MODUL PANDUAN PENGGUNAAN MINITAB 14 DALAM ANALISIS DATA
Created by :
Reza Septiani Pontoh
1
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DAFTAR ISI 1 PENGANTAR STATISTIKA & PENGENALAN MINITAB...........................................4 MINITAB...........................................4 1.1 Pendahuluan.................................. Pendahuluan......................................................... .............................................. ............................................ ................................. ............ 4 1.1.1 Input Data............................................ Data................................................................... .............................................. ....................................... ....................4 ....4 2 STATISTIKA DESKRIPTIF............... DESKRIPTIF...................................... .............................................. ................................................... ............................ .....6 2.1 Meringkas Data.................. Dat a......................................... .............................................. ............................................................. ...................................... ......6 ...... 6 2.2 Menyajikan Data.............................................................. Data.................................................................................................... ...................................... ......7 ...... 7 2.2.1 Histogram........................................ Histogram............................................................... .............................................. ....................................... ........................ ........7 7 2.2.2 Boxplot................................. Boxplot........................................................ .............................................. ......................................................... .................................. .8 2.2.3 Steam and Leaf................................................................ Leaf......................................................................................................9 ......................................9 2.2.4 Plot....................................... Plot.............................................................. ............................................... ............................................... ...................................9 ............9 3 STATISTIKA DASAR....................................... DASAR.............................................................. ...................................................... ............................... ........10 ........ 10 3.1 Satu Gugus Data Contoh................................................... Contoh............................................................................................ ......................................... 10 3.2 Dua Gugus Data Contoh ....................................................... .............................................................................. .....................................12 ..............12 4 ANALISIS REGRESI.................................. REGRESI.......................................................... ............................................... ..................................... ......................13 ........13 4.1 Regresi Linier............................................. Linier.................................................................... ................................................................13 .........................................13 4.2 Regresi Bertatar............................ Bertatar................................................... .............................................. .......................................................16 ................................16 4.2.1 Prosedur Stepwise........................................................ Stepwise....................................................................................... ............................... ........16 ........ 16 4.2.2 Forward Selection........................ Selection............................................... .............................................. .................................................17 ..........................17 4.2.3 Backward Elimination................... Elimination.......................................... .............................................. ............................................... ........................ 17 4.3 Regresi Terbaik (Best Regression).............................................................. Regression)............................................................................. ...............18 18 4.4 Regresi Percobaan Satu Faktor Bertaraf kualitatif.....................................................19 4.5 Regresi Percobaan Satu Faktor Faktor Bertaraf Kuantitatif................................................. Kuantitatif................................................. 21 4.6 Regresi Percobaan Tiga Faktor Bertaraf Kualitatif dan Kuantitatif.....................24 5 REGRESI DAN MASALAH PELANGGARAN ASUMSI.............................................28 ASUMSI.............................................28 5.1 Heteroskedastisitas (Heteroscedasticity)....................................................................28 5.2 Autokorelasi (Serial Independen).................................. Independen)......................................................... .............................................33 ......................33 5.2.1 Metode Kuadrat Terkecil Terboboti (Weighted Least Square)...........................34 5.2.2 Transformasi model....................................... model.............................................................. ......................................................35 ...............................35 5.3 MULTIKOLINIERITAS..... MULTIKOLINIERITAS............................ .............................................. .............................................. .........................................36 ..................36 6 PERANCANGAN PERCOBAAN............................... PERCOBAAN...................................................... .............................................. ..............................41 .......41 6.1 KLASIFIKASI PERLAKUAN................................. PERLAKUAN........................................................ ............................................ ..........................42 .....42 6.1.1 Rancangan Perlakuan............................................................ Perlakuan................................................................................. .............................. .........42 42 6.1.2 Rancangan Lingkungan............................ Lingkungan................................................... ...........................................................43 ....................................43 6.2 PERCOBAAN FAKTORIAL................................... FAKTORIAL.......................................................... ............................................ ..........................43 .....43 6.2.1 Percobaan Dua Faktor RAL...................................................... RAL.................................................................... .......................... ............43 43 6.2.2 Percobaan Dua Faktor RAK...................................... RAK............................................................. ..........................................45 ...................45 6.2.3 Percobaan Dua Faktor RBSL.................................................... RBSL.................................................................. .......................... ............46 46 6.3 RANCANGAN PETAK TERPISAH ( Split Plot Design).........................................46 Design).........................................46 6.4 RANCANGAN RANCANGAN BLOK TERPISAH ( Split Split Block Design or Strip Plot Design)......47 7 PENGUJIAN ASUMSI...................................... ASUMSI............................................................. .............................................. ...................................... .................48 ..48 7.1 Pengujian Keaditifan Model.......................................... Model................................................................. .............................................50 ......................50 7.2 Pengujian Kenormalan Galat................................................. Galat........................................................................ .....................................50 ..............50 7.3 Pengujian Kehomogenan Ragam..................................... Ragam....................................................................... .................................. ........50 ........ 50
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8.2 Transformasi untuk k buah data sample bebas...........................................................53 bebas...........................................................53 8.2.1 Transformasi logaritma ( log Y )................................................... ).........................................................................53 ......................53 8.2.2 Transformasi akar kuadrat ( √Y )........................................................................ )........................................................................ 53 8.2.3 Transformasi Arcsin ( Sin-1 √Y)..........................................................................54 8.3 Transformasi Dalam Regresi Linear Sederhana ........................................................55 ........................................................55 9 REGRESI LOGIT DAN PROBIT........................................ PROBIT............................................................... .............................................55 ......................55 9.1 Regresi Logit ......................................................... ................................................................................ ............................................... ........................ .....55 9.1.1 Model Logit....................................... Logit.............................................................. .............................................. ........................................... .................... 56 9.1.2 Pengujian Parameter....................... Parameter.............................................. .............................................. ..............................................56 .......................56 9.1.3 Intepretasi Koefisien........................ Koefisien............................................... .............................................. .............................................57 ......................57 9.2 Regresi Probit.......................................... Probit................................................................. .............................................. ......................................... .....................60 ...60 9.2.1 Intepretasi koefisien...................... koefisien............................................. .............................................. ................................................60 .........................60 9.2.2 Kriteria pemilihan Model Terbaik......................................... Terbaik.......................................................................60 ..............................60 9.3 Perbedaan Logit dan Probit........................................ Probit............................................................... ......................................... .........................61 .......61 9.4 Perbedaan Regresi Linier dan Logistik......................................................................61 10 ANALISIS MULTIVARIATE........ MULTIVARIATE............................... ............................................... ............................................... .................................63 ..........63 10.1 ANALISIS KOMPONEN UTAMA........................................ UTAMA............................................................... .................................63 ..........63 10.2 ANALISIS KORESPONDENSI....... KORESPONDENSI.............................. .............................................. ................................................63 .........................63 10.2.1 Analisis korespondensi sederhana...................................................... sederhana..................................................................... ...............63 63 10.2.2 Analisis korespondensi berganda...................................................................... berganda...................................................................... 65 10.3 ANALISIS GEROMBOL..................................................... GEROMBOL............................................................................. ........................ ...........68 ........... 68 10.3.1 Konsep Jarak.............................................................. Jarak............................................................................................. ............................... ........68 ........ 68 10.3.2 Metode Perbaikan Jarak....................................................................................68
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1
PENGAN PENGANTAR TAR STATIS STATISTIK TIKA A & PENGEN PENGENALA ALAN N MINI MINITAB TAB
Dalam Dalam berbag berbagai ai litera literatur tur,, statis statistik tik atau. atau. statistic dapat dapat diarti diartikan kan sebaga sebagaii penduga parameter, dimana parameter disini dapat berupa rata-rata, standar deviasi, proporsi dan lain-lain. Sementara itu Statistika atau statistics adalah suatu disiplin ilmu yang mempelajar mempelajarii metode metode pengumpulan pengumpulan data, data, menganalis menganalisis is (termasuk (termasuk pendugaan pendugaan parameter) dan menarik kesimpulan dari data tersebut. Data dibagi ke dalam kelompok menurut sumbernya, yaitu data primer dan data data seku sekund nder er.. Data Data prim primer er adal adalah ah data data yang yang bera berasa sall dari dari sumb sumber er asli asli dan dan diku dikump mpul ulka kan n seca secara ra khus khusus us untu untuk k menj menjaw awab ab pert pertan anya yaan an pene penelit litia ian n kita kita.. Data Data sekunder adalah data yang berasal dari hasil survey pihak lain. Statistika adalah salah satu alat untuk membantu para pengambil kebijakan dalam dalam membuat membuat keputusan keputusan.. Pengambilan Pengambilan keputusan keputusan ini umumnya umumnya didasarkan didasarkan atas informasi yang tersedia dari data contoh. Untuk mengetahui prosedur pengambilan keputusan tersebut terlebih dahulu diperlukan pengertian-pengertian dasar tentang konsep dan teori statistika. Konsep-konsep dan tahapan-tahapan yang harus dimiliki oleh seorang peneliti dalam melakukan penelitiannya, yaitu: 1. Pendefinisian masalah 2. Pendefinisian populasi 3. Penentuan peubah / variabel 4. Teknik penarikan contoh 5. Pembuatan alat ukur 6. Metode analisis 7. Interpretasi hasil analisis 8. Kesimpulan 9. Penyajian hasil analisis
1.1 Pend Pendah ahul ulua uan n Paket program MINITAB MINITAB merupakan merupakan perangkat lunak yang dapat digunakan digunakan sebagai media pengolahan data yang menyediakan berbagai jenis perintah sehingga memu memung ngki kink nkan an pros proses es pema pemasu sukk kkan an data data,, mani manipu pula lasi si data data,, pemb pembua uata tan n graf grafik ik,, peringkasan nilai-nilai numerik dan analisis statistika. MINITAB memiliki dua sesi primer yaitu worksheet (lembar kerja) untuk melihat dan dan meng menged edit it lemb lembar ar kerj kerja, a, sert serta a sesi sesi command yang yang merupa merupakan kan layar layar untuk untuk menampilk menampilkan an hasil. hasil. Perintah-p Perintah-perinta erintah h MINITAB MINITAB dapat dapat diakses diakses melalui melalui menu, kotak dialog dan perintah interaktif. Perintah interaktif ditulis pada sesi command.
1.1.1 Inp Input Data READ : Perintah READ selain digunakan untuk memanggil atau membaca data dari File ASCII juga j uga dapat digunakan untuk memasukkan data melalui keyboard. Tulislah setiap baris data pada satu baris baru dan pisahkan masing-masing angka dengan spasi atau koma dan akhiri dengan END. END. Contoh : MTB > READ C2 C3
SET Contoh
DATA> 2 4 DATA DATA> > 3.5 3.5 27 DATA> 1 12 DATA> END : Perintah SET digunakan untuk memasukkan data ke suatu kolom. : MTB > SET C6 DATA> 2 7 9
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DATA> 1 : 3 / .5 DATA> 3(1) DATA> 2(1:3) DATA> (1:3) 2 DATA> 2 (1: 3) 2 Contoh: MTB> SET C6 DATA> 1 : 3 / .5 DATA> END
1 1 1 1 2
1.5 1 2 1 1
2 1 3 2 2
2.5
3
1 2 3
2 3 2
3 3
: Perintah LET digunakan untuk perhitungan aritmatik, bisa juga digunakan untuk mengganti atau memperbaiki memperbaiki nilai dalam kolom. LET E = ekspresi aljabar Contoh: MTB> LET C1 (3) = 4 MTB> LET C4 = (C1-MEAN(Cl))**2 MTB> LET K2= SUM (ABSO(Cl-MEAN (ABSO(Cl-MEAN (C1))) LET
Ekspresi aljabar didalam format perintah tersebut adalah : 1. Oper Operas asii ari aritm tmat atik ik ( +, -, *, *, /, ** **,, =, =, ~=, ~=, >, <, <=, <=, => => ) 2. Fungsi Fungsi : ABSO ABSOLUT LUTE, E, EXPO, EXPO, MINI MINIMUN MUN,, ROUN ROUND, D, SUM, SUM, COUNT COUNT,, MEDI MEDIAN, AN, STDEV, RANK, SORT, MEAN, PARSUM, SQRT, SSQ Perintah fungsi selalu diikuti dengan tanda kurung → MTB> SQRT (C5) DELETE & ERASE : DELETE berfungsi untuk menghapus baris, sedangkan ERASE digunakan untuk menghapus kolom. Contoh: MTB> DELETE 2,4 C1-C3
MTB> ERASE C1
INSERT Contoh:
: berfungsi untuk menyisipkan baris data pada lembar kerja. MTB> INSERT 2,3 Cl-C2 DATA> 56 8 DATA> END
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SUSAN 2 JAMES 1
115 145
65 64
MTB> COPY 'NAMA' 'JK' 'BB' 'TB' C12-C14; SUBC> USE 'JK'=l. (perintah ini sama dengan OMIT 'JK'=2.) Hasilnya :
CODE : untuk menggandakan sekaligus mengganti beberapa nilai STACK DAN UNSTACK : UNSTACK : Perintah Perintah STACK digunakan untuk menggabungk menggabungkan an kolom atau konstanta konstanta diatas diatas kolo kolom m atau atau kons konsta tant nta a yang yang lain lain,, UN UNST STAC ACK K digu diguna naka kan n untu untuk k meme memeca cah h atau atau memisa memisahka hkan n isi sebuah sebuah atau atau bebera beberapa pa kolom kolom ke dalam dalam bebera beberapa pa kolom kolom atau atau konstanta. Subcommand yang digunakan untuk memecah adalah SUBSCRIPTS.
2
STA STATIS TISTIK TIKA DES DESKRIPT RIPTIF IF
Statis Statistik tika a deskri deskripti ptiff adalah adalah bidang bidang statis statistik tika a yang yang membi membicar caraka akan n metode metode mengumpulk mengumpulkan, an, meringkas/ meringkas/menye menyederha derhanakan nakan dan menyajikan menyajikan data sehingga sehingga dapat dapat memberikan informasi. Mengumpulkan data dapat dilakukan dengan cara: 1. Penelitian 2. Observasi Ukuran yang digunakan dalam meringkas data: 1. Ukuran pemusatan ( mean, median, modus, kuartil) 2. Ukuran penyebaran ( ragam, range, jarak antar kuartil) Penyajian data dapat berupa : 1. Tabulasi 2. Grafik ( histogram, histogram, boxplot (diagram kotak garis), garis), steam and leaf (diagram dahan daun), plot) Peringkasan dan penyajian data yang baik akan sangat membantu dalam menganalisis data selanjutnya. Dan membantu dalam mengambil kesimpulan secara deskriptif.
2.1 2.1 Meri Mering ngka kas s Data Data Tahapan menggunakan menu MINITAB : Klik Stat > Basic Statistics > Display Descriptive Statistics : isi dengan peubah yang akan dideskripsikan Variables klik Graph : pilih Histogram of Data with normal curve klik Statistics : checklist nilai-nilai statistic yang ingin ditampilkan Klik OK! Contoh : Data hasil yang diperoleh dari varietas padi lokal (ton/ha): 4.0, 4.0, 5.5, 6.0, 7.5, 4.8, 6.1, 4.5, 4.5, 5.0, 4.0, 5.3, 5.1, 5.8, 5.9. 6.5, 7.5, 7.5, 4.0, 4.5 Masukkan data diatas pada kolom C1 Kemudian ikuti tahapan menggunakan menu MINITAB diatas
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Descriptive Statistics: hasil Variable hasil
N 20
Mean 5.400
SE Mean 0.265
StDev 1.183
Minimum 4.000
Q1 4.500
Median 5.200
Keterangan : N : Banyak data Mean : Rataan Media edian n : Nil Nila ai ten teng gah set setelah lah da data teru teruru rutt dar darii ter terk kecil ecil hing hingga ga terbesar TrMean : Rataan Terpangkas, yaitu yaitu rataan setelah data terkecil terkecil dan terbesar dipotong masing-masing 5% StDev : Simpangan Baku/ akar dari ragam SE Mean ean : Rata Rataan an Gala Galatt Bak Baku/ u/ Sim Simpang pangan an Baku Baku bagi bagi N Min/ in/Max Max : Nil Nila ai ter terke kec cil/t il/te erbes rbesa ar set setel ela ah da data terur erurut ut Q1/Q3 : kuartill/kuartil3 Histogram Histogram (with Normal Curve) of hasil 4
3 y c n e u q 2 e r F
1
Mean 5.4 StDev 1.183 N 20
Q3 6.075
Maximum 7.500
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Tahapan menggunakan menu MINITAB : Klik Graph > Histogram Pilih bentuk histogram yang akan ditampilkan : isi dengan variabel yang akan dibuat histogram Graph Variables Klik Label jika ingin memberi judul histogram Klik Multiple Graph untuk memilih tampilan histogram (overlay lebih dari 1 variabel atau separate) Klik OK! Contoh : Data hasil yang diperoleh dari vareitas padi lokal (ton/ha): 4.0, 4.0, 5.5, 6.0, 7.5, 4.8, 6.1, 4.5. 4.5, 5.0, 4.0,5.3,5.1, 5.8 , 5,9,6.5, 7.5, 7.5, 4.0, 4.5 Masukkan data di atas pada kolom Cl, Cl, beri judul kolom Hasil (ton/ha) Kemudian ikuti tahapan menggunakan menu MINITAB diatas
histogramhasil (ton/ ha) 4
3 y c n e u q 2 e r F
1
0 4.0
2.2.2
4.5
5.0
5.5 6.0 hasil (ton/ ha)
6.5
7.0
7.5
Boxplot Peny Penyaj ajia ian n dala dalam m bent bentuk uk Boxp Boxplo lott tida tidak k mena menamp mpil ilka kan n data data asli asli,, teta tetapi pi menampilkan : Kesimetrisan penyebaran data, data , dapat dilihat dari apakah kotak terbagi dua oleh garis median sama besar atau tidak, dan apakah 'ekor' bawah dan 'ekor' atas sama panjang atau tidak. Keanehan data, data, jika data pengamatan berada di luar batas BB1 dan BA1, disebut pencilan minor, dan jika data pengamatan berada di luar batas BB2 dab BA2 disebut data ekstrim.
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50% data disebelah kiri Q2 dan 50% data di sebelah kanan Q2. Q3 : Nilai Kuartil 3, nilai yang menyekat kumpulan kumpulan data yang telah diurutkan, dimana data yang lebih kecil dari Q3 sebanyak 75% dan data yang lebih besar dari Q3 sebanyak 25%. BA1 = Q3 + 3/2(Q3-Q1) BB1 = Q1 - 3/2(Q3-Q1) BA2 = Q3 + 3(Q3-Q1) BB2 = Q1 - 3(Q3-Q1) Tahapan menggunakan menu MINITAB : Klik Graph > Boxplot Pilih bentuk histogram yang akan ditampilkan Graph Variables : isi dengan variabel yang akan dibuat boxplot Klik Label jika ingin memberi judul boxplot Klik OK! Contoh : Gunakan data contoh Histogram, kemudian ikuti tahapan di atas
BOXPLOT HASIL HASIL (TON/ HA) HA) 8
7
) a h / n 6 o t ( l i s a h 5
4
2.2 2.2.3
Stea Steam m and and Lea Leaf Memung Memungkin kinkan kan penggu pengguna na mendap mendapat at lebih lebih banyak banyak inform informasi asi diband dibandingk ingkan an dengan penyajian Histogram, karena diagram dahan daun menyajikan data asli dari setiap objek pengamatan. Tahapan menggunakan menu MINITAB : Klik Graph > Steam-and-leaf Graph Variables : isi variabel yang akan dibuat steam and-leaf Klik OK! Contoh: Gunakan data Histogram, ikuti langkah diatas
Stem-and-Leaf Display: hasil (ton/ha) Stem-and-leaf of hasil (ton/ha)
N
20
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Klik Graph > Plot Graph Variables
: isi Y dengan variabel yang akan diplotkan (Cl) dan X dengan peubah lain (C2) Klik Label jika ingin memberi judul plot serta klik pada bagian data labels jika ingin menampilkan tipe label pada plot (klik use y-value labels). labels ). Klik OK! Contoh : Gunakan data Histogram, dengan tambahan data lokasi penanaman padi : 1, 2, 3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 3. 1 = Jawa Barat, 2 = Jawa Tengah, 3 = J awa Timur Tempatkan data tambahan tambahan di atas pada kolom C2, kemudian ikuti tahapan tahapan di atas
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( x ± z (α / 2 )σ x )
dimana : x σ x
= nilai tengah contoh = galat baku nilai tengah x
Langkah-langkah menggunakan menu MINITAB untuk Uji Z : Klik Stat > BasicStatistcs > 1-Sample z : isi dengan peubah C1 atau hasil(ton/ha) Variables Test Mean : isi dengan nilai tengah populasi yang dihipotesiskan (5.2) Standar deviation : isi dengan simpangan baku populasi (1.2) Klik options : pilih taraf nyata pada confident level dan hipotesisnya pada alternative
One-Sample Z: hasil (ton/ha ) Test of mu = 5.2 vs not = 5.2 The assumed standard deviation = 1.2 Variable hasil (ton/ha)
N 20
Mean 5.40000
StDev 1.18322
SE Mean 0.26833
95% CI (4.87409, 5.92591)
Z 0.75
P 0.456
Intepretasi : Hipotesis yang diuji adalah Ho : μ = 5.0 vs H1 : μ ≠ 5.0. Setelah dilihat dari nilai P-value teryata nilainya lebih besar dari taraf nyata 0.05 yang berarti terima H0 atau μ = 5.0. Langkah-langkah menggunakan menu MINITAB untuk Uji T : Klik menu Stat > Basic Statistics > 1-Sample t Variables : isi dengan peubah C1 Test Mean : isi dengan nilai tengah populasi yang dihipotesiskan (5.2) Klik options : pilih taraf nyata pada confident level hipotesisnya pada alternative
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Hipotesis yang diuji adalah Ho : μ = 5.0 vs H1 : μ ≠ 5.0. Setelah dilihat dari nilai P-value teryata nilainya lebih besar dari taraf nyata yang kita tentukan yaitu sebesar 0.05 yang berarti terima H0.
3.2 Dua Gugus Gugus Data Data Cont Contoh oh Jika ingin melakukan uji kesamaan dua nilai tengah dan selang kepercayaan untuk beda nilai tengah populasi bagi dua gugus data contoh digunakan uji t dengan perintah TWOSAMPLE (jika 2 gugus data contoh diletakkan pada kolom terpisah) atau TWOT (jika 2 gugus data diletakkan pada satu kolom dan kolom lain dituliskan koding dari gugus contoh). Bentuk hipotesis yang diuji : 1. H0 : μ1 = μ2 vs H1 : μ1 ≠ μ2 2. H0 : μ1 = μ2 vs H1 : μ1 > μ2 3. H0 : μ1 = μ2 vs H1 : μ1 < μ2 Contoh: Kita ingin menguji apakah nilai tengah antara variabel di C1 dan C2 sama, dan kita mengujinya pada taraf taraf nyata sebesar 0.05 Langkah-langkah dalam MINITAB : Klik Stat > Basic statistics > 2-sample t Klik Samples in different columns: columns: Masukkan kolom mana yang akan diuji (C1 dan C2) Klik "Assume "Assume equal variances" variances " jika kita mennganggap bahwa ragam populasi sama Klik options : pilih taraf nyata pada confident level serta hipotesisnya pada alternative.
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4
ANALISIS REGRESI
Dalam MINITAB terdapat fasilitas fasilitas untuk analisis regresi yaitu regresi regresi linier, linier, regresi bertatar (stepwise regression), regresi terbaik (best regression), dan regresi kekar (robust regression). regression). Disini Disini hanya akan dibahas dibahas tentang tentang regresi regresi linier, linier, bertatar bertatar dan terbaik. Model : Y t = α + βXt + Ut Asumsi yang mendasari pendugaan model regresi linier dengan menggunakan metode kuadrat terkecil yaitu : Ut adalah random variable (peubah acak) dengan nilai rata-rata U t [E(Ut)] = 0, untuk semua t. Homoskedas Homoskedastisit tisitas as (kehomogen (kehomogenan an ragam) ragam) yang berarti berarti untuk setiap sisaan sisaan 2 ragamnya sama. Untuk semua t, Var (U t) = σ . Tidak ada auto korelasi dalam sisaan. sisaan. Tidak terdapat hubungan atau korelasi antara beberapa atau semua variabel bebas (multikolinearitas) Untuk setiap t, U t menyebar Normal (0, σ2)
4.1 4.1 Regr Regres esii Lini Linier er Regres Regresii Linier Linier adalah adalah persam persamaan aan regres regresii yang yang mengga menggamba mbarka rkan n hubung hubungan an antara antara peubah peubah bebas bebas (X, independen dengan an peub peubah ah tak tak beba bebas s (Y, (Y, independence ce variable variable) deng dependence variable) dimana hubungan keduanya dapat digambarkan sebagai suatu garis lurus. Berikut Berikut adalah adalah contoh contoh menganalis menganalisis is data menggunakan menggunakan regresi regresi linier. linier. Suatu telaah dilakukan untuk mengevaluasi sejauh mana pengaruh biaya yang dikeluarkan untuk iklan terhadap hasil penjualan, dikumpulkan data biaya iklan dan hasil penjualan sebagai berikut : No
Biaya
Hasil
No
Biaya
Hasil
1
40
385
7
40
490
2
20
400
8
20
420
3
25
395
9
50
560
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Regression Analysis: Hasil versus Biaya The regression equation is Hasil = 344 + 3.22 Biaya Predictor Constant Biaya S = 50.2257
Coef 343.71 3.221
SE Coef 44.77 1.240
R-Sq = 40.3%
Analysis of Variance Source DF SS Regression 1 17030
T 7.68 2.60
P 0.000 0.027
R-Sq(adj) = 34.3% MS 17030
F 6.75
P 0.027
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Residual Plots for Hasil Norm ormal Proba Probabi bili lity Plo Plott of the Resid Residua uals ls 99
Resi Residua duals Versu ersus s the Fitted itted Values lues 50
90
l a 0 u d i s e R -50
t n e c r 50 e P 10
1 -100
-50
0 Residual
50
100
Histog togram of the the Res Residuals
-100 400
450 475 Fitted Value
500
Res Residuals Versus the the Order of the the Data
3
50
y c 2 n e u q e r F 1
l a 0 u d i s e R -50
0
425
-100 -100
-75
-50 -25 0 Residual
25
50
1
2
3
4 5 6 7 8 9 Observation Order
10 11 12
Plot sisaan untuk hasil diatas dapat digunakan untuk pengujian asumsi. Plot digunakan kan untuk untuk uji asums asumsii kenorm kenormala alan; n; plot residual normal probability probability ... diguna versus versus the fitted fitted values values (plot sisaan dan dugaan) dugaan) menunjukka menunjukkan n kehomogenan kehomogenan ragam sisaan, jika plot membentuk suatu pola acak atau lebar selang homogen maka diindi diindikas kasika ikan n sisaa sisaan n bersif bersifat at homoge homogen; n; plot residuals versus the order ... menunj menunjukk ukkan an keacak keacakan an galat. galat. Dari Dari plot diatas diatas maka maka ketiga ketiga asumsi asumsi terseb tersebut ut telah telah terpenuhi. Kemudian bila ingin mengetahui apakah model regresi ordo berapa yang tepat bisa menggunakan menu fitted line plot, tapi hanya terbatas sampai model cubic. Dari langkah ini juga dipeloreh plot regresinya. Langkah-langkah dalam MINITAB: Klik Stat > Regression > Fitted Line Plot Response (Y) : Masukkan peubah respon (Hasil) Hasil) : Masukkan peubah bebas ( Biaya) Predictors (X) Biaya)
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Fitted Line Plot Hasil = - 337.4 +66.03 Biaya - 1.800 1.800 Biaya** Biaya** 2 + 0.01630 0.01630 Biaya* iaya** *3 S R-Sq R-Sq(adj)
550
53.9014 45.0% 24.4%
500 l i s a H 450
400
350 20
25
30
35 Biaya
40
45
50
interpretasi : Dari output terlihat bahwa hanya model linear yang nyata, sehingga untuk data ini model yang tepat adalah model regresi linear atau berordo satu. Ulangi langkah diatas ( Fitted Line Plot ) tapi pilih model linear. Regression Analysis: Hasil versus Biaya The regression equation is Hasil = 343.7 + 3.221 Biaya S = 50.2257 Analysis of Source Regression Error Total
R-Sq = 40.3% Variance DF SS 1 17030.0 10 25226.2 11 42256.2
R-Sq(adj) = 34.3%
MS 17030.0 2522.6
F 6.75
P 0.027
Fitted Line Plot Hasil = 343.7 343. 7 +3.221 Biaya 550
S R- Sq R-Sq(ad R-Sq(adj) j)
50.2257 40.3% 34.3%
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Nilai F yang paling besar dimasukkan kedalam model jika lebih besar dari Alpha to enter (A ENTER). Jika tidak ada lagi prediktor yang dapat ditambahkan ke dalam model, prosedur stepwise dihentikan.
4.2 4.2.2
Forw Forwar ard d Sel Selec ecti tion on Sama dengan prosedur stepwise, tetapi tetapi tanpa tanpa ada prediktor yang dikeluarkan dikeluarkan dari model. Untuk forward selection, gunakan nilai A to remove = 0.
4.2. 4.2.3 3
Back Backwa ward rd Elim Elimin inat atio ion n Tahapan untuk melakukan prosedur ini yaitu : Masukkan prediktor Keluarkan prediktor dengan menggunakan prosedur stepwise Tidak ada prediktor yang dimasukkan kembali kembali ke dalam model Gunaka Gunakan n nilai nilai A to enter enter = 10000 10000 dan tuliskan tuliskan semua semua predik prediktor tor pada sub perintah ENTER
Contoh kasus : Suatu survey dilakukan terhadap 17 rumah sakit di sekitar Jabotabek. Peubah-peubah
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Stepwise Regression: Y versus X1, X2, X3, X4, X5 Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15 Response is Y on 5 predictors, with N = 17 Step Constant X1 T-Value P-Value X2
1 -106.082
2 -118.446
3 2.008
4 1375.574
33.7 18.72 0.000
24.0 6.97 0.000
9.1 1.99 0.068
13.3 2.76 0.017
0.081
0.079
0.059
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Free Predictors
: masukkan peubah bebas (X) = C1 - C5
Best Subsets Regression: Y versus X1, X2, X3, X4, X5 Response is Y Vars 1 1 2
R-Sq 95.9 95.6 98.5
R-Sq(adj) 95.6 95.3 98.3
Mallows C-p 37.7 42.0 7.3
S 1163.2 1210.8 722.57
X X X X X 1 2 3 4 5 X X
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no
D1
D2
D3
keterangan
Y
taraf
ulangan
1
0
0
0
10
A
1
2
0
0
0
11
A
2
3
0
0
0
8
A
3
4
0
0
0
7
A
4
5
0
0
0
9
A
5
6
1
0
0
13
B
1
7
1
0
0
11
B
2
8
1
0
0
10
B
3
9
1
0
0
9
B
4
10
1
0
0
10
B
5
11
0
1
0
14
C
1
12
0
1
0
12
C
2
13
0
1
0
11
C
3
14
0
1
0
10
C
4
15
0
1
0
13
C
5
16
0
0
1
14
D
1
17
0
0
1
11
D
2
18
0
0
1
10
D
3
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Interpretasi : Dari output diatas untuk uji koefisien regresi diperoleh informasi bahwa ratarata perlakuan perlakuan B tidak dengan rata-rata rata-rata respon dari control control (A). tidak berbeda berbeda nyata nyata dengan sedangkan rata-rata respon untuk perlakuan C dan D pada taraf nyata 5% berbeda nyata dengan rata-rata respon perlakuan A. Dari Dari pers persam amaa aan n diat diatas as dipe dipero role leh h info inform rmas asii bahw bahwa a rata rata-ra -rata ta resp respon on dari dari perl perlak akua uan n C lebi lebih h ting tinggi gi sebe sebesa sarr 3.00 3.00 dari dari rata rata-ra -rata ta resp respon on perl perlak akua uan n A sert serta a perlakuan D mempunyai rata-rata respon lebih tinggi dari rata-rata respon perlakuan A sebesar 2.20.
4.5
Regres Regresii Percob Percobaa aan n Satu Satu Fakto Faktorr Berta Bertaraf raf Kuan Kuantit titat atif if
Model regresi percobaan percobaan satu faktor bertaraf bertaraf kuantitat kuantitatif if sering sering dirumuskan dirumuskan dalam bentuk fungsi fungsi polynomial. polynomial. Persamaan Persamaan regresi polynomial polynomial yang menyatak menyatakan an hubungan antara variable respon (Y) dan taraf-taraf kuantitatif (X) dengan ordo q dapat dinyatakan sebagai berikut : Y = β 0
+ β 1 X + β 2 X 2 + ... + β q X q + ε
contoh : Percobaan pengaruh temperatur terhadap daya aktifitas baterai yang diukur
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Langkah-langkah dengan minitab : Klik Graph > Scatter Plot Pilih bentuk histogram yang akan ditampilkan ( daya baterai). Y variables variables : isi dengan kolom Respon (daya baterai ). X variables : isi dengan kolom variabel bebas (temperatur)
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Dari model regresi ordo 3 ini terlihat bahwa dari anova untuk model diperoleh informasi bahwa model sudah sangat tepat dalam merepresentasikan data ( nilai-p < 0.05) 0.05) serta diperoleh R-Sq yang tinggi (93.9% (93.9%). ). Sementara itu untuk uji koefisien secara parsial terlihat bahwa semua koefisien untuk semua variabel nyata sehingga kita kita putusk putuskan an untuk untuk menggu menggunak nakan an model model regres regresii polyno polynomia miall ordo ordo 3 atau atau model regresi kubik . Usaha menemukan model regresi polynomial yang lebih cepat dan mudah bisa dengan dengan menggunaka menggunakan n contrast contrast polynomial polynomial orthogonal pada program program SAS. Dimana untuk n = 5, koefisien polynomial orthogonalnya sebagai berikut : koefisien kontras ortogonal
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D3
D 4
= =
1 jika jika penga pengama matan tan dari taraf c
2
0 untuk penga pengama matan tan lain (bukan dari c 2 )
1 jika jika penga pengama matan tan dari taraf c 3 0 untuk penga pengama matan tan lain (bukan dari c 3 )
tahap ketiga diperoleh model regresi :
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m3
v3 v1 v2 v3
8.662 6.215 6.953 9.112
8.526 7.106 6.914 9.14
8.594 6.661 6.934 9.126
Untuk entri data seperti pada kasus-kasu kasus-kasus s sebelumnya sebelumnya,, dimana dimana faktor faktor yang bertaraf kualitatif bertaraf kualitatif akan akan ditransformasi dan faktor bertaraf kualitatif bertaraf kualitatif akan akan di-dummy.
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Residual Plots for respon
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Pada faktor varietas diperoleh informasi bahwa varietas_2 memberikan nilai
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akan berakibat ketidakefisienan pengujian yang akan kita lakukan, karena keragaman
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Sort Column
: isi dengan peubah yang akan diurutkan
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Goodness-of-Fit Tests Method Chi-Square Pearson 40.5980 Deviance 51.0291 Hosmer-Lemeshow 5.8452
DF 47 47 8
P 0.733 0.318 0.665
Table of Observed and Expected Frequencies: (See Hosmer-Lemeshow Test for the Pearson Chi-Square Statistic) Group Value 1 2 3 4 5 6 7 8 9 10 Total Low Obs 4 5 7 9 8 7 11 8 9 2 70 Exp 4.5 5.7 6.9 7.3 6.9 8.9 11.8 7.9 8.3 1.9 High Obs 5 4 3 1 1 4 3 1 0 0 22 Exp 4.5 3.3 3.1 2.7 2.1 2.1 2.2 1.1 0.7 0.1 Total 9 9 10 10 9 11 14 9 9 2 92 Measures of Association: (Between the Response Variable and Predicted Probabilities) Pairs Concordant Discordant Ties Total
Number 1046 462 32 1540
Percent 67.9 30.0 2.1 100.0
Summary Measures Somers' D Goodman-Kruskal Gamma Kendall's Tau-a
0.38 0.39 0.14
Intepretasi : 1. Dili Diliha hatt dari dari nila nilaii ujiuji-G G denga engan n p-va p-valu lue e = 0.02 0.021 1 < nila nilaii α = 0.05 0.05,, dapa dapatt disimp disimpulk ulkan an bahwa bahwa peubah peubah bebas bebas (X) berpengaruh berpengaruh nyata terhadap terhadap peubah peubah respon (Y) 2. Diliha Dilihatt dari dari nilai nilai uji Wald Wald (Z), (Z), yait yaitu: u: Nilai parameter parameter β dengan nila nilai-p i-p = 0.03 0.030 0 < nila nilaii α = 0.05 0.05 dapa dapatt 1 disimpulkan bahwa koefisien peubah X1 (Smokes) nyata. Nilai parameter parameter β 2 dengan nilai nilai-p -p = 0.032 0.032 < nila nilaii α = 0.05, 0.05, dapa dapatt disimpulkan bahwa koefisien peubah X2(Weight) nyata. nyata. 3. Intepr Intepreta etasi si koefis koefisien ien model probit probit dilaku dilakukan kan dengan dengan meliha melihatt tanda tanda koefis koefisien ien itu sendiri. Untuk X1 koefisien yang diperoleh negatif , maka kecenderungan resting pulse
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Jadi fungsi linear dari peubah penjelas kasus diatas adalah g(x) = -1.2011 0.7038X1 + 0.015085X2 ; sehingga model peluang regresi probit dengan 2 faktor (peubah penjelas) adalah :
π 1 = F ( g ( x ) ) =
g ( x )
1 2π
∫ e
−
s2 2
ds
−∞
Kesimpulan : Regresi Logit dan probit digunakan untuk mengetahui hubungan antara peubah respon berskala biner ( dicotomous/binary ) dengan dengan beberapa beberapa peubah peubah penjelas penjelas yang bersifat kategorik atau kontinu. Model logit dan model probit merupakan dua model regresi yang saling dapat menggantikan satu dengan yang lain. Oleh karena itu sering hanya dibuat salah satu model tanpa mempertimbangkan model lain yang mungkin akan menghasilkan model yang lebih sesuai.
10 ANALISIS ANALISIS MULTIVARI MULTIVARIATE ATE 10.1 10.1 ANALIS ANALISIS IS KOMPO KOMPONEN NEN UTAMA UTAMA Telah dibahas pada bagian multikolinearitas multikolinearitas
10.2 10.2 ANALIS ANALISIS IS KORESPON KORESPONDEN DENSI SI Digunakan Digunakan jika kita ingin melihat hubungan antar peubah peubah kategorik kategorik secara visual dimensi ganda. Korespondensi digunakan untuk menggambarkan sebuah tabel Kontingensi. Analisis korespondensi dibagi 2 yaitu analisis korespondensi sederhana dan berganda.
10.2.1 10.2.1 Analisis Analisis korespo korespondens ndensii sederhana sederhana Digunakan bila hanya 2 peubah kategorik yang ingin dilihat hubungannya. Contoh: Seoran Seorang g peneli peneliti ti ingin ingin menget mengetahu ahuii bagaim bagaimana ana hubung hubungan an suatu suatu disipl disiplin in ilmu ilmu atau atau akadem akademik ik dengan dengan bebera beberapa pa katego kategori ri latar latar belaka belakang ng keuang keuangan an yaitu yaitu A sampai sampai E. Didapatkan hasil sebagai berikut: A
B 3 1 6 3 10 3 1 0 2 2
C 19 2 25 15 22 11 6 12 5 11
D 39 13 49 41 47 25 14 34 11 37
E 14 1 21 35 9 15 5 17 4 8
Akademik 10 12 29 26 26 34 11 23 7 20
Geologi biokimia kimia zoologi fisika teknik mikrobiologi biologi statistik matematika
Fund A B C D E
Tahapan menggunakan menu MINITAB: Klik Stat > Multivariate > Simples Correspondence Analysis Input Data : pilih Columns of a Contingency Table isi dengan data A-E (C1-C5) Rownames : isi dengan Akademik Colomn name : isi dengan Fund Klik Graph : klik Symmetric plot showing rows and columns. columns . klik Results: klik Contingency Table
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Simple Correspondence Analysis: CT1, CT2, CT3, CT4, CT5 Contingency Table Geology Biochemistry Output : Chemistry Zoology Physics Engineering Microbiology Botany Statistics Mathematics Total
A 3 1 6 3 10 3 1 0 2 2 31
B 19 2 25 15 22 11 6 12 5 11 128
C 39 13 49 41 47 25 14 34 11 37 310
D 14 1 21 35 9 15 5 17 4 8 129
E 10 12 29 26 26 34 11 23 7 20 198
Total 85 29 130 120 114 88 37 86 29 78 796
Analysis of Contingency Table Axis 1 2 3 4 Total
Inertia 0.0391 0.0304 0.0109 0.0025 0.0829
Proportion 0.4720 0.3666 0.1311 0.0303
Cumulative 0.4720 0.8385 0.9697 1.0000
Histogram ****************************** *********************** ******** *
Row Contributions ID 1 2 3 4 5 6 7 8 9 10
Name Geology Biochemistry Chemistry Zoology Physics Engineering Microbiology Botany Statistics Mathematics Mathematics
Qual 0.916 0.881 0.644 0.929 0.886 0.870 0.680 0.654 0.561 0.319
Mass 0.107 0.036 0.163 0.151 0.143 0.111 0.046 0.108 0.036 0.098
Inert 0.137 0.119 0.021 0.230 0.196 0.152 0.010 0.067 0.012 0.056 0.056
Component Coord Corr -0.076 0.055 -0.180 0.119 -0.038 0.134 0.327 0.846 -0.316 0.880 0.117 0.121 -0.013 0.009 0.179 0.625 -0.125 0.554 -0.107 0.240
1 Contr 0.016 0.030 0.006 0.413 0.365 0.039 0.000 0.088 0.014 0.029
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Column Contributions ID 1 2 3 4 5
Name A B C D E
Qual 0.587 0.816 0.465 0.968 0.990
Mass 0.039 0.161 0.389 0.162 0.249
Component Coord Corr -0.478 0.574 -0.127 0.286 -0.083 0.341 0.390 0.859 0.032 0.012
Inert 0.187 0.110 0.094 0.347 0.262
1 Contr 0.228 0.067 0.068 0.632 0.006
Component Coord Corr -0.072 0.013 -0.173 0.531 -0.050 0.124 -0.139 0.109 0.292 0.978
2 Contr 0.007 0.159 0.032 0.103 0.699
Symmetric Plot 0.50
Biochemistry
Engineering E
0.25 2 t n e n o 0.00 p m o C
Microbiology Mathematics Physics A
Botany
Statistics C Chemistry B
-0.25
Zoology D
Geology
-0.50 -0.50
-0.25
0.00 Component 1
0.25
0.50
interpretasi : Dari output juga ditampilkan lagi table kontingensi pada data dia atas, pada anal analis isis is tabl table e kont kontin inge gens nsii terl terlih ihat at dari dari tabe tabell ukur ukuran an 10x5 10x5 diri diring ngka kas s menj menjad adii 4 kompon komponen. en. Output Output selanj selanjutn utnya ya adalah adalah Row Contributi yang member memberika ikan n Contributions ons yang interp interpret retasi asi 2 kompon komponen. en. Coord Coord menunj menunjukk ukkan an koordi koordinat nat uatam uatama a dari dari baris. baris. Corr Corr menunjukkan kontribusi komponen pada inertia baris, dimana komponen 1 mampu merepresentasikan Zoology and Physics dengan baik. Contr menunjukkan kontribusi baris baris terhad terhadap ap masin masing-m g-masi asing ng sumbu, sumbu, dimana dimana Zoology Zoology and Physic Physics s mempun mempunyai yai kontribusi tertinggi pada komponen 1. Kecenderungan kedekatan antara peubah akademik dan fund dapat dilihat di symm symmet etri ric c plot plot diat diatas as.. Terl Terlih ihat at bahw bahwa a untu untuk k akad akadem emik ik zool zoolog ogy y cend cender erun ung g berhubungan kuat dengan fund D serta engeenering sangat dekat kaitannya dengan fund E. Selanjutnya untuk untuk peubah-peubah lainnya dapat dapat dilihat sendiri kedekatannya. kedekatannya.
10.2.2 10.2.2 Analisis Analisis korespo korespondens ndensii berganda berganda Digunakan bila lebih dari 2 peubah kategorik yang ingin dilihat hubungannya.
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Peubah punya/Tidak HP memiliki dua kategori yaitu punya dan tidak. Misalkan kita kita beri beri kode kode 1 untuk untuk punya punya dan 2 untuk untuk tidak. tidak. Peubah Peubah SMS/mi SMS/minggu nggu memiliki memiliki 3 kategori yaitu 0-5, 5-10, dan 10-15. Misalkan kita beri kode 1,2, dan 3 untuk ketiganya. Demikian Demikian juga dengan dengan peubah peubah biaya biaya hidup perbulan dan IPK yang masing-amsi masing-amsing ng memilki 4 kategori dengan kode masing-masing 1,2,3 dan 4. Cara entry data kedalam MINITAB adalah sebagai berikut: c1 1 1 2 1 1 2 2 2 1 2 1 1 1 1 1 1 2 1 1 2
c2 3 2 1 3 2 2 1 2 3 1 3 2 2 3 2 2 1 3 3 2
c3 4 3 2 3 2 2 1 3 4 3 4 3 3 3 4 3 2 4 3 3
c4 2 3 3 1 3 4 3 2 4 3 1 3 3 3 2 4 3 3 4 4
c5 punya tidak 0-5 10-15 15-20