Ordinal Path (3)
May 9, 2019 | Author: Ainul Ichsan Prihantono | Category: N/A
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TUGAS ANALISIS MULTIVARIAT FAKTOR-FAKTOR YANG MEMPENGARUHI SEMANGAT BELAJAR MAHASISWA DENGAN ANALISIS JALUR ORDINAL Studi Kasus Mahasiswa Prodi Statistika Angkatan 2009 Universitas Brawijaya
Oleh: FITRIYANI DEWI M. (0810950042) EVELLIN DEWI LUSIANA (0910950037) KASIH MAHARANI (0910950045) (0910950045) LELI DWI KORNIASARI (0910950047) OKKY ARISKA AMELIA (0910950057) RANIKA PERMATA ADI (0910950059) RISHA ARDASARI UTAMA (0910950063)
PROGRAM STUDI STATISTIKA JURUSAN MATEMATIKA FAKULTAS MATEMATIKA & ILMU PENGETAHUAN ALAM UNIVERSITAS BRAWIJAYA MALANG 2012
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BAB I PENDAHULUAN
1.1.Latar Belakang
Prestasi belajar dari mahasiswa biasanya dilihat berdasarkan nilai IPK. Hal penting dalam pencapaian nilai IPK yang baik tergantung pada proses belajar yang dialami oleh mahasiswa tersebut. Proses belajar tersebut akan berlangsung secara baik dan efektif apabila seorang mahasiswa memiliki penggerak/energi untuk mencapai tujuan pada apa akan yang dipelajari. Penggerak atau energi yang dimaksud inilah yang disebut dengan semangat belajar. Tanpa ada semangat belajar, seorang mahasiswa tidak akan mampu mengikuti proses belajar yang sedang berlangsung, akibatnya prestasi akademik pun akan menurun. Ada banyak hal yang dapat mempengaruhi semangat belajar dari mahasiswa di antaranya adalah kondisi perkuliahan yang baik, tempat tinggal mahasiswa, keaktifan di organisasi dan nilai IPK semester sebelumnya (hubungan 2). Selain itu pada faktor-faktor tersebut, terdapat pula hubungan antara kondisi perkuliahan
terhadap IPK semester lalu (hubungan 1). Hal ini
mengakibatkan terdapat lebih dari satu hubungan, sehingga analisis yang sesuai untuk digunakan adalah analisis jalur. Analisis jalur adalah analisis yang dapat menjelaskan keeratan hubungan antar variabel dengan cara menguraikan koefisien korelasi menjadi pengaruh langsung dan pengaruh tidak langsung. Untuk mengetahui besarnya sokongan pengaruh setiap variabel X terhadap variabel Y salah satu cara yang dapat digunakan adalah regresi dengan variabel dibakukan. Sehingga dapat dikatakan bahwa analisis regresi merupakan bagian dari analisis jalur. Ketika hanya ada satu variabel Y yang nilainya ditentukan oleh satu atau lebih variabel X, analisis jalur sama dengan analisis regresi. Tetapi analisis jalur tidak hanya terbatas pada kasus yang sederhana, pada kasus yang rumit analisis jalur masih dapat digunakan. Dalam analisis jalur, terdapat beberapa asumsi yang harus terpenuhi, salah satunya adalah skala pengukuran sekurang-kurangnya adalah interval. Dalam kehidupan nyata, tidak
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mengembangkan metode untuk menerapkan analisis jalur pada variabel dengan skala ordinal dan nominal. Metode yang digunakan adalah metode yang melibatkan model regresi nonlinier, yaitu model regresi logistik.
1.2. Masalah dan Batasan Masalah
Permasalahan yang dihadapi dalam makalah ini yaitu: -
Apa yang dimaksud dengan regresi logistik ordinal dan analisis jalur?
-
Bagaimana membentuk model regresi logistik ordinal untuk bentuk hubungan 1 dan 2?
-
Bagaimana menguji kesesuaian model logistik dan validitas dari model yang dihasilkan?
-
Bagaimana menentukan pengaruh langsung dan pengaruh tidak langsung pada masingmasing hubungan dan cara menginterpresikan pengaruh tersebut? Pada makalah ini, masalah dibatasi pada hubungan kondisi perkuliahan, keaktifan di
organisasi, tempat tinggal dan IPK semester lalu terhadap semangat belajar Mahasiswa Prodi Statistika Angkatan 2009 UB, yang dianalisis dengan analisis jalur ordinal. Software statistika yang digunakan adalah Minitab 15, SPSS 16 dan Ms. Excel 2007.
1.3.Tujuan
Tujuan dari makalah ini yaitu: -
Untuk mengetahui pengertian regresi logistik ordinal dan analisis jalur
-
Untuk mengetahui cara membentuk model regresi logistik ordinal
-
Untuk mengetahui pengujian kesesuaian model logistik dan validitas model yang dihasilkan
-
Untuk mengetahui penentuan pengaruh langsung dan pengaruh tidak langsung pada analisis jalur ordinal dan cara mengeinterpretasi pengaruh tersebut
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BAB II DASAR TEORI
2.1. Model Regresi Logistik Ordinal
Regresi logistik ordinal merupakan salah satu metode statistika untuk menganalisis variabel respon yang mempunyai skala ordinal yang terdiri atas tiga kategori atau lebih. Variabel prediktor yang dapat disertakan dalam model berupa data kategori atau kontinu yang terdiri atas dua variabel atau lebih. Model yang dapat dipakai untuk regresi logistik ordinal adalah model logit. Model tersebut adalah cumulative logit models. Pada model logit ini sifat ordinal dari respon Y dituangkan dalam peluang kumulatif sehingga cumulative logit models merupakan model yang didapatkan dengan membandingkan peluanga
kumulatif, yaitu peluang kurang dari aau sama dengan kategori respon ke-j pada p variabel prediktor yang dinyatakan dalam vector X,
P(Y≤j│X),
dengan peluang lebih besar dari
kategori respon ke- j, P(Y>j│X) (Hosmer dan Lemeshow, 2000).
Peluang kumulatif P(Y≤j│X) didefinisikan sebagai berikut (Agresti, 1990): p
P(Y
j )
exp( j
k X k )
1 exp( j
k 1 p
(1)
k X k ) k 1
di mana j=1,2,..J dan J = kategori respon Dalam hal klasifikasi, cumulative logit models logit models merupakan fungsi pembeda atau fungsi klasifikasi/ fungsi klasifikasi yang terbentuk bila terdapat J kategori respon adalah sejumlah J-1 fungsi. Jika j ( X ) P(Y
jX )
menyatakan peluang kategori respon ke-j pada p
variabel prediktor yang dinyatakan dalam vector X dan P (Y
j
) menyatakan peluang
kumulatif pada p variabel prediktor yang dinyatakan dalam vector X maka nilai didapatkan dengan persamaan berikut:
j ( X )
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Logit Logit[ P(Y
jX)] j β
T
(3)
X
di mana θ adalah vector intersep, β Jika j
T
=(β1, β2,...., βp ) adalah vector slope.
1 ( X) 2 ( X) ..... j ( X )
1
1 ( X)
j
1 ( X) 2 ( X)
, maka
.
(4)
. .
j
1 ( X) 2 ( X) ... j ( X )
Apabila terdapat J kategori respon, maka model logistik ordinal yang terbentuk adalah
1 1 1 X1 2 X 2 .... p X p 1 1
Logit( 1 ) ln
2 2 1 X1 2 X 2 .... p X p 1 2
Logit( 2 ) ln
. . .
j 1 Logit ( j 1 ) ln 1 j 1 j 1 1 X 1 2 X 2 .... p X p Untuk tiga kategori respon di mana j=1,2,3 maka nilai peluang kategori repon ke-j adalah sebagai berikut:
1
2
P(Y 1X)
exp(1
p
X ) k 1 p
1 exp(1
k
k
X ) k 1
k
P(Y 2X) 1 ( X) 2 ( X)
k
exp( 2
p
X ) k 1 p
k
k
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Dari kedua peluang kumulatif tersebut, maka akan didapatkan peluang untuk masingmasing kategori respon sebagai berikut:
1 ( X) P(Y 1X)
exp(1
p
X )
1 exp(1
2 ( X) P(Y 2X) 1 (X)
3 ( X) 1 P(Y 2X) 1
k
k 1 p
X ) k 1
k
k
p
X
exp( 2
exp( 2
k
k 1 p
1 exp(2
)
k
X k
k 1
k
)
exp(1
p
X ) k 1 p
1 exp(1
k
k
X ) k
k 1
k
p
X ) k
k 1 p
1 exp( 2
k
X ) k 1
k
k
p
1 exp( 2
X
k
)
1 exp( 2
X
k
)
k 1 p k 1
k
k
exp( 2
p
X ) k 1 p
1 exp( 2
k
k
X ) k 1
k
k
1
1 exp( 2
dengan demikian j
k
p
X ) k 1
k
k
1 ( X) 2 ( X) ..... j ( X )
Untuk klasifikasi, nilai j ( X)
pada persamaan tersebut akan dijadikan pedoman
pengklasifikasian. Suatu pengamatan akan masuk dalam respon kategori j berdasarkan nilai j ( X) yang terbesar.
2.2. Pendugaan Parameter Logistik Ordinal
Pendugaan parameter regresi logistik ordinal adalah dengan menggunakan metode kemungkinan maksimum. Metode yang digunakan untuk membentuk model regresi logistik ordinal adalah maximum likelihood . Misalkan
adalah nilai respon, y =1
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, } ∏ {0 1 0 1
(5)
dan log-likelihood nya nya adalah
, . / 86 8 6 7 6 79 86 8 6 7 6 79
(6)
Dengan menurunkan
terhadap setiap parameter dan menyamakan dengan nol, akan
didapatkan nilai duga untuk parameter tersebut.
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(8)
di mana
Melalui teknik iterasi Newton-Raphson
pada iterasi ke t (t =
dan
digunakan untuk menduga
0,1,2,…). Iterasi untuk pendugaan nilai
mencapai kondisi konvergen untuk setiap i.
| | untuk
terus dilakukan sampai
(9)
dan
2.3. Analisis Jalur
Analisis jalur erat sekali hubungannya dengan analisis regresi berganda, sehingga dapat dikatakan bahwa analisis regresi merupakan bagian dari analisis jalur. Ketika hanya ada satu variabel Y yang nilainya ditentukan oleh satu atau lebih variabel X, analisis jalur sama dengan analisis regresi. Tetapi analisis jalur tidak hanya terbatas pada kasus yang sederhana, pada kasus yang rumit analisis jalur masih mampu untuk digunakan. Beberapa peneliti mengatakan bahwa analisis jalur merupakan model bentuk hubungan sebab akibat, namun hubungan antar variabelnya harus searah (Lestari, 2005). Untuk mengetahui besarnya sokongan pengaruh setiap variabel X terhadap variabel Y
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Analisis jalur bukan merupakan metode untuk menemukan penyebab-penyebab, tetapi sebuah metode yang digunakan pada model yang dibuat peneliti berdasarkan pengetahuan dan pertimbangan teori (Kerlinger & Pedhazur, 1973). Menurut Wright (1934) dalam Kerlinger & Pedhazur (1973), metode analisis jalur tidak dimaksudkan untuk mencapai tugas yang tidak mungkin untuk memperoleh hubungan kausal dari nilai koefisien korelasi. Hal ini dimaksudkan untuk menggabungkan informasi kuantitatif yang diperoleh dari korelasi dengan informasi kualitatif tentang hubungan kausal yang telah diketahui untuk memperoleh interpretasi kuantitatif.
Asumsi Analisis Jalur
Menurut Dillon dan Goldstein (1984), setiap analisis yang digunakan untuk menyelesaikan permasalahan statistika harus memenuhi asumsi-asumsi yang mendasari agar kesimpulan yang diperoleh dapat dipertanggungjawabkan. Asumsi yang harus dipenuhi pada analisis jalur adalah: 1. Hubungan antar variabel adalah linier dan aditif 2. Hanya model rekursif yang dapat dipertimbangkan yaitu hanya system aliran kausal satu arah. Model dikatakan sebagai model rekursif jika antara error dan variabel penjelas saling bebas dan antar error saling bebas. 3. Skala pengukuran sekurang-kurangnya interval 4. Variabel penjelas diukur tanpa kesalahan 5. Model yang dianalisis dispesifikasikan dengan benar berdasarkan teori-teori dan konsep yang relevan.
Diagram Jalur Dan Model Analisis Jalur
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pertimbangan-pertimbangan dari penelitian sebelumnya atau berdasarkan perkiraanperkiraan dasar untuk mempermudah analisis jalur (Hair, 1995). Dalam menyusun diagram jalur, konvensinya antara lain untuk menggunakan garis dua arah untuk menunjukkan korelasi antar variabel dan garis berarah tunggal untuk menunjukkan pengaruh langsung dari satu variabel ke variabel lain (Li, 1986). Berikut ini adalah contoh dari diagram jalur:
Persamaan/model struktural bagi diagram jalur tersebut adalah:
di mana: i
: 1,2,…,n
Y, Z
: variabel endogen
Xi
: variabel eksogen
PY1X1,…,PY1Xk : koefisien jalur
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Pada analisis jalur variabel eksogen mempengaruhi variabel endogen tidak hanya melalui pengaruh langsung dan tak langsung tetapi juga melalui pengaruh tak teranalisis dan pengaruh semu (Dillon & Goldstein, 1984). Menurut Falkenbe (1999) dalam Ismaliana (2003), Diagram jalur model sebab akibat dapat digambarkan sebagai berikut:
X1 Py1x1
Py2x1
rx1x2
Py1x2
rx1x3
X2
rx2x3
Py2y1 Y1
Y2
Py2x3
Py1x3
Py2e2 X3 Py1e1 e1
e2
Secara ringkas, koefisien korelasi dapat didekomposisi menjadi empat komponen yaitu: 1. Pengaruh Langsung ( Direct Effect ) adalah pengaruh variabel eksogen terhadap
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Py2y1
Py1x1 X1
Y2
Y1
maka ry2x1= py1x1py1y2 3. Pengaruh tak teranalisis ( Unanalyzed Effect ) adalah pengaruh yang timbul karena adanya korelasi antara variabel eksogen Py1x1 X1 rx1x2
Y1 X2
Py1x2
maka r y1x1= py1x1 + py1x2 rx1x2 U 4. Pengaruh semu ( Spurious Effect ) adalah pengaruh variabel eksogen terhadap lebih dari satu variabel endogen yang saling berkorelasi Py1x1
Y1
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Y3 Y2
Y1 X1
Anggap terdapat empat variabel,
adalah variabel eksogen,
adalah variabel
endogen. Setiap variabel mempunyai jumlah kategorik masing-masing. Misal
*+
mempunyai kategorik
,
. Dari bentuk strukural yang ada, dapat dijelaskan dengan
persamaan model logistik:
β
β
β
β
β
β
β
β
β
β
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Log odds ratio dari
terhadap
adalah
β
Ketika dilakukan subtitusi
dan
dan
di atas dengan nilai harapan dari keduanya yaitu
akan didapatkan
Yang mana persamaan di atas adalah efek langsung
β
ke
. Jadi,
β
β
adalah efek langsung dari setiap variabel terhadap
Efek total dari
terhadap
β
β
.
adalah
β
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…
…
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…
di mana:
: variabel endogen ke- h kategori ke- j
: variabel eksogen ke- i kategori ke- k pada variabel endogen ke- h kategori j : pengaruh tidak langsung variabel eksogen ke- i dengan kategori k terhadap
variabel endogen ke- h kategori j h
: variabel endogen ke- h
i
: variabel eksogen ke- i
j
: kategori ke-j variabel endogen h
k
: kategori ke-k variabel eksogen i
2.5. Uji validitas model analisis jalur kategorik
Validitas menunjukkan sejauh mana suatu alat ukur itu mampu mengukur apa yang
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̂ ̂ ̂ ̂ Di mana :
2. Statistik Deviance Deviance
Di mana:
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̂
: peluang Y pada kategori ke- j dan x dengan kategori ke- i : pengamatan Y pada kategori ke- j dan x dengan kategori ke- i
p
: jumlah parameter pada model
g
: jumlah grup yang terbentuk
k : jumlah kategori peubah respon Hosmer-Lemeshow menyebar mengikuti sebaran Khi-kuadrat dengan derajat Statistik Hosmer-Lemeshow
bebas v = g(k -1)-1)- p. keputusan tolak Ho, jika
]
[ atau
lebih kecil dari peluang yang diinginkan(), sehingga
diambil kesimpulan bahwa model yang diperoleh tidak sesuai.
dapat
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BAB III METODOLOGI
3.1 Sumber Data 3.1.1 Lokasi dan Waktu Penelitian
Penelitian ini dilakukan pada mahasiswa statistika angkatan 2009 di Prodi Statistika Jurusan Matematika Fakultas MIPA Universitas Brawijaya Malang. Waktu penelitian dilakukan selama 1 hari dan penyebaran kuisioner dilakukan saat mahasiswa berada di kampus.
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Dalam penelitian ini, diketahui bahwa N (banyak populasi) sebesar 91 mahasiswa dan toleransi terjadinya galat yang digunakan (e) adalah 0.1. Dengan demikian diperoleh jumlah sampel sebanyak 48 mahasiswa Prodi Statistika angkatan 2009 Universitas Brawijaya. Berikut adalah perhitungan banyaknya sampel dalam penelitian ini dengan rumus Slovin :
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X2 = Aktif Organisasi
X3 = Tempat Tinggal
1 = Ya 2 = Tidak 1 = Rumah sendiri/family 2 = Kos / Kontrak
Nominal
Nominal
1 = Sangat Bersemangat 2 = Bersemangat Y2 = Semangat Belajar
3 = Kurang Bersemangat 4 = Tidak Bersemangat 5 = Sangat Tidak Bersemangat
Ordinal
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angka di belakang koma yang menunjukkan kualitas belajar keseluruhan dari materi program studi yang dihitung dengan menjumlah hasil perkalian nilai hasil belajar dengan bobot sks dibagi dengan jumlah kredit yang ditetapkan untuk program studi
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tinggal memiliki arti yang sama dengan rumah, kediaman, akomodasi, perumahan, dan arti-arti yang lain.
Mahasiswa rantau biasanya akan memilih rumah kos atau rumah kontrakan
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Bersemangat
Kurang bersemangat
Tidak bersemangat
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5.
Menghitung
efek
langsung
dan
tidak
langsung
variabel-variabel
eksogen
menggunakan Excel. a. Efek langsung diperoleh dengan menghitung trace dari hasil perkalian matriks
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BAB IV HASIL DAN PEMBAHASAN 4.1. Model Duga Regresi Logistik Ordinal
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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