]HUELoLPGH
1 S ( x)
P( y
(3)
0 x)
HúLWOL÷L\¶QLQ¶ÕDOPDNRúXOOXRODVÕOÕ÷ÕQÕJ|VWHUPHNWHGLU
Lojistik UHJUHV\RQ PRGHOLQGH SDUDPHWUH WDKPLQL \DSÕODELOPHVL LoLQ RODELOLUOLN
IRQNVL\RQX|QFHOLNOHROXúWXUXOPDOÕGÕr. \ED÷ÕPOÕGH÷LúNHQL S (x) EDúDUÕRODVÕOÕ÷ÕLOH
%HUQRXOOL GD÷ÕOPDNWDGÕU <XNDUÕGDNL HúLWOLN YH ¶GHQ i. birim için y i 1 ROGX÷XQGD
RODELOLUOLN IRQNVL\RQXQD NDWNÕVÕ S ( xi ) ve y i 0 ROGX÷XQGD RODELOLUOLN IRQNVL\RQXQD
NDWNÕVÕ 1 S ( xi ) NDGDUGÕUBuna göre i. ELULPLQRODELOLUOLNIRQNVL\RQXQDNDWNÕVÕ,
L( E i ) S ( xi ) yi (1 S ( xi )) (1 yi )
i=1,2,…,n
(4)
HúLWOL÷L LOH LIDGH HGLOLU *|]OHPOHULQ ELUELULQGHQ ED÷ÕPVÕ] ROGX÷X YDUVD\ÕOGÕ÷ÕQGD
RODELOLUOLNIRQNVL\RQXHúLWOLN¶WHNLKHUELUELULPLQoDUSÕOPDVÕ\ODHOGHHGLOLU
L( y E )
n
S (x )
i
yi
(1 S ( xi )) (1 yi )
i=1,2,…,n
(5)
i 1
Burada n ELULPVD\ÕVÕQÕJ|VWHUPHNWHGLU
Lojistik regresyon modelinin parametre tahmininde klasik yöntemler olarak; En Çok
Olabilirlik <HQLGHQ $÷ÕUOÕNODQGÕUÕOPÕú 7HNUDUOÕ (Q .oN .DUHOHU YH WHNUDUOÕ YHUL
45
Esin AVCI
durumunda Minimum lojit Ki-.DUH \|QWHPOHUL NXOODQÕOPDNWDGÕU 0XUDW YH ,úÕ÷ÕoRN
2007). Klasik yönteme alternatif olarak %D\HVFL\|QWHPGHNXOODQÕOPDNWDGÕU.
2.2 Lojistik Regresyon için Bayesci YDNODúÕP
%D\HVFL \DNODúÕP YHULOHUGHQ HOGH HGLOHQ \HQL ELOJL LOH |QFHGHn bilinen bilginin
GHUOHQPHVL LOH ROXúDQ ELU \|QWHPGLU (Wong vd., 2005). .ODVLN \DNODúÕPGD WDKPLQ
yöntemi sadece veriden derlenen en çok olabilirlik IRQNVL\RQXQD GD\DQÕUNHQ %D\HVFL
\DNODúÕPGD NODVLN \|QWHPGH HOGH HGLOHQ RODELOLUOLN SDUDPHWUH KDNNÕQGD ELOLQHQ |QVHO
bilgiyi ( p ( E ) ) GH÷LúWLUPHN LoLQ NXOODQÕOPDNWDGÕU %XQD göre BD\HVFL \DNODúÕPD J|UH
parametre tahmini;
p ( E / y ) v L( y E ) u p ( E )
(6)
HúLWOL÷L LOH YHULOHQ VRQVDO GD÷ÕOÕP LOH HOGH HGLOPHNWHGLU %XUDGD VRQVDO GD÷ÕOÕP
SDUDPHWUH KDNNÕQGDNL |QVHO ELOJL LOH YHULGHQ HOGH HGLOHQ RODELOLUOLN IRQNVL\RQXQXQ
ELUOHúPHVLQGHQROXúDQJQFHOELOJLGLU
%D\HVFL\DNODúÕPWHRULNoDWÕDOWÕQGDYHULQLQ|QVHOELOJLLOHELUOHúLPLQLQWHPHOYHGR÷DO
ELU \RO VD÷ODPDVÕ DVLPSWRWLN \DNODúÕP ROPDNVÕ]ÕQ YHULGHQ úDUWOÕ YH NHVLQ oÕNDUÕPODU
YHUPHVLNoN|UQHNE\NONOHULQGHNXOODQÕODELOPHVLSDUDPHWUHWDKPLQOHULYHKLSRWH]
WHVWOHULQGH GR÷UXGDQ oÕNDUÕPODU \DSPDVÕ ND\ÕS YHUL YH KL\HUDUúLN PRGHOOHU LoLQ
NROD\OÕNOD X\JXODQDELOPHVL JLEL ELUoRN DYDQWDMÕQÕQ \DQÕ VÕUD |QVHO ELOJLQLQ VHoLPLQGH
NHVLQ ELU \|QWHP ROPDPDVÕ YH |]HOOLNOH SDUDPHWUH VD\ÕVÕQÕQ ID]OD ROGX÷X GXUXPODUGD
KHVDSODPD]RUOX÷XGH]DYDQWDMODUÕDUDVÕQGDVD\ÕODELOPHNWHGLU&HQJL]YG
Bir parametrenin önsel bilgisi, veri elde edilmeden |QFHSDUDPHWUHKDNNÕQGDNLELOJLOHUL
LIDGH HGHU %X ELOJLOHU ELU GD÷ÕOÕP LOH LIDGH HGLOLU %X QHGHQOH %D\HVFL \DNODúÕPD J|UH
parametreler klasik \DNODúÕPGDNL JLEL VDELW RODUDN GH÷LO RODVÕOÕ÷D ED÷OÕ RODUDN
WDQÕPODQÕU gQVHO ELOJL ROPDNVÕ]ÕQ %D\HVFL oÕNDUVDPD \DSÕODPD] Önsel bilgiler; bilgi
verici (informative) ve bilgi verici olmayan (non-informative) olmak üzere
VÕQÕIODQGÕUÕOÕUODU%LOJLYHULFLROPD\DQ|QVHOOHUVRQVDOGD÷ÕOÕP]HULQGHPLQLPXPHWNL\H
sahiptir. Bu önseller daha objektiftiU ROGXNODUÕQGDQ ELUoRN LVWDWLVWLNoL WDUDIÕQGDQ
NXOODQÕOÕUODU $QFDN SDUDPHWUH KDNNÕQGDNL WRSODP EHOLUVL]OL÷LQ REMHNWLI |QVHOOH
YHULOPHVL KHU ]DPDQ X\JXQ GH÷LOGLU %D]Õ GXUXPODUGD REMHNWLI |QVHOOHU NXOODQÕFÕ\Õ
\DQOÕúVRQVDOODUD \|QOHQGLUHELOLU%LOJLYHULFL|QVHOOHURODELOLUOLNIRQNVL\RQXWDUDIÕQGDQ
HWNLVL D]DOWÕOPD\DQ |QVHO GD÷ÕOÕPGÕU %X WLS |QVHOOHU GHQH\LPOHUGHQ EHQ]HU JHoPLú
oDOÕúPDODUGDQ YH X]PDQ J|UúOHULnden elde edilen bilgi çerçevesinde belirlenirler.
6RQVDO GD÷ÕOÕP ]HULQGH ROGXNoD HWNLOL ROGXNODUÕ LoLQ |QVHO GD÷ÕOÕP EHOLUOHQLUNHQ oRN
GLNNDWOL ROXQPDOÕGÕU GHQHO RODUDN SDUDPHWUH KDNNÕQGDNL EHOLUVL]OL÷L HQ L\L DoÕNOD\DFDN
GD÷ÕOÕP
E j ~ N ( P j , V 2j )
j=1,2,...,p
(7)
%LOJL RODPDPDVÕ KDOLQGH HQ oRN VÕIÕU RUWDODPDOÕ YH RODELOGL÷LQFH E\N varyans
seçilmeliGLU 9DU\DQV LoLQ LOH DUDVÕQGD ELU DUDOÕN WHUFLK HGLOPHNWHGLU
(Rashwan vd., 2012).
46
/RMLVWLNUHJUHV\RQLoLQ%D\HVFL\DNODúÕP(úLWOL÷LQHJ|UHLIDGHHGLOLUVH
n gözlem için olabilirlik fonksiyonu,
L( y E )
(1 yi )
º
ª§ exp( E E x ... E x ) · yi §
exp( E 0 E 1 x1 ... E p x p ) ·
0
1 1
p p
¸
¸ ¨1 » (8)
«¨
»
«¨© 1 exp( E 0 E 1 x1 ... E p x p ) ¸¹ ¨© 1 exp( E 0 E 1 x1 ... E p x p ) ¸¹
¼
¬
Bilinmeyen E parametrelerine ait önsel bilginin E j ~ N ( P j , V 2j ) GD÷ÕOÕPOÕ ROGX÷X
YDUVD\ÕOGÕ÷ÕQGDVRQVDOGD÷ÕOÕP
p(E y)
p
u
j 1
(1 yi )
º
ª§ exp( E E x ... E x ) · yi §
exp( E 0 E1 x1 ... E p x p ) ·
0
1 1
p p
¸
¨
¸
¨
»
«
1

¸ ¨ 1 exp( E E x ... E x ) ¸
¨
»
i 1 «© 1 exp( E 0 E 1 x1 ... E p x p ) ¹ ©
0
1 1
p p ¹
¼
¬
n
1§E P
1
°
j
j
exp® ¨
¨
V
2
2SV i
j
°̄ ©
·
¸
¸
¹
2
½
°
¾
°¿
(9)
elde edilir (Acquah, 2013). (úLWOLN ¶XQ DQDOLWLN o|]P EXOXQPDPDNWDGÕU
øVWDWLVWLNVHO oÕNDUÕP \DSDELOPHN LoLQ Ker bir parametrenin PDUMLQDO GD÷ÕOÕPODUÕQÕQ elde
HGLOPHVLJHUHNPHNWHGLU %XQXQLoLQoRNOXLQWHJUDOOHUOHLúOHP \DSÕOPDVÕ JHUHNPHNWHGLU
U\JXODPDGD VRQVDO GD÷ÕOÕPODUGDQ LVWDWLVWLNVHO oÕNDUÕPODU yapmak için simülasyon
\|QWHPOHULQGHQ 0DUNRY =LQFLUL 0RQWH &DUOR 0&0& \|QWHPLQLQ NXOODQÕPÕQÕ
\D\JÕQODúWÕUPÕúWÕU0&0&\|QWHPLLOJLOHQLOHQVRQVDOGD÷ÕOÕPGDQEDúDUÕOÕELUúHNLOGH bir
|QFHNLQH ED÷OÕ |UQHNOHPOHU VHoPHNWHGLU MCMC yöntemi ücretsiz olarak indirilebilen
:LQ%8*6 SDNHW SURJUDPÕ \DUGÕPÕ\OD NROD\OÕNOD X\JXODQPDNWDGÕU %8*6
(Spiegelhalter vd |]HO RODUDN 0&0& \|QWHPLQLQ WDP RODVÕOÕN PRGHOOHULQH
X\JXODQPDVÕ LoLQ NXOODQÕODQ YH NRG \D]ÕPÕQD GD\DQDQ ELU SURJUDPGÕU %X SURJUDP
%D\HVFL DQDOL]L VD÷ODGÕ÷ÕQGDQ tüm parametreler rasgelH GH÷LúNHQ RODUDN HOH
DOÕQPDNWDGÕU
3. BULGULAR
%X oDOÕúPDGD 0DUPDUD hniversitesi’nde \ÕOÕQGD H÷LWLP J|UPHNWH RODQ
|÷UHQFLOHULQ NUHGL NDUWÕ VDKLSOLOL÷L EHOLUOHPHGH HWNLOL RODQ VRV\R-ekonomik ve
demografik faktörler Bayesci Lojistik UHJUHV\RQ \DUGÕPÕ\OD EHOLUOHQPH\H oDOÕúÕOPÕúWÕU
Bu amaçla, \ÕOÕQGD*D]LRVPDQSDúDYH øQ|QQLYHUVLWH|÷UHQFLOHULQLQNUHGLNDUWÕ
VDKLSOL÷LQL HWNLOH\HQ IDNW|UOHUL EHOLUOHPH oDOÕúPDVÕQGD X\JXOanan 24 soruluk anket,
Marmara ÜQLYHUVLWHVL +D\GDUSDúD YH *|]WHSH NDPSV¶QGH RNX\DQ |÷UHQFL\H
X\JXODQPÕúYHYHULOHUGHUOHQPLúWLU$QNHWLQJHQHOJYHQLUOLOLNNDWVD\ÕVÕCronbach Alfa
NDWVD\ÕVÕ RODUDN HOGH HGLOPLúWLU %X GH÷HU, NXOODQÕODQ DQNHWLQ ROGXNoD JYHQLOLU
ROGX÷XQXJ|VWHUPHNWHGLU
Marmara ÜQLYHUVLWHVL +D\GDUSDúD YH *|]WHSH NDPSV¶QGH RNX\DQ |÷UHQFLQLQ
NUHGLNDUWÕVDKLSOLOL÷LQLEHOLUOHPHGe etkili olabilecek de÷LúNHQOHU VÕQÕIG]H\LFLQVL\HW
\DúNDUGHúVD\ÕVÕ|÷UHWLPWUNUHGL-EXUVGXUXPXDQQHYHEDEDQÕQH÷LWLPGXUXPODUÕ
DQQHQLQ oDOÕúPD GXUXPX DLOHQLQ D\OÕN JHOLUL |÷UHQFLQLQ D\OÕN JHOLUL YH KDUFDPD WXWDUÕ
gibi sosyo-ekonomik ve demografik göstergeler HOHDOÕQPÕúWÕU$UDúWÕUPDGD \HUYHULOHQ
NUHGL NDUW VDKLSOLOL÷L %D÷ÕPOÕ GH÷LúNHQ YH HWNL HGHQ VRV\R-ekonomik ve demografik
47
Esin AVCI
J|VWHUJHOHUH ED÷ÕPVÕ] GH÷LúNHQOHUH ait
verilmektedir.
48
belirleyici
Tablo 1. Belirleyici istatistikler
De÷LúNHQOHU
KKS .UHGL.DUWÕ6DKLSOLOL÷L
1=Evet
+D\ÕU
6D\Õ
Yüzde
92
108
46
54
SNF 6ÕQÕI']eyi)
6ÕQÕI
6ÕQÕI
6ÕQÕI
6ÕQÕI
35
63
31
71
17.5
31.5
15.5
35.5
CNS (Cinsiyet)
0=Bayan
1=Erkek
117
83
58.5
41.5
YAS <Dú
1=17-20
2=21-23
3=24 ve üzeri
66
111
23
33
55.5
11.5
KRDS .DUGHú6D\ÕVÕ
1=1-2
2=3-4
YH]HULNDUGHú
134
66
0
67
33
0
OGT g÷UHWLP7U
g÷UHWLP
g÷UHWLP
178
22
89
11
KRDB (Kredi veya Burs)
+D\ÕU
2=Evet
BEGT %DED(÷LWLP'XUXmu)
øON|÷UHWLP
2=Lise
3=Önlisans
4=Lisans
5=Yüksek Lisans
80
120
40
60
73
65
13
45
4
36.5
32.5
6.5
22.5
2
AEGT $QQH(÷LWLP'XUXPX
øON|÷UHWLP
2=Lise
3=Önlisans
4=Lisans
5=Yüksek Lisans
114
58
4
22
2
57
29
2
11
1
ACLS $QQHdDOÕúPD'XUXPX
+D\ÕU
2=Evet
149
51
74.5
25.5
AGLR $LOHQLQ$\OÕN*HOLUL
1=500 TL’den az
2=501-1000 TL
3=1001-1500 TL
4=1501-2000 TL
10
39
43
38
5
19.5
21.5
19
istatistikler
Tablo
1’de
5=2001-2500 TL
6=2501 TL’den fazla
26
44
13
22
OGLR g÷UHQFLQLQ$\OÕN*HOLUL TL
1=300 TL’den az
2=301-600 TL
74
50
76
37
25
38
OHRC g÷UHQFLQLQ$\OÕN+DUFDPD7XWDUÕ) TL
1=300 TL’den az
2=301-600 TL
58
23
11
63
25
12
Bayesci \DNODúÕPÕQ X\JXODQPDVÕQGD HOH DOÕQDQ VRV\R-ekonomik ve demografik
göstergelere ait parametrelerin da÷ÕOÕPODUÕQÕQ |QVHO ELOJLQLQ EHOLUOHQPHVL DPDFÕ\OD,
\ÕOÕQGD<D\DUYHDUNDGDúODUÕWDUDIÕQGDQ *D]LRVPDQSDúDYHøQ|QhQLYHUVLWHOHULQGH
RNX\DQ |÷UHQFLOHULQ NUHGL NDUW VDKLSOLOL÷LQH HWNL HGHQ IDNW|UOHULQ VDSWDQPDVÕ
oDOÕúPDVÕQGDQ \DUDUODQÕOPÕúWÕU %X oDOÕúPDGD, her iki ünivHUVLWH YHULOHULQL ELUOHúWLUHQ
(Model-3) klasik Lojistik regresyon analizi sonucunda elde edilen parametre tahmin ve
2
bu tahminlere ait standart hatalar NXOODQÕOPÕú ( E ~ N ( Eˆ 2011 , V 2011
) ) ve Tablo 2’de
YHULOPLúWLU
Tablo 2. $oÕNOD\ÕFÕGH÷LúNHQOHULQSDUDPHWUHOHULLoLQ|QVHOELOJLOHU
'H÷LúNHQOHU
Sabit
SNF
CNS
YAS
KRDS
OGT
KRDB
BEGT
AEGT
ACLS
AGLR
OGLR
OHRC
Ê 2011
V Ê
-4.359
0.022
0.051
0.635
-0.173
0.277
0.291
-0.006
0.050
0.670
0.122
0.366
0.367
0.687
0.089
0.163
0.160
0.132
0.169
0.185
0.093
0.150
0.279
0.074
0.211
0.214
2011
Tablo 2’de verilen önsel bilgiler NXOODQÕODUDN %D\HVFL Lojistik regresyon yöntemi
:LQ%8*6 SURJUDPÕQGD PDFUR \D]ÕODUDN X\JXODQPÕúWÕU $oÕNOD\ÕFÕ GH÷LúNHQOHULQ
parametreOHULQHDLWVRQVDOGD÷ÕOÕPÕQ sonuçlarÕ 7DEOR¶WHYHULOPLúWLU
49
Esin AVCI
Tablo 3. Bayesci Lojistik UHJUHV\RQDQDOL]VRQXoODUÕ
Parametreler
MC
MC
V Eˆ
Ê
HATA Hata/ V
Eˆ
B.Sabit
B.SNF
B.CNS
B.YAS
B.KRDS
B.OGT
B.KRDB
B.BEGT
B.AEGT
B.ACLS
B.AGLR
B.OGLR
B.OHRC
-3.591
0.499
0.687
-0.344
0.0091
0.6514
0.8901
0.1316
-0.258
0.5046
0.1061
0.6807
0.2989
0.8186
0.1965
0.3487
0.346
0.3434
0.5493
0.3501
0.1679
0.2279
0.4372
0.1308
0.2
0.3430
0.0311
0.0063
0.0060
0.0117
0.0106
0.0060
0.0062
0.0041
0.0055
0.0076
0.0036
0.0047
0.0086
0.0380
0.0319
0.0173
0.0338
0.03078
0.0109
0.0177
0.0243
0.0243
0.0173
0.0271
0.0235
0.0247
Alt
VÕQÕU
%2.5
-5.206
0.1207
0.0053
-1.04
-0.6644
-0.4143
0.2048
-0.1956
-0.713
-0.3485
-0.1419
0.2911
-0.3734
Üst
VÕQÕU
%97.5
-1.964
0.8976
1.373
0.313
0.6762
1.748
1.577
0.4618
0.1885
1.368
0.3696
1.085
0.9712
%DúODPD
Örnek
Exp( Ê )
2500
2500
2500
2500
2500
2500
2500
2500
2500
2500
2500
2500
2500
17501
17501
17501
17501
17501
17501
17501
17501
17501
17501
17501
17501
17501
0.0276*
1.6471*
1.9877*
0.7089
1.0091
1.9182
2.4354*
1.1407
0.7726
1.6563
1.1119
1.9753*
1.3484
6RQVDO GD÷ÕOÕPÕQ \DNÕQVDPD VD÷ODPDVÕ YH EDúODQJÕo GH÷HUOHULQLQ HWNLVLQLQ HQ D]D
indirilmesi için, 0DUNRY ]LQFLULQGH HOGH HGLOHQ |UQHNOHPLQ EDúODQJÕo NÕVPÕ oÕNDUÕOÕU
Uygulamada 2 LWHUDV\RQXQ LON LWHUDV\RQX oÕNDUÕOPÕú YH HOGH HGLOHQ Bayesci
Lojistik regresyon VRQXoODUÕTablo 3’te YHULOPLúWLU7DEORLQFHOHQGL÷LQGH0&KDWDQÕQ
sonsal standart KDWD\D RUDQÕ ¶WHQ NoN ROGX÷X LoLQ 7KXPE NXUDOÕQD J|UH yeterli
LWHUDV\RQ VD\ÕVÕQD XODúÕOGÕ÷Õ V|\OHQLU YH JYHQ DUDOÕNODUÕ LQFHOHQGL÷LQGH
³´ GH÷HULQL LoHUHQ DUDOÕ÷D VDKLS RODQ SDUDPHWUHOHULQ NUHGL NDUW VDKLSOLOL÷L üzerine bir
HWNLVL ROPDGÕ÷Õ V|\OHQLU Buradan hareketOH NUHGL NDUW VDKLSOLOL÷LQH \Dú NDUGHú VD\ÕVÕ
|÷UHWLP WUEDEDH÷LWLPGXUXPXDQQHH÷LWLPGXUXPXDQQHoDOÕúPDGXUXPXDLOHQLQ
gelir durumunun önemli etNLOHULQLQROPDGÕ÷ÕVDSWDQPÕúWÕU
6ÕQÕI G]H\L FLQVL\HW NUHGL YH\D EXUV DOPD GXUXPX |÷UHQFLQLQ JHOLULQLQ NUHGL NDUW
VDKLSOLOL÷LQH|QHPVHYL\HVLQGH|QHPOLHWNLVLROGX÷XVDSWDQPÕúWÕU 6ÕQÕIG]H\LQGHNL
ELUELULPOLNDUWÕúNDWYH\DRUDQÕQGDHUNHN|÷UHQFLOHULQED\DQ|÷UHQFLOHUHJ|UH
\DNODúÕN NDW YH\D RUDQÕQGD NUHGL YH EXUV DODQODUÕQ DOPD\DQODUD J|UH \DNODúÕN
NDWYH\DRUDQÕQGDYH|÷UHQFLQLQJHOLULQGHNLELUELULPOLNDUWÕúÕQ\DNODúÕNNDW
YH\DRUDQÕQGDNUHGLNDUWVDKLSOLOL÷LQLDUWWÕUGÕ÷ÕJ|UOPHNWHGLU
%D\HVFL \DNODúÕPGD NXOODQÕODQ 0&0& \|QWHPLQLQ GR÷UX VRQXoODU YHULOGL÷LQH NDUDU
verebilmek için VRQVDO GD÷ÕOÕPD \DNÕQVDPDQÕQ JHUoHNOHúWL÷LQLQ VDSWDQPDVÕ JHUHNLU EX
amaçla; L] VRQVDO \R÷XQOXN YH RWRNRUHODV\RQ JUDILNOHULQGHQ \DUDUODQÕOÕU. Kredi kart
VDKLSOLOL÷LQHHWNLHGHQGH÷LúNHQOHUGHQkredi veya burs alma durumuna ait ilgili VÕUDVÕ\OD
iz (a), Kernel \R÷XQOXNEYHRWRNRUHODV\RQFgrafikleri ùHNLO¶GHYHULOPLúWLU
50
B.KRDB
3.0
2.0
1.0
0.0
-1.0
2500
5000
10000
15000
20000
iteration
(a)
B.KRDB
B.KRDB sample: 17501
1.0
0.5
0.0
-0.5
-1.0
1.5
1.0
0.5
0.0
-1.0
0.0
1.0
2.0
0
20
40
lag
(b)
ùHNLOø]Kernel yR÷XQOXN ve otokorelasyon grafikleri
(c)
ø]JUDIL÷LQGHVDOÕQÕPID]ODOÕ÷ÕYHVÕNOÕ÷ÕVRQVDOGD÷ÕOÕPD \DNÕQVDPDQÕQKÕ]OÕELUúHNLOGH
JHUoHNOHúWL÷LQL, Kernel \R÷XQOXNJUDIL÷LQGHLVHoDQúHNOLQGHROXúDQJ|UQPQsonsal
GD÷ÕOÕPD XODúÕOGÕ÷ÕQÕ 2WRNRUHODV\RQ JUDIL÷L LVH MDUNRY ]LQFLUL |UQHNOHUL DUDVÕQGDNL
ED÷ÕPOÕOÕ÷Õ |OoPHNWH EX QHGHQOH GúN NRUHODV\RQ \DNÕQVDPDQÕQ VD÷ODQGÕ÷ÕQÕ
belirtmektedir.
.ODVLNYH %D\HVFL \DNODúÕPÕQNDUúÕODúWÕUÕOPDVÕDPDFÕ\ODAkaike Bilgi Kriteri (AIC) ve
Bayesci Bilgi Kriteri (BIC) GH÷HUOHUL bütün veriler KHVDSODQPÕú YH 7DEOR ¶WH
YHULOPLúWLU
Tablo 4. Klasik ve Bayesci Lojistik regresyon için AIC ve B,&GH÷HUOHUL
Kriter
AIC
BIC
Klasik
301.9775
344.8556
Bayesci
267.44
310.3181
Tablo 4’te yer alan sonuçlar incelendL÷LQGH $,& YH %,& GH÷HUleri içinde en küçük
GH÷HULQ%D\HVFLLojistik UHJUHV\RQ\DNODúÕPÕQDDLWROGX÷XJ|UOU
4. SONUÇ
%X oDOÕúPDGD YDU RODQ ELOJLQLQ JQFHOOHQPHVLQL VD÷OD\DQ %D\HV \DNODúÕPÕQÕQ
NXOODQÕOPDVÕ DPDoODQPÕúWÕU %X DPDoOD, \ÕOÕQGD *D]LRVPDQSDúD YH øQ|Q
üniverVLWH |÷UHQFLOHULQH NUHGL NDUWÕ VDKLSOLOL÷LQH HWNL HGHQ IDNW|UOHULQ EHOLUOHQPHVL LoLQ
uygulanan anket formu MarmDUD hQLYHUVLWHVL |÷UHQFLOHULQH X\JXODQPÕúWÕU .UHGL NDUWÕ
VDKLSOLOL÷LQe etki eden faktörler, Bayesci Lojistik UHJUHV\RQ \DNODúÕPÕ LOH DQDOL]
edilPLúWLU gQVHO ELOJLOHU \ÕOÕQGD *D]LRVPDQSDúD YH øQ|Q hQLYHUVLWHOHUL LoLQ
uygulanan klasik Lojistik UHJUHV\RQ DQDOL]L VRQXoODUÕQGDQ HOGH HGLOPLúWLU. Bayesci
Lojistik UHJUHV\RQ \DNODúÕPÕQÕQ \HWHUOL \DNÕQVDPD VD÷ODPDVÕLOHLOJLOL7KXPENXUDOÕL]
51
Esin AVCI
grafL÷L.HUQHO \R÷XQOXN JUDIL÷LYHRWRNRUHODV\RQJUDILNOHULYHULOPLúWLU.ODVLN\|QWHP
LOHNDUúÕODúWÕUÕOPDVÕQGD$,&YH%,&NULWHUOHULNXOODQÕOPÕúWÕU.
\ÕOÕQGD<D\DUYHDUNDGDúODUÕWDUDIÕQGDQ*D]LRVPDQSDúDYHøQ|QhQLYHUVLWHOHULQGH
RNX\DQ |÷UHQFLOHULQ NUHGL NDUW VDKLSOLOL÷LQH HWNL HGHQ IDNW|UOHULQ VDSWDQPDVÕ
oDOÕúPDVÕQGDKHULNLQLYHUVLWHYHULOHULQLELUOHúWLUHQ0RGHO-3) klasik Lojistik regresyon
analizi sonucunda elde edilen parametre tahmin ve bu tahminlere ait standart hatalar
NXOODQÕOarak belirlenen |QVHO ELOJLOHU \DUGÕPÕ\OD X\JXODQDQ %D\HVFL Lojistik regresyon
VRQXFXQGD \HWHUOL \DNÕQVDPDQÕQ 7KXPE NXUDOÕ L] JUDIL÷L Kernel \R÷XQOXN JUDIL÷L YH
RWRNRUHODV\RQJUDILNOHULLOHVD÷ODQGÕ÷ÕJ|UOPúWU
.UHGLNDUWÕVDKLSOLOL÷LQHHWNLHGHQIDNW|UOHULQEHOLUOHQPHVLQGHVÕQÕIG]H\LFLQVL\HW\Dú
NDUGHú VD\ÕVÕ |÷UHWLP WU NUHGL-EXUV GXUXPX DQQH YH EDEDQÕQ H÷LWLP GXUXPODUÕ
DQQHQLQ oDOÕúPD GXUXPX DLOHQLQ D\OÕN JHOLUL |÷UHQFLQLQ D\OÕN JHOLUL YH KDUFDPD WXWDUÕ
gibi sosyo-ekonomik ve demografik göstergeler eOHDOÕQPÕúWÕU
%5 önem düzeyine göre belirlenen güven DUDOÕ÷ÕQGDVÕIÕU¶ÕLoHUPH\HQDQODPOÕEXOXQDQ
IDNW|UOHULQ VÕQÕI G]H\L FLQVL\HW NUHGL YH\D EXUV DOPD GXUXPX |÷UHQFLQLQ JHOLULQLQ
NUHGL NDUW VDKLSOLOL÷LQH |QHPOL HWNLVL ROGX÷X VDSWDQPÕúWÕU 6ÕQÕI G]H\LQGHNL DUWÕú NDW HUNHN |÷UHQFLOHULQ ED\DQ |÷UHQFLOHUH J|UH \DNODúÕN NDW NUHGL YH EXUV DODQODUÕQ
DOPD\DQODUDJ|UH\DNODúÕNNDWYH|÷UHQFLQLQJHOLULQGHNLELUELULPOLNDUWÕúÕQ\DNODúÕN
NDWYH\DNUHGLNDUWVDKLSOLOL÷LQLDUWWÕUGÕ÷ÕJ|UOPHNWHGLUg÷UHQFLQLQNUHGLNDUWÕVDKLS
olmaODUÕQGD NHQGL JHOLULQLQ HWNLOL DQFDN DLOH JHOLU GXUXPXQ HWNLOL ROPDPDVÕQÕQ QHGHQL
|÷UHQFLOHULQ\DUÕ]DPDQOÕoDOÕúPDVÕQDED÷ODQDELOLU
.ODVLN YH %D\HVFL \DNODúÕPÕQ NDUúÕODúWÕUÕOPDVÕ DPDFÕ\OD $,& YH %,& GH÷HUOHUL
heVDSODQPÕú YH HQ NoN $,& YH %,& GH÷HUOHULQLQ %D\HVFL \DNODúÕPD DLW ROGX÷X
VDSWDQPÕúWÕU 'ROD\ÕVÕ\OD EX oDOÕúPD LoLQ %D\HVFL \DNODúÕPÕQ WHUFLK HGLOHELOHFH÷L
J|VWHUPLúWLU
gQVHO ELOJL\H EDúYXUPD HVQHNOL÷L QHGHQL\OH %D\HVFL \DNODúÕP NODVLN \|QWHPOHUH J|UH
oldukça DYDQWDMOÕROGX÷X\RUXPX\DSÕODELOPHNWHGLU
5. KAYNAKLAR
Acquah, H. D., 2013. Bayesian Lojistic Regression Modelling Via Markov Chain
Monte Carlo Algorithm, Journal of Social and Development Sciences, 4(4), 193-197.
$NNXú = dHOLN 0 Y., 2004. Lojistik Regresyon ve Diskriminant Analizi
Yöntemlerinde Önemli Ölçütler. VII. Ulusal Biyoistatistik Kongresinde sunulan bildiri,
0HUVLQhQLYHUVLWHVL7ÕS)DNOWHVL%L\RLVWDWLVWLN$QDELOLP'DOÕ0HUVLQ
Anderson, H. A., 1983. Robust Inference Using Lojistic Models, Bulletin of
International Statistical Institute, 48, 25-53.
Anderson, J. A., 1979. Multivariate Lojistic Compounds. Biometrika, 66, 17-191.
%DúDUÕU*dRNDH÷LúNHQOLVerilerde A\UÕPVDPDSorunu ve Lojistik Regresyon
Analizi, (UygulaPDOÕøVWDWLVWLN'RNWRUD7H]L+DFHWWHSHhQLYHUVLWHVL-36, Ankara.
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Cengiz, M. $ 7HU]L ( ùHQHO 7 0XUDW 1 Lojistik Regresyonda Parametre
Tahmininde Bayesci bir YDNODúÕP, Afyon Kocatepe Üniversitesi Fen Bilimleri Dergisi,
12, 15-22.
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dDYXú0 F., 2006. Bireysel FLQDQVPDQÕQTemininde Kredi KDUWODUÕ Türkiye’de Kredi
KDUWÕ KXOODQÕPÕ Üzerine bir AUDúWÕUPD, Selçuk Üniversitesi Sosyal Bilimler Enstitüsü
Dergisi, 15, 173-187.
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<D\ÕQ'D÷ÕWÕP
Keskin, D., Koparan, E., 2010. Üniversite Ö÷UHQFLOHULQLQ Kredi KDUWÕ SDKLSOLOL÷LQL
Belirleyen Faktörler, (VNLúHKLU 2VPDQJD]L hQLYHUVLWHVL øNWLVDGL YH øGDUL %LOLPOHU
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53
Esin AVCI
54
INVESTIGATION OF FACTORS EFFECTIVE ON CREDIT CARD
OWNERSHIP OF MARMARA UNIVERSITY STUDENTS BY
BAYESIAN LOGISTIC REGRESSION
ABSTRACT
Bayesian approach is a method which combines new information
obtained from data and previous knowledge. In this study, Bayesian
approach, which is an alternative to classical approach, is applied to
logistic regression, which explores the effects of covariate(s) on binary
dependent variable. To this aim, socioeconomic and demographic factors
that are effective on credit card ownership of Marmara University
students are examined.
Keywords: Credit card ownership, Bayesian logistic regression, WinBUGS
55
Cilt: 10 Sayı: 02 Kategori: 02 Sayfa: 56-72 ISSN No: 1303-6319 Volume: 10 Number: 02 Category: 02 Page: 56-72
56
Modelling the low Birth Weight of New Born Babies with
Binary Logistic Regression Based on Mars Method
Yeni Doğan Bebeklerin Düşük Doğum Ağırlığının Mars
Yöntemine Dayalı İkili Lojistik Regresyonla Modellenmesi
yöntemlerden biri çRNGH÷LúNHQOLX\DUOD\ÕFÕ regresyon X]DQÕPODUÕMARS (Multivariate
Adaptive Splines) yöntemidir.
'úNGR÷XPD÷ÕUOÕNOÕolarak dünyaya gelen EHEHNOHULOHUOH\HQ\ÕOODUGDVD÷OÕNDoÕVÕQGDQ
ED]Õ VRUXQODUOD NDUúÕODúPDNWDGÕU Bu nedenle, ELU EHEH÷LQ GR÷XP D÷ÕUOÕ÷ÕQÕQ GúN
GH÷HUGHROXSROPD\DFD÷ÕQÕQWDKPLQLLoLQX\JXQELUPRGHOHLKWL\DoYDUGÕU. Bu modelin
ROXúWXUXOPDVÕQÕ LoHUHQ X\JXODPD oDOÕúPDVÕQGD EHEH÷LQ GR÷XP D÷ÕUOÕ÷ÕQÕ
etkileyebilecek faktörler, 0$56\|QWHPLQHGD\DOÕORMLVWLNUHJUHV\RQPRGHOLROXúWXUPDN
LoLQNXOODQÕOPÕúWÕUdDOÕúPDGD'DQLPDUND8OXVDO'R÷XP*UXEX’QXQKDPLOHOLNWHDWHúYH
EXQDED÷OÕ|OGR÷XPODUODLOJLOLoDOÕúPDVÕQDDLWYHULOHULQELUNÕVPÕNXOODQÕOPÕúWÕU. Bebek
GR÷XP D÷ÕUOÕNODUÕ ED÷ÕPOÕ GH÷LúNHQ RODUDN VHoLOPLúWLU dDOÕúPDGD GR÷XP \DSDFDN
NDGÕQODUGDQ HOGH HGLOHQ YHULOHU NXOODQÕODUDN, EHEH÷LQ GúN GR÷XP D÷ÕUOÕNOÕ ROXS
ROPD\DFD÷ÕQD LOLúNLQ PRGHO ROXúWXUXOPXúWXU 2OXúWXUXODQ PRGHO \DUGÕPÕ\OD GR÷XP
öncesi bebek do÷XP D÷ÕUOÕ÷Õ LoLQ WDKPLQOHU \DSÕODELOPHVL ve gerekli önlemlerin
DOÕQDELOPHVLDPDoODQPÕúWÕU
2. MARS <g17(0ø
dRN GH÷LúNHQOL X\DUOD\ÕFÕ regresyon X]DQÕPODUÕ 0$56 )ULHGPDQ WDUDIÕQGDQ ¶OÕ
\ÕOlDUÕQ EDúÕQGD JHOLúWLULOPLú, parametrik olmayan bir regresyon yöntemidir. MARS
NHOLPHVLDoÕOÕPÕDúD÷ÕGDNLNDYUDPODUÕQEDúKDUIOHULQGHQROXúWXUXOPXúWXU
Multivariate (çRNGH÷LúNHQOL: ÇRNER\XWOXYHULOHU]HULQGHLúOHP\DSÕODELOLU|]HOOLNOH
ED÷ÕPVÕ]GH÷LúNHQVD\ÕVÕID]ODROGX÷XGXUXPODUGDWHUFLKVHEHELGLU
Adaptive (u\DUOD\ÕFÕ): Y|QWHPLQ EDVDPDNODUÕ ILQDO PRGHOLQH XODúDQD NDGDU HOHPH YH
VHoPHDúDPDODUÕQGDQROXúXU
Regression (regresyon): BD÷ÕPOÕ YH ED÷ÕPVÕ] GH÷LúNHQOHU DUDVÕQGDNL IRQNVL\RQHO
LOLúNL\LLIDGHHWPHNWHGLU
Splines (X]DQÕPODU): 5HJUHV\RQ HúLWOL÷L Güz bir reJUHV\RQ GR÷UXVX \HULQH ENOPú
bir \DSÕ\DVDKLSWLU
0$56 \|QWHPL EDQNDFÕOÕN VLJRUWDFÕOÕN HNRQRPLQLQ \DQÕ VÕUD \DúDP DQDOL]L VRV\DO
bilimler gibi ELUoRN DODQGD NXOODQÕOPDNWDGÕU /LWHUDWUGH 0$56 \|QWHPL LOH LOJLOL
olarak, Kim (2000JHQoOHULQX\XúWXUXFXNXOODQÕPÕLOHLOJLOL\DSWÕ÷ÕoDOÕúPDVRQXFXQGD
ED÷ÕPOÕ GH÷LúNHQLQ NDWHJRULN ROGX÷X GXUXPODUGD GD 0$56¶ÕQ L\L VRQXoODU YHUGL÷LQL
J|VWHUPLúWLU Kuhnert, Do vd. (2000) parametrik olmayan CART ve MARS
yöntemlerini, parametrik loMLVWLN UHJUHV\RQOD NDUúÕODúWÕUPÕúWÕU 0RWRU ND]DODUÕQGDNL
\DUDODQPD YHULOHULQH X\JXODQPÕú RODQ EX oDOÕúPD LoLQ 0$56 PRGHOLQLQ GL÷HU LNLVLQH
J|UH GDKD L\L SHUIRUPDQV J|VWHUGL÷L EHOLUWLOPLúWLU $PHULNDQ dHYUH .RUXPD .XUXOXúX
(EPA)’ndan 1DVK YH %UDGIRUG ¶XQ \DSWÕ÷Õ oDOÕúPDGD EHOLrli bir bölgedeki bir
NXUED÷DWUQQYDUOÕ÷Õ ORMLVWLNUHJUHV\RQYH0$56 \|QWHPL\OHWDKPLQHGLOPLúYHLNL
yöntemin soQXoODUÕ GH÷HUOHQGLULOPLúWLU Kolyshkina ve Brookes (2002) sigorta riskini
YHUL PDGHQFLOL÷L \DNODúÕPODUÕQÕ 0$56 ve klasik lojistik regresyonla tahmin etmeye
oDOÕúPÕúWÕU Dieterle ( ]DPDQD ED÷OÕ DQDOLWLN YHULOHU ]HULQH KD]ÕUODGÕ÷Õ GRNWRUD
WH]LQGH \DSD\ VLQLU D÷ODUÕ JHQHWLN DOJRULWPDODU &$57 YH 0$56¶Õ NDUúÕODúWÕUPÕúWÕU
Lee, Chiu vd. ( NUHGL VNRUODPD LOH LOJLOL oDOÕúPDODUÕQGD diskriminant analizi,
57
Soner ÖZTÜRK, Volkan SEVİNÇ
ORMLVWLN UHJUHV\RQ &$57 YH 0$56¶ÕQ GR÷UX VÕQÕIODPD RUDQODUÕQÕ YH KDWDODUÕQÕ
NDUúÕODúWÕUPÕúODUGÕU Stokes ve Lattyak ( 0$56 \|QWHPLQL HNRQRPHWULN ED]Õ
VLVWHP YH \D]ÕOÕPODU LOH JHOLúWLUPLú YH NXOODQPÕúODUGÕU Kriner (2007), \DúDP DQDOizini
MARS \|QWHPLQL NXOODQDUDN \DSPÕúWÕU Quiros, Felicisimo vd. (2009) MARS
yöntemini, arazi örtüsünün uydu görüntülerinden yararlanarak sÕQÕIODQGÕUÕOPDVÕ LoLQ
NXOODQPÕúODUGÕU Mina (2009), yoksulluk profilinin belirlenmesinde MARS yöntemini
NXOODQPÕú ve ED]ÕNRúXOODUDOWÕQGDparametrik lojistik regresyondan dahDHWNLOLROGX÷XQX
EHOLUWPLútir. Mina ( |]UO NLúLOHULQ Lú VHoLPL LOH LOJLOL \DSWÕ÷Õ oDOÕúPDGD,
parametrik ORMLVWLN UHJUHV\RQ YH 0$56 \|QWHPOHULQL NXOODQPÕú YH VRQXoODUÕ
GH÷HUOHQGLUPLúWLU Samui ve Kothari ( GHSRODUGDNL EXKDUODúPD ND\ÕSODUÕQÕQ
WDKPLQLQL0$56LOH\DSPÕúYHVRQXoODUÕ\DSD\VLQLUD÷ODUÕLOHNÕ\DVODPÕúWÕUTürkiye’de
ise Tunay ( 7UNL\H¶GH SDUDQÕQ JHOLU GRODúÕP KÕ]ODUÕQÕQ MARS yöntemiyle
tahmini oDOÕúPDVÕQÕ JHUoHNOHúWLUPLúWLU Yerlikaya ( 0$56 ]HULQGH ELU WDNÕP
düzenlemeler yaparak, ROXúWXUGX÷X \HQL PRGHOL YHUL PDGHQFLOL÷L X\JXODPDODUÕ LoLQ
NXOODQPÕúWÕU .DQ YH <D]ÕFÕ (201 \DNÕW WNHWLPL LoLQ IDNW|UL\HO GHQH\OHUL UHJUHV\RQ
D÷DoODUÕ ve MARS yönteminin VRQXoODUÕQÕ NDUúÕODúWÕUPÕúODUGÕU Kayri (2010) internet
ED÷ÕPOÕOÕ÷Õ |OoH÷LQL 0$56 \|QWHPLQL NXOODQDUDN DQDOL] HWPLúWLU Tunay (2010)
EDQNDFÕOÕNNUL]OHULni ve Türkiye’deki durgunlXNODUÕ0$56 yöntemi LOHWDKPLQHWPLúWLU
Topak ( 7UNL\H¶GH NXUXPVDO EDúDUÕVÕ]OÕ÷Õ Podellemek için MARS yöntemini
NXOODQPÕúWÕU
2.1 Mars Yöntemi ile Tahmin
Parametrik olmayan regresyon yöntemleri, Kernel tahmini, yerel polinom regresyonu
veya düzlHúWLUPH X]DQÕPODUÕ \|QWHPOHULQL NXOODQÕU 0$56 \|QWHPL ED÷ÕPOÕ GH÷LúNHQ
YH ED÷ÕPVÕ] GH÷LúNHQ NPHVL DUDVÕQGDNL LOLúNL\L G]OHúWLUPH X]DQÕPODUÕQÕ NXOODQDUDN
belirleyen bir yöntemdir (Friedman, 1991).
0$56\|QWHPLKHUED÷ÕPVÕ]GH÷LúNHQLQED÷ÕPOÕGH÷LúNHQOHRODQLOLúNLVLQL incelemenin
\DQÕ VÕUD ED÷ÕPVÕ] GH÷LúNHQOHULQ ELUELUOHUL DUDVÕQGDNL HWNLOHúimlerini de belirler ve bu
HWNLOHúLPOHULQ ED÷ÕPOÕ GH÷LúNHQ ]HULQGHNL HWNLVLQL RUWD\D NR\DU 7XQD\ %X
QHGHQOH J|]OHP VD\ÕVÕQÕQ YH ED÷ÕPVÕ] GH÷LúNHQ VD\ÕVÕQÕQ oRN ROGX÷X GXUXPODUGD
MARS yöntemi iyi sonuç vermektedir.
Matematikte iki tU H÷UL EXOXQPDNWDGÕU %LULQFL WU LQWHUSRODV\RQ H÷ULOHULGLU %X
H÷ULOHUGH H÷UL WP YHUL QRNWDODUÕQGDQ JHoPHNWHGLU %X NODVLN H÷UL oL]LPLGLU 'L÷HUL LVH
G]OHúWLUPH H÷ULOHULGLU ']OHúWLUPH H÷ULOHULQGH LVH H÷UL YHUL QRNWDODUÕQD \DNÕQ
ROPDNWDGÕU 7DP RODUDN bu noktalardan geçmesi gerekmez. Bu anlamda en basit
G]OHúWLUPHH÷ULOHULQHSDUoDOÕGR÷UXVDOUHJUHV\RQH÷ULVLdir.
ùHNLO 1. 3DUoDOÕGR÷UXVDOUHJUHV\RQH÷ULVL
58
ùHNLO1¶GHVD÷GDNLUHVLPRULMLQDOYHUL\HDLWWLU0RGHOOHPH \DSÕOÕUNHQYHUL\H ait düz bir
GR÷UX ROXúWXUPDN yeriQH UHJUHV\RQ GR÷UXVXQXQ bükülmesi VD÷ODQPÕúWÕU Soldaki
resimdeNLGR÷UXoQRNWDGDQENOPúWUYHELU0$56X]DQÕPÕ KDOLQHJHOPLúWLU
0$56 SDUoDOÕ GR÷UXVDO UHJUHV\RQ X\JXOD\DUDN HVQHN PRGHOOHU ROXúWXUXU %D÷ÕPVÕ]
GH÷LúNHQLQ IDUNOÕ DUDOÕNODUÕQGD D\UÕ UHJUHV\RQ H÷LP GH÷HUOHUL kullanDUDN GR÷UXVDOOÕ÷Õ
korur 5HJUHV\RQ GR÷UXVXQXQ H÷LPLQLQ GH÷LúWL÷L YH ELU DUDOÕNWDQ GL÷HULQH JHoLOGL÷L
QRNWDODUD G÷P GHQLU .XOODQÕODFDN GH÷LúNHQOHU YH KHU GH÷LúNHQ LoLQ DUDOÕNODUÕQ ELWLú
QRNWDVÕDUDúWÕUPDVRQXFXEXOXQXU/HH, Chiu vd., 2004).
ùHNLO2øNLG÷PQRNWDOÕSDUoDOÕGR÷UXVDOUHJUHV\RQ|UQH÷L
gUQH÷LQ, ùHNLO ¶GH \HU DODQ ve aUDOÕNODUÕ ELUELUiQGHQ D\ÕUDQ LNL WDQH G÷P QRNWDVÕ
ROGX÷X J|UOPHNWHGLU 5HJUHV\RQ GR÷UXVXQXQ H÷LPL ELU DUDOÕNWDQ GL÷HULQH JHoWL÷LQGH
GH÷LúPHNtedir.
Modeldeki dH÷LúNHQOHU HWNLOHúLPOHUL YH G÷P QRNWDODUÕQÕQ NRQXPX, kaba kuvvet
\DNODúÕPÕ\OD EUXWH IRUFH DSURDFK NDWVD\ÕODU LVH en küçük kareler (EKK) yöntemiyle
EXOXQXU .DED NXYYHW \DNODúÕPÕ WP RODVÕ o|]POHU EXOXQGXNWDQ VRQUD HQ L\L RODQD
karar verilen bir yöntemdir (Dieterle, 2003).
MARS yöntemLQGH GL÷HU ELU |QHPOL NRQX, temel fonksL\RQODUGÕU Temel fonksiyonlar,
GH÷LúNHQ DUDOÕNODUÕQÕQ WDQÕPODQGÕ÷Õ E|OJHVHO PRGHOOHUGLU 7HPHO IRQNVL\RQODU WHN ELU
H÷UL IRQNVL\RQX \D GD ELUGHQ ID]OD GH÷LúNHQLQ HWNLOHúLP WHULPL RODELOLU 7HPHO
IRQNVL\RQODU G÷P QRNWDODUÕQÕQ EHOLUOHQPHVL DoÕVÕQGDQ |QHPOLGLU 3XW, Massart vd.,
7HPHO IRQNVL\RQODU ED÷ÕPOÕ GH÷LúNHQ < YH ED÷ÕPVÕ] GH÷LúNHQOHU
( X 1 , X 2 ,..., X m ) DUDVÕQGDNL LOLúNL\L WHPVLO HGHQ IRQNVL\onlar olarak görev yapar. Bu
fonksiyonlar, E0 VDELW SDUDPHWUHVL YH GL÷HU WHPHO IRQNVL\RQODUÕQ D÷ÕUOÕNODQGÕUÕOPÕú
WRSODPÕQGDQROXúXU
Ŷ
M
f x E0 ¦ E m Bm X m 1
(1)
E0 : sabit terim
Bm X : m. temel fonksiyon
Em : m. temel fonksiyonun NDWVD\ÕVÕ
M 7HPHOIRQNVL\RQVD\ÕVÕ
59
Soner ÖZTÜRK, olkan SEİNÇ
+RNH\ VRSDVÕ WHPHO IRQNVL\RQODUÕ - HSTF (hockey stick basis functions - HSBF),
MARS modelinde önemli bir unsurdur. HSTF’ler sürekli ;GH÷LúNHQLQLQG|QúPúKDOL
olan X* GH÷HULQL
X*
max 0, X c , veya
max 0, c X (2)
biçiminde WDQÕPODU X* ;¶LQ HúLN GH÷HUL RODUDN WDQÕPODQDQ F GH÷HUinden küçük tüm
GH÷HUOHULLoLQGH÷HULQLF¶GHQE\N;GH÷HUOHULLoLQVH;GH÷HUOHULQLDOÕUgUQHNRODUDN
X, YHDUDVÕQGDGH÷HUOHUDODQED÷ÕPVÕ] bir GH÷LúNHQROVXQ c 10, 20,30,..., 70,80
için X* ¶LQDOGÕ÷ÕGH÷HUOHUùHNLO 3¶GHJ|VWHULOPLúWLU
ùHNLO )DUNOÕHúLNGH÷HUOHULFLoLQROXúWXUXODQWemel fonksiyonlar
MARS, YHUL VHWLQLQ WP GH÷HUOHUL LoLQ GH÷LúLN F GH÷HUOHULQH NDUúÕOÕN JHOHQ ook VD\ÕGD
WHPHOIRQNVL\RQROXúWXUabilir.
0$56LONRODUDNVUHNOLED÷ÕPOÕGH÷LúNHQOHULQWDKPLQLLoLQWDVDUODQPÕúWÕU'DKDVRQUD
ikilLNDWHJRULNED÷ÕPOÕGH÷LúNHQOHULoLQGHNXOODQÕOPÕúWÕU6DOIRUG6\VWHPV7XQD\
oDOÕúPDVÕQGD, MARS yönteminin geleneksel regresyon yöntemleri ile parametrik
ROPD\DQPRGHOOHULQVWQONOHULQLEDúDUÕ\ODELUOHúWLUHQYHHNRQRPLNGXUJXQOXNODUJLEL
LNLOL\DSÕGDNLNDWHJRULNED÷ÕPOÕGH÷LúNHQOHUHUDKDWOÕNODX\JXODQDELOHQ\DSÕVÕQÕQ|QHPOL
ELUDYDQWDMROGX÷XQXEHOLUWPLúWLU.
.ODVLN PRGHOOHPHGH NDWHJRULN GH÷LúNHQLQ KHU ELU NDWHJRULVL LoLQ NXNOD GH÷LúNHQOHU
kümeVLROXúWXUXOXU%XNPHUHJUHV\RQDQDOL]LQGHJLUGLOHURODUDNNXOODQÕOÕU%D÷ÕPVÕ]
GH÷LúNHQOHULQ NDWHJRULN ROGX÷X GXUXPODUGD 0$56 GD D\QÕ úHNLOGH NXNOD GH÷LúNHQOHU
ROXúWXUXU$QFDNEXNXNODGH÷LúNHQOHULOJLOLGH÷LúNHQLQG]H\OHULQLQEWQGU
Hastie ve Tibshirani (1990) MARS yöntemini, ED÷ÕPOÕ GH÷LúNHQLQ LNLOL ROPDVÕ
durumunda, DúD÷ÕGDNLformül LOHORJLVWLNUHJUHV\RQDQDOL]LQHX\DUODPÕúWÕU
lojit P Y 1 f x H
60
(3)
Bu HúLWOLNte f x MARS yöntemi tarafÕQGDQWDKPLQHGLOHQWHPHOIRnksiyonlar kümesini
ifade eder. MARS yöntemi ile formül (3) birlikte HOH DOÕQÕUVD DúD÷ÕGDNL formül elde
edilir.
lojit Pi M
E0 ¦ E m Bm X (4)
m 1
Takip eden DOW EDúOÕNWD 0$5S yöntemiyle modellemede yer alan basamaklar
LQFHOHQPLúWLUø\LELUPRGHOEXoEDVDPDNVRQXFXQGDROXúXU
2.2 MARS Yönteminin BDVDPDNODUÕ
Put, Massart vd. (2003)’e göre MARS \|QWHPLQGHHQL\LPRGHOoEDVDPDNWDQROXúDQ
bir süreç sonunda elde edilir. %X EDVDPDNODU \DSÕP EXGDPD YH \XPXúDWPD DúDPDODUÕ
RODUDNDGODQGÕUÕOÕU
2.2.1 <DSÕPDúDPDVÕFonstructive phase)
%XDúDPDGDVDELWWHULPOHEDúOD\DUDNYHVUHNOLRODUDNWemel fonksiyonlar eklenir. Temel
IRQNVL\RQODU HNOHQGLNoH NDUÕúÕN YH GH HVQHN ELU PRGHO ROXúXU %X LúOHP NXOODQÕFÕ
WDUDIÕQGDQ VUHFLQ EDúÕQGD EHOLUOHQHQ PDNVLPXP WHULP VD\ÕVÕQD XODúDQD NDGDU GHYDP
eder.
2.2.2 %XGDPDDúDPDVÕ (pruning phase)
øNLQFLDúDPDJHULGR÷UXHOHPHDúDPDVÕGÕU%LULQFLDúDPDGDID]ODVD\ÕGDWHPHOIRQNVL\RQ
HNOHQGL÷LLoLQROXúWXUXOan modeODúÕUÕWDKPLQOHPHproblemi\OHNDUúÕNDUúÕ\DNDODFDNWÕU
$úÕUÕtahminlemeyi RUWDGDQNDOGÕUPDNLoLQJHULGR÷UXHOHPHLúOHPLQHEDúODQÕU0RGHOLQ
NDUPDúÕNOÕ÷ÕQÕQD]DOWÕOGÕ÷ÕDúDPDGÕU0RGHOHHQD]NDWNÕVÕRODQWHPHOIRQNVL\RQODUDWÕOÕU
%X DúDPDGD &UDYHQ YH 9DKED WDUDIÕQGDQ JHOLúWLULOHQ *HQHOOHúWLULOPLú dDSUD]
Geçerlilik (GÇG) (Generalized Cross-Validation - GCV |OoW NXOODQÕOÕU *d* D\QÕ
]DPDQGDX\XPHNVLNOL÷LNULWHULGLU
Y fˆ X 1 ¦
GÇG M N 1 C M N N
i
M
2
i
i 1
(5)
2
Formül (5)’de,
1|UQHNOHPYHULVD\ÕVÕ
C M PRGHOGHX\GXUXODQJHoHUOLSDUDPHWUHVD\ÕVÕ
(÷HUPRGHOGH0WDQHGR÷UXVDOED÷ÕPVÕ]WHPHOIRQNVL\RQYDUVD
CM
M dc
(6)
HúLWOL÷L VD÷ODQÕU Formül (6)’da, c ilHUL GR÷UX RODQ VUHoWH VHoLOHQ G÷P VD\ÕVÕ G LVH
her bir WHPHO IRQNVL\RQ RSWLPL]DV\RQXQGDNL PDOL\HWL LIDGH HGHQ ELU GH÷HUGLU 0$56
61
modellerinde genelOLNOHG DOÕQÕU)ULHGPDQWP\DSÕODQoDOÕúPDODUVRQXFXQGD
GLoLQHQL\LGH÷HUOHULQ 2 d d d 4 DUDOÕ÷ÕQGDROGX÷XQXEHOLUWPLúWLU
2.2.3 <XPXúDWPDDúDPDVÕ (smoothing phase)
6RQ RODUDN E|OJHVHO VÕQÕUODU LoLQGHNL VUHNVL]OL÷LQ JLGHULOPHVL ve birincil ve ikincil
WUHYOHULQVUHNOLOL÷LQLQVD÷ODQPDVÕLoLQ\XPXúDWPDJHUHNOLGLU4XLURVYG., 2009).
0$56¶ÕQ EHOLUWLOHQ EX EDVDPDNODUÕQÕQ |]HWL YH PRGHOOHQPHVL ùHNLO YH ùHNLO ’de
gösterilPLútir. ;HNVHQLED÷ÕPVÕ]GH÷LúNHQL<HNVHQLED÷ÕPOÕGH÷LúNHQLJ|VWHUPHNWHGLU.
ùHNLO4. OriMLQDOYHULQLQGD÷ÕOÕPJUDIL÷L
ùHNLO ¶GH yer alan soldaki fonksiyon, düz tepeli fonksiyon olarak bilinir ve X=30 ile
[ G÷P QRNWDODUÕQD VDKLSWLU 6D÷GDNL IRQNVL\RQ LVH J|]OHQHQ YHULQLQ GD÷ÕOÕPÕQÕ
göstermektedir.
ùHNLO5. MARS modeli YHROXúWXUGX÷XG÷POHU
0$56 LON RODUDN WHN ELU G÷P QRNWDVÕQÕ ; EHOLUOHPLúWLU øOHUL DúDPDGD G÷m
VD\ÕVÕDUWWÕNoD0$56ùHNLO5’deNL\DNODúÕPODPRGHOLWDKPLQHWPLúWLU6DOIRUG6\VWHPV
2001).
3. 'hùh.'2ö80$ö,5/,ö,1$ø/øù.ø10$56<g17(0ø1('$<$/,
ø.ø/ø/2-ø67ø.5(*5(6<2102'(/ø
'R÷XP D÷ÕUOÕ÷Õ ELU EHEH÷LQ GR÷GX÷X DQGD |OoOHQ D÷ÕUOÕ÷ÕGÕU 'R÷XP D÷ÕUOÕ÷Õ \HQL
GR÷DQ |OPOHUL ve bebeklLN G|QHPL KDVWDOÕNODUÕQÕ HWNLOH\HQ faktörlerden biridir. Bu
QHGHQOH GR÷XP D÷ÕUOÕ÷Õ YH EXQX HWNLOH\HQ IDNW|UOHU KHU ]DPDQ |QHPOL ELU NOLQLN
DUDúWÕUPDNRQXVXROPXúWXU
62
'úN GR÷XP D÷ÕUOÕ÷Õ 'Q\D 6D÷OÕN gUJW WDUDIÕQGDQ JUDPGDQ GDKD D] RODQ
GR÷XPD÷ÕUOÕ÷ÕRODUDNWDQÕPODQPÕúWÕU 'R÷XPD÷ÕUOÕ÷ÕYHJHEHOLNKDIWDVÕFenin veya yeni
GR÷DQ |OPOHULQLQ HWNHQOHULQGHQ ELULGLU. %X QHGHQOH GúN GR÷XP D÷ÕUOÕNODUÕ veya
SUHPDWUH GR÷XPODU KDIWDGDQ |QFH JHUoHNOHúHQ GR÷XP EX NRQXGDNL DUDúWÕUPDODU
için önemli birer veridirler. %HEH÷LQ GúN GR÷XP D÷ÕUOÕNOÕ RODUDN GR÷PDVÕQGD HWNLVL
olan etkenler, Kramer (WDUDIÕQGDQDúD÷ÕGDNLPDGGHOHUOHYHULOPLúWLU
x
x
x
x
x
x
x
x
'R÷XPNXVXUXGDGHQLOHQEeEH÷LQGR÷XPVDOJHQHWLNYH\DSÕVal anomalileri
$QQHQLQGR÷XPJHoPLúLönceki öOGR÷XPVD\ÕVÕGúNVD\ÕVÕ
$QQHQLQ\NVHNNDQEDVÕQFÕúHNHUNDOSFL÷HUYHE|EUHNKDVWDOÕNODUÕ
$QQHQLQDONROX\XúWXUXFXYHVLJDUDDOÕúNDQOÕ÷Õ
$QQHQLQJHoLUGL÷LHQIHNVL\RQODU
Plasentada görülen sorunlar (bHEH÷H NDQ YH EHVLQ XODúÕPÕQGD VRUXQODUD VHEHS
ROPDNWDGÕU
Annenin yeterince beslenememesi ve X\JXQNLORGDROPDPDVÕ
Sosyoekonomik faktörler (az geliri olan, H÷LWLP G]H\L GúN \DúÕQGDQ
küçük veya \DúÕQGDQ E\N DQQHOHrin bu konuda artan riske sahip ROGX÷X
VDSWDQPÕúWÕU
8\JXODPD oDOÕúPDVÕ \HQL GR÷DQ EHEHNOHULQ GúN GR÷XP D÷ÕUOÕ÷ÕQD VDKLS ROPD
RODVÕOÕNODUÕQÕQ, ikili lojistik regresyon analizi ile 0$56\|QWHPLQHGD\DOÕRODUDNmodel
ROXúWXUXOPDVÕQÕ NDSVDPDNWDGÕU
%X oDOÕúPD VRQXFXQGD ROXúWXUXODFDN PRdel bir annenin EHEH÷LQLQ GR÷XP D÷ÕUOÕ÷ÕQÕQ
GúN ROPDVÕ \D GD ROPDPDVÕ LOH LOJLOL WDKPLQGH EXOXQXODELOHFHNWLU %|\OHFH GúN
GR÷XP D÷ÕUOÕNOÕ RODUDN PH\GDQD JHOPH RODVÕOÕ÷Õ EXOXQDQ EHEHN LoLQ |QOHPOHU
DOÕQÕODELOHFHNWLU
ÇDOÕúPDGD 'DQLPDUND 8OXVDO 'R÷XP *Uubu (DUDG)’nun KDPLOHOLNWH DWHú YH EXQD
ED÷OÕ |O GR÷XPODUOD LOJLOL oDOÕúPDVÕQD DLW YHULOHULQ ELU NÕVPÕ NXOODQÕOPÕúWÕU '8'*
YH \ÕOODUÕ DUDVÕQGD .000 NDGÕQ ile KDPLOHOL÷LQ LON 12–16 KDIWDVÕQGD ve
sonunda J|UúS JHUHNOL YHULOHUL GHUOHPLúWLU Andersen, Vastrup vd. (2002) GHUOHQPLú
olan bu verilerden 31 0DUW |QFHVLQH DLW RODQODUÕ kullanarak, |O GR÷XP ED÷ÕPOÕ
GH÷LúNHQLQLHWNLOH\HQ, ED÷ÕPVÕ]GH÷LúNHQOHULVDSWDPDoDOÕúPDVÕJHUoHNOHúWLUPLúWLU
dDOÕúPDPÕ]GD $QGHUVHQ, Vastrup vd. (¶QLQ HOGH HWWL÷L ED÷ÕPVÕ] GH÷LúNHQOHUGHQ
\DUDUODQÕOPÕúWÕU $QFDN EX oDOÕúPDGDQ IDUNOÕ biçimde, ED÷ÕPOÕ GH÷LúNHQ, bebeklerin
GR÷XPD÷ÕUOÕ÷Õolarak VHoLOPLúWLU<HQLGR÷DQELUEHEH÷LQD÷ÕOÕ÷Õortalama olarak 2500–
4000 JU DUDVÕQGDGÕU $÷ÕUOÕ÷Õ JUDPdan daha az olan bir bebek GúN GR÷XP
D÷ÕUOÕNOÕ RODUDN nitelendirilmektedir. %X QHGHQOH JUDPGDQ GúN RODQ ED÷ÕPOÕ
GH÷LúNHQOHU GL÷HUOHUL RODUDN NRGODQPÕúWÕU. dDOÕúPDPÕ]GD \HU DODQ GH÷LúNHQOHUGen
annede ilk 17 haftada gözlenen \NVHN DWHú VD\ÕVÕ DQQHQLQ \DúÕ |QFHNL |O GR÷XP
VD\ÕVÕ |QFHNL FDQOÕ GR÷XP VD\ÕVÕ JHEHOLN KDIWDVÕ VLJDUD DONRO YH NDKYH NXOODQÕPÕ YH
EHEH÷LQ GR÷XP ER\X GH÷LúNHQOHULQLQ GR÷XP D÷ÕUOÕ÷ÕQÕ HWNLOH\HELOHFH÷L GúQFHVL\OH
GH÷HUOHQGLUPH\H DOÕQPÕúWÕU ELUH\H DLW YHUL NXOODQÕOPÕúWÕU .XOODQÕODn aday
GH÷LúNHQOHUYH NRGODPDLúOHPL Tablo 1¶GHYHULOPLúWLU
63
Tablo 1. Uygulama verileri ve kodlDPDLúOHPL
%D÷ÕPOÕGH÷LúNHQ
$OGÕ÷ÕGH÷HUOHUYHNRGODUÕ
0=DA t 2500 gr
Y
'R÷XPD÷ÕUOÕ÷Õ
1=DA<2500 gr
%D÷ÕPVÕ]GH÷LúNHQOHU
øON KDIWDGDNL yüksek 0,1,2,......
x1
DWHúVD\ÕVÕ
$QQHQLQ\DúÕ
0=yas<35
x
2
x3
x4
*HoPLúWHNLGúNVD\ÕVÕ
gQFHNL
VD\ÕVÕ
FDQOÕ
1=yas t 35
GúüN\DSPDPÕúVD
GL÷HUG
GR÷XP FDQOÕGR÷XP\DSPDPÕúVD
x5
'R÷XPGDJHEHOLNKDIWDVÕ
GL÷HUG
6UHNOLGH÷LúNHQGLU
x6
Sigara NXOODQÕPÕ
0=Yok
x7
Alkol tüketimi
1=Var
0=Yok
x8
Kahve tüketimi
1=Var
0=Yok
Boy
1=Var
6UHNOLGH÷LúNHQGLUFP
x9
.ÕVDOWPDVÕ
DA
DWHú
yer
Gúük
FDQOÕ
hafta
sigara
alkol
kahve
boy
3.1 Verilerin MARS Yöntemi ile Analizi
Verilerin çok dH÷LúNHQOL X\DUOD\ÕFÕ UHJUHV\RQ X]DQÕPODUÕ 0$56 \|QWHPL\OH DQDOL]L
D\QÕ DGÕ WDúÕ\DQ 0$56 SDNHW SURJUDPÕ LOH \DSÕOPÕúWÕU %X SURJUDPÕQ 0$56 sürümü, 6DOIRUG6\VWHPVWDUDIÕQGDQJHOLúWLULOHQ6DOIRUG3UHGLFWLYH0RGHOHU%XLOGHUDGOÕ
paket program içinde yer DOPDNWDGÕU %X SURJUDP &$57 0$56 &RPELQH
5DQGRPIRUHVW 7UHHQHW JLEL GH÷LúLN SDUDPHWULN ROPD\DQ DQDOL] \|QWHPlerini içeren
SDNHW SURJUDPGÕU 0$56 SURJUDPÕQGD YHULOHU JLULOGLNWHQ VRQUD D\QÕ YHULOHUOH ID]OD
VD\ÕGDPRGHOROXúWXUXODELOLU%XPRGHOOHUGHQHQL\LRODQÕHQD]X\XPHNVLNOL÷LQHVDKLS
RODQGÕU 8\XP HNVLNOL÷L HQ D] RODQ PRGHO LVH HQ GúN *d* GH÷HULQH VDKLS RODQGÕU
MARS program oÕNWÕODUÕQGDPRGHOHDLW*d*GH÷HULYHULOPHNWHGLU
Maksimum temel IRQNVL\RQ VD\ÕVÕ PDNVLPXP HWNLOHúLP G÷POHU DUDVÕQGDNL en az
J|]OHP VD\ÕVÕ G÷P RSWLPL]DV\RQX LoLQ VHUEHVWOLN GHUHFHVL JLEL GH÷HUOHU ROXúWXUXODQ
modeli belirler (Tablo 2)%XGH÷HUOHUWDPDPHQX\JXOD\ÕFÕ\DEÕUDNÕOPÕúWÕU
64
Tablo 0$56¶WDPRGHOROXúXPXQGDHWNLOLRODQGH÷HUOHUYHNÕVDOWPDODUÕ
MARS’taki iONGH÷HUL
.ÕVDOWPDVÕ
0DNVLPXPWHPHOIRQNVL\RQVD\ÕVÕ
15
TFmax
0DNVLPXPHWNLOHúLPVD\ÕVÕ
1
Emax
'÷POHUDUDVÕQGDNLHQD]J|]OHPVD\ÕVÕ
0
Omin
'÷PRSWLPL]DV\RQXLoLQVHUEHVWOLNGHUHFHVL
3
SD
Model tahmininde öncelikle serbestlik derecesi 3 olarak beOLUOHQPLúWLU )DUNOÕ Tfmax,
(PD[YH2PLQGH÷HUOHULLoLQ*d*GH÷HUOHULND\GHGLOPLúWLUTablo 3-6).
Tablo (PD[ LoLQ*d*GH÷HUOHUL
TFmax
Omin
15
0
0.04499
20
0.04499
30
0.04508
40
0.04719
50
0.04719
100
0.04912
25
0.04506
0.04512
0.04508
0.04734
0.04734
0.04931
50
0.04542
0.04580
0.04553
0.04802
0.04802
0.04966
TFmax
Omin
15
0
0.04522
20
0.04522
30
0.04504
40
0.04735
50
0.04735
100
0.04914
25
0.04476
0.04517
0.04521
0.04734
0.04735
0.04920
50
0.04432
0.04527
0.04531
0.04742
0.04748
0.04938
TFmax
Omin
15
0
0.04522
20
0.04522
30
0.04504
40
0.04735
50
0.04735
100
0.04914
25
0.04476
0.04517
0.04521
0.04734
0.04734
0.04918
50
0.04369
0.04546
0.04513
0.04749
0.04754
0.04922
65
TFmax
Omin
15
0
0.04522
20
0.04522
30
0.04504
40
0.04735
50
0.04735
100
0.04914
25
0.04476
0.04517
0.04506
0.04734
0.04734
0.04918
50
0.04320
0.04530
0.04525
0.04753
0.04753
0.04922
<DSÕODQ X\JXODPDODU VRQXFXQGD *d* GH÷HULQLQ 7)PD[¶ÕQ YH 2PLQ¶LQ ¶D \DNÕQ
ROGX÷XQGD HQ NoN GH÷HUOHUH VDKLS ROGX÷X J|UOPúWU (PD[ GH÷HULQLQ ROGX÷X
GXUXPGDLVHEXGH÷HUHQNoNROPDNWDGÕUTablo 6).
Tfmax=50, Omin= (PD[ ROGX÷XQGD GH÷LúLN 6' GH÷HUOHUL için hesaplanan GÇG
GH÷HUOHUL LQFHOHQGL÷LQGH 6' GH÷HUL D]DOGÕNoD *d* GH÷HULQLQ D]DOGÕ÷Õ YH RSWLPXP
modele yakODúÕOGÕ÷ÕJ|UOPúWUTablo7).
Tablo'H÷LúLN6'GH÷HUOHULLoLQKHVDSODQDQ*d*GH÷HUOHUL
*HQHOOHúWLULOPLú dDSUD]
SD
*d*'H÷HUL
0
0.04018
1
0.04141
2
0.04234
3
0.04320
*HoHUOLOLN
2OXúWXUXODQ EX PRGHOOHUGHQ HQ NoN *d* GH÷HULQH VDKLS RODQ PRGHO X\JXQ PRGHO
RODUDNVHoLOPLúYHEXPRGHOHDLWoÕNWÕODU DúD÷ÕGD\RUXPODQPÕúWÕU
7IPD[ 2PLQ (PD[ YH6' GH÷HUOHULLOH0$56X\JXODQPÕúWÕU
MARS yönteminde sÕUDGDQ en küçük kareler (EKK) \|QWHPLQGH ROGX÷X JLEL WHPHO
IRQNVL\RQODULOHED÷ÕPOÕGH÷LúNHQDUDVÕQGDNLLOLúNinin derecesini gösteren üç tür R-kare
GH÷HULNXOODQÕOPDNWDGÕU:
R-Kare%D÷ÕPOÕGH÷LúNHQYHWHPHOIRQNVL\RQODUDUDVÕQGDNLLOLúNLQLQGHUHFHVL%XGH÷HU
EDVLW VÕUDGDQ HQ NoN NDUHOHU UHJUHV\RQ DQDOL]L LoLQ KHVDSODQDQ R 2 LOH D\QÕ úHNLOGH
KHVDSODQÕU
']HOWLOPLú5-Kare7HPHOIRQNVL\RQODUÕQVD\ÕVÕLoLQG]HOWLOPLúWLU
Merkezsel olmayan R-Kare 8\XP L\LOL÷LQL WHVW HWPHN LoLQ NXOODQÕOPDPDOÕGÕU 1DVK
ve Bradford, 2001).
Modele ait R-kDUHGH÷HUL\DNODúÕNRODUDNEXOXQPXúWXU']HOWLOPLú5-kare 0.64 ve
merkezsel olmayan R-NDUHRODUDNEXOXQPXúWXU
8\JXODPDVRQXFXQGDLOHULYHJHUL\HGR÷UXLúOH\HQ\|QWHPOHWHPHOfonksiyonlardan 17
WDQHVLQLQ PRGHO ROXúXPXQD NDWNÕ VD÷ODGÕ÷Õ J|UOPHNWHGLU %X IRQNVL\RQODUÕQ NDWVD\Õ
WDKPLQOHULVWDQGDUWKDWDODUÕWYHSGH÷HUOHULJ|UOPHNWHGLUTablo 8).
66
Tablo 0RGHOHNDWNÕVÕRODQWHPHOIRQNVL\RQODUYHGH÷HUOHUL
Parametre
Tahmin
S.H
t-'H÷HUL p-'H÷HUL
Sabit
Temel Fonksiyon 5
Temel Fonksiyon 9
Temel Fonksiyon 11
Temel Fonksiyon 13
Temel Fonksiyon 20
Temel Fonksiyon 21
Temel Fonksiyon 23
Temel Fonksiyon 27
Temel Fonksiyon 29
Temel Fonksiyon 31
Temel Fonksiyon 33
Temel Fonksiyon 35
Temel Fonksiyon 39
Temel Fonksiyon 41
Temel Fonksiyon 43
Temel Fonksiyon 45
Temel Fonksiyon 49
1.00630
0.60325
-0.37637
0.42711
-0.04797
-0.03822
-0.57432
-0.61113
-0.92835
0.44079
0.60825
-0.14866
0.04525
0.02204
0.04300
-0.03130
-0.02329
-0.08965
0.03693
0.08294
0.04868
0.06110
0.01628
0.01477
0.08996
0.08769
0.13938
0.09552
0.09022
0.06755
0.01951
0.00688
0.01453
0.01698
0.00655
0.01298
27.25161
7.27300
-7.73197
6.99032
-2.94576
-2.58753
-6.38400
-6.96930
-6.66081
4.61449
6.74164
-2.20068
2.31965
3.20159
2.95862
-1.84370
-3.55531
-6.90723
0.00000
0.00000
0.00000
0.00000
0.00330
0.00981
0.00000
0.00000
0.00000
0.00000
0.00000
0.02800
0.02057
0.00141
0.00317
0.06553
0.00040
0.00000
0RGHO GH÷HUOHQGLUPHVL LoLQ NXOODQÕODQ ) LVWDWLVWL÷L YH S GH÷HUL RODUDN
KHVDSODQPÕúWÕU
7DKPLQ HGLOHQ PRGHO VRQXFXQGD EHEH÷LQ GR÷XP D÷ÕUOÕ÷ÕQÕ, KDIWD GH÷LúNHQLQLQ HWNLOHGL÷LVÕUDVÕ\ODFDQOÕDONRO\DúYHVLJDUDGH÷LúNHQOHULQLQoRNWDQD]DGR÷UXHWNLVLQLQ
ROGX÷X J|UOPHNWHGLU Tablo 9). Tablodaki VRQ VWXQ LOJLOL GH÷LúNHQLQ \RNOX÷XQGD
PRGHOLQJHQHOX\XPL\LOL÷LQGHQHNDGDUOÕNDzDOPDRODFD÷Õ\HUDOPDNWDGÕU
Tablo 9øOLúNLOLGH÷LúNHQOHULQ|QHPL
'H÷LúNHQ
Önemi
*HQHOOHúWLULOPLú dDSUD]
Geçerlilik
(GÇG)
'H÷HUL
Hafta
100.00000
0.11055
&DQOÕ
18.24723
0.04253
Alkol
17.01658
0.04253
<Dú
16.85816
0.04253
9.55560
0.04253
Sigara
0RGHOLQ ROXúXPXQGD HWNLVL RODQ WHPHO IRQNVL\RQODU YH DoÕNODPDODUÕ Tablo 10’da
YHULOPLúWLU
67
Tablo 10. Temel fonksiyonlar
BF5 = max( 0, HAFTA - 38)
BF7 = (SIGARA in ( 0 ))
BF8 = (SIGARA in ( 1 ))
BF9 = max( 0, HAFTA - 35)
BF11 = max( 0, HAFTA - 36)
BF13 = max( 0, HAFTA - 37) * BF8
BF15 = (CANLI in ( 0 ))
BF17 = (YAS in ( 0 ))
BF18 = (YAS in ( 1 ))
BF20 = max( 0, 35 - HAFTA) * BF17
BF21 = max( 0, HAFTA - 38) * BF17
BF23 = max( 0, HAFTA - 37) * BF18
BF25 = (ALKOL in ( 1 )) * BF15
BF27 = max( 0, HAFTA - 36) * BF25
BF29 = max( 0, HAFTA - 38) * BF25
BF31 = max( 0, HAFTA - 35) * BF25
BF33 = max( 0, HAFTA - 39) * BF25
BF35 = max( 0, HAFTA - 40) * BF17
BF36 = max( 0, 40 - HAFTA) * BF17
BF39 = (CANLI in ( 0 )) * BF36
BF40 = (CANLI in ( 1 )) * BF36
BF41 = (ALKOL in ( 1 )) * BF40
BF43 = (SIGARA in ( 0 )) * BF41
BF45 = max( 0, HAFTA - 32) * BF7
BF49 = max( 0, HAFTA - 32) * BF17
68
%XWHPHOIRQNVL\RQODUYHNDWVD\ÕODUÕ\ODHOGHHGLOHQmodel LVHDúD÷ÕGDNLJLELRODFDNWÕU
Y = 1.00629 + 0.60322 * BF5 - 0.376346 * BF9 + 0.42706 * BF11
-0.0479715 * BF13 - 0.0382146 * BF20 - 0.574262 * BF21
-0.611076 * BF23 - 0.928152 * BF27 + 0.440698 * BF29
+ 0.608121 * BF31- 0.148641 * BF33 + 0.0452413 * BF35
+ 0.0220369 * BF39 + 0.0430005 * BF41 - 0.0313029 * BF43
- 0.023289 * BF45 - 0.0896456 * BF49;
'R÷UX VÕQÕIODPD RUDQÕ LVH PRGHOLQ X\JXQOX÷XQX \D GD YHULOHULQ ED÷ÕPOÕ GH÷LúNHQLQL
DoÕNODPD \]GHVLQL J|VWHUHQ ELU GH÷HUGLU 0RGHOH DLW '62 GH÷HUL RODUDN
KHVDSODQPÕúWÕU øOJLOL GH÷LúNHQOHU ED÷ÕPOÕ GH÷LúNHQLQ \DNODúÕN %95’ini DoÕNODPÕúWÕU
(Tablo11).
'R÷UX VÕQÕIODPD oL]HOJHOHULQGHQ PRGHOLQ GX\DUOÕOÕ÷Õ VHQVLWLYLW\ YH |]JOO÷
VSHFLILFLW\KDNNÕQGDGD\RUXP\DSÕODELOLU
Tablo 11. MARS için sÕQÕIODQGÕUPDoL]HOJHVL
Tahmin edilen
Gözlenen
DA
0.00
$GÕP'$ 850
0.00
1.00
37
toplam
1.00
8
87
858
124
95.418
'X\DUOÕOÕN UHIHUDQV G]H\LQ GR÷UX WDKPLQ HGLOPH GHUHFHVLGLU 0$56 LoLQ UHIHUDQV
G]H\ GR÷XP D÷ÕUOÕ÷ÕQÕQ JUDP YH VWQ LIDGH HGHQ YH RODUDN NRGODQPÕú
düzeydir. Bu durumda GX\DUOÕOÕN GH÷HULQHVDKLSWLU
Özgüllük ise RODUDN NRGODQDQ GR÷XP D÷ÕUOÕ÷ÕQÕQ JUDPGDQ NoN ROGX÷X
GH÷HUOHULLIDGHHGHQG]H\LQGR÷UXWDKPLQHGLOPHVLQLQ derecesidir. MARS yöntemiyle
\DSÕODQDQDOL]LoLQ|]JOON, 87/124=0.7016 RODUDNKHVDSODQPÕúWÕU
4. SONUÇ VE YORUMLAR
5HJUHV\RQ DQDOL]LQGH DPDo ED÷ÕPOÕ GH÷LúNHQL ED÷ÕPVÕ] GH÷LúNHQOHULQ ELU IRQNVL\RQX
úHNOLQGH \D]PDNWÕU 3DUDPHWULN UHJUHV\RQ DQDOL]LQGH |QFHden bilinen modele ait
IRQNVL\RQXQSDUDPHWUHOHULQLWDKPLQHWPHNDPDoODQÕU3DUDPHWULNROPD\DQ\|QWHPGHLVH
SDUDPHWUH \HULQHGR÷UXGDQIRQNVL\RQODUWDKPLQHGLOLU0RGHOLVHEXIRQNVL\RQODUÕQELU
kombinasyonudur.
*HUoHNOHúWLULOHQ DUDúWÕUPDGD verilerin MARS yöQWHPLQH GD\DOÕ LNLOL ORMLVWLN UHJUHV\RQ
LOH \DSÕODQ DQDOL]i sonucunda GR÷UX VÕQÕIODQGÕUPD RUDQÕ %95.418 olarak EXOXQPXúWXU
%X RUDQ ROGXNoD \NVHN ELU RUDQGÕU 2OXúWXUXODQ PRGHOGH JHEHOLN KDIWDVÕ HQ ID]OD
HWNL\H VDKLS GH÷LúNHQ RODUDN RUWD\D oÕNPÕúWÕU Tablo $\QÕ úHNLOGH Tablo 10
69
LQFHOHQGL÷L ]DPDQ KDIWD GH÷LúNHQLQLQ WHN EDúÕQD ELU WHPHO IRQNVL\RQ ROXúWXUGX÷X JLEL
GL÷HU ELUoRN WHPHO IRQNVL\RQXQ ROXúXPXQGD GD \HU DOGÕ÷Õ J|UOPHNWHGLU +DIWD
GH÷LúNHQL VUHNOL GH÷LúNHQ ROGX÷X LoLQ ELUoRN G÷P QRNWDVÕ RUWD\D oÕNPÕúWÕU $QDOL]
VRQXFXQGD EHEH÷LQ GR÷XP D÷ÕUOÕ÷ÕQÕ - DUDVÕQGDNL KDIWDODUÕQ HWNLOHGL÷L WHPHO
IRQNVL\RQODUGDQ DQODúÕOPDNWDGÕU 0RGHOLQ WHPHO IRQNVL\RQODUÕQÕQ NDWVD\ÕODUÕ
LQFHOHQGL÷LQGH HQ E\N NDWVD\ÕODUGDQ ELULVLQLQ \LQH EX GH÷LúNHQH DLW ROGX÷X
görülmektedir.
%HEH÷LQ GR÷XP D÷ÕUOÕ÷ÕQÕ HWNLOH\HQ ELU GL÷HU |QHPOL GH÷LúNHQ LVH, DQQH DGD\ÕQÕQ GDKD
|QFH FDQOÕ GR÷XP \DSÕS \DSPDPDVÕGÕU 'DKD |QFH FDQOÕ GR÷XP \DSPDPÕú ELU ELUH\LQ
D\QÕ ]DPDQGD DONRO GH NXOODQÕ\RU ROPDVÕQÕQ EHEH÷LQ GúN GR÷XP D÷ÕUOÕNOÕ ROPDVÕQD
VHEHSROGX÷X, QXPDUDOÕWHPHOIRQNVL\RQGDQDQODúÕOPDNWDGÕUTablo 10).
<DúGH÷LúNHQLQLQGHEHEH÷LQGR÷XPD÷ÕUOÕ÷ÕQGDHWNLOLROGX÷X DQODúÕOPDNWDGÕU\DúÕQD
kadar RODQ \DúODUD DLW QXPDUDOÕ WHPHO IRQNVL\RQXQ GúN GR÷XP D÷ÕUOÕ÷Õna etki
HWPHGL÷L 35’LQ ]HULQGHNL \DúODUD DLW QXPDUDOÕ WHPHO IRQNVL\RQGD LVH \Dú
GH÷LúNHQLQLQWHPHOIRQNVL\RQNDWVD\ÕVÕNDGDUHtkili ROGX÷XDQODúÕOPDNWDGÕUTablo10).
$UDúWÕUPDGD NXOODQÕODQ LON RQ \HGL KDIWDGD DWHúOL KDVWDOÕN JHoLUPH YH NDKYH NXOODQPD
GXUXPODUÕ çok düúNPLNWDUGDHWNL\HVDKLSROGXNODUÕLoLQLoLQROXúWXUXODQPRGHOGH \HU
DOPDPÕúODUGÕU %XoDOÕúPDQÕQELUGHYDPÕQLWHOL÷LQGHoDOÕúPDGDNXOODQÕODQGH÷LúNHQOHUH
DOWHUQDWLI\HQLGH÷LúNHQOHULQYDUOÕ÷ÕYHEHEHNGR÷XPD÷ÕUOÕ÷ÕQDHWNLOHULDUDúWÕUÕODELOLU
5. KAYNAKLAR
Andersen A. M. N., Vastrup P., Wohlfahrt J., Amdersen P. K., Olsen J., Melbye M.,
2002. Fever in pregnancy and risk of fetal death: a cohort study. Lancet, 360: 15521556.
Craven, P., Wahba, G., 1979. Smoothing Noisy Data with Spline Functions. Numer.
Math, 31: 377-403.
Dieterle, F. J., 2003. Multianalyte Quantifications by Means of Integration of Artificial
Neural Networks, Genetic Algorithms and Chemometrics for Time-Resolved Analytical
Data. Ph.D. thesis, Universität Tübingen, Tübingen, 183 s.
Friedman, J. H., 1991. Multivariate Adaptive Regression Splines (with discussion).
Ann. of Statistics, 19: 1-141.
Hastie, T. J., Tibshirani, R. J., 1990. Generalized Additive Models. Chapman &
Hall/CRC, New York, 335s.
Kan, B., <D]ÕFÕ %, 2010. Comparison of the Results of Factorial Experiments,
Fractional Factorial Experiments, Regression Trees and MARS for Fuel Consumption
Data. WSEAS TRANS. on MAT., 2:110-119.
Kayri, M., 2010. The Analysis of Internet Addiction Scale Using Multivariate Adaptive
Regression Splines. Iranian J Publ Health, 39: 51-63.
70
Kim, J. H., 2000. MARS Modeling for Ordinal Categorical Response Data: A Case
Study. Soongsil University.
Kolyshkina, I., Brookes, R., 2002. Data Mining Approaches to Modelling Insurance
Risk. Electronic Version, Pricewaterhouse Coopers, New York, 20 s.
Kramer, M. S., 1987. Determinants of Low Birth Weight: Methodological Assessment
and Meta-analysis. Bull World Health Organ. 1987;65(5):663-737.
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AdditiveRegression Splines) Logistic Regressions for Prediction of A Dichotomous
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States Environmental Protection Agency (EPA), USA, 40s.
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Regression Splines (MARS) in QSRR, IEJMD. BioChemPress.com.
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Modeling Using the B34S ProSeries Econometric System and SCA WorkBench.
6FLHQWL¿F&RPSXWLQJ$VVRFLDWHV&RUSV
71
Topak, M. S., 2011. An Empirical Study to Model Corporate Failures In Turkey: A
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Kemal Üniversitesi Sos. Bilim. Metinleri, 15 s.
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Tahmini. 2'7h*HOLúWLUPH'HUJLVL-454.
Tunay, K. B., 2010. %DQNDFÕOÕN .UL]OHUL YH (UNHQ 8\DUÕ 6LVWHPOHUL 7UN %DQNDFÕOÕN
SeNW|U øoLQ%LU0RGHOgQHULVL%''.%DQNDFÕOÕNYH)LQDQVDO3L\DVDODU'HUJLVL46.
Tunay, K. B., 2011. 7UNL\H¶GH 'XUJXQOXNODUÕQ 0$56 <|QWHPL\OH 7DKPLQ YH
Kestirimi. 0DUPDUDhQLYHUVLWHVLøø%)'HUJLVL;;;-91.
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MODELLING THE LOW BIRTH WEIGHT OF NEW BORN
BABIES WITH BINARY LOGISTIC REGRESSION BASED ON
MARS METHOD
ABSTRACT
Babies with low birth weight have some health problems in later years.
Therefore, it is important to estimate before the birth whether a new born
baby will have a low birth weight or not. In order to obtain this estimation,
logistic regression model is a suitable choice. Logistic regression analysis is
a modelling technique which is used when the dependent variable is
categorical. It is also easily interpreted. When the dependent variable has
only two categories, the logistic regression is called binary logistic
regression. Logistic regression has parametric and nonparametric solutions.
MARS method is a nonparametric method which can be used as an
alternative to the parametric solutions in the analysis of logistic regression.
The nonparametric models require fewer assumptions compared to the
parametric ones and they are also more flexible. In the application, a binary
logistic regression model has been fitted to estimate whether a new born
baby will have a low birth weight or not. The model has been estimated
based on the MARS method. In the analysis, data belonging to 982 subjects
have been investigated by applying the MARS software. In the conclusion
part, the findings are interpreted.
Keywords: Low birth weight, Logistic regression, Maximum likelihood, MARS.
72
73
Yasemin KAYHAN, Süleyman GÜNAY
represents the distribution of the p-dimensional observations over two dimensional
space. On the second graph, the relations between variables are shown by vectors, and
the location of each vector is determined by mapped observations. With this main
feature, CoPlot gives researcher an opportunity to make deeper and richer
interpretations of the multivariate data (Lipshitz and Raveh, 1998).
Several disciplines such as econometrics (Huang and Liao, 2012), medicine (Bravata et
al., 2008), computer science (Talby et al., 1999), management science (Weber et al.,
1996) require the analysis of complex multivariate data often coming from large data
sets describing numerous variables for many subjects, and the multivariate nature of
these data make it difficult to assess the associations of the predictors and the outcomes
of interest. So, CoPlot can be seen as a valuable exploratory analysis tool in such
analyses.
The objectives of this paper are simply trying to introduce the CoPlot and presenting an
application of this method on a demographic data. Although CoPlot have been used
previously in several disciplines, it has not been used on demographic data sets. So, this
study can be considered as useful in the sense that there are not many applications of
CoPlot available in the literature. Analyzing Turkey Demographic and Health Survey
2008 data set with only CoPlot map does not give enough evidence to make any
conclusion in general. To get deeper understanding on the issues investigated, more
works on detailed statistical analysis and inferences would be needed.
In the following section, general description and methodology of CoPlot are given. In
section 3, an application of CoPlot on a part of Turkey Demographic and Health Survey
2008 data set is presented. By using an original data set, interpretive superiority of
CoPlot over MDS is displayed. Some concluding remarks are given in the final part.
2. GENERAL DESCRIPTION OF COPLOT
The final product of CoPlot is a simple picture of the multidimensional dataset. With
this plot, one can observe: the similarity between the observations, the correlations
among the variables and the mutual relationships between the observations and the
variables. The main advantage of CoPlot over a MDS is that the clusters of observations
which are highly characterized by a particular variable is mapped together and located
in the same direction as that of the variable’s vector (Bravata et al., 2008). CoPlot
analysis is performed in four steps. MDS analysis is executed in the first three steps, and
in the last step the variables’ vectors are placed on top of the obtained MDS graph.
Let’s assume that X is an nxp data matrix. The rows of this matrix are assumed as the
observations and will be denoted as n points on two dimensional space. The columns of
the data matrix correspond to the variables which are exhibited by p vectors on CoPlot
map.
Step 1: Standardization of the Data Matrix
The dissimilarities between the observations are transformed into distances by using
City-Block distance in CoPlot. That is why the variables measured on different scales
affect the distance values evaluated by City-Block in such a way that the largest
variance variable will dominate the distance measure (Borg, 2005). So the data matrix
X is transformed into matrix Z as follows,
74
An Application of CoPlot on Turkey Demographic
and Health Survey 2008 Data
Türkiye Nüfus ve Sağlık Araştırması 2008 Verisi
Üzerinde bir CoPlot Uygulaması
Zij
x
Xj
ij
Sj
,
i
1, 2, , n ; j 1, 2, , p
(1)
where X j and S j represent mean and standard deviation of the j-th column of X ,
respectively.
Step 2: Creating distance matrix
At this step, dissimilarities between the observations are transformed into distances by
using a proper distance function. The distance between each pair of observations is
denoted in a symmetric Dnxn d ij
matrix. To generate Dnxn , 2-combination of n,
C n, 2 , different dij distances are calculated as follows,
p
d ij
¦Z
ik
Z jk ! 0
(2)
k 1
Step 3: Mapping Distances
The representation of n observations on two-dimensional space is generated during this
step. If the rank-order of the proximities implies that the distances must have the same
rank-order as proximities, we speak of non-metric MDS. In non-metric MDS, it is
accepted that rank-order of the distances between the observations are informative. In
this case, disparities (dˆij ) which are obtained from Euclidean distances of MDS
coordinate matrix Y are assigned to the proximities in such a way that these values
display the same rank-order as the data (Kruskal, 1964a; Borg, 2005).
The aim of non-metric MDS is to obtain a coordinate matrix Y such that the distance
between rows i and j of Y, dij Y , matches disparities as closely as possible. Namely,
MDS tries to minimize the following cost function,
ˆ Y
V 2 d,
¦¦ w dˆ
n
i 1
i 2
j
ij
ij
d ij Y 2
(3)
This function is known as raw-Stress introduced by Kruskal (1964a), and the value of
the function measures the quality of MDS representation. The wij is a user defined
weight that should be non-negative and relates to missing information. The
minimization of this function has no closed form solution, so it must be solved by
iterative algorithms. Within these iterative algorithms, one of them is known as
SMACOF (De Leeuw, 1977). By using this iterative majorization algorithm, optimal Y
can be obtained.
Step 4: Adding Vectors
At this step, vectors corresponding to the variables are drawn onto the map obtained
from Step 3. For each variable, CoPlot produces a vector coming up from the center of
mass of the points denoted on the MDS map.
75
To find the direction of the vector j, initial angle between the j-th vector and the x axis is
assumed to be zero. Then projections of all points on to the vector are calculated. The
goal is to find the angle which maximizes the correlation between n projected scores
and the z-score values for variable j. The angle which makes the correlation maximum
decides the direction of the vector for variable j. This procedure is separately performed
for all variables in the data set (Lipshitz and Raveh, 1998).
This vector representation has some advantages for the interpretation of the data. By
considering the correlation values of the vectors, it may be decided that which variables
should be kept in graphical representation or discarded. The vectors for highly
correlated variables are located in the same direction. The vectors for highly negatively
correlated variables are located along the same axis but in opposite directions. Two
vectors which are orthogonal to each other imply that corresponding variables are not
correlated. Obviously, the main advantage of this representation is to allow the
simultaneous consideration of both variables and observations.
3. APPLICATION
For a better explanation of the procedure, some applications of the use of CoPlot are
given in this section. The used data set, Turkey Demographic and Health Survey - 2008,
is taken from Hacettepe University Institute of Population Studies (TDHS-2008, with
the permission number 2012/8). 2,473 women who have terminated their pregnancy
within the duration of their marriage are selected. Respondents (observations) are
separated with respect to 12 regions. Table 1 represents the number of respondents
within corresponding regions.
Table 1. Number of respondents from each region
Regions
Number of Respondents
Istanbul
187
West Marmara
134
East Marmara
199
Aegean
199
Central Anatolia
191
Mediterranean
333
West Black Sea
232
East Black Sea
124
West Anatolia
160
North East Anatolia
188
Central East Anatolia
204
South East Anatolia
322
Ten variables such as; respondent’s current age (1), number of household member (2),
wealth index which is an indicator of the level of wealth (3), total children ever born (4),
age of respondent at 1st birth (5), age of respondent at first marriage (6), years since
first marriage (7), partner’s educational attainment (8), partner’s age (9), number of
living children (10) are selected from the survey data set. Observations are classified on
the MDS and CoPlot maps with respect to the respondent’s educational attainment as
76
follows; not educated women – square shaped points, highest educated women – cross
shaped points, and the rest of the women – circle shaped points. Obtained Figure 1 and
Figure 2 for the first region (Istanbul) represent distinction between MDS and CoPlot,
and display what is gained in interpretability. Figure 1 shows the map produced by
MDS of the ten variables describing 187 respondents using City-Block distance to
calculate the dissimilarities between the observations. Figure 1(a) shows the embedding
of 187 observations in a two dimensional space, and high educated women and not
educated women generate two different clusters. In Figure 1(b), each variable is shown
as a point and the points are arranged in such a way that their distances correspond to
the correlations. That is, two points are close to each other (such as Wealth Index and
Number of Household Members), if their corresponding correlation is high. However,
one cannot decide the direction of this correlation. Conversely, Wealth Index and
Number of Household Members are found as highly negatively correlated variables, see
Figure 2. With MDS analysis, it is possible to obtain a graph that displays the relations
between either the observations or the relations between variables, but seeing the
observations and variables at the same graph simultaneously is not possible.
MDS (V1=0.15482)
MDS (V1=0.01912)
Age of Respondent at 1 st Birth
Age at 1 st M arriage
Y ears Since 1 st M arriage
Number of Household M embers
Wealth Index
Partner's Educational Attainment
Total Children Ever Born
Number of Living Children
Current Age
Partner's Age
Figure 1. MDS representations of observations (a) and variables (b) for Istanbul
The map of Coplot is a simple picture of the multivariate data. From Figure 2, the
following results can be concluded: Ages of the women who live in Istanbul are
generally the same with the ages of their husband (1 and 9). Number of living children
is highly correlated with the number of total children ever born (4 and 10). So it can be
thought that child mortality is not high in this region. Age of 1-st marriage and age of 1st birth are close to each other (5 and 6). So it can be said that women got pregnant
when they got married. From the vectors 2 and 8, it is said that well educated husband
do not prefer to live with big families. As the distribution of the observations and the
directions of the vector are considered together, it can be said that, well educated
women prefer to become mother at later age, and well educated women are richer. Not
educated women have more children, and they live with crowded families. It is obvious
that CoPlot map of the data set gives much more information.
77
CoPlot (V1=0.15482)
1(0.93558)
9(0.89522)
7(0.91047)
3(0.65769)
5(0.66184)
6(0.62672)
8(0.52303)
10(0.90381)
4(0.90025)
2(0.73542)
Figure 2. CoPlot representation of Istanbul
The following findings with CoPlot about rest of the regions are indicated by Figure 3
and Figure 4: For Aegean region, obtained map, therefore the comment, is nearly same
as Istanbul. For Central Anatolia, variables 1, 7 and 9 are correlated. It may be
concluded that not educated women got married at an early age with peers. For Central
East Anatolia, not educated women rate is high relative to Istanbul, Aegean and Central
Anatolia. For East Marmara, high educated women’s number of living children and
number of total children ever born are low (4 and 10 are in the opposite direction of
cross-shaped cluster). Similar to Central East Anatolia, in Mediterranean region, the
number of living children and the number of total children ever born are high for not
educated women (4 and 10 are in the same direction of square-shaped cluster). For East
Blacksea, high educated women get married and be mother at a late age. Similar
comments can be given for the rest of the regions. We do not claim that a single map
can be sufficient for these comments. These results need to be discussed sociologically
and/or psychologically in details and need to be extended with more comparative
statistical analysis. However, CoPlot is useful to give a researcher insight into
understanding the possible relations in the data and for further analysis.
CoPlot (V1=0.16481)
9(0.9098)
1(0.9014)
CoPlot (V1=0.17752)
3(0.70608)
8(0.78411)
3(0.76993)
2(0.71411)
9(0.91484)
1(0.89821)
7(0.955)
8(0.66242)
6(0.60104)
7(0.9087)
CoPlot (V1=0.15787)
10(0.90694)
4(0.90813)
5(0.30491)
5(0.6123)
6(0.27918)
6(0.66882)
4(0.88739)
2(0.31934)
4(0.81346)
10(0.78742)
10(0.85019)
5(0.64563)
8(0.51342)
3(0.61925)
7(0.93969)
1(0.90639)
9(0.88252)
2(0.62877)
Figure 3. CoPlot maps of Aegean, Central Anatolia and Central East Anatolia
78
East Marmara (V1=0.17094)
Mediterrenean (V1=0.16736)
5(0.72167)
2(0.59963)
6(0.66288)
3(0.56584)
East Blacksea (V1=0.16464)
2(0.67658)
9(0.89828)
1(0.93556)
8(0.51308)
7(0.92481)
10(0.84737)
10(0.85925)
4(0.87435)
4(0.83631)
4(0.87891)
6(0.66387)
5(0.64301)
8(0.6391)
3(0.62658)
8(0.30979)
10(0.87723)
7(0.9425)
9(0.90151)
1(0.90858)
5(0.80678)
2(0.68818)
7(0.95493)
3(0.52456)
6(0.83655)
9(0.88283)
1(0.93568)
South East Anatolia (V1=0.17139)
North East Anatolia (V1=0.16425)
1(0.94826)
9(0.90533)
3(0.80409)
2(0.65414)
8(0.67391)
5(0.57594)
7(0.94282)
West Anatolia (V1=0.16976)
2(0.7423)
10(0.87342)
6(0.55491)
10(0.89602)
4(0.90149)
4(0.86959)
10(0.88091)
4(0.88083)
6(0.51167)
2(0.73907)
1(0.89558)
3(0.68467)
10(0.828)
7(0.94185)
5(0.63106)
9(0.9013)
3(0.47927)
1(0.90515)
9(0.92552)
3(0.72243)
West Blacksea (V1=0.16954)
4(0.82587)
6(0.55792)
8(0.68832)
9(0.90104)
West Marmara (V1=0.18282)
8(0.49258)
6(0.5713)
7(0.94191)
8(0.59184)
2(0.61163)
7(0.94968)
5(0.57231)
5(0.46709)
1(0.9207)
10(0.85445)
4(0.86133)
2(0.5725)
7(0.89344)
8(0.58641)
9(0.92443)
3(0.4966)
1(0.91408)
5(0.66641)
6(0.65308)
Figure 4. CoPlot maps of 8 regions
Kruskal-stress, V1 , describes how well the map represents the observations in lower
dimension. Generally MDS representations having Kruskal stress value 0.10 are
accepted as fair and over 0.20 as poor (Kruskal, 1964b). To improve the goodness of fit,
investigating the scree plot may be helpful. For example, from Figure 4, West Marmara
has the highest V1 value. Figure 5 is the scree plot of this region for 30, 90 and 134
observations. For 30 observations, since V1 is almost 0.15, two dimensional MDS
representation of the region can be accepted as fair. For 134 observations, to reach a fair
representation at least 3 dimensions are needed. It is obvious that, V1 decreases with
decreasing number of observations.
From Figure 4, it can be seen that, some of these ten variables do not have high
correlation coefficient values for every region. For example, at East Black Sea,
correlation coefficient values of vectors 3 and 8 are low (0.52456 and 0.30979,
respectively). In CoPlot map, low correlation coefficient value, namely less than 0.70,
implies that that variable is not interpretive. From Figure 6, when these two vectors are
79
discarded from the CoPlot analysis, it is seen that correlation coefficient values of the
rest of the variables are increasing and the V1 is reducing. Therefore, it can be
suggested that vectors with low correlation coefficients should be omitted from the
analysis.
Scree Plot - West Marmara
0.35
n=30
n=90
n=134
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
Figure 5. Scree plot for West Marmara
East Blacksea (V1=0.16464)
East Blacksea Reduced (V1=0.11648)
2(0.67658)
6(0.84574)
9(0.88955)
5(0.81543)
7(0.95115)
10(0.85925)
4(0.87435)
8(0.30979)
5(0.80678)
4(0.88367)
7(0.95493)
10(0.87847)
3(0.52456)
6(0.83655)
9(0.88283)
1(0.93631)
2(0.70636)
1(0.93568)
Figure 6. Effect of reducing the low correlated variables on CoPlot for East Black Sea
4. CONCLUSION
CoPlot is a graphical representation of a multivariate data set by projecting onto the
two-dimensional space. It provides the possibility to analyze the observations and
variables on the same graph. Previously, CoPlot has been applied to data sets from
different disciplines (Raveh, 2000; Bravata et al., 2008). In this study, it has been
applied to a demographic data for the first time. The purpose of this study is to show
how to use CoPlot for a demographic data set. Our analysis of the demographic data
suggests that CoPlot may be a useful analyzing tool for exploring the relation between
observations and variables in multivariate data. At the application part of this study, it
can be seen that with a simple analysis one can roughly get the picture of the regions,
80
and with these pictures similarities or dissimilarities between the regions would be
observed. In CoPlot analysis, there are two goodness of fit measures such as Kruskall
raw stress value and Pearson correlation coefficient. These measures enable the user
whether to keep or delete specific variables and observations from the map. Possible
problems and solutions of the problems such as low goodness of fit or correlation
coefficient value are also discussed in the application part. Besides, the superiority of
CoPlot on the visual interpretation of multivariate data is emphasized.
5. REFERENCES
Borg, I., Groenen, P. J. F., 2005. Modern Multidimensional Scaling, 2nd edition,
Springer.
Bravata, D. M., Shojania, K. G., Olkin, I., Raveh, A., 2008. CoPlot: A Tool for
Visualizing Multivariate Data in Medicine, Statistics in Medicine, 27, 2234-2247.
De Leeuw, J., 1977. Application of Convex Analysis to Multidimensional Scaling, In: J.
Barra et al. Recent Developments in Statistics, 133-145.
Hastie, T., Tibshirani, R., Friedman, J., 2008. The Elements of Statistical Learning, 2nd
edition.
Huang, H., Liao, W., 2012. A CoPlot-based Efficiency Measurement to Commercial
Banks, Journal of Software 7:10, 2247-2251.
Kruskal, J. B., 1964a. Nonmetric Multidimensional Scaling: A Numerical Method,
Psychometrika, 29:2, 115-129.
Kruskal, J. B., 1964b. Multidimensional Scaling by Optimizing Goodness of Fit to A
Nonmetric Hypothesis, Psychometrika, 29:1, 1-27.
Lipshitz, G., Raveh, A., 1998. Socio-economic Differences Among Localities: A New
Method of Multivariate Analysis, Regional Studies, 32:8, 747-757.
Raveh, A., 2000. CoPlot: A Graphic Display Method for Geometrical Representations
of MCDM, European Journal of Operational Research, 125, 670-678.
Talby, D., Feitelson, D. G., Raveh, A., 1999. Comparing Logs and Models of Parallel
Workloads Using the CoPlot Method, Lecture Notes in Computer Science 1659, 43-66.
Weber, Y., Shenkar, O., Raveh, A., 1996. National and Corporate Cultural Fit in
Mergers/Acquisitions: An Exploratory Study, Management Science 42:8, 1215-1227.
81
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VD\ÕGD oRN GH÷LúNHQOL J|]OHPLQ LNL ER\XWOX X]D\GDNL GD÷ÕOÕPÕQÕ J|VWHULU
øNLQFLJUDILNKHUELULELUGH÷LúNHQLWHPVLOHGHQSVD\ÕGDRNWDQROXúXU. CoPlot
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ROGX÷X LoLQ VRV\R-HNRQRPL HNRQRPL WÕS JLEL oHúLWOi disiplinlerde
NXOODQÕOPÕúWÕU, ancak GHPRJUDILN oDOÕúPDODUGD NXOODQÕOPDPÕúWÕU Bu
oDOÕúPDGD &R3ORW NÕVDFD DoÕNODQDFDN YH “Türkiye Demografik YH 6D÷OÕN
Anketi 2008” YHUL NPHVLQLQ ELU SDUoDVÕ ]HULQGH EDVLW ELU X\JXODPDVÕ
VXQXODFDNWÕU &R3ORW \|QWHPLQLQ oRN GH÷LúNHQOL YHUi kümesini görsel olarak
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Anahtar Kelimeler: Kruskal raw stress, Çok boyutlu ölçekleme, Scree plot.
82
DANIŞMA KURULU ÜYELERİ - ADVISORY BOARD MEMBERS
İSTATİSTİK
Research
Temmuz-July 2013
ISSN 1303-6319
Turkish Statistical Institute
Ali YAZICI
Alper GÜVEL
Asaf Savaş AKAT
Aşır GENÇ
Aydın ÖZTÜRK
Ayşe GÜNDÜZ HOŞGÖR
Bedriye SARAÇOĞLU
Coşkun Can AKTAN
Deniz GÖKÇE
Ekrem ERDEM
Ercan UYGUR
Erdem BAŞCI
Erinç YELDAN
Erol TAYMAZ
Eser KARAKAŞ
Fatih ÖZATAY
Fatin SEZGİN
Fikri AKDENİZ
Fikri ÖZTÜRK
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İSTATİSTİK ARAŞTIRMA DERGİSİ Journal of Statistical Research İ l

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