1 PENDAHULUAN

1.1 Latar Belakang

Hepatitis C merupakan masalah kesehatan masyarakat baik di dunia maupun di Indonesia. Hepatitis C adalah penyakit peradangan hati yang disebabkan oleh infeksi virus hepatitis C (Saraswati et al., 2022). Penyakit ini dapat berkembang menjadi kondisi kronis yang berisiko menyebabkan sirosis, gagal hati, atau kanker hati. Menurut laporan Organisasi Kesehatan Dunia (WHO), sekitar 58 juta orang di seluruh dunia hidup dengan infeksi HCV, dengan 1,5 juta kasus baru setiap tahunnya (Yang et al., 2023). Dalam konteks pengelolaan dan penelitian medis, analisis data laboratorium memegang peranan penting dalam diagnosis dan pemantauan pasien Hepatitis C. Data laboratorium biasanya mencakup berbagai parameter, seperti enzim hati (ALT, AST), kadar bilirubin, albumin, dan indikator fungsi ginjal. Namun, banyaknya variabel ini sering kali menyulitkan proses interpretasi, terutama jika terdapat hubungan antarvariabel yang saling memengaruhi. Oleh karena itu, diperlukan metode statistik yang mampu mereduksi dimensi data tanpa kehilangan informasi penting. Principal Component Analysis (PCA) dan Factor Analysis (FA) adalah dua pendekatan statistik yang sering digunakan dalam analisis data multivariat. PCA bertujuan untuk mereduksi dimensi data dengan mengubah variabel asli menjadi sejumlah kecil komponen utama yang mencakup sebagian besar variabilitas dalam data. Sementara itu, FA bertujuan untuk mengidentifikasi struktur laten di balik data dengan menggali hubungan antarvariabel melalui faktor-faktor tersembunyi.

1.2 Rumusan Masalah

Bagaimana penerapan PCA untuk mereduksi dimensi data laboratorium pasien Hepatitis C tanpa kehilangan informasi penting?
Bagaimana FA dapat digunakan untuk mengidentifikasi faktor-faktor laten yang berkontribusi terhadap variasi dalam data laboratorium pasien Hepatitis C?
Apakah hasil analisis PCA dan FA dapat memberikan wawasan yang mendalam tentang pola dan hubungan antarvariabel dalam data laboratorium?

1.3 Tujuan

Mengaplikasikan PCA untuk mereduksi dimensi data laboratorium pasien Hepatitis C, sehingga mempermudah interpretasi data.
Menggunakan FA untuk mengidentifikasi faktor-faktor laten yang memengaruhi parameter laboratorium pasien Hepatitis C.
Membandingkan hasil dari PCA dan FA untuk mengevaluasi efektivitas kedua metode dalam menganalisis data laboratorium pasien Hepatitis C.

2 METODOLOGI PENELITIAN

2.1 Data

Data yang digunakan dalam analisis ini bersumber dari Kaggle dengan tautan berikut: https://www.kaggle.com/datasets/fedesoriano/hepatitis-c-dataset. Data tersebut mencakup berbagai parameter laboratorium pasien hepatitis C.

2.2 Variabel

Variabel yang digunakan dalam analisis ini terdiri dari dua kelompok utama, yaitu data pasien dan data laboratorium.

Data Pasien
- X: ID/Nomor Pasien
- Usia: Umur pasien dalam tahun
Data Laboratorium
- ALB: Albumin, protein utama dalam darah yang diproduksi oleh hati
- ALP: Alkaline Phosphatase, enzim yang berhubungan dengan fungsi hati dan tulang
- ALT: Alanine Aminotransferase, indikator kesehatan hati
- AST: Aspartate Aminotransferase, menunjukkan kerusakan sel hati
- BIL: Bilirubin, mengindikasikan fungsi hati dan saluran empedu
- CHE: Cholinesterase, enzim yang berhubungan dengan fungsi hati
- CHOL: Kolesterol, terkait dengan metabolisme lemak dalam tubuh
- CREA: Kreatinin, digunakan untuk menilai fungsi ginjal
- GGT: Gamma-glutamyl transferase, enzim yang berhubungan dengan fungsi hati
- PROT: Protein total dalam darah

3 PEMBAHASAN

3.1 Load dan Pre-processing Data

Load data

data <- read.csv("~/HepatitisCdata.csv", header = TRUE)
head(data)

##   X      Category Age Sex  ALB  ALP  ALT  AST  BIL   CHE CHOL CREA  GGT PROT
## 1 1 0=Blood Donor  32   m 38.5 52.5  7.7 22.1  7.5  6.93 3.23  106 12.1 69.0
## 2 2 0=Blood Donor  32   m 38.5 70.3 18.0 24.7  3.9 11.17 4.80   74 15.6 76.5
## 3 3 0=Blood Donor  32   m 46.9 74.7 36.2 52.6  6.1  8.84 5.20   86 33.2 79.3
## 4 4 0=Blood Donor  32   m 43.2 52.0 30.6 22.6 18.9  7.33 4.74   80 33.8 75.7
## 5 5 0=Blood Donor  32   m 39.2 74.1 32.6 24.8  9.6  9.15 4.32   76 29.9 68.7
## 6 6 0=Blood Donor  32   m 41.6 43.3 18.5 19.7 12.3  9.92 6.05  111 91.0 74.0

Cek tipe data

str(data)

## 'data.frame':    615 obs. of  14 variables:
##  $ X       : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ Category: chr  "0=Blood Donor" "0=Blood Donor" "0=Blood Donor" "0=Blood Donor" ...
##  $ Age     : int  32 32 32 32 32 32 32 32 32 32 ...
##  $ Sex     : chr  "m" "m" "m" "m" ...
##  $ ALB     : num  38.5 38.5 46.9 43.2 39.2 41.6 46.3 42.2 50.9 42.4 ...
##  $ ALP     : num  52.5 70.3 74.7 52 74.1 43.3 41.3 41.9 65.5 86.3 ...
##  $ ALT     : num  7.7 18 36.2 30.6 32.6 18.5 17.5 35.8 23.2 20.3 ...
##  $ AST     : num  22.1 24.7 52.6 22.6 24.8 19.7 17.8 31.1 21.2 20 ...
##  $ BIL     : num  7.5 3.9 6.1 18.9 9.6 12.3 8.5 16.1 6.9 35.2 ...
##  $ CHE     : num  6.93 11.17 8.84 7.33 9.15 ...
##  $ CHOL    : num  3.23 4.8 5.2 4.74 4.32 6.05 4.79 4.6 4.1 4.45 ...
##  $ CREA    : num  106 74 86 80 76 111 70 109 83 81 ...
##  $ GGT     : num  12.1 15.6 33.2 33.8 29.9 91 16.9 21.5 13.7 15.9 ...
##  $ PROT    : num  69 76.5 79.3 75.7 68.7 74 74.5 67.1 71.3 69.9 ...

Hapus kolom non-numerik

data_numeric <- data[, sapply(data, is.numeric)]
str(data_numeric)

## 'data.frame':    615 obs. of  12 variables:
##  $ X   : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ Age : int  32 32 32 32 32 32 32 32 32 32 ...
##  $ ALB : num  38.5 38.5 46.9 43.2 39.2 41.6 46.3 42.2 50.9 42.4 ...
##  $ ALP : num  52.5 70.3 74.7 52 74.1 43.3 41.3 41.9 65.5 86.3 ...
##  $ ALT : num  7.7 18 36.2 30.6 32.6 18.5 17.5 35.8 23.2 20.3 ...
##  $ AST : num  22.1 24.7 52.6 22.6 24.8 19.7 17.8 31.1 21.2 20 ...
##  $ BIL : num  7.5 3.9 6.1 18.9 9.6 12.3 8.5 16.1 6.9 35.2 ...
##  $ CHE : num  6.93 11.17 8.84 7.33 9.15 ...
##  $ CHOL: num  3.23 4.8 5.2 4.74 4.32 6.05 4.79 4.6 4.1 4.45 ...
##  $ CREA: num  106 74 86 80 76 111 70 109 83 81 ...
##  $ GGT : num  12.1 15.6 33.2 33.8 29.9 91 16.9 21.5 13.7 15.9 ...
##  $ PROT: num  69 76.5 79.3 75.7 68.7 74 74.5 67.1 71.3 69.9 ...

Cek missing value

sum(is.na(data_numeric))

## [1] 31

Menampilkan jumlah missing value per kolom

colSums(is.na(data_numeric))

##    X  Age  ALB  ALP  ALT  AST  BIL  CHE CHOL CREA  GGT PROT 
##    0    0    1   18    1    0    0    0   10    0    0    1

Imputasi missing value dengan median

data_numeric_imputed <- data_numeric
data_numeric_imputed[] <- lapply(data_numeric_imputed, function(x) ifelse(is.na(x), median(x, na.rm = TRUE), x))

3.2 Summary

3.2.1 Statistika Deskriptif

summary_stats <- summary(data_numeric_imputed)
print(summary_stats)

##        X              Age             ALB             ALP        
##  Min.   :  1.0   Min.   :19.00   Min.   :14.90   Min.   : 11.30  
##  1st Qu.:154.5   1st Qu.:39.00   1st Qu.:38.80   1st Qu.: 52.95  
##  Median :308.0   Median :47.00   Median :41.95   Median : 66.20  
##  Mean   :308.0   Mean   :47.41   Mean   :41.62   Mean   : 68.22  
##  3rd Qu.:461.5   3rd Qu.:54.00   3rd Qu.:45.20   3rd Qu.: 79.30  
##  Max.   :615.0   Max.   :77.00   Max.   :82.20   Max.   :416.60  
##       ALT              AST              BIL             CHE        
##  Min.   :  0.90   Min.   : 10.60   Min.   :  0.8   Min.   : 1.420  
##  1st Qu.: 16.40   1st Qu.: 21.60   1st Qu.:  5.3   1st Qu.: 6.935  
##  Median : 23.00   Median : 25.90   Median :  7.3   Median : 8.260  
##  Mean   : 28.44   Mean   : 34.79   Mean   : 11.4   Mean   : 8.197  
##  3rd Qu.: 33.05   3rd Qu.: 32.90   3rd Qu.: 11.2   3rd Qu.: 9.590  
##  Max.   :325.30   Max.   :324.00   Max.   :254.0   Max.   :16.410  
##       CHOL            CREA              GGT              PROT      
##  Min.   :1.430   Min.   :   8.00   Min.   :  4.50   Min.   :44.80  
##  1st Qu.:4.620   1st Qu.:  67.00   1st Qu.: 15.70   1st Qu.:69.30  
##  Median :5.300   Median :  77.00   Median : 23.30   Median :72.20  
##  Mean   :5.367   Mean   :  81.29   Mean   : 39.53   Mean   :72.04  
##  3rd Qu.:6.055   3rd Qu.:  88.00   3rd Qu.: 40.20   3rd Qu.:75.40  
##  Max.   :9.670   Max.   :1079.10   Max.   :650.90   Max.   :90.00

Dari hasil statika deskriptif tersebut didapatkan informasi sebagai berikut:

Informasi Umum
- X: 1 - 615 (ID)
- Age: 19 - 77 tahun, median 47 tahun
Parameter Biokimia Darah
- ALB: 14.90 - 82.20 g/L, median 41.95 g/L
- ALP: 11.30 - 416.60 U/L, median 66.20 U/L
- ALT: 0.90 - 325.30 U/L, median 23.00 U/L
- AST: 10.60 - 324.00 U/L, median 25.90 U/L
- BIL: 0.80 - 254.00 µmol/L, median 7.30 µmol/L
Metabolisme dan Fungsi Organ
- CHE: 1.42 - 16.41, median 8.26
- CHOL: 1.43 - 9.67, median 5.30
- CREA: 8.00 - 1079.10 µmol/L, median 77.00 µmol/L
- GGT: 4.50 - 650.90 U/L, median 23.30 U/L
- PROT: 44.80 - 90.00 g/L, median 72.20 g/L

3.2.2 Distribusi Data Numerik

library(tidyr)

## Warning: package 'tidyr' was built under R version 4.4.3

library(ggplot2)

dist_data <- gather(data_numeric_imputed, key = "Variable", value = "Value")

ggplot(dist_data, aes(x = Value)) +
  geom_histogram(bins = 30, fill = "orange", alpha = 0.7) +
  facet_wrap(~Variable, scales = "free") +
  theme_minimal() +
  ggtitle("Distribusi Data Numerik")

Interpretasi Distribusi Data Numerik:

Age: Distribusi normal dengan puncak sekitar 40-50 tahun, mayoritas usia 30-60 tahun.
ALB: Distribusi mendekati normal, puncak sekitar 40-50 g/L, tanpa outlier mencolok.
ALP: Skewed ke kanan, beberapa individu memiliki nilai sangat tinggi (outlier).
ALT: Skewed ke kanan, mayoritas nilai rendah, beberapa kasus dengan nilai tinggi (indikasi gangguan hati).
AST: Skewed ke kanan, sebagian besar individu memiliki nilai rendah, ada beberapa outlier tinggi.
BIL: Sangat skewed ke kanan, mayoritas kadar bilirubin rendah, sebagian kecil dengan nilai tinggi.
CHE: Distribusi mendekati normal, puncak sekitar 7-10, tanpa outlier signifikan.
CHOL: Distribusi mendekati normal, puncak sekitar 5-6, tanpa skewness signifikan.
CREA: Sangat skewed ke kanan, mayoritas kadar creatinine rendah, beberapa individu dengan nilai sangat tinggi.
GGT: Skewed ke kanan, mayoritas nilai rendah, beberapa kasus dengan nilai tinggi (gangguan hati atau alkohol).
PROT: Distribusi mendekati normal, puncak sekitar 70-75 g/L, tanpa banyak nilai ekstrem.

Kesimpulan:

ALP, ALT, AST, BIL, CREA, dan GGT memiliki distribusi skewed ke kanan, dengan beberapa outlier.
ALB, CHE, CHOL, dan PROT lebih mendekati distribusi normal.
Variabel skewed dapat memerlukan transformasi (misalnya log) untuk analisis lebih lanjut.

3.2.3 Correlation Matrix

cor_matrix <- cor(data_numeric_imputed, use = "complete.obs")

library(corrplot)

## corrplot 0.95 loaded

corrplot::corrplot(cor_matrix, order = "hclust", 
                   tl.col = "black", tl.srt = 45, tl.cex = 0.5,
                   addCoef.col = "black", number.cex = 0.7, 
                   method = "color", diag = FALSE)

Interpretasi Heatmap Korelasi:

Korelasi Positif Kuat:
- ALB dan PROT (0.55): Keduanya merupakan indikator status protein dalam tubuh.
- CHE dan CHOL (0.42): Berhubungan dengan metabolisme lemak dan hati.
- AST dan GGT (0.49): Indikator fungsi hati, sering meningkat pada gangguan hati.
- AST dan ALT (0.27): Kedua enzim ini sering meningkat bersamaan dalam kondisi kerusakan hati.
Korelasi Negatif Kuat:
- BIL dan CHE (-0.33): Bilirubin tinggi dapat mengindikasikan gangguan hati yang menurunkan kadar CHE.
- X dan ALB (-0.31): Korelasi negatif yang mungkin berasal dari skala atau indeks tertentu.
- X dan PROT (-0.27): Mirip dengan hubungan antara X dan ALB.
Korelasi dengan Usia:
- Age dan X (0.42): X mungkin terkait dengan indeks atau ID pasien yang berkorelasi dengan usia.
- Age dan ALB (-0.20): Albumin cenderung menurun dengan bertambahnya usia.
- Age dan PROT (-0.15): Sedikit korelasi negatif dengan protein total.

Kesimpulan:

Beberapa variabel memiliki hubungan erat yang dapat digunakan untuk analisis lebih lanjut.
Faktor hati (AST, ALT, GGT, BIL) menunjukkan pola korelasi khas pada gangguan hati.
Variabel X kemungkinan besar merupakan indeks atau variabel yang tidak biologis langsung.

3.3 Asumsi

Hitung matriks korelasi antar variabel numerik

library(psych)

## Warning: package 'psych' was built under R version 4.4.3

## 
## Attaching package: 'psych'

## The following objects are masked from 'package:ggplot2':
## 
##     %+%, alpha

cor_matrix <- cor(data_numeric_imputed, use = "complete.obs")

3.3.1 KMO (Kaiser-Meyer-Olkin) Test

kmo_result <- KMO(cor_matrix)
print(kmo_result)

## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = cor_matrix)
## Overall MSA =  0.65
## MSA for each item = 
##    X  Age  ALB  ALP  ALT  AST  BIL  CHE CHOL CREA  GGT PROT 
## 0.67 0.60 0.68 0.53 0.62 0.63 0.75 0.75 0.66 0.54 0.63 0.59

Interpretasi KMO:

KMO > 0.8 → Sangat Baik
KMO 0.7-0.8 → Baik
KMO 0.6-0.7 → Cukup
KMO < 0.6 → Kurang Baik

Interpretasi Hasil KMO:

Nilai KMO secara keseluruhan = 0.65 Nilai ini tergolong sedang, menunjukkan bahwa data cukup layak untuk analisis faktor tetapi tidak optimal.
MSA (Measure of Sampling Adequacy) untuk setiap variabel:
- Tertinggi: BIL (0.75), CHE (0.75) → Variabel ini paling cocok untuk analisis faktor.
- Sedang: ALB (0.68), CHOL (0.66), X (0.67), AST (0.63), ALT (0.62), GGT (0.63) → Bisa dipertimbangkan dalam analisis faktor.
- Terendah: ALP (0.53), CREA (0.54), PROT (0.59), Age (0.60) → Variabel ini kurang cocok untuk analisis faktor.

Kesimpulan:

Data cukup layak untuk analisis faktor, tetapi ada beberapa variabel dengan nilai MSA rendah (<0.6) yang mungkin perlu dipertimbangkan untuk dihapus atau diperbaiki.
Variabel BIL dan CHE memiliki kecocokan terbaik dalam analisis faktor.
Jika ingin meningkatkan kelayakan analisis faktor, dapat mempertimbangkan untuk menghapus variabel dengan MSA rendah atau melakukan transformasi data.

3.3.2 Bartlett’s Test

bartlett_result <- cortest.bartlett(cor_matrix, n = nrow(data_numeric_imputed))
print(bartlett_result)

## $chisq
## [1] 1449.894
## 
## $p.value
## [1] 1.944117e-259
## 
## $df
## [1] 66

Interpretasi Bartlett’s Test:

p-value < 0.05 → Korelasi antar variabel signifikan (FA/PCA layak dilakukan)
p-value > 0.05 → Tidak ada korelasi signifikan (FA/PCA kurang cocok)

Interpretasi Hasil Bartlett’s Test:

Statistik uji chi-square = 1449.894 Nilai ini menunjukkan adanya hubungan yang kuat antara variabel.
p-value = 1.94e-259 (sangat kecil)
- p-value jauh lebih kecil dari 0.05, sehingga H0 (hipotesis nol) ditolak.
- Artinya, terdapat korelasi yang signifikan antar variabel dalam dataset.
Derajat kebebasan (df) = 66

Kesimpulan:

Hasil uji Bartlett menunjukkan bahwa korelasi antar variabel cukup kuat.
Dataset memenuhi syarat untuk analisis faktor, karena korelasi antar variabel signifikan.
Analisis faktor dapat dilanjutkan karena asumsi korelasi antar variabel terpenuhi.

3.4 Principal Component Analysis (PCA)

Standardisasi data

scaled.X <- scale(cor_matrix)

Membuat matriks covariance

cov.matrix <- cov(scaled.X)

Menampilkan hasil matriks kovarian

cov.matrix

##                X        Age         ALB         ALP         ALT         AST
## X     1.00000000  0.6586955 -0.73350172  0.03135162 -0.16980952  0.56549861
## Age   0.65869551  1.0000000 -0.58141035  0.19558613 -0.19126316  0.14830928
## ALB  -0.73350172 -0.5814104  1.00000000 -0.43014326 -0.09119108 -0.60625392
## ALP   0.03135162  0.1955861 -0.43014326  1.00000000  0.14979867  0.06586815
## ALT  -0.16980952 -0.1912632 -0.09119108  0.14979867  1.00000000  0.27757595
## AST   0.56549861  0.1483093 -0.60625392  0.06586815  0.27757595  1.00000000
## BIL   0.44184185  0.1111139 -0.58951497  0.03958679 -0.15047204  0.58054273
## CHE  -0.71902213 -0.4050105  0.70139110 -0.18891926  0.08252408 -0.68558391
## CHOL -0.47387553 -0.0923347  0.44422713 -0.05067309 -0.06556057 -0.68255525
## CREA -0.12234015 -0.1479253 -0.11764162  0.15952559 -0.23347769 -0.13837952
## GGT   0.43223526  0.2102836 -0.64076190  0.56336112  0.25052271  0.70818065
## PROT -0.57819013 -0.5935009  0.83313265 -0.43165424 -0.02226938 -0.36606150
##               BIL         CHE        CHOL         CREA         GGT        PROT
## X     0.441841847 -0.71902213 -0.47387553 -0.122340147  0.43223526 -0.57819013
## Age   0.111113936 -0.40501052 -0.09233470 -0.147925252  0.21028358 -0.59350088
## ALB  -0.589514966  0.70139110  0.44422713 -0.117641624 -0.64076190  0.83313265
## ALP   0.039586793 -0.18891926 -0.05067309  0.159525593  0.56336112 -0.43165424
## ALT  -0.150472035  0.08252408 -0.06556057 -0.233477685  0.25052271 -0.02226938
## AST   0.580542731 -0.68558391 -0.68255525 -0.138379522  0.70818065 -0.36606150
## BIL   1.000000000 -0.76957203 -0.59774482  0.003514015  0.41997782 -0.40686327
## CHE  -0.769572030  1.00000000  0.69960124 -0.148664917 -0.58789123  0.54995942
## CHOL -0.597744823  0.69960124  1.00000000 -0.232873066 -0.46941591  0.33377905
## CREA  0.003514015 -0.14866492 -0.23287307  1.000000000  0.04409571 -0.23225278
## GGT   0.419977821 -0.58789123 -0.46941591  0.044095711  1.00000000 -0.49870902
## PROT -0.406863275  0.54995942  0.33377905 -0.232252783 -0.49870902  1.00000000

Visualisasi menggunakan corrplot

library(corrplot)
corrplot::corrplot(cov.matrix, order = "hclust", 
                   tl.col = "black", tl.srt = 45, tl.cex = 0.5,
                   addCoef.col = "black", number.cex = 0.7, 
                   method = "color", diag = FALSE)

Gambar tersebut menunjukkan matriks korelasi antar variabel. Beberapa variabel, seperti ALT dan AST, memiliki korelasi positif tinggi (biru gelap), menunjukkan hubungan erat. Sebaliknya, ada juga korelasi negatif kuat seperti antara ALP dan CHOL (merah gelap). Selain itu, terdapat hubungan yang lemah antara beberapa variabel, seperti CHOL dan ALT (lingkaran kecil atau hampir putih), menunjukkan bahwa variabel-variabel tersebut tidak memiliki hubungan signifikan. Hubungan lemah ini menunjukkan variabel yang mungkin tidak saling memengaruhi secara langsung.

Hitung nilai eigen dan vektor eigen

eigen.decomp <- eigen(cov.matrix)

# Lihat hasil
eigen.vals <- eigen.decomp$values
eigen.vecs <- eigen.decomp$vectors

Urutkan nilai eigen dan vektor eigen

sorted.idx <- order(eigen.vals, decreasing = TRUE)
sorted.eigen.vals <- eigen.vals[sorted.idx]
sorted.eigen.vecs <- eigen.vecs[, sorted.idx]

cat("NILAI EIGEN (terurut):\n")

## NILAI EIGEN (terurut):

print(sorted.eigen.vals)

##  [1] 5.407315e+00 1.605082e+00 1.536091e+00 1.351642e+00 6.568304e-01
##  [6] 4.878820e-01 3.269650e-01 2.130554e-01 1.846384e-01 1.292751e-01
## [11] 1.012241e-01 9.580077e-17

cat("VEKTOR EIGEN:\n")

## VEKTOR EIGEN:

print(sorted.eigen.vecs)

##              [,1]        [,2]        [,3]        [,4]        [,5]        [,6]
##  [1,] -0.34141333  0.19523725  0.23252968 -0.25326311 -0.14228598 -0.26698600
##  [2,] -0.22761856  0.53054785  0.01915496 -0.32500816 -0.15296021 -0.15965198
##  [3,]  0.39037562 -0.15234327  0.13197779  0.03051427  0.08859005 -0.28040653
##  [4,] -0.15387274  0.15623881 -0.61491106  0.18945998  0.44587722 -0.13081412
##  [5,] -0.01623089 -0.43600737 -0.48539888 -0.27919477 -0.49025302  0.26587227
##  [6,] -0.33290464 -0.39425614  0.04641564 -0.15416872 -0.05051698 -0.18934669
##  [7,] -0.30559328 -0.18948729  0.29554847  0.11854896  0.40400106  0.58148716
##  [8,]  0.38318339  0.06078883 -0.19480906 -0.09577711 -0.07821014  0.02116628
##  [9,]  0.29816095  0.32037934 -0.20293007 -0.22152751  0.22922952  0.27443297
## [10,] -0.03708940  0.12448682 -0.03564054  0.78358184 -0.39841120 -0.08261326
## [11,] -0.32481068 -0.18894019 -0.33003250  0.02590317  0.25868692 -0.35796177
## [12,]  0.32935320 -0.30627281  0.18865581 -0.06863340  0.24346034 -0.38999868
##              [,7]         [,8]        [,9]        [,10]       [,11]       [,12]
##  [1,] -0.15732500  0.098987799  0.63199423 -0.189007063 -0.14574828 -0.38032564
##  [2,]  0.31034388  0.111979575 -0.54112834  0.239724184 -0.11099472 -0.20091251
##  [3,]  0.19962915 -0.023932926 -0.24919232 -0.538203115  0.22483050 -0.52615624
##  [4,]  0.35634510 -0.020067355  0.28348253  0.140357421  0.23380719 -0.19967528
##  [5,]  0.19605933  0.298010536  0.03325202 -0.063095063 -0.12571700 -0.19208139
##  [6,] -0.32674507 -0.166964212 -0.17199124  0.376388045  0.56238887 -0.21734874
##  [7,]  0.09310469  0.029834265 -0.10584675  0.066380108 -0.25122662 -0.42388342
##  [8,] -0.19880281 -0.636704710  0.05802069  0.301862565 -0.36505636 -0.35088109
##  [9,] -0.57647518  0.437432755 -0.05398514 -0.009763631  0.19163206 -0.16413052
## [10,] -0.18969242  0.238286636 -0.07418086  0.154128150 -0.05338646 -0.27825903
## [11,] -0.37099709  0.009255004 -0.32004333 -0.297228880 -0.46804091  0.08119185
## [12,]  0.11297592  0.453323963  0.09593072  0.494038208 -0.27225141 -0.01965979

Filter nilai eigen yang lebih besar dari 1

selected.idx <- which(sorted.eigen.vals > 1)
selected.eigen.vals <- sorted.eigen.vals[selected.idx]

cat("NILAI EIGEN DI ATAS 1:\n")

## NILAI EIGEN DI ATAS 1:

print(selected.eigen.vals)

## [1] 5.407315 1.605082 1.536091 1.351642

Hitung total nilai eigen

tot.eigen <- sum(eigen.vals)

Proporsi variansi per komponen utama

var.pc <- sapply(sorted.eigen.vals, function(x) x / tot.eigen) * 100
print(var.pc)

##  [1] 4.506095e+01 1.337568e+01 1.280076e+01 1.126368e+01 5.473587e+00
##  [6] 4.065683e+00 2.724708e+00 1.775462e+00 1.538653e+00 1.077292e+00
## [11] 8.435339e-01 7.983398e-16

Penjelasan:

Proporsi variansi per komponen utama menunjukkan persentase keragaman (variansi) yang dijelaskan oleh masing-masing komponen utama (Principal Components, PC).
PC1 menjelaskan 45.06% dari total keragaman.
PC2 menjelaskan 13.38%.
PC3 menjelaskan 12.80%.
PC4 menjelaskan 11.26%.
Dan seterusnya untuk komponen berikutnya, dengan kontribusi variansi yang lebih kecil.

cat("Total keragaman (4 PC):", sum(var.pc[1:4]))

## Total keragaman (4 PC): 82.50108

Jumlah variansi yang dijelaskan oleh empat komponen utama pertama, yaitu 82.50%. Ini berarti bahwa empat komponen utama pertama cukup untuk menjelaskan 82.50% dari total keragaman dalam data. Dengan kata lain, sebagian besar informasi dalam data dapat diringkas menggunakan hanya empat komponen utama ini.

Pilih vektor eigen untuk komponen utama

selected.eigen.vecs <- sorted.eigen.vecs[, 1:4]

Proyeksikan data asli ke ruang berdimensi lebih rendah

pca.result <- scaled.X %*% selected.eigen.vecs

Ubah menjadi data frame

pca.X <- as.data.frame(pca.result)
colnames(pca.X) <- paste("PC", 1:4, sep = "")

Tampilkan hasil

pca.X

##              PC1        PC2        PC3        PC4
## X    -2.45858642  0.9785520  1.3693435 -0.9113667
## Age  -1.46924079  2.3073412  0.3893402 -1.1445311
## ALB   3.47189449 -0.5528809  1.0428763  0.2287895
## ALP  -0.96916488  0.7150075 -2.2181796  0.8039450
## ALT   0.03567852 -1.6940538 -1.6652987 -0.9862123
## AST  -2.41176514 -1.6479851  0.4298469 -0.5497593
## BIL  -2.07777887 -0.6611421  1.6426087  0.5808725
## CHE   3.28148618  0.3824464 -0.4850123 -0.3112630
## CHOL  2.34802094  1.4304900 -0.5246411 -0.7817552
## CREA -0.07295453  0.6469090  0.2422504  3.1039320
## GGT  -2.26882691 -0.7179845 -1.2544124  0.1755504
## PROT  2.59123741 -1.1866997  1.0312781 -0.2082019

Visualisasi Scree plot untuk semua komponen utama

library(factoextra)

## Warning: package 'factoextra' was built under R version 4.4.3

## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

library(gridExtra)

## Warning: package 'gridExtra' was built under R version 4.4.3

pca <- prcomp(scaled.X)

screeplot <- fviz_eig(pca, 
                      addlabels = TRUE,           
                      barfill = "skyblue",        
                      barcolor = "darkblue",      
                      linecolor = "red")          

# Menampilkan scree plot
plot(screeplot)

Berdasarkan scree plot:

PC1 menjelaskan 45.1% varians, diikuti oleh PC2 (13.4%), PC3 (12.8%), dan PC4 (11.3%).
Penurunan signifikan terjadi setelah PC4, dengan kontribusi varians semakin kecil (<6%).
Elbow criterion menunjukkan bahwa 4 komponen utama sudah cukup untuk merepresentasikan data.

Correlation circle untuk kontribusi (contrib)

contrib_circle <- fviz_pca_var(pca, col.var = "contrib",
                               gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), 
                               repel = TRUE) + 
  ggtitle("Kontribusi Variabel")

Buat correlation circle untuk cosinus kuadrat (cos2)

cos2_circle <- fviz_pca_var(pca, col.var = "cos2", 
                            gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), 
                            repel = TRUE) + 
  ggtitle("cos2")

Tampilkan kedua correlation circle dalam satu grid

grid.arrange(contrib_circle, cos2_circle, ncol = 2)

Interpretasi Correlation :

Dim1 (45.1%) dan Dim2 (13.4%) menjelaskan sebagian besar varians.
Kontribusi variabel (grafik kiri):
- CHE, CHOL, ALB, PROT, Age (merah) berkontribusi tinggi.
- CREA, ALP (biru/hijau) kontribusinya kecil.
cos2 (grafik kanan):
- CHE, CHOL, ALB, PROT, Age (oranye/merah) direpresentasikan baik dalam PC1-PC2.
- CREA, ALP kurang terwakili.

Kesimpulan: CHE, CHOL, ALB, PROT, dan Age dominan di PC1-PC2, sedangkan CREA dan ALP kurang signifikan.

Plot kontribusi variabel untuk PC1, PC2, PC3, dan PC4

pca_result <- prcomp(scaled.X)

Kesimpulan PCA

Reduksi Dimensi yang Efektif Empat komponen utama (PC1–PC4) mampu menjelaskan 82.50% variansi data, sehingga sebagian besar informasi dapat dirangkum dalam empat dimensi ini.
Berdasarkan scree plot, kontribusi variansi mulai berkurang signifikan setelah PC4, sehingga empat PC cukup untuk representasi data.

Interpretasi Komponen Utama:
- PC1 (45.1%) → Dipengaruhi oleh ALB, CHE, X, AST, PROT, menunjukkan keterkaitan dengan fungsi hati dan metabolisme protein.
- PC2 (13.4%) → Didominasi Age, ALT, AST, CHOL, PROT, yang berhubungan dengan usia dan enzim hati.
- PC3 (12.8%) → ALP sangat dominan, diikuti ALT, GGT, BIL, menunjukkan hubungan dengan enzim hati dan bilirubin.
- PC4 (11.3%) → CREA mendominasi, menunjukkan keterkaitan dengan fungsi ginjal.
Hubungan antar Variabel:
- ALT & AST memiliki korelasi tinggi, menunjukkan hubungan erat dalam fungsi hati.
- ALP dan CHOL korelasi negatif, kemungkinan hubungan terbalik dalam kondisi medis tertentu.
- CREA dan ALP memiliki kontribusi kecil, kurang signifikan di PC1-PC2.

3.5 Factor Analysis (FA)

Melakukan Factor Analysis dengan jumlah faktor optimal

library(psych)
library(nFactors)

## Warning: package 'nFactors' was built under R version 4.4.3

## Loading required package: lattice

## 
## Attaching package: 'nFactors'

## The following object is masked from 'package:lattice':
## 
##     parallel

fa_result <- fa(data_numeric_imputed, nfactors = 8, rotate = "varimax")
fa_result

## Factor Analysis using method =  minres
## Call: fa(r = data_numeric_imputed, nfactors = 8, rotate = "varimax")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        MR5   MR6   MR2   MR4   MR3   MR7   MR8   MR1   h2     u2 com
## X     0.89  0.10 -0.06 -0.11  0.00  0.19  0.02 -0.09 0.86 0.1400 1.2
## Age   0.49  0.03  0.12  0.11 -0.02  0.01 -0.13 -0.04 0.28 0.7172 1.4
## ALB  -0.21 -0.08 -0.07  0.09  0.01 -0.25  0.46  0.78 0.94 0.0589 2.1
## ALP   0.07  0.17  0.97  0.07  0.08 -0.01 -0.04 -0.05 1.00 0.0049 1.1
## ALT  -0.06  0.49  0.09  0.05 -0.04 -0.15  0.05 -0.04 0.28 0.7179 1.4
## AST   0.19  0.71 -0.07 -0.14 -0.01  0.37  0.05 -0.02 0.70 0.2958 1.8
## BIL   0.05  0.10  0.03 -0.05  0.02  0.59  0.01 -0.07 0.37 0.6252 1.1
## CHE  -0.15  0.07  0.02  0.31  0.00 -0.53  0.24  0.11 0.47 0.5275 2.4
## CHOL  0.07 -0.03  0.07  0.97 -0.03 -0.19  0.12  0.05 1.00 0.0050 1.1
## CREA -0.03 -0.01  0.08 -0.02  0.99  0.02 -0.01  0.00 1.00 0.0050 1.0
## GGT   0.19  0.55  0.33  0.02  0.10  0.24 -0.03  0.03 0.52 0.4759 2.5
## PROT -0.11  0.09 -0.03  0.11 -0.02 -0.06  0.77  0.20 0.67 0.3250 1.3
## 
##                        MR5  MR6  MR2  MR4  MR3  MR7  MR8  MR1
## SS loadings           1.19 1.13 1.10 1.09 1.01 0.99 0.91 0.68
## Proportion Var        0.10 0.09 0.09 0.09 0.08 0.08 0.08 0.06
## Cumulative Var        0.10 0.19 0.28 0.38 0.46 0.54 0.62 0.68
## Proportion Explained  0.15 0.14 0.14 0.14 0.12 0.12 0.11 0.08
## Cumulative Proportion 0.15 0.29 0.42 0.56 0.68 0.80 0.92 1.00
## 
## Mean item complexity =  1.5
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  66  with the objective function =  2.38 with Chi Square =  1449.89
## df of  the model are -2  and the objective function was  0 
## 
## The root mean square of the residuals (RMSR) is  0 
## The df corrected root mean square of the residuals is  NA 
## 
## The harmonic n.obs is  615 with the empirical chi square  0.03  with prob <  NA 
## The total n.obs was  615  with Likelihood Chi Square =  0.07  with prob <  NA 
## 
## Tucker Lewis Index of factoring reliability =  1.05
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    MR5  MR6  MR2  MR4  MR3  MR7
## Correlation of (regression) scores with factors   0.92 0.82 0.99 0.99 1.00 0.73
## Multiple R square of scores with factors          0.84 0.68 0.98 0.97 0.99 0.54
## Minimum correlation of possible factor scores     0.68 0.36 0.97 0.95 0.99 0.08
##                                                    MR8  MR1
## Correlation of (regression) scores with factors   0.79 0.88
## Multiple R square of scores with factors          0.63 0.77
## Minimum correlation of possible factor scores     0.26 0.53

Melihat loadings dari setiap faktor

print(fa_result$loadings, cutoff = 0.3)

## 
## Loadings:
##      MR5    MR6    MR2    MR4    MR3    MR7    MR8    MR1   
## X     0.888                                                 
## Age   0.487                                                 
## ALB                                             0.460  0.779
## ALP                 0.972                                   
## ALT          0.490                                          
## AST          0.711                       0.371              
## BIL                                      0.594              
## CHE                        0.306        -0.531              
## CHOL                       0.965                            
## CREA                              0.993                     
## GGT          0.555  0.330                                   
## PROT                                            0.775       
## 
##                  MR5   MR6   MR2   MR4   MR3   MR7   MR8   MR1
## SS loadings    1.191 1.126 1.102 1.095 1.006 0.995 0.909 0.678
## Proportion Var 0.099 0.094 0.092 0.091 0.084 0.083 0.076 0.057
## Cumulative Var 0.099 0.193 0.285 0.376 0.460 0.543 0.619 0.675

Visualisasi Faktor Loadings dengan Diagram

fa.diagram(fa_result)

Interpretasi Hasil Faktor Loadings:

Faktor MR5 memiliki loading tertinggi untuk variabel X (0.888), menunjukkan kontribusi besar terhadap variabel tersebut.
Faktor MR6 memiliki loading tertinggi untuk variabel AST (0.711), berkaitan erat dengan enzim hati.
Faktor MR2 memiliki loading tertinggi untuk variabel ALP (0.972), terkait dengan enzim alkalin fosfatase.
Faktor MR4 memiliki loading signifikan untuk CHE (0.306) dan CHOL (0.965), berkaitan dengan metabolisme kolesterol.
Faktor MR3 memiliki korelasi negatif dengan CHE (-0.531) dan loading signifikan untuk BIL (0.594), terkait fungsi hati.
Faktor MR1 memiliki loading tinggi pada PROT (0.775) dan X (0.779), berhubungan dengan kadar protein tubuh.
Proporsi varians yang dijelaskan oleh masing-masing faktor kecil, dengan MR5 menjelaskan 9.9% dan MR1 5.7%.
Secara kumulatif, delapan faktor pertama menjelaskan sekitar 67.5% dari total varians data.

Menampilkan faktor skor untuk tiap observasi

fa_scores <- fa_result$scores
fa_scores

##                 MR5          MR6          MR2          MR4           MR3
##   [1,] -1.902506061 -0.425270139 -0.381029509 -1.661426880  0.4273714999
##   [2,] -1.773062999  0.040584674  0.209448692 -0.494221355 -0.1984906292
##   [3,] -1.727394112  0.587928544  0.337372001 -0.155206981  0.0373470642
##   [4,] -1.817967525 -0.115163954 -0.453197615 -0.373594526 -0.0491604537
##   [5,] -1.753924622  0.078229312  0.372055636 -0.814060498 -0.2006770657
##   [6,] -1.673385145  0.212840963 -1.037222954  0.878788508  0.6595233516
##   [7,] -1.709533482 -0.398553563 -0.801602801 -0.367670825 -0.2241918037
##   [8,] -1.776443656 -0.061233727 -0.948798473 -0.310560856  0.5558823705
##   [9,] -1.614184531 -0.246008908  0.206452763 -1.197372065 -0.0540063859
##  [10,] -1.750853422 -0.550332941  1.005919439 -0.597154124 -0.1762675492
##  [11,] -1.870965505 -0.475480897 -0.274783250 -1.397417862 -0.1425219958
##  [12,] -1.592418913 -0.417490150  0.285348437 -0.880880679 -0.1460574092
##  [13,] -1.622532945 -0.112981808  0.474038529  0.210117001 -0.1475446511
##  [14,] -1.829026651 -0.364349559 -0.361897043 -1.566592335 -0.3931217812
##  [15,] -1.736066870 -0.359272831 -0.953570773  0.004314572 -0.3560296293
##  [16,] -1.712422659  0.270226518  0.006007924 -0.745722359 -0.2624154604
##  [17,] -1.617265177 -0.324082946  0.337444992  0.043592307  0.0968011552
##  [18,] -1.556080001  0.274184438  0.860499997  0.095072194  0.3233462604
##  [19,] -1.723297501  0.408061107 -0.542877116  0.468870294  0.6232932367
##  [20,] -1.703797011  0.032341092 -0.068118022 -0.969287109  0.4039388737
##  [21,] -1.600587245 -0.122125541 -0.646009053  0.406252252  0.4498904318
##  [22,] -1.540889089 -0.065114239  1.030488210 -0.774774138 -0.1984300723
##  [23,] -1.618529335  0.697511619 -0.112923238 -0.126016382 -0.1590822277
##  [24,] -1.809245529  0.983902049 -0.819093810 -1.420944972 -0.2100108529
##  [25,] -1.684741487  0.184303439 -1.359395784 -0.541281677  0.2587736025
##  [26,] -1.571132392  0.219089025  0.475157960  0.166164582 -0.0829932427
##  [27,] -1.421775048 -0.185843515 -1.660480274  1.643993240  0.1886611501
##  [28,] -1.666356270 -0.251197861 -0.997849822 -0.648602958 -0.4047180639
##  [29,] -1.477842986 -0.187742496  0.791128281 -0.221938957  0.1633247879
##  [30,] -1.569674496 -0.017030268 -0.062934768 -0.685183700 -0.0115248188
##  [31,] -1.450796198  0.396176844  0.266177062  0.620121993  0.1157825048
##  [32,] -1.631208509 -0.026459854  0.324397475 -1.086438253  0.1548701407
##  [33,] -1.560390288  0.225111760 -0.311655668 -0.179878417  0.2529817406
##  [34,] -1.488588405  0.289841920  0.018852035  0.425339792  0.0387443470
##  [35,] -1.621929245  0.594975305  0.020625567 -0.549835694  0.3107161552
##  [36,] -1.351069000  0.728742250  0.314632630  1.492757841  0.1319445402
##  [37,] -1.462209226  0.026920003  2.045358137 -1.044948007 -0.2236090896
##  [38,] -1.662042726 -0.134127508  0.485264839 -1.393103999  0.0576844647
##  [39,] -1.334412635  0.212734934  1.044898179  0.289894623 -0.0733896194
##  [40,] -1.405892884  0.007947496  0.222793931 -0.586239100 -0.0543639784
##  [41,] -1.343379911 -0.219335922 -0.405225414  0.882798910 -0.2979052473
##  [42,] -1.323068211  0.176815458  0.722046252 -0.176040249  0.0839089532
##  [43,] -1.778494136  0.042712923  1.331425771 -0.836404525 -0.4818749568
##  [44,] -1.490023328 -0.259666309 -0.151824360 -0.231743220 -0.2349351706
##  [45,] -1.514696075 -0.260212777 -0.170663476 -1.011224504  0.0696308234
##  [46,] -1.436981849  1.252262965  0.177137510  0.228664857  0.1177190695
##  [47,] -1.441443062 -0.224288956 -0.199376292 -0.240147684  0.1897416756
##  [48,] -1.133346440 -0.418973859 -0.028576110  1.860656664 -0.2452584001
##  [49,] -1.274257499  0.045219872  0.105365860  0.286369949 -0.1535047035
##  [50,] -1.339309671  0.294608595  0.899844865  0.162047389  0.2025445583
##  [51,] -1.485860885  0.197379428 -0.017610743 -0.838540099  0.2481524066
##  [52,] -1.424198426 -0.045415956 -0.734949581  0.047726011 -0.0175408250
##  [53,] -1.347302186 -0.255055229  0.805674374 -0.382559958  0.2203795380
##  [54,] -1.417280482 -0.422333608  1.611775585  0.104993592 -0.4384832472
##  [55,] -1.249408526  0.054762327  0.064820449  1.461081265  0.2253161058
##  [56,] -1.314122204  0.110008074 -0.403088610  0.775877579  0.0623588197
##  [57,] -1.483007019 -0.299196528  0.052679609 -1.048325648  0.2918811597
##  [58,] -1.286280775  0.508930773  0.022517090  0.895272516  0.0012082019
##  [59,] -1.374183879  0.210470309  1.163736339 -1.224183150  0.4939442868
##  [60,] -1.321590116 -0.178905066 -0.186116529 -0.284471884 -0.2540702931
##  [61,] -1.106209137 -0.203175749  0.766212294  1.019066227  0.1880192231
##  [62,] -1.371340145 -0.113106109  0.540974751 -0.895965819 -0.0414402562
##  [63,] -1.338890622  0.236432115 -0.853276969 -0.281344286 -0.1048325200
##  [64,] -1.331028923 -0.317621558 -0.543971104 -0.283161107  0.1635254535
##  [65,] -1.542976835  0.941194485 -0.349521422  0.251100390  0.2913721455
##  [66,] -1.400892833  0.889966942 -0.318099107 -0.104171376  0.0079246357
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## [452,]  0.721698829 -0.316842492 -0.295323159  0.646089632 -0.1810031119
## [453,]  0.641210769  0.973313288  0.009939330 -0.799920886 -0.2223318113
## [454,]  0.780217690 -0.273434911  0.493321709 -0.183740817 -0.2911516350
## [455,]  0.804808266 -0.700954221 -0.542938422  0.046893101 -0.4717217026
## [456,]  0.931397191 -0.525704201 -0.268866461  1.459545409 -0.1178740448
## [457,]  0.798650433 -0.628214073  0.044963684 -0.449987801 -0.1861970487
## [458,]  0.951146171 -0.493137185 -0.662757241  1.226975245 -0.3315021626
## [459,]  1.050320398  0.124778306  2.640715874  0.391807464 -0.2449054761
## [460,]  0.953242413 -0.068235856  0.409138346  0.917860847 -0.2067871627
## [461,]  0.544362257 -0.897187400  0.444802124 -1.955820186 -0.0772473673
## [462,]  0.729181397 -0.510136402 -0.960649632 -0.293845120  0.1132335682
## [463,]  1.125518046  0.389556868  1.810784377  0.681841153  0.0132323699
## [464,]  0.938958408 -0.387225081 -0.265861298  0.400385202 -0.1609217006
## [465,]  1.034578945 -0.278652087  0.406844314  0.388884443 -0.3247722394
## [466,]  0.902024596 -0.548790713 -0.571455076 -0.076817896 -0.1818028413
## [467,]  0.842623778 -0.402391163 -0.924161609  0.372246777  0.1732844474
## [468,]  0.734024886 -0.365269374 -0.282800523 -1.142449098 -0.0938300639
## [469,]  1.127839380 -0.302163532  0.721252009  1.384961236 -0.1288601605
## [470,]  1.008416603 -0.643089637  0.691149155  0.100620624  0.1189048883
## [471,]  1.041317401 -0.053823865  0.565181272  1.411675420 -0.1766103280
## [472,]  1.181356586 -0.567733950  0.249775335  2.132308340 -0.2607679498
## [473,]  1.003241736 -0.338240809  0.015706368  0.582065340 -0.2812821542
## [474,]  0.805135905 -0.258553131 -0.430964474 -1.057906860 -0.2616988497
## [475,]  1.059761796 -0.088566739  0.188475945  0.722407919 -0.2645576874
## [476,]  0.914389199 -0.155313600  0.663895044 -0.382245092 -0.2552560844
## [477,]  1.104123875 -0.461705776  0.654429442  0.837084298 -0.4981189928
## [478,]  1.150766519 -0.644798482 -0.304729650  1.401733186 -0.0762725719
## [479,]  0.979498003 -0.342018881  0.974372494 -0.147363255 -0.2316204540
## [480,]  0.934590171 -0.614037373  0.743587689 -0.614806592 -0.4449197958
## [481,]  1.056273419 -0.146666331 -0.068984541  0.335003651  0.0470590979
## [482,]  1.046316271  0.205375819  0.676493429 -0.083751130 -0.4091432063
## [483,]  0.891627100 -0.653210277 -1.425258502 -0.343309985 -0.1460797087
## [484,]  0.901945491 -0.214823912  0.038675566 -0.201196696 -0.4745734214
## [485,]  1.061005588 -0.694010997 -0.496765792  0.023932291 -0.2446575082
## [486,]  1.056724974 -0.148939448 -0.344482776  0.031176570 -0.1956157824
## [487,]  1.173410157 -0.514191866  0.304584029  1.090172566 -0.2579702646
## [488,]  1.018702857 -0.463943969  0.797837683 -0.552349077 -0.2479619728
## [489,]  1.094706958 -0.637987229  1.318642767  0.317924679 -0.3850151718
## [490,]  0.978367199 -0.527905125  0.078790985 -0.737654567 -0.2743368591
## [491,]  1.242572918 -0.412121343  0.461348078  1.961793977 -0.1276633492
## [492,]  1.151209787 -0.658146763 -0.086128951  0.403219304 -0.2743587791
## [493,]  1.150168126 -0.420344138  0.719791879  0.351129187 -0.3179877066
## [494,]  1.075569387 -0.513132682  0.858088324 -0.057078952 -0.3287513606
## [495,]  1.111758160 -0.794891074  1.419629848 -0.049676844 -0.4199748126
## [496,]  0.894478328 -0.667769821 -0.461571089 -1.224504164 -0.5203978410
## [497,]  1.136475939  0.288023544  0.394585247 -0.345738657 -0.3689395543
## [498,]  1.222066672 -0.529350639  0.586640900  0.509918506 -0.2012906685
## [499,]  1.215059541 -0.427194911  1.149848944 -0.471941638 -0.0449869542
## [500,]  1.366003438  0.134042862  0.320851634  2.582488937 -0.1609756301
## [501,]  1.162536813 -0.546353099 -0.409810822  0.321245201 -0.3311679066
## [502,]  1.078120190 -0.454739211  0.551330055 -0.371550401 -0.4765085621
## [503,]  1.202695748 -0.355983980  0.427424110  1.252378753 -0.2514738109
## [504,]  1.076828046 -0.760430980 -0.661401607 -0.283199586 -0.1430655676
## [505,]  1.231208377 -0.472142452 -0.338299308  1.579677166 -0.1858705193
## [506,]  1.364247239 -0.684338847 -0.424210469  1.187624934 -0.2774717849
## [507,]  1.205293533 -0.390224230 -0.238781140 -0.677304240 -0.2268117157
## [508,]  1.280888695 -0.466723389 -0.232149562  0.516974539 -0.3096285806
## [509,]  1.128499243 -0.751496472  0.963076635  0.066201552 -0.4874480438
## [510,]  0.873027906  0.612567474 -0.824013429 -1.234980523 -0.8064165215
## [511,]  1.335207084 -0.098130883  0.196351447  0.880970957 -0.1705363582
## [512,]  1.375648276 -0.129580890  0.428130530  1.626313116 -0.3593899958
## [513,]  1.440016937 -0.214985578  0.261019706  1.539919847 -0.1737509111
## [514,]  1.295084616 -0.555381191 -0.029138814  0.767327461 -0.3137793424
## [515,]  1.205276717 -0.346949556  0.766408157 -0.684985310 -0.5067775631
## [516,]  1.117710384 -0.344950049  0.300608444 -0.924999442 -0.2293442839
## [517,]  1.291328448 -0.018393043  0.893227766 -0.805221341 -0.1248493169
## [518,]  1.302777010 -0.112162084  0.356640335  1.217574961 -0.2201308923
## [519,]  1.412917402 -0.477867223 -0.735452655  1.063307619  0.0615866453
## [520,]  1.287298812 -0.691005073  0.691600676 -0.243334587  0.4089775701
## [521,]  1.251267751 -0.698820312 -0.092294510  0.978365131 -0.2636646848
## [522,]  1.336378048 -0.752283411 -0.785739931  0.307254332 -0.1128976425
## [523,]  1.443833649 -0.386258027  0.417174431  1.477510841 -0.5149995529
## [524,]  1.456509704 -0.342908270 -0.565316367  1.915401779 -0.3318273288
## [525,]  1.176758860 -0.444941109 -0.794839852 -0.822800979  0.2713338978
## [526,]  1.412623268 -0.383776830 -0.302712734  0.368365283 -0.3123053623
## [527,]  1.481055633 -0.618780566 -0.163030204  1.325502086 -0.2363958689
## [528,]  1.159802265 -0.128731458  0.601739356 -1.122701295 -0.4365887916
## [529,]  1.568324040 -0.504223536 -0.158441413  1.981501452 -0.4294024878
## [530,]  1.377676623  0.147590062 -0.736196633  0.330881404 -0.2441257382
## [531,]  1.392215528 -0.729287081  0.200839321  0.350018421 -0.5774074045
## [532,]  1.594338105 -0.257216468  1.254189599  1.022490950 -0.4852631540
## [533,]  1.537666018 -0.507366931  1.209694526 -0.608752430 -0.4538154136
## [534,]  0.929569780  2.005189576  1.545909441 -0.688686848  1.6716132062
## [535,]  1.242623443 -0.135508052  1.776004351  0.153766576 -1.2102016512
## [536,]  0.980328618 -0.903405942 -1.041659777 -1.610352157 -0.3370758816
## [537,]  1.324512180  3.849253276  0.641430867  1.931398076  0.0123616200
## [538,]  1.022697227  1.185368879 -0.261464798 -2.202015760 -1.4600866611
## [539,]  1.199082969  0.700143854  0.357669199 -1.510452090 -0.6651298190
## [540,]  0.903041377  5.717117952  4.425205088 -0.570193822 -1.2874411279
## [541,]  1.293848957 -0.212665436 -0.552959578  0.475780608  0.0095818543
## [542,]  0.619721924  1.077056893 -0.172217892 -1.454909169 -0.3648777464
## [543,]  0.504403123  2.459235877 -2.249804599 -1.709164027  0.1195948014
## [544,]  0.541331170  2.840257606 -1.655559765 -0.681472157 -0.1497500937
## [545,]  0.777246527 -0.224677068 -1.414940079 -2.098300605 -0.4285832970
## [546,]  0.753945190  0.118763849  0.111480998 -2.076958354 -0.0341369486
## [547,]  1.063622377  0.654607618 -0.273889264  0.354800232  0.1897804606
## [548,]  0.909820089  1.296790930 -1.644956996  0.774866436  0.2039780621
## [549,]  1.048287039  0.585309240 -1.498391628 -0.034040697 -0.5489300914
## [550,]  1.143018980 -0.310101993 -1.244708671 -0.118690631  0.1497413595
## [551,]  0.993276304  0.234263787 -1.153728909 -1.321363039 -0.1347531175
## [552,]  1.169878004 -0.139100205 -1.457248983  0.226209112 -0.0958649191
## [553,]  0.971618154  0.286233550 -2.010857125 -0.346503769  0.1919571368
## [554,]  1.299394908  1.195178205 -1.684599464  4.158688458 -0.2179620122
## [555,]  1.123027123 -0.159274990 -1.526311043 -1.365528741  0.1334718693
## [556,]  1.306874192  0.857273505 -0.443845440 -1.083775266 -0.2126908370
## [557,]  1.384886033  0.101497213 -1.275619668  1.539631802  0.0502998460
## [558,]  1.087253762  1.901858073 -1.616330096 -0.101585116  0.0805956475
## [559,]  0.407127001  6.412164646  0.803248783 -1.403132484  0.2536299283
## [560,]  1.462255065  2.197537077  0.627119048  0.945799495  1.3164351067
## [561,]  1.124425580 -0.868946180 -1.424644158  0.239302862 -0.3025800167
## [562,]  1.011816689 -0.340962173 -1.472191653 -0.072559237 -0.2540518817
## [563,]  1.199316792 -0.223717631 -1.400821507 -0.600026195 -0.3736314577
## [564,]  1.112464796  1.916028165 -1.559755411 -0.789369747 -0.2195077449
## [565,]  1.094067427  1.209924028 -1.278680654  0.099156004 -0.3844128440
## [566,]  0.833367033  1.802433161 -1.286276553 -1.040609141 -0.0715912293
## [567,]  0.972540599  1.491185426 -1.826889583 -1.090527234 -0.2692893114
## [568,]  1.175217280  0.189545081 -0.867468723 -1.114639103 -0.0244719286
## [569,]  0.925776120  1.261060806 -0.218588408 -1.921298828 -0.1430239036
## [570,]  1.149456203  1.552588718 -0.330224349 -0.502908952 -0.0890662724
## [571,]  0.857474785  3.448069748 -0.666310560 -0.917748529  0.0560660224
## [572,]  1.244952893  0.798117329 -0.188815044 -1.424307333  0.6509068133
## [573,]  1.029687494  3.073187965 -1.714186144  0.680351133  0.4008173686
## [574,]  1.267019526  0.071292500 -1.492329959 -0.753022909 -0.0656856575
## [575,]  1.360637380  0.740556290 -1.469291612 -0.862064636 -0.1891825001
## [576,]  1.396919610 -0.232990432 -1.330823885 -0.667780420 -0.1152116592
## [577,]  1.261691650  2.050371901 -0.516578892 -0.756373923  0.0589687940
## [578,]  0.955339709  2.666488316 -1.590500681 -0.460201548 -0.1522732587
## [579,]  1.035004559  0.388174143 -1.174755050 -1.001426255 -0.3045427380
## [580,]  1.445263219 -0.378526839 -0.665250576  0.565606500 -0.0817386187
## [581,]  1.458677169 -0.447850889 -1.780460926 -0.469336423  0.1616811356
## [582,]  1.103658207  0.254760096 -0.056108974 -2.370510495  0.1741291364
## [583,]  1.073829890  1.480890936 -0.355580664 -1.097127744 -0.2332281594
## [584,]  1.347201422  0.135079463 -0.164758467 -0.498432905  0.0297008930
## [585,]  1.343594353  2.829209568 -0.679906637  0.072957678 -0.3751108832
## [586,]  0.975772686  0.898223842 -0.070287406 -2.120896147 -0.4081392271
## [587,]  1.410976377 -0.353974540  1.998429402 -0.179667869  8.6414028462
## [588,]  0.542924555 -0.407117290  0.624232094  0.950539865  0.3835437017
## [589,]  0.727207097  3.594403122 -0.538540355 -0.707201933 -0.6108918736
## [590,]  0.808296205  0.589061012 -2.458593213 -0.858120171  0.0721924571
## [591,]  0.102173181  0.879774183 -0.473049673  1.402852264  0.6185553566
## [592,]  1.302658948 -0.781047848  0.100918492 -0.677573926 20.0904979133
## [593,]  1.128234669  2.914562876 -0.611899896  0.058806421 -0.3457296202
## [594,]  0.853111305  3.881570834 -0.654351154 -1.153094040 -0.4112780335
## [595,]  0.975445419 -0.357927901 -1.560679837 -0.517869808 -0.0387925851
## [596,]  0.699082308  4.690305399 -0.371338276 -0.768132328 -0.2847846498
## [597,]  0.945953652  0.712890126  1.356900819 -2.232844990 -0.5504148068
## [598,]  0.747559565 -0.464888499 -0.963126694 -2.734656651 -0.1037918440
## [599,]  0.956189448  2.939147135  2.504748876 -0.965984633 -0.5077851380
## [600,]  1.090447166  0.589533185 -0.902317244 -1.160197352  0.7922725659
## [601,]  0.669537533 -0.910277675  0.335404432 -0.555434742 -0.3406492810
## [602,]  0.909277747  0.002547039  0.743724245 -1.074140081 -0.5511082346
## [603,]  1.345894722  3.307978641  0.777783095 -0.234434734 -0.2094627339
## [604,]  1.551242002  0.533951152 -0.218135277 -0.227546943 -0.1573530640
## [605,]  1.297058688  0.309161695 -1.550069035 -1.257504728 -0.5301684210
## [606,]  0.672843190 -0.543768058  0.442059482  0.761991240  0.0439468675
## [607,]  1.641566949 -0.316102693  4.230188310 -1.469058042  7.7966330907
## [608,]  1.201105658 -0.182915131 -1.302911930 -1.399031138  1.6613488049
## [609,]  1.115910674  1.764549582 -1.178857482 -1.100398753 -0.3878953214
## [610,]  1.005326737  4.331167845 -1.577558847 -0.138984216  1.2465773437
## [611,]  1.981923368  3.450978484 13.273898890  0.290291929 -1.5302116167
## [612,]  1.285990341 -0.688257368  1.465127274 -1.975378102 -0.5209388017
## [613,]  1.090310046  0.346683481  0.704131778 -1.307938760 -0.3664870611
## [614,]  1.221028354  0.171132869 -0.139748247 -0.910131000 -0.5805909364
## [615,]  1.506151186  1.346111209 -0.459855754 -0.122982218 -0.1931207673
##                  MR7           MR8          MR1
##   [1,]  0.1707681543 -0.6033424673 -0.604515840
##   [2,] -0.4725312489  0.4778806886 -1.323908379
##   [3,]  0.0210611646  0.8996816931  0.301153925
##   [4,]  0.3822214047  0.3175431810 -0.156933873
##   [5,] -0.2458454428 -0.5931483969 -0.539144596
##   [6,]  0.0627737624 -0.0253927473 -0.473089885
##   [7,]  0.2247118849  0.1656709006  0.470053043
##   [8,]  0.4707378054 -0.9065994239  0.246893501
##   [9,] -0.2970401763 -0.0485824172  1.521887807
##  [10,]  0.9012587355 -0.3679098654  0.260721631
##  [11,]  0.8242268707  0.2720709645  0.275765458
##  [12,]  0.0016518656 -0.5448517509  0.932866762
##  [13,] -0.0795531319 -0.6944690440 -1.096645372
##  [14,]  0.2051223311 -0.4269101956 -0.565945431
##  [15,]  0.5858675565 -0.4099308819 -0.696657051
##  [16,] -0.1414239957 -0.0422465923 -0.273758545
##  [17,]  0.2899808386 -0.1484458760 -0.366988246
##  [18,] -0.1662373258  0.5399465722  0.113720477
##  [19,]  0.1724565562 -0.5753607589 -1.110186118
##  [20,]  0.1436289410 -0.6715409301  0.148439834
##  [21,]  0.2452377736  0.0632510487  0.105799027
##  [22,] -0.2816671265  0.3200880597  0.560554204
##  [23,] -0.1955156574 -0.5234682983  0.133970188
##  [24,] -0.3526897233 -0.4709902484  0.230601980
##  [25,] -0.4236527875 -0.1836573863 -0.653911578
##  [26,] -0.1162343874  0.9093945559 -0.184921677
##  [27,]  0.0930672168 -0.9012684593  0.060379887
##  [28,] -0.1055276067 -2.9706568404 -1.449815476
##  [29,]  0.0357054426 -0.1040470458  0.371836727
##  [30,] -0.2386445798  0.2074999150  0.557496531
##  [31,] -0.2411880987 -0.2687030701 -0.228356933
##  [32,] -0.0034076148  0.8555347438  0.205904758
##  [33,] -0.1139215522 -0.4530573771  0.236675677
##  [34,] -0.6022573592 -0.3061399720 -1.187581090
##  [35,] -0.0548608232  0.1286517965 -0.253665799
##  [36,] -0.4145602356  0.5277942527 -0.129555908
##  [37,] -0.3656824903 -0.0948468304 -0.168791184
##  [38,]  0.4502955633  0.6930604555  0.949412453
##  [39,] -0.5795063820  0.8038536068  0.322983826
##  [40,] -0.5410988522 -0.6878436429  0.517765367
##  [41,]  0.1525819705  0.1120587167  0.702631321
##  [42,] -0.7317647128  0.8493482277  1.009638226
##  [43,]  0.4189408513  0.8218768650 -3.259522675
##  [44,]  0.2561180584  0.8713519820  0.174415272
##  [45,]  0.4071918504 -0.2666428389  0.749295666
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## [330,] -0.2540420030  0.5856119411  0.013224576
## [331,]  0.1958048211  1.1053475280  0.418203527
## [332,]  0.2589648913 -0.1004544676 -1.182617457
## [333,]  0.0098249239  0.3302551996 -0.681814852
## [334,] -0.0423503077 -0.0550044271  0.070644741
## [335,] -0.0086863466 -0.0388794266 -0.272629775
## [336,]  0.6354929331 -0.2999894993 -0.689760333
## [337,]  0.0126996755  0.1026630955 -0.040396426
## [338,] -0.0899245612 -1.0546498459 -0.596741859
## [339,] -0.2860779321  0.3637538383 -1.363701533
## [340,] -0.1064595132  1.3079027358  0.172143800
## [341,] -0.1687422632  1.2438753629  0.161490094
## [342,]  0.4705966160 -0.0969373512 -0.300222060
## [343,] -0.3365610393  0.6998471947 -0.641157551
## [344,] -0.4312026391 -0.0768530116 -0.202765126
## [345,] -0.5171795207  0.3118867573  0.078219190
## [346,] -0.0506220912 -0.1094462770  0.789428207
## [347,] -0.1205006246  0.4006671782  4.070654569
## [348,] -0.1894992839 -0.8836076439  0.528654642
## [349,] -0.2432792238  0.0281644449 -0.838751439
## [350,]  0.0627543984  1.3318410586  0.322376200
## [351,] -0.1561381914  1.2560781758  0.346266663
## [352,] -0.1428955423  0.7801275820  0.216026501
## [353,]  0.2199160768  0.6104263312  0.698582050
## [354,] -0.3222706586  0.9910619662 -0.912813172
## [355,]  0.2567038440  1.2759984477 -0.975922302
## [356,]  0.1517622429  0.2018024832  0.117922632
## [357,]  0.3278703416 -0.2084983661 -0.373759029
## [358,] -0.2151649410  0.3025508904 -0.090312442
## [359,]  0.2122514887 -0.3530030588  0.024786964
## [360,]  0.4317089441 -0.1383106410 -0.137906538
## [361,] -0.4241357279  0.9916048510  0.727916469
## [362,] -0.6962269452  1.1338017928  1.231016741
## [363,] -0.1730164214 -0.5411433546  0.260507195
## [364,]  0.0098111631  0.3627776182 -0.390294036
## [365,] -0.2222963357 -0.3606866322 -0.084850750
## [366,]  0.2097596859 -0.0191735926 -1.862575179
## [367,] -0.1318134909  0.9215887407  0.524409459
## [368,]  0.5541396785  0.2011068822  0.018906926
## [369,] -0.1012973171  0.2734595706 -0.119847803
## [370,]  0.2005685094  0.1787566686  0.107037599
## [371,] -0.1125395701  0.3751391586  0.036053302
## [372,]  0.1768994516  0.1133296175 -0.360741220
## [373,] -0.2343938334  0.2303080640 -0.398929915
## [374,] -0.0062944389  0.0762886724  0.174534242
## [375,]  0.1861746800  0.7000695314 -0.640363066
## [376,]  0.4080432974  0.2724989060  0.857754587
## [377,]  0.0053525303 -0.7973986873  0.096663632
## [378,]  0.1870028325  1.4737434584 -0.268391479
## [379,] -0.2317905400  0.0865901614 -0.617765924
## [380,] -0.0524288788 -0.5995943670 -0.363122018
## [381,]  0.0656760742  0.1942798103  0.327458190
## [382,]  0.2726905642 -0.1631758667 -0.006295097
## [383,] -0.7320580674  0.1843911314 -0.944465107
## [384,]  0.1279166860  0.3676254243 -0.631398523
## [385,] -0.0564925023 -0.5333422109 -1.064593331
## [386,] -0.1810071937  0.2397898214  0.457276891
## [387,]  0.0368059113 -0.5730393856 -1.268963822
## [388,]  0.3403436986 -0.0597832154  1.409256707
## [389,]  0.0040430231 -0.8207221437 -0.440768990
## [390,] -0.5029403134  0.4641789205  0.388305094
## [391,]  0.0937126865 -0.9509177934 -0.716964701
## [392,]  0.1114620652 -0.0311826620  0.690884441
## [393,] -0.1000000399  1.0323018673  1.039274587
## [394,] -0.4892236233 -0.3304063911 -1.119824496
## [395,] -0.2688892162  0.2418974874  0.191210347
## [396,] -0.0591168043 -0.4869240489 -0.679731621
## [397,] -0.1137936589 -0.5458328719 -0.829801167
## [398,]  0.0176559844 -0.6500882121 -0.644428455
## [399,] -0.2768203226  0.1517845157 -0.426503038
## [400,]  0.2138183739 -0.2682128106 -0.412129797
## [401,]  0.3638813386 -0.1985003816  0.255702225
## [402,] -0.5584314081 -0.3072120401  0.305979380
## [403,] -0.2680478708  1.0095967877  0.580532120
## [404,]  0.1092917223 -0.3160334905 -0.600614547
## [405,]  0.0080880269 -0.3068000825 -0.614284617
## [406,]  0.1030295653 -0.4471779873  3.897550312
## [407,] -0.6343127459  0.3159399303  1.197503268
## [408,]  0.0556882441 -0.3894427018 -0.636118621
## [409,] -0.1561449603 -0.1049737673 -1.162519689
## [410,]  0.3223856442 -0.3917398447 -0.088866578
## [411,] -0.1061405135 -0.0488443430 -1.806105331
## [412,] -0.0760128703  0.6723203837 -0.624067001
## [413,]  0.1185649452 -0.8733809305  0.229094747
## [414,] -0.1747450833  0.0837608313  0.252053123
## [415,] -0.1666561003 -0.1961353007  0.038819767
## [416,] -0.0189757399  0.4701879415 -0.143630467
## [417,]  0.1169614701 -0.3326066031 -0.479725229
## [418,] -0.5784435634  0.2906201426 -0.967682719
## [419,]  0.1115422924  0.8943480789  1.731114674
## [420,]  0.4570998251 -0.9879725390 -0.649541389
## [421,]  0.2071110547 -0.6084501009 -0.163699498
## [422,]  0.4225180989 -0.7868187632 -0.573482089
## [423,]  0.1614629658 -0.6293831446 -1.010877967
## [424,] -0.1418046052 -0.6788153439  0.184488457
## [425,] -0.2709477521  0.7368747097  0.640332776
## [426,]  0.0311843366 -0.0153308011 -1.179019819
## [427,] -0.1900493438 -0.0283194983 -0.564476173
## [428,]  0.4070264144 -1.0157893139 -1.222048044
## [429,] -0.2492062478  0.9632888753  0.493911020
## [430,] -0.1137250200 -0.0557744579  0.830658219
## [431,] -0.4903060417 -0.3862640689 -0.522308194
## [432,] -0.4102407576  0.5506488155 -0.707350889
## [433,] -0.0981550941  0.2766613203  0.319078461
## [434,] -0.9685528666  0.4382556126  0.833572054
## [435,]  0.9286270662  0.3430269608  0.284463796
## [436,] -0.2507625261  0.2491684203  0.344226851
## [437,]  0.4467756065  0.0641814244  0.823570055
## [438,]  0.2159713938 -1.2318307836  0.133169995
## [439,] -0.0851895032 -0.0592751827 -0.382737290
## [440,] -0.4370045457  0.3572963712 -0.363372979
## [441,] -0.3737904236 -0.4886737861 -0.365389278
## [442,] -0.2379291243  0.4168507594 -0.594365387
## [443,]  0.5119376064  0.1241768406 -0.159033126
## [444,]  0.3155402882 -1.4291921795 -0.595265396
## [445,] -0.1794181666  0.9388350827  0.199698544
## [446,] -0.0649538852  0.7812260723  0.039843573
## [447,] -0.4291357054 -0.5176750690 -0.828826489
## [448,] -0.4526630029  0.4201818197  0.986136373
## [449,] -0.4004224996 -0.2796638259  0.313226991
## [450,] -0.2991278097 -0.0562717777 -0.356036750
## [451,] -0.0312508856 -0.5358225970  0.163507008
## [452,]  0.3281099216 -0.3125213043 -1.096954794
## [453,] -0.1657157770  1.0209178783  0.670440879
## [454,] -0.1144042491 -0.1380308746 -0.522686760
## [455,] -0.0621934608 -0.2407120470 -0.475560923
## [456,]  0.1767381342 -0.0337625746 -0.335438988
## [457,] -0.1737716070 -0.3231974005 -0.380044466
## [458,]  0.0604857552  0.0206526435  0.306530784
## [459,] -0.5931812090  0.2054013997  0.091869040
## [460,] -0.3092610615  0.1539859941 -0.146011176
## [461,]  1.0459783584  0.5957551748  1.102647356
## [462,]  0.3695743014  0.0571128173 -0.233988157
## [463,] -0.8535040800  0.1321958250  1.180348743
## [464,] -0.3922582995  0.5890854685  0.127291717
## [465,] -0.3494959187 -0.0250249933  0.882885254
## [466,] -0.1486802180  0.1834723934  0.420485949
## [467,] -0.0796636348 -0.4270210888 -0.868794221
## [468,] -0.2777204763  0.9026736496 -0.295922781
## [469,] -0.3566082487 -0.5488158160  0.035125841
## [470,]  0.0574486780  1.6430053041  1.350747311
## [471,] -0.2989971251 -0.6713922104 -0.602583145
## [472,] -0.0001869065 -0.2546237954  0.086984388
## [473,] -0.5319393863 -0.0545501304 -0.654767107
## [474,]  0.0875629120 -0.0163796520  0.416928364
## [475,] -0.7249795502 -0.0694871555 -0.507143901
## [476,] -0.1586186308  0.4327417842 -0.272997408
## [477,] -0.4336258569  0.8706555960  0.025415904
## [478,]  0.0164302675 -0.3247682281  0.581808679
## [479,] -0.0674549622  0.7973230573 -0.233352242
## [480,] -0.1202275370 -0.3400245694 -0.322171367
## [481,] -0.3766628362  0.6945847152  0.875562828
## [482,]  0.0600978016  0.9121322595  1.797137508
## [483,] -0.0172746964 -0.1606751446 -0.186732153
## [484,] -0.2362222149 -2.1542410948 -1.840419480
## [485,] -0.1688759231  0.1175622782  0.607675606
## [486,] -0.4501362825 -0.7729615332  0.128498229
## [487,] -0.1102769060  0.2873650093  0.290128260
## [488,] -0.2169539052  0.6090251659 -0.012323074
## [489,]  0.0215170105 -0.1893185763 -0.729589788
## [490,]  0.0957482421 -1.5965047868 -0.196079410
## [491,] -0.0385327269  0.2197874127 -0.381601813
## [492,] -0.1859876759 -0.5959808142  0.020544340
## [493,] -0.4665868873  0.2222046689 -0.429014382
## [494,] -0.3200388094 -0.2930267625 -1.060008030
## [495,] -0.0097361806 -0.0483371963 -0.694830250
## [496,]  0.1225373450 -1.2400041139 -0.782569489
## [497,] -0.0028393755 -0.7034546450  1.415809190
## [498,] -0.2115547856 -0.5758888534  0.104501606
## [499,] -0.3004132380  0.9811790639  1.129823230
## [500,] -0.4391859385  0.5648345695 -0.400680648
## [501,] -0.2956451550  0.4898211188  0.025276266
## [502,] -0.0541995101 -0.0730091126 -0.308091181
## [503,] -0.3776996187 -1.3944988240 -2.103565684
## [504,] -0.0513260464 -0.3560357771 -0.416106917
## [505,]  0.2432946161 -0.2202506146 -0.344909251
## [506,] -0.1565916680  0.2908623878  1.183557505
## [507,] -0.7660022112 -0.2076073951  0.788178888
## [508,] -0.8542115033  0.2567898516  0.027210392
## [509,]  0.2738635145 -0.1491932761 -0.860379371
## [510,]  0.2076188988 -3.0156818099 -1.151556696
## [511,] -0.5084060691  0.8217932168  0.732097723
## [512,] -0.2740595575  0.5562824053  0.088663059
## [513,] -0.4103318641  0.0309300597  0.729368366
## [514,] -0.2710428010  0.0787540078 -0.268202803
## [515,] -0.4189275593  0.1358480255 -0.047683331
## [516,] -0.5770195737  0.0644858982 -1.017391964
## [517,] -0.4696581600  1.0234495168  1.918610510
## [518,] -0.2058476950  0.7118561233 -0.529463394
## [519,] -0.6118791782  0.2862495518  0.557265065
## [520,] -0.2602555386  0.1266679189  0.110775321
## [521,]  0.3126824375  0.8870867577 -0.933077453
## [522,] -0.1629949867 -0.2049490874  0.563266377
## [523,] -0.3616663781 -0.5201831862 -0.658387710
## [524,] -0.3465517524 -0.9786722701 -0.763227727
## [525,] -0.0128063720  0.4033725763  0.520417591
## [526,] -0.3661059168  0.5416360688  1.043649482
## [527,] -0.4627200300 -0.9613397970 -0.598591967
## [528,] -0.0722625709 -2.1482232371 -1.277500342
## [529,] -0.1005161912 -0.8803353066  0.160852687
## [530,] -0.7993864549  0.3123309121  0.149367478
## [531,] -0.2335238694 -1.1110326730 -0.985151572
## [532,] -0.4380939455 -0.3263498913  0.319261940
## [533,] -1.0266185427 -0.6697650724  0.176085076
## [534,]  0.0535153840 -2.1856320859 -2.073430968
## [535,]  0.2184781709 -3.4565214961 -1.143617302
## [536,]  0.1554248155 -3.9013819996 -1.742882293
## [537,] -1.8096870247  1.1587486017  0.487407063
## [538,] -1.0009822745 -3.4258089101 -3.318498944
## [539,]  0.3103185825 -3.7719135916 -1.681209152
## [540,] -0.8710858413 -2.9283223442 -2.158525403
## [541,] -0.4601661520 -0.8787853914  1.157397795
## [542,] -0.0206748921  0.8005486320 -0.194746740
## [543,]  0.9609837539  0.1351104222  1.612755939
## [544,]  1.0857967944  0.0945472606  0.639233899
## [545,] -0.0737187570  0.7859350141  0.514330793
## [546,] -0.1000518263  3.0061271730  0.294009637
## [547,] -0.6969129455  1.1462616963  0.117296591
## [548,]  0.2717840359  1.6881901591  0.020017687
## [549,] -0.7603062645  0.2785716732 -0.284662854
## [550,] -0.2127411929  0.3162533666  0.802107136
## [551,] -0.2614898514  0.9559650450  0.816293091
## [552,] -0.3693697634  0.2090085403  0.392180643
## [553,]  0.1356499477  0.1633494879 -0.027869772
## [554,]  1.1696287193  0.5248591204  0.042064515
## [555,] -0.3640055458  0.5046673213  1.361818250
## [556,] -1.3260565977  0.5569289176  1.101440408
## [557,]  0.1829763538  0.1231843970  0.167886795
## [558,]  0.2308099079  0.8116133046  0.263652032
## [559,]  3.6078438646  0.7036486014  0.687996381
## [560,]  1.1908790927 -1.2504717774  1.837068173
## [561,]  0.2683697036  1.0696819118 -0.138044734
## [562,]  0.4283162692  1.5038694945 -1.445784661
## [563,]  0.1552486179 -0.1014720229 -0.010760625
## [564,]  0.6576387450  2.0635526891  1.250828842
## [565,] -0.0896376772 -0.4733190483  0.376852088
## [566,]  0.4621403756  1.4441177579 -0.727860948
## [567,]  0.8170589688  0.3665140844  1.060979682
## [568,]  0.4928541080  0.4761692370  1.684860061
## [569,] -0.2994360803  1.4045349093 -0.839613865
## [570,] -0.5667007127  1.7083218516  0.249508477
## [571,] -0.9823146198  1.8219060219 -0.641271998
## [572,]  0.0379120772  0.7204605408  1.032497286
## [573,]  1.1353958158  1.0429963343  0.009739678
## [574,]  0.5271534219  1.3571144741  0.816521189
## [575,] -0.0783707607 -0.1589704317  0.916794913
## [576,]  0.2086772512 -0.6808536937 -0.034488642
## [577,] -0.6179115231  0.5986224027 -0.902874737
## [578,]  0.3520290391  0.6308561304  1.029033524
## [579,]  0.2304416320  0.8993590133 -0.422897172
## [580,]  0.2674458810  1.5467369489 -0.232211452
## [581,]  0.2174067096  0.6480897025  0.228734413
## [582,] -0.5858735447  1.2970427349 -0.845608605
## [583,] -0.1298620874  0.0470879337 -0.569538512
## [584,] -0.0089139556  0.9134364352 -0.621075200
## [585,]  0.3666299593 -0.2320052431 -0.369584049
## [586,]  0.3325941473  0.9379054187  0.582550429
## [587,]  0.7092433560 -1.7798956496 -0.046911425
## [588,]  5.4116649489  0.3878165980 -0.783690917
## [589,]  2.6834843246 -0.2503775469  0.283971349
## [590,]  2.0676640718 -0.4025979030 -1.566987489
## [591,]  6.6072903257  0.3511652226 -2.753603714
## [592,]  0.1523370140 -0.0287420326 -1.137591128
## [593,]  0.1565782182  0.9266882196  0.123105583
## [594,]  2.5756387066  0.6755620980  0.578896424
## [595,]  2.4584133448 -1.0882272360 -0.367569981
## [596,]  3.3054496541 -0.9963939772 -0.830156311
## [597,]  2.0540742091 -1.1939768689 -1.938258832
## [598,]  3.1556989313 -2.0699790463 -0.090723639
## [599,]  3.0063596850  0.7973040596 -0.857469575
## [600,]  2.0091877561  0.4224688521 -0.415379087
## [601,]  5.0028946498 -1.0338326805 -0.893186242
## [602,]  3.5489305398 -0.2069093766 -0.668437672
## [603,]  1.5895589477 -0.0804851609  0.750918109
## [604,]  0.2221373114  1.0186069382  0.041596768
## [605,]  1.3611886502 -2.3247451716 -1.790132127
## [606,]  5.3362743092  0.1725799065 -0.274720800
## [607,] -0.0766697544 -1.4241580242 -0.307333359
## [608,]  0.8241057755  1.5313785399 -0.736513894
## [609,]  1.0434303975  1.1973898468 -1.367745919
## [610,]  2.4693279899 -0.0681687108  0.734636220
## [611,]  2.2251574790 -0.4520641719  0.739677890
## [612,]  1.2973388442 -0.1827989633 -2.515112160
## [613,]  2.3821521141  1.3383168575 -2.047317437
## [614,]  0.9768322642 -0.0589444229 -1.044677520
## [615,] -0.4861427184 -0.3931944822 -0.639898357

Melihat korelasi antar faktor

fa_correlation <- cor(fa_scores)
print(fa_correlation)

##              MR5          MR6           MR2           MR4          MR3
## MR5  1.000000000  0.039746629  0.0048417324  0.0302704360 -0.005611317
## MR6  0.039746629  1.000000000  0.0709800634  0.0039602668 -0.010929002
## MR2  0.004841732  0.070980063  1.0000000000 -0.0008965788  0.002942664
## MR4  0.030270436  0.003960267 -0.0008965788  1.0000000000  0.002479219
## MR3 -0.005611317 -0.010929002  0.0029426637  0.0024792191  1.000000000
## MR7  0.113566619  0.198287523 -0.0414165538 -0.1152708240  0.018201735
## MR8  0.006324929  0.082505752 -0.0218212131  0.0377878158 -0.003506808
## MR1 -0.033200532 -0.016282086  0.0019847051 -0.0313872059  0.004801317
##             MR7          MR8          MR1
## MR5  0.11356662  0.006324929 -0.033200532
## MR6  0.19828752  0.082505752 -0.016282086
## MR2 -0.04141655 -0.021821213  0.001984705
## MR4 -0.11527082  0.037787816 -0.031387206
## MR3  0.01820173 -0.003506808  0.004801317
## MR7  1.00000000 -0.065038837 -0.127461816
## MR8 -0.06503884  1.000000000  0.289582948
## MR1 -0.12746182  0.289582948  1.000000000

Kesimpulan FA

Hasil Factor Analysis (FA) menunjukkan bahwa ada 8 faktor utama (MR1-MR8) yang terbentuk.
Variabel dengan loading tinggi pada faktor tertentu:
- MR1: ALB (0.779), ALP (0.460)
- MR2: CHE (0.306, -0.531)
- MR3: BIL (0.594)
- MR4: ALT (0.490)
- MR5: X (0.888), Age (0.487)
- MR6: AST (0.711)
- MR7: Tidak ada variabel dengan loading tinggi
- MR8: GGT (0.775)
Faktor dengan kontribusi terbesar berdasarkan SS loadings: MR5 (1.191), MR6 (1.126), MR2 (1.102) Faktor ini menjelaskan sebagian besar varians dalam data.
Proporsi varians total yang dijelaskan oleh 8 faktor: Cumulative Var = 0.675 (67.5%), menunjukkan faktor-faktor ini cukup baik dalam menjelaskan variabilitas data.
Faktor utama dan interpretasi kemungkinan
- MR1: Faktor terkait protein dan enzim hati (ALB, ALP)
- MR2: Faktor terkait kesehatan enzim (CHE)
- MR3: Faktor terkait bilirubin (BIL)
- MR5: Faktor terkait usia dan parameter X
- MR6: Faktor terkait enzim hati AST

4 KESIMPULAN

Berdasarkan hasil analisis menggunakan PCA dan FA pada data laboratorium pasien Hepatitis C, beberapa kesimpulan dapat diambil sebagai berikut:

Penerapan PCA (Principal Component Analysis): PCA berhasil mereduksi dimensi data laboratorium pasien Hepatitis C dengan efektif. Empat komponen utama pertama (PC1–PC4) dapat menjelaskan 82,50% dari total variansi dalam data. Komponen-komponen ini menunjukkan hubungan kuat antara berbagai parameter laboratorium seperti albumin (ALB), enzim hati (AST, ALT, GGT), dan faktor-faktor yang terkait dengan fungsi hati dan ginjal. Hasil ini menunjukkan bahwa PCA dapat merangkum informasi penting dalam data tanpa kehilangan esensi variabilitas, memungkinkan pemahaman yang lebih mudah dan cepat dalam menganalisis data laboratorium yang kompleks.
Penerapan FA (Factor Analysis): Analisis faktor mengidentifikasi delapan faktor utama yang menjelaskan sekitar 67,5% dari total varians dalam data. Faktor-faktor utama ini mencerminkan struktur laten di balik hubungan antar variabel, seperti faktor terkait dengan fungsi hati (ALT, AST, ALP), metabolisme protein (ALB, PROT), dan bilirubin (BIL). Faktor MR5, yang berhubungan dengan variabel X (ID pasien) dan usia, menunjukkan pentingnya parameter demografis dalam mempengaruhi data laboratorium. Hasil FA memberikan wawasan lebih lanjut tentang faktor-faktor yang mendasari variasi dalam data, yang bisa membantu dalam interpretasi dan pemantauan kondisi pasien.
Perbandingan PCA dan FA: Meskipun kedua metode ini digunakan untuk mereduksi dimensi dan mengidentifikasi pola dalam data, PCA lebih efektif dalam mengurangi dimensi data dan menjelaskan variansi yang lebih besar dalam jumlah komponen utama yang lebih sedikit. Sementara itu, FA memberikan wawasan yang lebih dalam mengenai faktor-faktor yang mempengaruhi variabilitas data, dengan menunjukkan struktur laten yang tidak langsung terlihat melalui PCA.
Implikasi Klinis: Hasil analisis ini dapat membantu dalam mempermudah interpretasi data laboratorium pasien Hepatitis C, yang sangat penting dalam diagnosis dan pemantauan kondisi medis pasien. Dengan menggunakan PCA dan FA, praktisi medis dapat lebih cepat memahami hubungan antarvariabel dan mengidentifikasi faktor-faktor yang mempengaruhi kondisi pasien, yang dapat mempercepat proses pengambilan keputusan klinis.

Secara keseluruhan, penerapan PCA dan FA dalam analisis data laboratorium pasien Hepatitis C terbukti efektif dalam mereduksi dimensi data dan mengidentifikasi faktor-faktor yang memengaruhi variabilitas dalam data. Kedua metode ini memiliki potensi besar dalam mendukung analisis dan diagnosis klinis pada pasien Hepatitis C.

5 DAFTAR PUSTAKA

Saraswati, A., Larasati, T., & Suharmanto. (2022). Faktor Risiko Terjadinya Penyakit Hepatitis C. Jurnal Penelitian Perawat Profesional, 4(1), 649–654. http://jurnal.globalhealthsciencegroup.com/index.php/JPPP

Yang, J., Qi, J. L., Wang, X. X., Li, X. H., Jin, R., Liu, B. Y., Liu, H. X., & Rao, H. Y. (2023). The burden of hepatitis C virus in the world, China, India, and the United States from 1990 to 2019. Frontiers in Public Health, 11, 1041201. https://doi.org/10.3389/FPUBH.2023.1041201/FULL

Principal Component Analysis (PCA) dan Factor Analysis (FA) pada Data Laboratorium Pasien Hepatitis C

Nailah Masruroh (23031554100), Yulia Eka Restania (23031554199)

2025-03-24

1 PENDAHULUAN

1.1 Latar Belakang

1.2 Rumusan Masalah

1.3 Tujuan

2 METODOLOGI PENELITIAN

2.1 Data

2.2 Variabel

3 PEMBAHASAN

3.1 Load dan Pre-processing Data

3.2 Summary

3.2.1 Statistika Deskriptif

3.2.2 Distribusi Data Numerik

3.2.3 Correlation Matrix

3.3 Asumsi

3.3.1 KMO (Kaiser-Meyer-Olkin) Test

3.3.2 Bartlett’s Test

3.4 Principal Component Analysis (PCA)

3.5 Factor Analysis (FA)

4 KESIMPULAN

5 DAFTAR PUSTAKA