ANALISIS FUNGSI TAHAN HIDUP DENGAN MENGGUNAKAN PERLUASAN MODEL REGRESI COX PADA KASUS FUNGSI KEGAGALAN TAK PROPORSIONAL - Diponegoro University | Institutional Repository (UNDIP-IR)

DAFTAR ISI
Halaman
HALAMAN JUDUL ................................................................................................

i

HALAMAN PENGESAHAN ..................................................................................

ii

KATA PENGANTAR .............................................................................................. iv
ABSTRAK ................................................................................................................ vi
ABSTACT ................................................................................................................ vii
DAFTAR ISI ............................................................................................................. viii
DAFTAR SIMBOL .................................................................................................. xi
DAFTAR TABEL ..................................................................................................... xiii
DAFTAR GAMBAR ................................................................................................ xiv
DAFTAR LAMPIRAN ............................................................................................. xv

BAB I

PENDAHULUAN ....................................................................................

1

1.1 Latar Belakang ...................................................................................

1

1.2 Perumusan Masalah ...........................................................................

3

1.3 Pembatasan Masalah ..........................................................................

3

1.4 Tujuan Penulisan ................................................................................

3

1.5 Manfaat Penulisan ..............................................................................

4

1.6 Sistematika Penulisan ........................................................................

4

BAB II TINJAUAN PUSTAKA ...........................................................................

5

2.1 Konsep Dasar Uji Tahan Hidup ........................................................

5

2.1.1 Fungsi Tahan Hidup ( Fungsi Survival )..................................

6

2.1.2 Fungsi Densitas Peluang ..........................................................

7

2.1.3 Fungsi Kegagalan ( Fungsi Hazard ) .......................................

7

2.2 Hubungan antara Fungsi – Fungsi Tahan Hidup ..............................

8

2.3 Tipe – Tipe Penyensoran...................................................................

9

2.3.1 Sensor Tipe I ............................................................................ 10
2.3.2 Sensor Tipe II ........................................................................... 11
2.3.3 Sensor Tipe III ......................................................................... 13

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2.4 Metode Maksimum Likelihood ( Maximum Likelihood Estimator
/MLE ) ............................................................................................. 14
2.5 Matriks .............................................................................................. 16
2.5.1 Definisi Matriks ....................................................................... 16
2.5.2 Matriks Transpose .................................................................... 17
2.5.3 Matriks Invers ( Kebalikan ) .................................................... 17
2.5.4 Matriks Hessian ....................................................................... 17
2.6 Prosedur Newton - Raphson ............................................................ 18
2.7 Asumsi Fungsi Kegagalan Proporsional .......................................... 19
2.7.1 Model Fungsi Kegagalan Proporsional .................................... 19
2.8 Uji Logrank ...................................................................................... 20
2.9 Kaplan Meier Estimator ................................................................... 21

BAB III PERLUASAN MODEL REGRESI COX ................................................ 23
3.1 Model Fungsi Kegagalan Tak Proporsional ..................................... 23
3.2 Perluasan Model Regrsi Cox ............................................................ 24
3.2.1 Estimasi Parameter .................................................................. 24
3.2.2 Taksiran Fungsi Dasar Tahan Hidup dan Fungsi Dasar

Kegagalan ............................................................................... 32
3.3 Pengujian Asumsi Fungsi Kegagalan Proporsional ........................... 33
3.3.1 Menilai Variabel (Waktu) Bebas yang Tidak Memenuhi
Asumsi Fungsi Kegagalan Proporsional ................................ 34
3.4 Pengujian Parameter .......................................................................... 35
3.4.1 Pengujian secara Serentak ........................................................ 35
3.4.2 Pengujian secara Parsial ........................................................... 36
3.5 Flowchart ............................................................................................ 37
3.6 Aplikasi Metode Perluasan Model Regresi Cox ................................ 38
3.6.1 Contoh Kasus ........................................................................... 38
3.6.2 Identifikasi Variabel ................................................................. 38
3.7 Analisis Deskriptif secara Keseluruhan ............................................. 40
3.8 Analisis Deskriptif Tiap Variabel ...................................................... 41
3.8.1 Klinik ........................................................................................ 41

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3.8.2 Prison Record ........................................................................... 43
3.8.3 Dosis Methadone ..................................................................... 45
3.9 Uji Log Rank..................................................................................... 47

3.9.1 Waktu Ketahanan Hidup Pasien ............................................. 48
3.10 Asumsi Pemodelan ............................................................................ 48
3.10.1 Pemeriksaan Variabel (waktu) Bebas yang Tidak
Memenuhi Asumsi Fungsi Kegagalan Proporsional ............. 51
3.11 Pengujian Perluasan Model Regresi Cox .......................................... 54
3.11.1 Pengujian Serentak ................................................................. 54
3.11.2 Pengujian Parsial .................................................................... 55

BAB IV KESIMPULAN ....................................................................................... 60
DAFTAR PUSTAKA ............................................................................................... 62
LAMPIRAN .............................................................................................................. 64

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DAFTAR SIMBOL

T

: Variabel random waktu tahan hidup

t

: Waktu tahan hidup

n

: Jumlah komponen uji

r

: Jumlah komponen gagal dalam penyensoran

n0

: Banyaknya komponen uji pada kondisi tertentu

n f t 

: Banyaknya komponen yang gagal

ns t 

: Banyaknya komponen yang bertahan hidup

S(t)

: Fungsi tahan hidup

f(t)

: Fungsi densitas peluang dari T

F(t)

: Fungsi distribusi kumulatif

h(t)

: Fungsi kegagalan dari waktu tahan hidup T

H(t)

: Fungsi kegagalan kumulatif

�(t)

: Estimasi fungsi tahan hidup yang terjadi

ℎ( )

: Estimasi fungsi kegagalan



: Tingkat signifikansi

WL

: Statistik uji Log Rank

nj

: Jumlah kesesuaian pasien yang masih hidup pada waktu ke t(j)

dj

: Jumlah total kematian pasien pada waktu ke t(j)

ekj

: Nilai ekspektasi kematian pada waktu ke t(j) pada kelompok ke-k

dkj

: Jumlah kematian dari kelompok ke-k pada waktu ke t(j)

Vkj

: Varians dari dkj

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hi(t)

: Model fungsi kegagalan proporsional untuk individu ke-i

h0(t)

: Fungsi dasar kegagalan

β

: Parameter beta
: Parameter gamma

L(β)

: Fungsi likelihood

x(j)

: Variabel penjelas dari individu yang gagal pada saat ke-j
: Indikator sensoring
dimana, i =

�� �
�

0,
1,

U( )

: Vektor penaksir maksimum likelihood

H(β)

: Matrik Hessian

I ()

: Matrik informasi yang diamati

u (β s )

: vektor skor efisien

� −1 (β s )

: Invers matriks informasi yang diamati
: Estimasi schoenfeld residual dari variabel j untuk individu ke i

2

: Statistik Uji Chi-square

Wj

: Statistik Uji Wald

Exp(B)

: Hazard ratio atau odds ratio

-2LL

: - 2 Log Likelihood

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DAFTAR TABEL

Tabel 3.1 Statistik Deskriptif Pasien Pecandu Heroin ......................................... 40
Tabel 3.2 Deskriptif Pasien Pecandu Heroin Variabel Klinik ............................. 41
Tabel 3.3 Deskriptif Pasien Pecandu Heroin Variabel Prison Record ................. 43
Tabel 3.4 Deskriptif Pasien Pecandu Heroin Variabel Dosis Methadone ........... 45
Tabel 3.5 Statistik Uji Log Rank Tiga Faktor ...................................................... 47
Tabel 3.6 Uji Shcoenfeld Residual ...................................................................... 51
Tabel 3.7 Uji Parsial Model ................................................................................. 56

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DAFTAR GAMBAR

Gambar 2.1 Contoh Grafik Tersensor Tipe I ................................................................ 11
Gambar 2.2 Contoh Grafik Tersensor Tipe II ............................................................... 12
Gambar 2.3 Contoh Grafik Tersensor Tipe III ............................................................. 13
Gambar 3.1 Prosedur Analisis Data .............................................................................. 37
Gambar 3.2 Fungsi Tahan Hidup dan Fungsi Kegagalan Variabel Klinik ................... 42
Gambar 3.3 Fungsi Tahan Hidup dan Fungsi Kegagalan Variabel Prison Record........ 44
Gambar 3.4 Fungsi Tahan Hidup dan Fungsi Kegagalan Variabel Dosis
Methadone.................................................................................................. 46
Gambar 3.5 Plot elog [-elog S(t)] Terhadap Waktu Survival......................................... 49

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DAFTAR LAMPIRAN

Lampiran 1. Data Ketahanan Hidup Pecandu Heroin .......................................... 64
Lampiran 2. Taksiran Fungsi Tahan Hidup Tiap Variabel .................................. 68
Lampiran 3. Uji Logrank Variabel Bebas ........................................................... 81
Lampiran 4. Taksiran Fungsi Tahan Hidup dan Fungsi Kegagalan
Metode Kaplan-Meier ...................................................................... 83
Lampiran 5. Uji Schoenfeld Residual .................................................................. 88
Lampiran 6. Model Regresi Cox Fungsi Kegagalan Proporsional ..................... 89
Lampiran 7. Perluasan Model Regresi cox .......................................................... 90
Lampiran 8. Taksiran Fungsi Dasar Kegagalan .................................................. 92
Lampiran 9. Taksiran Fungsi Kegagalan dari Fungsi Dasar Kegagalan ............. 97

 

Lampiran 10. Tabel Distribusi Chi-Kuadrat  2 .............................................. 102

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