Library:

> # install.packages("knitr")
> # install.packages("rmarkdown")
> # install.packages("prettydoc")
> # install.packages("equatiomatic")

PENDAHULUAN

Latar Belakang

Penyakit kardiovaskular adalah penyebab utama kematian di seluruh dunia. Ini mencakup berbagai kondisi seperti penyakit jantung koroner, stroke, dan penyakit pembuluh darah lainnya. Studi epidemiologi menunjukkan bahwa banyak faktor berkontribusi terhadap risiko seseorang terkena penyakit kardiovaskular. Beberapa faktor ini termasuk tekanan darah, kolesterol, obesitas, merokok, minum minuman keras, umur, dan banyak lainnya.

Identifikasi faktor-faktor risiko yang berkontribusi terhadap penyakit kardiovaskular adalah langkah baik dalam upaya pencegahan dan pengelolaan penyakit ini. Memahami faktor-faktor ini memungkinkan untuk mengembangkan strategi pencegahan yang lebih efektif dan merancang program intervensi yang tepat.

Dalama penelitian kasus ini akan menggunakan metode PCA. Principal Component Analysis (PCA) adalah teknik statistik yang digunakan untuk mereduksi dimensi data, mengidentifikasi pola, dan mengidentifikasi hubungan antarvariabel. Dalam konteks penyakit kardiovaskular, PCA dapat membantu mengidentifikasi faktor-faktor yang paling kuat mempengaruhi risiko penyakit dan hubungan antara variabel-variabel ini.

Tujuan dari penelitian ini adalah untuk menjalankan analisis PCA pada data faktor risiko penyakit kardiovaskular. PCA akan membantu kita mengidentifikasi faktor-faktor utama yang berkorelasi dengan risiko penyakit kardiovaskular dan memberikan pemahaman lebih dalam tentang bagaimana variabel-variabel ini berinteraksi.

Tinjauan Pustaka

Konsep dan Dasar PCA

Principal Component Analysis (PCA) atau disebut Analisis Komponen Utama (AKU) adalah suatu teknik statistik multivariat yang secara linear mengubah bentuk sekumpulan variabel asli menjadi kumpulan variabel yang lebih kecil yang tidak berkorelasi yang dapat mewakili informasi dari kumpulan variabel asli (Radiarta, dkk. 2013). Metode PCA menjadi efektif saat dihadapkan pada data dengan jumlah variabel yang besar dan memiliki korelasi yang signifikan antara variabel-variabelnya. Metode analisis komponen utama ini didasarkan pada perhitungan nilai eigen dan vektor eigen yang menyatakan penyebaran data dari suatu dataset. Nilai eigen mengukur sejauh mana komponen utama menjelaskan variasi dalam data, sedangkan vektor eigen menggambarkan arah komponen utama dalam ruang data asli. Menurut Johnson dan Wichern, dengan menggunakan PCA, variabel yang tadinya sebanyak n variabel akan direduksi menjadi k variabel baru (principal component) dengan jumlah k lebih sedikit dari n dan dengan hanya menggunakan k principal component akan menghasilkan nilai yang sama dengan menggunakan n variabel. Untuk penentuan banyaknya principal component yang digunakan terdapat 2 cara:

  1. Memilih komponen utama yang merupakan hasil dari nilai eigen dimana nilai eigen tersebut bernilai lebih dari 1.
  2. Memilih k buah komponen utama pertama dimana k buah komponen utama dapat menjelaskan keragaman sebaran minimal 75%

\[ \frac {\left(\sum_{i=1}^{k} \lambda_i\right)} {\left(\sum_{i=1}^{p} \lambda_i\right)} \geq 0.75 \]

Terdapat beberapa tujuan dari dilakukannya Principal Component Analysis (PCA), diantaranya:

  • Menggali esensi informasi yang paling krusial dari sebuah dataset.
  • Mereduksi ukuran data dengan mengekstrak hanya informasi penting saja.
  • Menyederhanakan deskripsi dari dataset.
  • Menganalisis struktur observasi dan variabel
  • Mengurangi dimensi data tanpa mengurangi informasi yang terkandung pada dataset.

Algoritma dan Teknik PCA

Berikut adalah beberapa algoritma dan teknik terkait Principal Component Analysis (PCA):

  1. Mempersiapkan data yang akan dianalisis menggunakan metode Principal Component Analysis (PCA).
  2. Melakukan standarisasi data jika varianel dalam skala yang berbeda.
  3. Melakukan eksplorasi korelasi yang dapat memberikan pemahaman yang lebih mendalam tentang hubungan antara variabel asli dalam dataset.
  4. Melakukan perhitungan matriks kovarians.
  5. Melakukan perhitungan nilai eigen dan vektor eigen dari matriks kovarians.
  6. Membentuk scree plot yang digunakan sebagai alat visual dalam metode Principal Component Analysis (PCA) untuk membantu menentukan berapa banyak komponen utama yang harus dipertahankan.
  7. Melakukan perhitungan nilai kumulatif eigen yang mencerminkan jumlah variasi yang dijelaskan oleh komponen-komponen utama tertentu dan komponen-komponen sebelumnya
  8. Melakukan perhitungan persamaan PC
  9. Melakukan perhitungan nilai loading yang menggambarkan sejauh mana setiap variabel berkontribusi terhadap pembentukan setiap komponen utama.
  10. Membentuk plot PCA sebagai bentuk visual dari hasil Principal Component Analysis (PCA)

Data

Data yang digunakan berasal dari kaggle yang merupakan kumpulan data 462 individu yang memiliki resiko penyakit kardiovaskular di Western Cape, Afrika Selatan. Namun, dalam penelitian mengenai faktor-faktor yang menyebabkan seseorang terkena penyakit kardiovaskular ini hanya digunakan 150 sampel. Dataset yang digunakan mengandung 9 variabel, diantaranya:

  • X1 sbp (Systolic Blood Pressure)

  • X2 tobacco (Cumulative tobacco (kg))

  • X3 ldl (Low density lipoprotein cholesterol)

  • X4 adiposity

  • X5 typea (Type A behavior)

  • X6 obesity

  • X7 alcohol (Current alcohol consumption)

  • X8 age (Age on onset)

  • X9 chd (response, cardiovaskular disease)

SOURCE CODE

Library yang Dibutuhkan

> library(readxl)
> library(corrplot)
> library(factoextra)
> library(FactoMineR)

Import Data

> library(readxl)
> kardiovaskular = read_excel("C:/Users/user/Documents/AA/kardiovaskular.xlsx", 
+                        col_types = c("numeric", "numeric", "numeric", 
+                                      "numeric", "numeric", "text", 
+                                      "numeric", "numeric", "numeric",
+                                      "numeric", "numeric"))
> kardiovaskular = subset(kardiovaskular, select = -c(1,6))
> kardiovaskular = head(kardiovaskular, 150)
> View(kardiovaskular)

Statistika Deskriptif

> summary(kardiovaskular)

Eksplorasi Korelasi

> library(corrplot)
> korelasi = cor(kardiovaskular)
> corrplot(korelasi, method="number")

Matriks Kovarians

Nilai Eigen dan Vektor Eigen

> sc = scale(kardiovaskular)
> sc
> 
> s = cov(sc)
> s_eigen = eigen(s)
> s_eigen

Scree Plot

> plot(s_eigen$values, xlab="Eigenvalue Number", 
+      ylab="Eigenvalue Size", main="Scree Plot")
> lines(s_eigen$values)

Nilai Kumulatif Eigen

> for (eg in s_eigen$values)
+ {
+   print(eg/sum(s_eigen$values))
+ }

Persamaan PC

> s_eigen$vectors[,1:5]

Matriks Korelasi

Nilai Eigen dan Vektor Eigen

> kor_eigen = eigen(korelasi)
> kor_eigen

Scree Plot

> plot(kor_eigen$values, xlab="Eigenvalue Number", 
+      ylab="Eigenvalue Size", main="Scree Plot")
> lines(kor_eigen$values)

Nilai Kumulatif

> for (cor in kor_eigen$values)
+ {
+   print(cor/sum(kor_eigen$values))
+ }

Persamaan PC

> kor_eigen$vectors[,1:5]

Fungsi PCA

Fungsi PCA dengan library prcomp

> pca1 = prcomp(x=kardiovaskular, scale=T, center=T)
> pca1
> print(pca1$rotation[,1:3],digits=4)
> summary(pca1)

Fungsi PCA dengan library factoextra dan FactoMineR

> library(factoextra)
> library(FactoMineR)
> pca2 = princomp(x=kardiovaskular, cor=T)
> summary(pca2)

Nilai Loading

> print(pca2$loadings, digits=4, cutoff=0.1)

Plot PCA

> plot(pca2)

Nilai Kontribusi

> pca_contrib = get_pca_var(pca2)
> pca_contrib$contrib

PCA dengan function PCA

> pca3 = PCA(kardiovaskular, scale.unit=T, graph=FALSE)
> pca3$eig

Plot Kontribusi Variabel

> fviz_pca_var(pca3, col.var="contrib", gradient.cols=c("#00AFBB",
+                                                   "#E7B800","#FC4E07"),
+              axes=c(1,2), repel=T)

Plot Individu

> fviz_pca_ind(pca3, title="Individual ~ PCA", axes=c(1,2))

Biplot

> fviz_pca_biplot(pca3, axes=c(1,2), repel=TRUE, col.var = "#2E9FDF",
+                 col.ind = "#696969")

HASIL DAN PEMBAHASAN

Import Data dan Struktur Data

tibble [150 × 9] (S3: tbl_df/tbl/data.frame)
 $ sbp      : num [1:150] 160 144 118 170 134 132 142 114 114 132 ...
 $ tobacco  : num [1:150] 12 0.01 0.08 7.5 13.6 6.2 4.05 4.08 0 0 ...
 $ ldl      : num [1:150] 5.73 4.41 3.48 6.41 3.5 6.47 3.38 4.59 3.83 5.8 ...
 $ adiposity: num [1:150] 23.1 28.6 32.3 38 27.8 ...
 $ typea    : num [1:150] 49 55 52 51 60 62 59 62 49 69 ...
 $ obesity  : num [1:150] 25.3 28.9 29.1 32 26 ...
 $ alcohol  : num [1:150] 97.2 2.06 3.81 24.26 57.34 ...
 $ age      : num [1:150] 52 63 46 58 49 45 38 58 29 53 ...
 $ chd      : num [1:150] 1 1 0 1 1 0 0 1 0 1 ...

Statistika Deskriptif

      sbp           tobacco            ldl           adiposity    
 Min.   :103.0   Min.   : 0.000   Min.   : 1.070   Min.   : 9.39  
 1st Qu.:121.2   1st Qu.: 0.325   1st Qu.: 3.380   1st Qu.:19.45  
 Median :131.0   Median : 2.560   Median : 4.520   Median :25.66  
 Mean   :133.7   Mean   : 3.917   Mean   : 4.958   Mean   :25.17  
 3rd Qu.:142.0   3rd Qu.: 5.997   3rd Qu.: 6.082   3rd Qu.:30.90  
 Max.   :206.0   Max.   :31.200   Max.   :15.330   Max.   :42.49  
     typea          obesity         alcohol             age       
 Min.   :13.00   Min.   :17.75   Min.   :  0.000   Min.   :15.00  
 1st Qu.:49.00   1st Qu.:22.88   1st Qu.:  0.680   1st Qu.:32.00  
 Median :55.00   Median :25.80   Median :  6.575   Median :45.00  
 Mean   :54.68   Mean   :26.16   Mean   : 14.887   Mean   :42.91  
 3rd Qu.:61.00   3rd Qu.:28.72   3rd Qu.: 21.575   3rd Qu.:54.75  
 Max.   :78.00   Max.   :46.58   Max.   :108.000   Max.   :64.00  
      chd      
 Min.   :0.00  
 1st Qu.:0.00  
 Median :0.00  
 Mean   :0.38  
 3rd Qu.:1.00  
 Max.   :1.00

Berdasarkan data yang digunakan terlihat bahwa setiap variabel memiliki rentang nilai yang jauh, sehingga akan diperlukan standarisasi.

Eksplorasi Korelasi

Dari ouput yang telah disajikan di atas dapat diketahui nilai korelasi antar variabel dalam dataset. Terdapat korelasi positif moderat dengan nilai korelasi antara 0.40 hingga 0.70, seperti pada antara variabel X1 dan X8, X2 dan X8, X3 dan X4, dan lain-lain. Terdapat korelasi positif lemah dengan nilai korelasi antara 0.20 hingga 0.40, seperti pada antara variabel X1 dan X2, X1 dan X4, X1 dan X7, dan lain-lain. Terdapat korelasi positif sangat lemah dengan nilai korelasi antara 0 hingga 0.20, seperti pada antara variabel X1 dan X3, X1 dan X5, X1 dan X6, dan lain-lain. Selain itu, terdapat juga korelasi negatif yang sangat lemah dengan nilai korelasi antara -0.20 hingga 0, seperti X3 dan X7, X4 dan X5, dan lain-lain.

Matriks Kovarians

Nilai Eigen dan Vektor Eigen

               sbp      tobacco         ldl   adiposity       typea
  [1,]  1.54388947  1.731302084  0.34318888 -0.26642955 -0.59773052
  [2,]  0.60392632 -0.836884779 -0.24350943  0.44394593  0.03367496
  [3,] -0.92351379 -0.821891194 -0.65686507  0.91796011 -0.28202778
  [4,]  2.13136643  0.767428783  0.64542741  1.66062538 -0.38726202
  [5,]  0.01644936  2.074012591 -0.64797570  0.33674381  0.55984619
  [6,] -0.10104604  0.488976496  0.67209552  1.42555568  0.77031468
  [7,]  0.48643093  0.028459252 -0.70131191 -1.15891947  0.45461194
  [8,] -1.15850458  0.034885074 -0.16350512 -1.36557416  0.77031468
  [9,] -1.15850458 -0.839026720 -0.50130112 -0.74561010 -0.59773052
 [10,] -0.10104604 -0.839026720  0.37430167  0.74747000  1.50695440
 [11,]  4.24628351  0.446137682 -0.89243333  0.91666852  1.82265714
 [12,]  0.01644936  2.181109625 -0.23017538 -0.35942416  1.08601742
 [13,] -0.92351379 -0.839026720 -1.36801454 -1.95324842  0.45461194
 [14,] -0.10104604 -0.839026720 -1.37245922 -1.02846870 -0.59773052
 [15,] -1.27599997  1.227946027 -1.18578249 -1.02976029 -0.07155929
 [16,] -0.98226149 -0.511309797 -1.11911222  0.48786005 -2.07100996
 [17,] -0.80601840  0.767428783  4.61008573 -0.40979624  0.55984619
 [18,]  0.72142171  1.410010984  1.48102804  1.31577038  2.45406261
 [19,]  1.42639407 -0.282122145  1.11211925  1.14915504  0.66508043
 [20,] -0.57102761  2.159690218  0.56542310  1.39326589 -1.01866750
 [21,] -1.62848615 -0.494174272 -1.43024012 -1.66005708  2.03312563
 [22,] -0.10104604  0.853106410 -0.93688017  0.17142006 -0.38726202
 [23,]  0.95641250 -0.774768499  0.63209336  1.13882230  0.77031468
 [24,]  0.25144014 -0.710510279 -0.51019049  0.45040389 -0.07155929
 [25,]  0.48643093  3.059305299 -0.27462222 -0.10239740  0.66508043
 [26,] -0.57102761  0.017749548  3.31668263  0.79009252 -0.07155929
 [27,] -0.92351379  0.446137682  2.08550510  1.12848957  0.55984619
 [28,]  0.66267402  1.110139290  0.12539936  0.30703720  0.45461194
 [29,]  0.60392632  0.037027014  0.26318457  0.80430003  0.55984619
 [30,]  0.72142171 -0.839026720  0.73876578  0.06680113  0.55984619
 [31,]  0.13394475 -0.299257671 -0.44796491  0.05905158 -0.38726202
 [32,]  1.42639407 -0.620548771  0.60986994 -0.16697699  1.19125166
 [33,] -0.68852300  0.574654122  0.27651862  1.39197429 -0.17679353
 [34,] -0.45353222  1.035171366  0.69876362  1.14269708 -0.59773052
 [35,]  0.83891711  0.339040649  0.95211062  0.01772064  0.13890920
 [36,] -0.68852300  0.073440006 -0.23017538 -1.56706248  0.24414345
 [37,]  0.36893553 -0.003669858  1.04989368 -0.01586075 -0.80819901
 [38,] -1.39349536  0.154833751 -0.18128385  0.68289041 -0.70296476
 [39,] -0.21854143 -0.839026720 -0.95021422 -0.71590349  1.61218865
 [40,]  0.13394475  1.559946831  0.37874636  0.86242166  2.13835988
 [41,] -0.92351379 -0.779052381  0.37430167  1.10136614  0.55984619
 [42,]  0.60392632 -0.830458957 -0.70131191 -0.20184996 -2.59718119
 [43,] -0.80601840 -0.839026720 -1.72803396 -1.18216812 -0.80819901
 [44,] -0.21854143 -0.279980205 -0.99466107 -0.28192865 -0.38726202
 [45,] -1.15850458 -0.839026720 -0.87465459 -1.99328776 -0.07155929
 [46,] -0.33603682  0.156975692 -0.73242470 -0.31421845  0.77031468
 [47,]  1.66138486  0.746009376  1.59658983 -0.06752442  0.98078317
 [48,] -1.04100918 -0.429916052  1.15656610  0.16496210 -0.28202778
 [49,] -1.15850458 -0.839026720 -1.34134643 -1.82796401 -0.07155929
 [50,] -0.45353222 -0.025089265 -0.47907770  0.85467211  0.24414345
 [51,] -0.68852300 -0.839026720  0.35207825  0.73972044 -0.91343325
 [52,]  0.01644936 -0.303541552 -0.57686075  0.73972044 -0.28202778
 [53,]  1.07390789 -0.646252059  1.84993683  0.65318380  0.13890920
 [54,]  0.01644936  0.891661342 -1.51468912 -0.99101254  0.13890920
 [55,]  1.30889868 -0.196444519 -1.39468265  0.30703720  0.55984619
 [56,]  1.07390789  0.443995742  1.34768752  0.94379195 -1.01866750
 [57,] -0.92351379 -0.839026720 -0.87465459 -1.16279425 -0.59773052
 [58,] -0.45353222  0.253363022 -0.88798864  0.17142006  0.03367496
 [59,] -1.80472924 -0.832600898 -0.33240312 -0.80244014 -0.70296476
 [60,] -0.74727070 -0.667671466  0.14762278 -0.80373173 -0.80819901
 [61,]  0.48643093 -0.779052381 -1.40357201 -0.53508064  0.24414345
 [62,]  0.25144014 -0.592703543  0.05872909  0.34836813  0.66508043
 [63,]  1.07390789  1.324333357 -0.11016891 -0.06752442  1.08601742
 [64,]  0.36893553 -0.742639389 -0.29240096 -0.10885536 -1.43960448
 [65,] -0.21854143 -0.839026720 -1.39468265 -1.90158475  0.24414345
 [66,]  0.13394475  0.737441613 -1.23022933  0.37936634  0.66508043
 [67,] -0.57102761  0.193388683 -0.76353749 -0.52603950 -0.70296476
 [68,] -1.27599997 -0.751207152 -1.36801454 -1.92225021 -1.65007298
 [69,] -0.92351379  0.116278819  1.02767025  0.51110870 -0.70296476
 [70,] -0.68852300 -0.839026720 -0.70575659 -1.17183539  1.29648591
 [71,] -0.92351379 -0.839026720 -0.57241606 -1.68459733 -0.38726202
 [72,] -0.21854143 -0.470612924 -1.02132917 -1.91062589  1.40172016
 [73,] -0.21854143  0.360460055 -0.70575659 -0.04815054  0.34937770
 [74,] -0.45353222 -0.819749253  0.03206099 -1.53735586 -0.49249627
 [75,] -0.33603682 -0.753349093  0.53875499  0.15204618  0.98078317
 [76,]  0.13394475 -0.839026720 -0.37240528 -1.00134528 -0.28202778
 [77,]  0.01644936 -0.839026720  0.41874852  0.73197089 -0.59773052
 [78,]  0.36893553 -0.710510279  0.26762925  1.06132680  0.34937770
 [79,]  2.01387104  0.124846582  0.76543388  0.42586364 -1.22913599
 [80,] -1.51099075 -0.753349093  0.42319320 -0.29096980  0.24414345
 [81,] -1.15850458 -0.196444519  0.92544252 -0.32713437  0.03367496
 [82,]  0.36893553  0.904512986 -0.01238585  2.23667532 -0.17679353
 [83,]  0.83891711  0.189104802  0.50319752  1.46946980  0.87554893
 [84,]  0.83891711  1.774140897 -0.51907985  1.15948777  0.24414345
 [85,] -0.33603682 -0.839026720 -1.12355691 -1.55285497  0.87554893
 [86,] -0.21854143 -0.719078042 -0.73686938  0.73455408 -0.59773052
 [87,] -0.45353222  1.410010984 -0.20795196 -1.01296960  1.29648591
 [88,]  0.36893553 -0.839026720  0.05428441  0.27862218 -1.43960448
 [89,] -0.45353222 -0.646252059  0.30318673 -0.95484797  0.03367496
 [90,] -0.68852300 -0.684806991 -0.40796275  0.93087603 -2.17624420
 [91,] -1.04100918 -0.618406831 -0.94576954 -1.84992107 -1.01866750
 [92,] -0.80601840 -0.046508672 -0.41685212  1.87115484  0.66508043
 [93,]  0.54517862 -0.740497449 -1.13689096 -0.29742776  0.77031468
 [94,] -0.92351379  0.017749548 -0.44796491 -0.80244014 -0.07155929
 [95,]  3.54131115 -0.474896806  0.60542525  1.09749137 -0.80819901
 [96,]  0.01644936 -0.196444519 -0.26128817 -0.27159592  0.13890920
 [97,]  0.25144014 -0.376367535 -0.02571991 -0.04427577 -1.65007298
 [98,]  0.13394475 -0.839026720  0.01872694  0.31091197 -0.59773052
 [99,] -0.68852300 -0.153605705  2.82776736  1.31577038  0.03367496
[100,]  1.77888025  1.731302084 -0.46574364 -0.72106986 -0.38726202
[101,]  0.13394475  0.874525816  1.28546194 -0.17601813 -0.38726202
[102,]  1.89637564 -0.824033135 -0.41240743  0.53177417 -0.17679353
[103,] -0.92351379 -0.839026720 -0.27462222  0.63897629 -0.28202778
[104,] -0.33603682 -0.749065211 -0.15906043  0.19466871 -1.43960448
[105,] -0.92351379 -0.517735619  0.18762494  0.08617500  0.98078317
[106,]  1.42639407 -0.067928078 -0.88354396  0.63768469  0.87554893
[107,] -1.51099075 -0.517735619 -0.27906691 -0.02361030  1.19125166
[108,]  2.13136643  0.788848189  0.24096115  1.63479355 -1.33437024
[109,] -0.92351379 -0.624832653  0.35652294 -0.39688032  0.77031468
[110,] -0.57102761 -0.839026720 -0.85243117 -1.01296960 -0.59773052
[111,] -1.15850458 -0.839026720  1.35657689 -0.45629354  1.19125166
[112,]  2.01387104  1.088719883  1.58770047 -0.08948148  1.50695440
[113,]  0.01644936 -0.410638586 -0.57686075 -1.35394983 -0.28202778
[114,]  2.36635722 -0.839026720  1.55658768  1.28218899 -2.07100996
[115,] -1.04100918  5.843828170 -0.79465028 -1.31520208 -0.80819901
[116,] -0.33603682 -0.839026720  2.49886073  0.85725530 -0.91343325
[117,]  0.36893553  0.124846582 -0.16350512 -0.92514136  0.87554893
[118,]  1.19140329 -0.689090873  0.42319320 -0.02231871 -4.38616337
[119,]  0.95641250 -0.089347485  0.90321910  0.02805337 -0.49249627
[120,] -0.21854143 -0.839026720 -0.46129896  0.04871884  1.40172016
[121,] -0.33603682 -0.410638586  0.52097625 -0.49891607  1.19125166
[122,] -0.80601840 -0.539155026  0.57431246 -0.60740978  0.55984619
[123,] -0.80601840 -0.839026720  0.02317162  0.12363117  0.98078317
[124,]  0.25144014  0.124846582 -0.93688017  0.63768469  0.03367496
[125,]  1.13265559  0.831687003 -0.44352022  0.07196749 -0.07155929
[126,] -0.62977531  1.003042256  2.76109710  1.30543764  1.61218865
[127,]  0.83891711  0.026317311 -0.43018617 -0.57899476  0.55984619
[128,]  0.13394475  0.009181786 -0.97688233  0.65964175 -0.49249627
[129,]  0.01644936  1.045881070  1.08989583  0.21533418 -2.07100996
[130,]  1.07390789  1.769857016 -0.40796275  1.63479355  0.87554893
[131,]  1.42639407  2.052593185  0.03650567  0.72551293 -0.07155929
[132,] -0.10104604 -0.410638586 -0.83465243  1.31964515 -1.01866750
[133,]  0.01644936 -0.517735619 -0.54574796 -0.47050105 -1.43960448
[134,]  0.48643093  0.754577139  0.24985052  1.13623912 -0.80819901
[135,]  0.01644936  0.446137682 -0.73686938  0.42328046  1.08601742
[136,] -0.68852300  0.056304480  1.81882404  0.52919098 -1.12390175
[137,] -1.04100918 -0.260702739 -0.56352670 -1.50506607  0.03367496
[138,] -0.33603682 -0.731929686 -0.55908201 -1.59676909  1.19125166
[139,] -0.80601840 -0.839026720 -0.56797138 -1.67038982 -0.38726202
[140,] -0.57102761 -0.839026720 -0.44796491  1.44363796  0.45461194
[141,]  1.54388947  2.159690218  0.41874852  1.54309053  0.34937770
[142,] -0.21854143 -0.243567213 -0.03016459 -2.03849347  0.87554893
[143,] -0.33603682 -0.239283332  0.25429520 -1.40561350  0.98078317
[144,] -0.21854143  0.124846582  0.40096978  1.58312988  0.66508043
[145,] -1.45224306 -0.581993839  0.52542094  0.52789939 -0.80819901
[146,]  0.60392632 -0.839026720 -0.49685643 -0.83343834  0.13890920
[147,] -0.92351379 -0.614122949 -0.79909496 -1.57481203 -0.91343325
[148,]  0.13394475 -0.097915248  0.63209336  0.91408533 -1.22913599
[149,]  0.13394475 -0.517735619  0.48986346  0.17658643 -0.07155929
[150,] -0.57102761  2.480981318  0.04095036 -0.14372833 -0.91343325
             obesity     alcohol          age        chd
  [1,] -0.1871027961  3.94949329  0.661775355  1.2730679
  [2,]  0.5936464232 -0.61546116  1.462310058  1.2730679
  [3,]  0.6526946835 -0.53149364  0.225120062 -0.7802674
  [4,]  1.2759818754  0.44972686  1.098430647  1.2730679
  [5,] -0.0362016865  2.03695299  0.443447708  1.2730679
  [6,]  1.0091712178 -0.03584534  0.152344180 -0.7802674
  [7,] -1.1690534949 -0.58859156 -0.357086995 -0.7802674
  [8,] -0.6660497962 -0.39186764  1.098430647  1.2730679
  [9,] -0.2833295906 -0.59482914 -1.012069934 -0.7802674
 [10,]  0.8648310260 -0.71430294  0.734551237  1.2730679
 [11,]  0.1431300670  1.97553674  1.243982411  1.2730679
 [12,] -0.6704237414 -0.71430294 -0.211535231  1.2730679
 [13,] -1.0028435771 -0.71430294 -1.885380519 -0.7802674
 [14,] -0.5523272208 -0.66776094 -2.030932283 -0.7802674
 [15,] -0.5741969468 -0.68167555  0.734551237 -0.7802674
 [16,] -0.0580714125  0.72657979  0.225120062 -0.7802674
 [17,] -0.1849158235  0.94057702  0.443447708 -0.7802674
 [18,]  1.4378178480 -0.04784070  0.734551237  1.2730679
 [19,]  0.6876862451  1.84214833  1.389534176  1.2730679
 [20,]  0.8604570808 -0.71430294  1.171206529  1.2730679
 [21,] -1.1449967963 -0.07279105 -1.667052873  1.2730679
 [22,]  0.0009768478  0.51929996  0.079568298 -0.7802674
 [23,] -0.3314429879 -0.71430294  0.516223590 -0.7802674
 [24,]  0.5564678889 -0.64425003  1.098430647 -0.7802674
 [25,]  0.0075377656 -0.71430294  0.516223590 -0.7802674
 [26,] -0.6398061249 -0.61546116 -0.065983466  1.2730679
 [27,]  2.7653102181 -0.71430294  0.370671826 -0.7802674
 [28,] -1.1362489058  0.32209623  1.316758294  1.2730679
 [29,]  0.7161168890 -0.44800593  0.952878883 -0.7802674
 [30,]  0.4186886150 -0.31941566  1.462310058  1.2730679
 [31,] -0.9394213716 -0.71430294  0.152344180  1.2730679
 [32,] -0.8803731113  0.48475332  0.225120062  1.2730679
 [33,]  0.4186886150 -0.11213583  1.171206529  1.2730679
 [34,]  0.8954486424 -0.71430294 -0.138759349  1.2730679
 [35,]  0.8057827657 -0.54156974  0.370671826 -0.7802674
 [36,] -1.4577338785  1.63630794 -1.084845816  1.2730679
 [37,]  0.2634135601  1.04997471 -0.793742287 -0.7802674
 [38,]  1.0376018616  0.01597462  0.225120062 -0.7802674
 [39,] -0.2833295906 -0.71430294 -1.012069934 -0.7802674
 [40,]  0.3333966834  0.38639136  1.098430647  1.2730679
 [41,]  1.0550976425 -0.71430294 -0.138759349  1.2730679
 [42,] -0.5260835496 -0.49070941 -0.939294052 -0.7802674
 [43,] -0.8759991661 -0.71430294 -2.030932283 -0.7802674
 [44,]  0.0294074916 -0.07279105  0.588999472  1.2730679
 [45,]  4.4667749033 -0.71430294 -1.885380519 -0.7802674
 [46,] -0.7010413578 -0.68983240  0.370671826 -0.7802674
 [47,] -0.0974369194 -0.43313168  1.098430647  1.2730679
 [48,]  0.8429613000 -0.54156974 -0.720966405  1.2730679
 [49,] -1.3090197415  1.15601370 -1.958156401 -0.7802674
 [50,]  0.9566838753 -0.71430294 -0.939294052 -0.7802674
 [51,]  0.6242640396 -0.51709921 -0.065983466 -0.7802674
 [52,]  0.2262350259  0.42093800  0.443447708 -0.7802674
 [53,]  0.5433460533 -0.69654980 -0.065983466  1.2730679
 [54,] -0.7666505359  2.48366022 -0.866518170  1.2730679
 [55,] -0.4910919879  1.87669497  0.734551237 -0.7802674
 [56,]  0.0906427245  4.09919539  0.370671826 -0.7802674
 [57,] -0.5085877688 -0.55980269 -1.084845816 -0.7802674
 [58,] -0.1389893988 -0.12221193 -0.357086995  1.2730679
 [59,] -0.7032283304 -0.58859156 -1.812604637 -0.7802674
 [60,] -0.7972681524 -0.71430294  1.316758294 -0.7802674
 [61,] -0.5479532756 -0.57371731 -0.720966405 -0.7802674
 [62,] -0.1105587550 -0.60202636  0.079568298 -0.7802674
 [63,]  0.0119117108  0.46268185  1.025654765 -0.7802674
 [64,]  0.2349829163 -0.23064999 -0.357086995 -0.7802674
 [65,] -0.8934949469 -0.61546116 -1.885380519 -0.7802674
 [66,] -0.2527119742  2.24663189  0.807327119 -0.7802674
 [67,]  0.5105414643 -0.31029919 -0.939294052 -0.7802674
 [68,] -0.8913079743  0.29234773 -1.157621698 -0.7802674
 [69,]  0.6242640396 -0.18122911 -0.720966405 -0.7802674
 [70,] -1.1143791798 -0.71430294 -0.793742287  1.2730679
 [71,] -1.5320909470 -0.68551407 -2.030932283 -0.7802674
 [72,] -1.8251452758 -0.18170892 -1.230397580 -0.7802674
 [73,] -0.0865020564  1.35849539 -0.502638759 -0.7802674
 [74,] -1.8382671114 -0.49214886 -1.667052873 -0.7802674
 [75,]  0.3727621903 -0.18122911 -0.648190523 -0.7802674
 [76,] -0.9831608236 -0.09726158 -0.211535231 -0.7802674
 [77,]  0.6570686287 -0.71430294  0.880103001 -0.7802674
 [78,]  0.2262350259 -0.71430294  0.880103001  1.2730679
 [79,] -0.4167349194  0.45548464  0.952878883  1.2730679
 [80,] -0.0952499468  2.74036094 -0.284311113 -0.7802674
 [81,] -0.7797723715 -0.71430294  0.152344180  1.2730679
 [82,]  4.2786952594 -0.40578226  0.734551237  1.2730679
 [83,] -0.1564851797 -0.67207927  0.880103001  1.2730679
 [84,]  0.0490902450 -0.02337016  1.025654765  1.2730679
 [85,] -1.1821753305 -0.71430294 -1.885380519 -0.7802674
 [86,]  0.2984051218  0.88491855  0.152344180 -0.7802674
 [87,] -1.4839775497 -0.71430294  0.443447708  1.2730679
 [88,]  0.3662012725 -0.65432613 -0.357086995 -0.7802674
 [89,] -0.9219255907 -0.71430294 -0.138759349 -0.7802674
 [90,]  0.4777368752 -0.71430294  0.880103001 -0.7802674
 [91,] -0.9984696318 -0.63033541 -1.594276991 -0.7802674
 [92,]  0.9654317657 -0.71430294  1.535085940  1.2730679
 [93,]  0.6592556013  0.02605072 -1.012069934 -0.7802674
 [94,] -0.2199073851 -0.31461752  0.443447708  1.2730679
 [95,]  0.8757658890 -0.70518646  0.952878883 -0.7802674
 [96,] -1.2281017552 -0.25128201  1.389534176 -0.7802674
 [97,] -0.0208928783  0.64309208 -1.012069934 -0.7802674
 [98,]  0.3137139300 -0.64377021 -0.284311113 -0.7802674
 [99,]  0.1999913546 -0.71430294  0.588999472  1.2730679
[100,] -0.5938797003  0.23333056 -0.284311113 -0.7802674
[101,] -0.7579026455 -0.58091453  0.516223590 -0.7802674
[102,]  0.4842977930 -0.71430294 -1.157621698 -0.7802674
[103,]  1.3175343548 -0.52669549  0.225120062 -0.7802674
[104,]  1.0529106698 -0.21865463 -0.866518170 -0.7802674
[105,]  0.5411590807 -0.52765512 -1.012069934 -0.7802674
[106,]  0.1059515327  4.46769287  1.535085940 -0.7802674
[107,] -0.8453815496  0.32209623  1.316758294  1.2730679
[108,]  2.4613210262 -0.41825743  0.807327119  1.2730679
[109,] -0.5851318099 -0.34436601 -0.065983466 -0.7802674
[110,] -0.9000558647 -0.71430294 -1.812604637 -0.7802674
[111,] -0.1411763714 -0.59482914 -1.958156401 -0.7802674
[112,]  0.0053507930 -0.49214886  0.807327119  1.2730679
[113,] -1.1209400976 -0.61546116 -0.429862877 -0.7802674
[114,] -0.1936637139 -0.71430294  1.316758294  1.2730679
[115,] -1.4774166319  1.63966664  1.171206529  1.2730679
[116,]  0.4930456835 -0.01089499  0.370671826 -0.7802674
[117,] -0.9284865086  0.34560713 -0.793742287  1.2730679
[118,] -1.2149799196 -0.71430294 -0.065983466 -0.7802674
[119,] -0.6135624537  0.41230134  1.316758294  1.2730679
[120,]  0.4077537519 -0.68167555 -1.157621698 -0.7802674
[121,] -0.7207241113 -0.14668247  1.243982411 -0.7802674
[122,] -0.0668193029 -0.30598086 -1.084845816 -0.7802674
[123,]  0.0119117108 -0.12701008 -0.720966405 -0.7802674
[124,] -0.3008253715  0.47995517  0.952878883  1.2730679
[125,] -0.0536974673  0.58263546  0.152344180 -0.7802674
[126,]  1.5274837247 -0.71430294  1.171206529  1.2730679
[127,]  0.3552664095 -0.63033541 -1.084845816 -0.7802674
[128,]  1.8074162179  0.17383357 -0.357086995 -0.7802674
[129,]  0.7183038616  0.70210925  1.243982411  1.2730679
[130,]  1.8402208070 -0.51422032  1.535085940 -0.7802674
[131,] -0.2986383989  0.31729808  1.389534176 -0.7802674
[132,]  1.1556983822  3.11557582  1.098430647  1.2730679
[133,] -0.3183211523 -0.18122911 -0.939294052  1.2730679
[134,]  0.6854992725  0.45020668  0.807327119 -0.7802674
[135,] -0.0143319605  2.07389870 -0.211535231 -0.7802674
[136,] -0.4604743715  0.21365817  0.661775355  1.2730679
[137,] -1.0990703716  0.17383357 -0.793742287 -0.7802674
[138,] -1.0728267004  0.37631526 -1.084845816 -0.7802674
[139,] -1.2324757004 -0.68983240 -1.667052873 -0.7802674
[140,]  1.4596875740 -0.25416090  0.807327119 -0.7802674
[141,]  1.6871327247 -0.54540826  0.807327119  1.2730679
[142,] -1.4992863580  0.12393287 -1.303173462  1.2730679
[143,] -0.2592728920 -0.68983240 -0.357086995 -0.7802674
[144,]  1.1053980123  0.83549766  1.098430647 -0.7802674
[145,] -0.3139472071 -0.21241705 -0.211535231 -0.7802674
[146,] -0.8869340291 -0.48399201 -0.211535231 -0.7802674
[147,] -0.8891210017  0.07019365 -0.866518170 -0.7802674
[148,]  0.5630288068 -0.56412102  0.006792416  1.2730679
[149,]  0.7051820260 -0.01857202 -0.720966405  1.2730679
[150,] -0.6419930975 -0.71430294  1.316758294  1.2730679
attr(,"scaled:center")
       sbp    tobacco        ldl  adiposity      typea    obesity    alcohol 
133.720000   3.917133   4.957867  25.172800  54.680000  26.155533  14.887067 
       age        chd 
 42.906667   0.380000 
attr(,"scaled:scale")
       sbp    tobacco        ldl  adiposity      typea    obesity    alcohol 
17.0219440  4.6686634  2.2498786  7.7423844  9.5026100  4.5725310 20.8413908 
       age        chd 
13.7408159  0.4870125 
eigen() decomposition
$values
[1] 2.9170705 1.4118707 1.1160130 0.8881532 0.7544710 0.6896814 0.5952564
[8] 0.4500630 0.1774207

$vectors
             [,1]        [,2]        [,3]        [,4]         [,5]         [,6]
 [1,] -0.31973708 -0.14262025  0.34461445  0.12455766  0.773491598  0.198441710
 [2,] -0.30996648 -0.43526564 -0.05557033 -0.13412055 -0.428116965  0.344570801
 [3,] -0.30990366  0.26461176 -0.35483022 -0.31694635  0.143035499 -0.640891920
 [4,] -0.48970888  0.30954926  0.12746058  0.12731536 -0.067679308 -0.008954996
 [5,] -0.05041004 -0.29167622 -0.56381468  0.74378096  0.113142417 -0.130737116
 [6,] -0.32518859  0.47402587  0.10231868  0.39951587 -0.349989459  0.142905789
 [7,] -0.13137526 -0.48735217  0.48813492  0.13164914 -0.238112413 -0.601909041
 [8,] -0.49127221 -0.09438679  0.02994198 -0.08463022  0.003883837  0.035546448
 [9,] -0.31454298 -0.26108737 -0.41045561 -0.33543413  0.037155563  0.173196143
             [,7]        [,8]         [,9]
 [1,] -0.10094403  0.30416870  0.052699907
 [2,] -0.53373277  0.29783460 -0.133412610
 [3,] -0.31610477  0.26739028  0.070423465
 [4,]  0.09764729 -0.22536704 -0.752942956
 [5,] -0.06173799 -0.06933312 -0.052674791
 [6,]  0.16851093  0.37921619  0.429364966
 [7,]  0.24446864  0.10237158  0.019604333
 [8,] -0.11443341 -0.71225197  0.469049515
 [9,]  0.70029635  0.17347413  0.004533299

Scree Plot

Pada gambar scree plot diatas, dapat dilihat bahwa titik 5 kurva mulai melandai. Maka analisis kasus ini akan menggunakan 5 Principal Component (PC).

Nilai Kumulatif Eigen

[1] 0.3241189
[1] 0.1568745
[1] 0.1240014
[1] 0.09868369
[1] 0.08383011
[1] 0.07663127
[1] 0.0661396
[1] 0.050007
[1] 0.01971341

Berdasarkan nilai kumulatif yang didapatkan, 5 Principal Component (PC) sudah menangkap sekitar 78% keragaman. Sehingga dapat kita susun 5 buah PC.

Persamaan PC

             [,1]        [,2]        [,3]        [,4]         [,5]
 [1,] -0.31973708 -0.14262025  0.34461445  0.12455766  0.773491598
 [2,] -0.30996648 -0.43526564 -0.05557033 -0.13412055 -0.428116965
 [3,] -0.30990366  0.26461176 -0.35483022 -0.31694635  0.143035499
 [4,] -0.48970888  0.30954926  0.12746058  0.12731536 -0.067679308
 [5,] -0.05041004 -0.29167622 -0.56381468  0.74378096  0.113142417
 [6,] -0.32518859  0.47402587  0.10231868  0.39951587 -0.349989459
 [7,] -0.13137526 -0.48735217  0.48813492  0.13164914 -0.238112413
 [8,] -0.49127221 -0.09438679  0.02994198 -0.08463022  0.003883837
 [9,] -0.31454298 -0.26108737 -0.41045561 -0.33543413  0.037155563

Hasil di atas dapat dituliskan dalam bentuk persamaan:

PC1 = -0.319X1 - 0.309X2 - 0.309X3 - 0.489X4 - 0.050X5 - 0.325X6 - 0.131X7 - 0.491X8 - 0.019X9

PC2 = -0.142X1 - 0.435X2 + 0.264X3 + 0.309X4 - 0.291X5 + 0.474X6 - 0.487X7 - 0.094X8 - 0.261X9

PC3 = 0.344X1 - 0.055X2 - 0.354X3 + 0.127X4 - 0.563X5 + 0.102X6 + 0.488X7 + 0.029X8 - 0.410X9

PC4 = 0.124X1 - 0.134X2 - 0.316X3 + 0.127X4 + 0.743X5 + 0.399X6 + 0.131X7 - 0.084X8 - 0.335X9

PC5 = 0.773X1 - 0.4282 + 0.143X3 - 0.067X4 + 0.113X5 - 0.3499X6 - 0.238X7 + 0.003X8 + 0.037X9

Matriks Korelasi

Nilai Eigen dan Vektor Eigen

eigen() decomposition
$values
[1] 2.9170705 1.4118707 1.1160130 0.8881532 0.7544710 0.6896814 0.5952564
[8] 0.4500630 0.1774207

$vectors
             [,1]        [,2]        [,3]        [,4]         [,5]         [,6]
 [1,] -0.31973708 -0.14262025  0.34461445  0.12455766  0.773491598  0.198441710
 [2,] -0.30996648 -0.43526564 -0.05557033 -0.13412055 -0.428116965  0.344570801
 [3,] -0.30990366  0.26461176 -0.35483022 -0.31694635  0.143035499 -0.640891920
 [4,] -0.48970888  0.30954926  0.12746058  0.12731536 -0.067679308 -0.008954996
 [5,] -0.05041004 -0.29167622 -0.56381468  0.74378096  0.113142417 -0.130737116
 [6,] -0.32518859  0.47402587  0.10231868  0.39951587 -0.349989459  0.142905789
 [7,] -0.13137526 -0.48735217  0.48813492  0.13164914 -0.238112413 -0.601909041
 [8,] -0.49127221 -0.09438679  0.02994198 -0.08463022  0.003883837  0.035546448
 [9,] -0.31454298 -0.26108737 -0.41045561 -0.33543413  0.037155563  0.173196143
             [,7]        [,8]         [,9]
 [1,] -0.10094403 -0.30416870 -0.052699907
 [2,] -0.53373277 -0.29783460  0.133412610
 [3,] -0.31610477 -0.26739028 -0.070423465
 [4,]  0.09764729  0.22536704  0.752942956
 [5,] -0.06173799  0.06933312  0.052674791
 [6,]  0.16851093 -0.37921619 -0.429364966
 [7,]  0.24446864 -0.10237158 -0.019604333
 [8,] -0.11443341  0.71225197 -0.469049515
 [9,]  0.70029635 -0.17347413 -0.004533299

Scree Plot

Pada gambar scree plot diatas, dapat dilihat bahwa titik 5 kurva mulai melandai. Maka analisis kasus ini akan menggunakan 5 Principal Component (PC).

Nilai Kumulatif Eigen

[1] 0.3241189
[1] 0.1568745
[1] 0.1240014
[1] 0.09868369
[1] 0.08383011
[1] 0.07663127
[1] 0.0661396
[1] 0.050007
[1] 0.01971341

Berdasarkan nilai kumulatif yang didapatkan, 5 Principal Component (PC) sudah menangkap sekitar 78% keragaman. Sehingga dapat kita susun 5 buah PC.

Persamaan PC

             [,1]        [,2]        [,3]        [,4]         [,5]
 [1,] -0.31973708 -0.14262025  0.34461445  0.12455766  0.773491598
 [2,] -0.30996648 -0.43526564 -0.05557033 -0.13412055 -0.428116965
 [3,] -0.30990366  0.26461176 -0.35483022 -0.31694635  0.143035499
 [4,] -0.48970888  0.30954926  0.12746058  0.12731536 -0.067679308
 [5,] -0.05041004 -0.29167622 -0.56381468  0.74378096  0.113142417
 [6,] -0.32518859  0.47402587  0.10231868  0.39951587 -0.349989459
 [7,] -0.13137526 -0.48735217  0.48813492  0.13164914 -0.238112413
 [8,] -0.49127221 -0.09438679  0.02994198 -0.08463022  0.003883837
 [9,] -0.31454298 -0.26108737 -0.41045561 -0.33543413  0.037155563

Hasil di atas dapat dituliskan dalam bentuk persamaan:

PC1 = -0.319X1 - 0.309X2 - 0.309X3 - 0.489X4 - 0.050X5 - 0.325X6 - 0.131X7 - 0.491X8 - 0.019X9

PC2 = -0.142X1 - 0.435X2 + 0.264X3 + 0.309X4 - 0.291X5 + 0.474X6 - 0.487X7 - 0.094X8 - 0.261X9

PC3 = 0.344X1 - 0.055X2 - 0.354X3 + 0.127X4 - 0.563X5 + 0.102X6 + 0.488X7 + 0.029X8 - 0.410X9

PC4 = 0.124X1 - 0.134X2 - 0.316X3 + 0.127X4 + 0.743X5 + 0.399X6 + 0.131X7 - 0.084X8 - 0.335X9

PC5 = 0.773X1 - 0.4282 + 0.143X3 - 0.067X4 + 0.113X5 - 0.3499X6 - 0.238X7 + 0.003X8 + 0.037X9

Fungsi PCA

Fungsi PCA dengan library prcomp

Standard deviations (1, .., p=9):
[1] 1.7079433 1.1882217 1.0564152 0.9424188 0.8686029 0.8304706 0.7715286
[8] 0.6708674 0.4212134

Rotation (n x k) = (9 x 9):
                  PC1         PC2         PC3         PC4          PC5
sbp       -0.31973708  0.14262025 -0.34461445 -0.12455766 -0.773491598
tobacco   -0.30996648  0.43526564  0.05557033  0.13412055  0.428116965
ldl       -0.30990366 -0.26461176  0.35483022  0.31694635 -0.143035499
adiposity -0.48970888 -0.30954926 -0.12746058 -0.12731536  0.067679308
typea     -0.05041004  0.29167622  0.56381468 -0.74378096 -0.113142417
obesity   -0.32518859 -0.47402587 -0.10231868 -0.39951587  0.349989459
alcohol   -0.13137526  0.48735217 -0.48813492 -0.13164914  0.238112413
age       -0.49127221  0.09438679 -0.02994198  0.08463022 -0.003883837
chd       -0.31454298  0.26108737  0.41045561  0.33543413 -0.037155563
                   PC6         PC7         PC8          PC9
sbp       -0.198441710 -0.10094403  0.30416870 -0.052699907
tobacco   -0.344570801 -0.53373277  0.29783460  0.133412610
ldl        0.640891920 -0.31610477  0.26739028 -0.070423465
adiposity  0.008954996  0.09764729 -0.22536704  0.752942956
typea      0.130737116 -0.06173799 -0.06933312  0.052674791
obesity   -0.142905789  0.16851093  0.37921619 -0.429364966
alcohol    0.601909041  0.24446864  0.10237158 -0.019604333
age       -0.035546448 -0.11443341 -0.71225197 -0.469049515
chd       -0.173196143  0.70029635  0.17347413 -0.004533299
               PC1      PC2      PC3
sbp       -0.31974  0.14262 -0.34461
tobacco   -0.30997  0.43527  0.05557
ldl       -0.30990 -0.26461  0.35483
adiposity -0.48971 -0.30955 -0.12746
typea     -0.05041  0.29168  0.56381
obesity   -0.32519 -0.47403 -0.10232
alcohol   -0.13138  0.48735 -0.48813
age       -0.49127  0.09439 -0.02994
chd       -0.31454  0.26109  0.41046
Importance of components:
                          PC1    PC2   PC3     PC4     PC5     PC6     PC7
Standard deviation     1.7079 1.1882 1.056 0.94242 0.86860 0.83047 0.77153
Proportion of Variance 0.3241 0.1569 0.124 0.09868 0.08383 0.07663 0.06614
Cumulative Proportion  0.3241 0.4810 0.605 0.70368 0.78751 0.86414 0.93028
                           PC8     PC9
Standard deviation     0.67087 0.42121
Proportion of Variance 0.05001 0.01971
Cumulative Proportion  0.98029 1.00000

Fungsi PCA dengan library princomp

Importance of components:
                          Comp.1    Comp.2    Comp.3     Comp.4     Comp.5
Standard deviation     1.7079433 1.1882217 1.0564152 0.94241880 0.86860290
Proportion of Variance 0.3241189 0.1568745 0.1240014 0.09868369 0.08383011
Cumulative Proportion  0.3241189 0.4809935 0.6049949 0.70367860 0.78750871
                           Comp.6    Comp.7    Comp.8     Comp.9
Standard deviation     0.83047062 0.7715286 0.6708674 0.42121337
Proportion of Variance 0.07663127 0.0661396 0.0500070 0.01971341
Cumulative Proportion  0.86413998 0.9302796 0.9802866 1.00000000

Nilai Loading


Loadings:
          Comp.1  Comp.2  Comp.3  Comp.4  Comp.5  Comp.6  Comp.7  Comp.8 
sbp        0.3197  0.1426  0.3446  0.1246  0.7735  0.1984  0.1009  0.3042
tobacco    0.3100  0.4353         -0.1341 -0.4281  0.3446  0.5337  0.2978
ldl        0.3099 -0.2646 -0.3548 -0.3169  0.1430 -0.6409  0.3161  0.2674
adiposity  0.4897 -0.3095  0.1275  0.1273                         -0.2254
typea              0.2917 -0.5638  0.7438  0.1131 -0.1307                
obesity    0.3252 -0.4740  0.1023  0.3995 -0.3500  0.1429 -0.1685  0.3792
alcohol    0.1314  0.4874  0.4881  0.1316 -0.2381 -0.6019 -0.2445  0.1024
age        0.4913                                          0.1144 -0.7123
chd        0.3145  0.2611 -0.4105 -0.3354          0.1732 -0.7003  0.1735
          Comp.9 
sbp              
tobacco   -0.1334
ldl              
adiposity -0.7529
typea            
obesity    0.4294
alcohol          
age        0.4690
chd              

               Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
SS loadings    1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Proportion Var 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111
Cumulative Var 0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889 1.0000

Perhitungan nilai loading yang menggambarkan sejauh mana setiap variabel berkontribusi terhadap pembentukan setiap komponen utama.

Plot PCA

Nilai Kontribusi

               Dim.1      Dim.2       Dim.3      Dim.4        Dim.5
sbp       10.2231797  2.0340535 11.87591168  1.5514612 59.828925233
tobacco    9.6079216 18.9456178  0.30880613  1.7988322 18.328413582
ldl        9.6040280  7.0019384 12.59044882 10.0454988  2.045915401
adiposity 23.9814784  9.5820742  1.62462000  1.6209202  0.458048878
typea      0.2541172  8.5075017 31.78869933 55.3210122  1.280120656
obesity   10.5747618 22.4700524  1.04691114 15.9612930 12.249262116
alcohol    1.7259459 23.7512141 23.82757006  1.7331496  5.669752128
age       24.1348385  0.8908867  0.08965224  0.7162275  0.001508419
chd        9.8937288  6.8166613 16.84738059 11.2516054  0.138053586
                 Dim.6      Dim.7      Dim.8       Dim.9
sbp        3.937911236  1.0189697  9.2518597  0.27772802
tobacco   11.872903716 28.4870673  8.8705452  1.77989245
ldl       41.074245291  9.9922226  7.1497563  0.49594644
adiposity  0.008019196  0.9534993  5.0790303 56.69230950
typea      1.709219346  0.3811580  0.4807082  0.27746336
obesity    2.042206465  2.8395935 14.3804922 18.43542742
alcohol   36.229449368  5.9764918  1.0479941  0.03843299
age        0.126354994  1.3095005 50.7302864 22.00074475
chd        2.999690387 49.0414973  3.0093275  0.00205508

PCA dengan function PCA

       eigenvalue percentage of variance cumulative percentage of variance
comp 1  2.9170705              32.411894                          32.41189
comp 2  1.4118707              15.687453                          48.09935
comp 3  1.1160130              12.400144                          60.49949
comp 4  0.8881532               9.868369                          70.36786
comp 5  0.7544710               8.383011                          78.75087
comp 6  0.6896814               7.663127                          86.41400
comp 7  0.5952564               6.613960                          93.02796
comp 8  0.4500630               5.000700                          98.02866
comp 9  0.1774207               1.971341                         100.00000

Plot Kontribusi Variabel

Variabel age (X8) dan adiposity (X4) memiliki nilai kontribusi yang sangat tinggi, dimana menunjukkan representasi variabel baik pada Principal Component (PC). Dalam hal ini, variabel ditempatkan dekat dengan lingkaran pada lingkaran korelasi. Sedangkan, variabel typea (X5) memiliki nilai kontribusi terendah, dimana hal ini menunjukkan bahwa variabel tersebut tidak sepenuhnya direpresentasikan oleh Principal Component (PC). Variabel tersebut dekat dengan pusat lingkaran dan variabel tersebut tidak terlalu penting untuk Principal Component (PC).

Plot Individu

Biplot

Hasil Uji

Berdasarkan hasil analisis yang telah dilakukan mengenai faktor-faktor yang menyebabkan seseorang terkena penyakit kardiovaskular dapat disimpulkan bahwa terdapat lima principal component yang terbentuk. Berikut adalah persamaan principal component yang terbentuk:

PC1 = -0.319X1 - 0.309X2 - 0.309X3 - 0.489X4 - 0.050X5 - 0.325X6 - 0.131X7 - 0.491X8 - 0.019X9

PC2 = -0.142X1 - 0.435X2 + 0.264X3 + 0.309X4 - 0.291X5 + 0.474X6 - 0.487X7 - 0.094X8 - 0.261X9

PC3 = 0.344X1 - 0.055X2 - 0.354X3 + 0.127X4 - 0.563X5 + 0.102X6 + 0.488X7 + 0.029X8 - 0.410X9

PC4 = 0.124X1 - 0.134X2 - 0.316X3 + 0.127X4 + 0.743X5 + 0.399X6 + 0.131X7 - 0.084X8 - 0.335X9

PC5 = 0.773X1 - 0.4282 + 0.143X3 - 0.067X4 + 0.113X5 - 0.3499X6 - 0.238X7 + 0.003X8 + 0.037X9

Dengan kata lain, penelitian ini berhasil mengurangi jumlah variabel dari 9 variabel menjadi 5 variabel baru (principal component) dengan mempertahankan 78% dari varians atau keragaman. Principal component yang terbentuk itu dapat diberi nama sebagai berikut:

  • Principal component 1 yang mencakup variabel adiposity (X4) dan age (X8) sebagai faktor kondisi tubuh

  • Principal component 2 yang mencakup variabel tobacco (X2), obesity (X6), dan alkohol (X7) sebagai faktor gaya hidup

  • Principal component 3 yang mencakup variabel typea (X5) sebagai faktor perilaku

  • Principal component 4 yang mencakup variabel ldl (X3) dan chd (X9) sebagai faktor penyakit

  • Principal component 5 yang mencakup variabel sbp (X1) sebagai faktor genetik

DAFTAR PUSTAKA

  1. Johnson dan Wichern. (2007). Applied Multivariate Statistical Analysis. Edisi keenam. Pearson Prentice Hall.

  2. Radiarta, I Nyoman, Hasnawi, dan A. M. (2013). Kondisi Kualitas Perairan Di Kabupaten Morowali Provinsi Sulawesi Tengah, 299–309.