Modul Anmul 1 PCA dan FA
Kelompok 9
2026-02-24
library(psych)
library(dplyr)
library(FactoMineR)
library(factoextra)
library(GPArotation)
library(reshape2)
library(ggplot2)Import Data
Dataset yang digunakan dalam penelitian ini adalah Spotify Recommendation Dataset yang diperoleh dari platform Kaggle dan dapat diakses melalui tautan berikut, https://www.kaggle.com/datasets/bricevergnou/spotify-recommendation.
Dimana dataset ini memiliki 13 variabel, yaitu 12 variabel fitur audio seperti acousticness, danceability, energy, tempo, valence, dan lainnya, serta 1 variabel target bernama “liked” yang bersifat biner, yaitu 1 (disukai) dan 0 (tidak disukai).
1. Dataset
# Import Data
data <- read.csv("data.csv")
knitr::kable(head(data))| danceability | energy | key | loudness | mode | speechiness | acousticness | instrumentalness | liveness | valence | tempo | duration_ms | time_signature | liked |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.803 | 0.6240 | 7 | -6.764 | 0 | 0.0477 | 0.4510 | 7.34e-04 | 0.1000 | 0.6280 | 95.968 | 304524 | 4 | 0 |
| 0.762 | 0.7030 | 10 | -7.951 | 0 | 0.3060 | 0.2060 | 0.00e+00 | 0.0912 | 0.5190 | 151.329 | 247178 | 4 | 1 |
| 0.261 | 0.0149 | 1 | -27.528 | 1 | 0.0419 | 0.9920 | 8.97e-01 | 0.1020 | 0.0382 | 75.296 | 286987 | 4 | 0 |
| 0.722 | 0.7360 | 3 | -6.994 | 0 | 0.0585 | 0.4310 | 1.20e-06 | 0.1230 | 0.5820 | 89.860 | 208920 | 4 | 1 |
| 0.787 | 0.5720 | 1 | -7.516 | 1 | 0.2220 | 0.1450 | 0.00e+00 | 0.0753 | 0.6470 | 155.117 | 179413 | 4 | 1 |
| 0.778 | 0.6320 | 8 | -6.415 | 1 | 0.1250 | 0.0404 | 0.00e+00 | 0.0912 | 0.8270 | 140.951 | 224029 | 4 | 1 |
2. Struktur dan Tipe Data
# Struktur dan Tipe Data
str(data)## 'data.frame': 195 obs. of 14 variables:
## $ danceability : num 0.803 0.762 0.261 0.722 0.787 0.778 0.666 0.922 0.794 0.853 ...
## $ energy : num 0.624 0.703 0.0149 0.736 0.572 0.632 0.589 0.712 0.659 0.668 ...
## $ key : int 7 10 1 3 1 8 0 7 7 3 ...
## $ loudness : num -6.76 -7.95 -27.53 -6.99 -7.52 ...
## $ mode : int 0 0 1 0 1 1 0 1 0 1 ...
## $ speechiness : num 0.0477 0.306 0.0419 0.0585 0.222 0.125 0.324 0.171 0.0498 0.447 ...
## $ acousticness : num 0.451 0.206 0.992 0.431 0.145 0.0404 0.555 0.0779 0.143 0.263 ...
## $ instrumentalness: num 7.34e-04 0.00 8.97e-01 1.18e-06 0.00 0.00 0.00 3.96e-05 2.24e-03 0.00 ...
## $ liveness : num 0.1 0.0912 0.102 0.123 0.0753 0.0912 0.114 0.175 0.0944 0.104 ...
## $ valence : num 0.628 0.519 0.0382 0.582 0.647 0.827 0.776 0.904 0.308 0.745 ...
## $ tempo : num 96 151.3 75.3 89.9 155.1 ...
## $ duration_ms : int 304524 247178 286987 208920 179413 224029 146053 161800 247460 165363 ...
## $ time_signature : int 4 4 4 4 4 4 4 4 4 4 ...
## $ liked : int 0 1 0 1 1 1 1 1 0 1 ...Preprocessing
1. EDA
Untuk mengetahui distribusi data, sehingga dapat dilakukan analisis lanjutan
summary(data)## danceability energy key loudness
## Min. :0.1300 Min. :0.0024 Min. : 0.000 Min. :-42.261
## 1st Qu.:0.4625 1st Qu.:0.5335 1st Qu.: 2.000 1st Qu.: -9.962
## Median :0.7050 Median :0.6590 Median : 6.000 Median : -7.766
## Mean :0.6367 Mean :0.6384 Mean : 5.497 Mean : -9.482
## 3rd Qu.:0.7990 3rd Qu.:0.8375 3rd Qu.: 8.000 3rd Qu.: -5.829
## Max. :0.9460 Max. :0.9960 Max. :11.000 Max. : -2.336
## mode speechiness acousticness instrumentalness
## Min. :0.0000 Min. :0.0278 Min. :3.050e-06 Min. :0.0000000
## 1st Qu.:0.0000 1st Qu.:0.0568 1st Qu.:4.220e-02 1st Qu.:0.0000000
## Median :1.0000 Median :0.0962 Median :2.130e-01 Median :0.0000076
## Mean :0.5385 Mean :0.1490 Mean :3.191e-01 Mean :0.1923373
## 3rd Qu.:1.0000 3rd Qu.:0.2305 3rd Qu.:5.040e-01 3rd Qu.:0.0975000
## Max. :1.0000 Max. :0.5400 Max. :9.950e-01 Max. :0.9690000
## liveness valence tempo duration_ms
## Min. :0.0331 Min. :0.0353 Min. : 60.17 Min. : 77203
## 1st Qu.:0.0840 1st Qu.:0.2690 1st Qu.:100.24 1st Qu.:178301
## Median :0.1050 Median :0.5250 Median :124.90 Median :204000
## Mean :0.1485 Mean :0.4936 Mean :121.09 Mean :213409
## 3rd Qu.:0.1770 3rd Qu.:0.7175 3rd Qu.:142.46 3rd Qu.:242374
## Max. :0.6330 Max. :0.9800 Max. :180.04 Max. :655213
## time_signature liked
## Min. :1.000 Min. :0.0000
## 1st Qu.:4.000 1st Qu.:0.0000
## Median :4.000 Median :1.0000
## Mean :3.913 Mean :0.5128
## 3rd Qu.:4.000 3rd Qu.:1.0000
## Max. :5.000 Max. :1.0000Visualisasi Distribusi
par(mar = c(8,4,4,2))
boxplot(data[, c("danceability","energy","key","loudness","mode","speechiness",
"acousticness","instrumentalness",
"liveness","valence","tempo","duration_ms","time_signature","liked")],
col="lightblue",
las=2,
main="Boxplot Variabel")
2. Hapus Missing Value
sum(is.na(data))## [1] 03. Feature Selection
kolom_dihapus <- c(
"key",
"mode",
"time_signature",
"liked"
)
data_clean <- data[, !(names(data) %in% kolom_dihapus)]
knitr::kable(head(data_clean))| danceability | energy | loudness | speechiness | acousticness | instrumentalness | liveness | valence | tempo | duration_ms |
|---|---|---|---|---|---|---|---|---|---|
| 0.803 | 0.6240 | -6.764 | 0.0477 | 0.4510 | 7.34e-04 | 0.1000 | 0.6280 | 95.968 | 304524 |
| 0.762 | 0.7030 | -7.951 | 0.3060 | 0.2060 | 0.00e+00 | 0.0912 | 0.5190 | 151.329 | 247178 |
| 0.261 | 0.0149 | -27.528 | 0.0419 | 0.9920 | 8.97e-01 | 0.1020 | 0.0382 | 75.296 | 286987 |
| 0.722 | 0.7360 | -6.994 | 0.0585 | 0.4310 | 1.20e-06 | 0.1230 | 0.5820 | 89.860 | 208920 |
| 0.787 | 0.5720 | -7.516 | 0.2220 | 0.1450 | 0.00e+00 | 0.0753 | 0.6470 | 155.117 | 179413 |
| 0.778 | 0.6320 | -6.415 | 0.1250 | 0.0404 | 0.00e+00 | 0.0912 | 0.8270 | 140.951 | 224029 |
4. Mengganti Nama Fitur
colnames(data_clean) <- paste0("X", 1:ncol(data_clean))
knitr::kable(head(data_clean))| X1 | X2 | X3 | X4 | X5 | X6 | X7 | X8 | X9 | X10 |
|---|---|---|---|---|---|---|---|---|---|
| 0.803 | 0.6240 | -6.764 | 0.0477 | 0.4510 | 7.34e-04 | 0.1000 | 0.6280 | 95.968 | 304524 |
| 0.762 | 0.7030 | -7.951 | 0.3060 | 0.2060 | 0.00e+00 | 0.0912 | 0.5190 | 151.329 | 247178 |
| 0.261 | 0.0149 | -27.528 | 0.0419 | 0.9920 | 8.97e-01 | 0.1020 | 0.0382 | 75.296 | 286987 |
| 0.722 | 0.7360 | -6.994 | 0.0585 | 0.4310 | 1.20e-06 | 0.1230 | 0.5820 | 89.860 | 208920 |
| 0.787 | 0.5720 | -7.516 | 0.2220 | 0.1450 | 0.00e+00 | 0.0753 | 0.6470 | 155.117 | 179413 |
| 0.778 | 0.6320 | -6.415 | 0.1250 | 0.0404 | 0.00e+00 | 0.0912 | 0.8270 | 140.951 | 224029 |
Assumptions
1. Corrrelation
Masing-masing variabel memiliki korelasi dan tidak bernilai 0
cor(data_clean)## X1 X2 X3 X4 X5 X6
## X1 1.0000000 0.1371879 0.45507806 0.38859635 -0.23417647 -0.80705280
## X2 0.1371879 1.0000000 0.81356741 0.12282548 -0.77258324 -0.24144369
## X3 0.4550781 0.8135674 1.00000000 0.27971003 -0.66498947 -0.53826636
## X4 0.3885964 0.1228255 0.27971003 1.00000000 -0.07971033 -0.34324228
## X5 -0.2341765 -0.7725832 -0.66498947 -0.07971033 1.00000000 0.29432036
## X6 -0.8070528 -0.2414437 -0.53826636 -0.34324228 0.29432036 1.00000000
## X7 -0.1370687 0.1665082 0.07809341 -0.00666515 -0.14098782 0.05573045
## X8 0.6123443 0.3194087 0.36353233 0.18070753 -0.31380646 -0.57222362
## X9 0.2235217 0.2149051 0.27446222 0.31391823 -0.25509744 -0.29949290
## X10 -0.2326207 -0.1345274 -0.20633365 -0.38839658 0.13879339 0.24968279
## X7 X8 X9 X10
## X1 -0.13706866 0.61234430 0.22352170 -0.2326207
## X2 0.16650819 0.31940867 0.21490513 -0.1345274
## X3 0.07809341 0.36353233 0.27446222 -0.2063337
## X4 -0.00666515 0.18070753 0.31391823 -0.3883966
## X5 -0.14098782 -0.31380646 -0.25509744 0.1387934
## X6 0.05573045 -0.57222362 -0.29949290 0.2496828
## X7 1.00000000 -0.01300429 -0.01055516 -0.1439661
## X8 -0.01300429 1.00000000 0.21801724 -0.1148418
## X9 -0.01055516 0.21801724 1.00000000 -0.2562498
## X10 -0.14396607 -0.11484180 -0.25624981 1.0000000corrplot::corrplot(cor(data_clean), tl.col = "black", tl.srt = 45, tl.cex = 0.5)
2. MSA
KMO/MSA memilki nilai > 0.5
r <- cor(data_clean)
KMO(r)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = r)
## Overall MSA = 0.71
## MSA for each item =
## X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
## 0.67 0.58 0.70 0.76 0.80 0.79 0.60 0.71 0.81 0.773. Bartlett test
Dengan syarat hasil p-value < 0.05
bartlett.test(data_clean)##
## Bartlett test of homogeneity of variances
##
## data: data_clean
## Bartlett's K-squared = 36677, df = 9, p-value < 2.2e-16Principal Component Analysis
Principal Component Analysis (PCA) merupakan teknik reduksi dimensi yang melalukan transformasi variabel asli menjadi sejumlah komponen utama baru yang tidak berkorelasi
1. Standarisasi
Data diubah agar memiliki skala yang sama, sehingga semua variabel memiliki kontribusi yang seimbang
Z <- scale(data_clean)2. PCA
Melakukan PCA untuk mendapatkan hasil loading dan eigen value, sehingga walaupun komponen yang dihasilkan sedikit tetapi informasi yang dimiliki tidak hilang
pca_full <- principal(Z,
nfactors = ncol(Z),
rotate = "none")
pca_full## Principal Components Analysis
## Call: principal(r = Z, nfactors = ncol(Z), rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 h2 u2 com
## X1 0.73 -0.51 -0.23 0.18 -0.05 0.02 -0.15 -0.12 0.28 0.08 1 0.0e+00 2.8
## X2 0.69 0.65 -0.05 -0.12 -0.11 0.02 0.11 0.17 -0.02 0.20 1 2.2e-15 2.5
## X3 0.84 0.34 -0.08 -0.09 -0.19 0.09 -0.18 0.24 0.09 -0.17 1 6.7e-16 2.0
## X4 0.48 -0.39 0.49 -0.15 -0.27 0.48 0.23 -0.07 -0.05 -0.01 1 1.1e-16 5.2
## X5 -0.68 -0.57 0.08 0.11 0.01 0.09 -0.03 0.43 0.04 0.05 1 1.2e-15 2.8
## X6 -0.78 0.39 0.19 -0.17 -0.02 0.00 0.31 -0.01 0.29 -0.03 1 -1.3e-15 2.5
## X7 0.06 0.41 0.45 0.71 0.26 0.20 -0.08 -0.01 0.03 0.00 1 8.9e-16 3.0
## X8 0.66 -0.22 -0.34 0.27 0.25 -0.11 0.49 0.09 -0.03 -0.06 1 -8.9e-16 3.7
## X9 0.48 -0.11 0.35 -0.46 0.65 0.00 -0.08 0.04 0.04 0.01 1 6.7e-16 3.5
## X10 -0.40 0.18 -0.68 -0.08 0.23 0.53 -0.03 -0.03 0.00 0.00 1 2.2e-16 3.1
##
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10
## SS loadings 3.81 1.69 1.25 0.91 0.73 0.59 0.47 0.30 0.17 0.08
## Proportion Var 0.38 0.17 0.12 0.09 0.07 0.06 0.05 0.03 0.02 0.01
## Cumulative Var 0.38 0.55 0.68 0.77 0.84 0.90 0.94 0.97 0.99 1.00
## Proportion Explained 0.38 0.17 0.12 0.09 0.07 0.06 0.05 0.03 0.02 0.01
## Cumulative Proportion 0.38 0.55 0.68 0.77 0.84 0.90 0.94 0.97 0.99 1.00
##
## Mean item complexity = 3.1
## Test of the hypothesis that 10 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0
## with the empirical chi square 0 with prob < NA
##
## Fit based upon off diagonal values = 13. Scree Plot
Untuk menentukan komponen PCA yang paling memiliki pengaruh
pca_vis <- prcomp(Z, scale. = FALSE)
fviz_eig(pca_vis,
addlabels = TRUE,
barfill = "skyblue",
barcolor = "darkblue",
linecolor = "red")## Warning in geom_bar(stat = "identity", fill = barfill, color = barcolor, :
## Ignoring empty aesthetic: `width`.
4. Mengambil 4 Komponen
pca_result <- principal(Z,
nfactors = 4,
rotate = "none")
pca_result## Principal Components Analysis
## Call: principal(r = Z, nfactors = 4, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 PC3 PC4 h2 u2 com
## X1 0.73 -0.51 -0.23 0.18 0.88 0.123 2.2
## X2 0.69 0.65 -0.05 -0.12 0.91 0.093 2.1
## X3 0.84 0.34 -0.08 -0.09 0.82 0.175 1.4
## X4 0.48 -0.39 0.49 -0.15 0.63 0.366 3.1
## X5 -0.68 -0.57 0.08 0.11 0.80 0.197 2.0
## X6 -0.78 0.39 0.19 -0.17 0.82 0.179 1.7
## X7 0.06 0.41 0.45 0.71 0.88 0.116 2.4
## X8 0.66 -0.22 -0.34 0.27 0.68 0.323 2.2
## X9 0.48 -0.11 0.35 -0.46 0.57 0.428 3.0
## X10 -0.40 0.18 -0.68 -0.08 0.66 0.338 1.8
##
## PC1 PC2 PC3 PC4
## SS loadings 3.81 1.69 1.25 0.91
## Proportion Var 0.38 0.17 0.12 0.09
## Cumulative Var 0.38 0.55 0.68 0.77
## Proportion Explained 0.50 0.22 0.16 0.12
## Cumulative Proportion 0.50 0.72 0.88 1.00
##
## Mean item complexity = 2.2
## Test of the hypothesis that 4 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0.05
## with the empirical chi square 52.99 with prob < 1.8e-07
##
## Fit based upon off diagonal values = 0.985. Menghitung Eigen Vector dari Loading
Fungsi principal() memberikan loading dan bukan eigenvector, sehingga untuk mendapatkan hasil final skor PCA dibutuhkan eigenvector agar sesuai dengan rumus aslinya
L <- as.matrix(pca_result$loadings)
lambda <- pca_result$values[1:4]
V <- sweep(L, 2, sqrt(lambda), "/")
V##
## Loadings:
## PC1 PC2 PC3 PC4
## X1 0.372 -0.395 -0.209 0.187
## X2 0.351 0.498 -0.126
## X3 0.428 0.259
## X4 0.245 -0.296 0.435 -0.157
## X5 -0.349 -0.436 0.113
## X6 -0.397 0.304 0.167 -0.178
## X7 0.319 0.405 0.743
## X8 0.339 -0.171 -0.305 0.286
## X9 0.244 0.313 -0.480
## X10 -0.203 0.142 -0.610
##
## PC1 PC2 PC3 PC4
## SS loadings 1.0 1.0 1.0 1.0
## Proportion Var 0.1 0.1 0.1 0.1
## Cumulative Var 0.1 0.2 0.3 0.46. Menghitung Skor PCA
Skor menunjukkan posisi setiap objek pada komponen utama, dimana ketika memiliki nilai tinggi maka objek punya karakteristik tertentu
scores <- Z %*% V
head(scores)## PC1 PC2 PC3 PC4
## [1,] -0.003877716 -0.29701663 -2.00275192 0.5130620
## [2,] 1.247996752 -0.66125321 0.10228409 -1.0215715
## [3,] -5.621553101 -0.82116628 -0.01689206 0.1259920
## [4,] 0.202192105 -0.01384421 -1.02648135 0.7033173
## [5,] 1.418859682 -0.92545428 0.18695078 -0.8128683
## [6,] 1.457208404 -0.30579038 -0.88004258 -0.2837756Visualisasi Principal Component Analysis
1. Biplot PCA
fviz_pca_biplot(pca_vis,
geom.ind = "point",
addEllipses = TRUE)## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## ℹ The deprecated feature was likely used in the ggpubr package.
## Please report the issue at <https://github.com/kassambara/ggpubr/issues>.
## This warning is displayed once per session.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
Interpretasi :
- Pada biplot PCA ini terdapat 2 komponen utama yaitu PC1 (38,1%) yang positif kearah kanan dan negatif kearah kiri, serta PC2 (16,9) yang positif kearah atas dan negatif kearah bawah
- Semakin panjang panah maka semakin berpengaruh variabel tersebut pada komponen utama atau PC1 PC2
- Pada PC1, variabel X5 dan X6 berkontribusi ke arah positif dan variabel X1, X2, X3 cenderung berpengaruh ke arah negatif
- Pada PC2, variabel X2 dan X6 cenderung kontribusi kearah positif dan variabel X1, X5 cenderung berpengaruh ke arah negatif
2. Contribution Plot
p1 <- fviz_contrib(pca_vis, choice = "var", axes = 1, top = 5) + ggtitle("PC1")
p2 <- fviz_contrib(pca_vis, choice = "var", axes = 2, top = 5) + ggtitle("PC2")
p3 <- fviz_contrib(pca_vis, choice = "var", axes = 3, top = 5) + ggtitle("PC3")
p4 <- fviz_contrib(pca_vis, choice = "var", axes = 4, top = 5) + ggtitle("PC4")
gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 1)
Interpretasi :
- Berbeda dengan biplot, contribution plot hanya melihat secara garis besar pengaruh masing-masing variabel pada komponen utama (PC1, PC2, PC3, PC4)
- Terlihat dari visualisasi tersebut, PC1 sangat dipengaruhi oleh variabel X3, PC2 sangat dipengaruhi oleh variabel X2, PC3 sangat dipengaruhi variabel X10, dan PC4 sangat dipengaruhi variabel X7
Factor Analysis
Factor Analysis (FA) adalah teknik statistik untuk Mencari faktor laten (tersembunyi) yang menyebabkan variabel-variabel saling berkorelasi.
1. Scree Plot Penentuan Jumlah Faktor
Menentukan jumlah faktor yang optimal dengan membandingkan eigenvalue pada data aktual, data simulasi acak, dan data hasil resampling.
fa.parallel(Z, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors = 3 and the number of components = NAHasilnya menunjukkan jumlah faktor yang layak dipertahankan dalam analisis adalah 3 faktor, karena hanya tiga faktor tersebut yang memiliki eigenvalue lebih besar dari eigenvalue data simulasi.
2. Ekstraksi Faktor
fa_pa <- fa(Z,
nfactors = 3,
rotate = "none",
fm = "pa")
fa_pa## Factor Analysis using method = pa
## Call: fa(r = Z, nfactors = 3, rotate = "none", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 PA3 h2 u2 com
## X1 0.74 -0.58 -0.21 0.925 0.07452 2.1
## X2 0.73 0.69 -0.03 0.999 0.00084 2.0
## X3 0.82 0.29 -0.02 0.760 0.23951 1.2
## X4 0.41 -0.27 0.44 0.444 0.55623 2.7
## X5 -0.65 -0.46 0.06 0.632 0.36778 1.8
## X6 -0.75 0.41 0.13 0.754 0.24596 1.6
## X7 0.05 0.21 0.16 0.072 0.92841 2.0
## X8 0.59 -0.20 -0.22 0.436 0.56399 1.5
## X9 0.39 -0.06 0.24 0.214 0.78589 1.7
## X10 -0.34 0.13 -0.51 0.387 0.61339 1.9
##
## PA1 PA2 PA3
## SS loadings 3.52 1.45 0.65
## Proportion Var 0.35 0.15 0.07
## Cumulative Var 0.35 0.50 0.56
## Proportion Explained 0.63 0.26 0.12
## Cumulative Proportion 0.63 0.88 1.00
##
## Mean item complexity = 1.9
## Test of the hypothesis that 3 factors are sufficient.
##
## df null model = 45 with the objective function = 5.06 with Chi Square = 960.01
## df of the model are 18 and the objective function was 0.28
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic n.obs is 195 with the empirical chi square 6.6 with prob < 0.99
## The total n.obs was 195 with Likelihood Chi Square = 53.07 with prob < 2.6e-05
##
## Tucker Lewis Index of factoring reliability = 0.903
## RMSEA index = 0.1 and the 90 % confidence intervals are 0.07 0.132
## BIC = -41.84
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA2 PA3
## Correlation of (regression) scores with factors 0.98 0.98 0.75
## Multiple R square of scores with factors 0.97 0.97 0.56
## Minimum correlation of possible factor scores 0.93 0.93 0.11Pada tahap ekstraksi tanpa rotasi menunjukkan bahwa dua faktor utama memiliki eigenvalue > 1, sedangkan faktor ketiga memiliki eigenvalue < 1 sehingga kontribusinya relatif kecil. Ketiga faktor mampu menjelaskan 56% total variansi data. Nilai RMSR sebesar 0,03 menunjukkan model cukup baik, namun nilai RMSEA sebesar 0,10 mengindikasikan bahwa model masih kurang optimal. Selain itu, beberapa variabel seperti X7 dan X9 memiliki nilai communalities yang rendah sehingga kurang representatif dalam pembentukan faktor.
3. Rotasi Varimax
fa_varimax <- fa(Z,
nfactors = 3,
rotate = "varimax",
fm = "pa",
scores = TRUE)
fa_varimax## Factor Analysis using method = pa
## Call: fa(r = Z, nfactors = 3, rotate = "varimax", scores = TRUE, fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA2 PA1 PA3 h2 u2 com
## X1 0.06 0.92 0.28 0.925 0.07452 1.2
## X2 0.99 0.07 0.08 0.999 0.00084 1.0
## X3 0.75 0.36 0.24 0.760 0.23951 1.7
## X4 0.03 0.21 0.63 0.444 0.55623 1.2
## X5 -0.77 -0.17 -0.09 0.632 0.36778 1.1
## X6 -0.19 -0.79 -0.31 0.754 0.24596 1.4
## X7 0.18 -0.17 0.10 0.072 0.92841 2.5
## X8 0.25 0.60 0.11 0.436 0.56399 1.4
## X9 0.19 0.16 0.39 0.214 0.78589 1.8
## X10 -0.09 -0.04 -0.61 0.387 0.61339 1.0
##
## PA2 PA1 PA3
## SS loadings 2.33 2.10 1.20
## Proportion Var 0.23 0.21 0.12
## Cumulative Var 0.23 0.44 0.56
## Proportion Explained 0.41 0.37 0.21
## Cumulative Proportion 0.41 0.79 1.00
##
## Mean item complexity = 1.5
## Test of the hypothesis that 3 factors are sufficient.
##
## df null model = 45 with the objective function = 5.06 with Chi Square = 960.01
## df of the model are 18 and the objective function was 0.28
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic n.obs is 195 with the empirical chi square 6.6 with prob < 0.99
## The total n.obs was 195 with Likelihood Chi Square = 53.07 with prob < 2.6e-05
##
## Tucker Lewis Index of factoring reliability = 0.903
## RMSEA index = 0.1 and the 90 % confidence intervals are 0.07 0.132
## BIC = -41.84
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA2 PA1 PA3
## Correlation of (regression) scores with factors 1.00 0.95 0.77
## Multiple R square of scores with factors 1.00 0.90 0.59
## Minimum correlation of possible factor scores 0.99 0.79 0.19Hasil setelah rotasi Varimax menunjukkan bahwa terbentuk tiga faktor utama. Faktor pertama terdiri dari variabel X2, X3, X5, dan X6. Faktor kedua terdiri dari X1 dan X8. Faktor ketiga terdiri dari X4, X9, dan X10. Ketiga faktor mampu menjelaskan 56% total variansi data. Nilai RMSR sebesar 0,03 dan TLI sebesar 0,903 menunjukkan model cukup baik, meskipun nilai RMSEA sebesar 0,10 mengindikasikan model belum sepenuhnya optimal. Variabel X7 memiliki nilai communalities rendah sehingga kontribusinya terhadap model relatif kecil.
4. Perbaikan FA Pertama
Z_new <- Z[, !colnames(Z) %in% c("X7")]Variabel X7 dihapus dari model karena tidak memiliki factor loading yang signifikan pada ketiga faktor (seluruh loading < 0,30). Selain itu, nilai communalitiesnya menunjukkan bahwa variabel tersebut tidak mampu dijelaskan oleh faktor yang terbentuk. Oleh karena itu, X7 dianggap tidak representatif dalam struktur faktor dan dikeluarkan dari analisis untuk meningkatkan kualitas model.
- Asumsi Ulang
- MSA
r_baru <- cor(Z_new)
KMO(r_baru)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = r_baru)
## Overall MSA = 0.71
## MSA for each item =
## X1 X2 X3 X4 X5 X6 X8 X9 X10
## 0.67 0.57 0.69 0.76 0.80 0.79 0.71 0.82 0.80- Uji Bartlett
cortest.bartlett(r_baru)## Warning in cortest.bartlett(r_baru): n not specified, 100 used## $chisq
## [1] 472.2902
##
## $p.value
## [1] 1.856749e-77
##
## $df
## [1] 36Scree Plot Penentuan Jumlah Faktor Baru
fa.parallel(Z_new, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors = 3 and the number of components = NARotasi Varimax Baru
fa_varimax <- fa(Z_new,
nfactors = 3,
rotate = "varimax",
fm = "pa",
scores = TRUE)## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected. Examine the results carefullyfa_varimax## Factor Analysis using method = pa
## Call: fa(r = Z_new, nfactors = 3, rotate = "varimax", scores = TRUE,
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA2 PA1 PA3 h2 u2 com
## X1 0.06 0.91 0.27 0.90 0.0996 1.2
## X2 1.00 0.06 0.10 1.01 -0.0098 1.0
## X3 0.76 0.36 0.25 0.76 0.2369 1.7
## X4 0.03 0.20 0.69 0.52 0.4848 1.2
## X5 -0.77 -0.17 -0.09 0.62 0.3755 1.1
## X6 -0.19 -0.80 -0.29 0.77 0.2341 1.4
## X8 0.24 0.62 0.09 0.45 0.5510 1.4
## X9 0.19 0.16 0.40 0.22 0.7783 1.8
## X10 -0.08 -0.09 -0.54 0.31 0.6930 1.1
##
## PA2 PA1 PA3
## SS loadings 2.30 2.08 1.18
## Proportion Var 0.26 0.23 0.13
## Cumulative Var 0.26 0.49 0.62
## Proportion Explained 0.41 0.37 0.21
## Cumulative Proportion 0.41 0.79 1.00
##
## Mean item complexity = 1.3
## Test of the hypothesis that 3 factors are sufficient.
##
## df null model = 36 with the objective function = 4.96 with Chi Square = 943.75
## df of the model are 12 and the objective function was 0.24
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic n.obs is 195 with the empirical chi square 3.64 with prob < 0.99
## The total n.obs was 195 with Likelihood Chi Square = 44.46 with prob < 1.3e-05
##
## Tucker Lewis Index of factoring reliability = 0.892
## RMSEA index = 0.118 and the 90 % confidence intervals are 0.082 0.156
## BIC = -18.81
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA2 PA1 PA3
## Correlation of (regression) scores with factors 1.00 0.94 0.77
## Multiple R square of scores with factors 1.00 0.89 0.60
## Minimum correlation of possible factor scores 0.99 0.77 0.20Hasil rotasi Varimax menunjukkan bahwa terbentuk tiga faktor utama. Faktor pertama terdiri dari variabel X2, X3, X5, dan X6. Faktor kedua terdiri dari X1 dan X8. Faktor ketiga terdiri dari X4 dan X10. Nilai RMSR sebesar 0,02 dan TLI sebesar 0,892 menunjukkan model cukup baik, meskipun nilai RMSEA sebesar 0,118 mengindikasikan model belum sepenuhnya optimal. Variabel X9 memiliki nilai communalities rendah sehingga kontribusinya terhadap model relatif kecil.
5. Perbaikan FA Kedua
Z_new3 <- Z_new[, -which(colnames(Z_new)=="X9")]Variabel X9 dihapus dari model karena memiliki nilai communalities < 0.30 menunjukkan bahwa variabel tersebut tidak mampu dijelaskan oleh faktor yang terbentuk. Oleh karena itu, X9 dianggap tidak representatif dalam struktur faktor dan dikeluarkan dari analisis untuk meningkatkan kualitas model.
- Asumsi Ulang
- MSA
r_baru1 <- cor(Z_new3)
KMO(r_baru1)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = r_baru1)
## Overall MSA = 0.7
## MSA for each item =
## X1 X2 X3 X4 X5 X6 X8 X10
## 0.67 0.56 0.69 0.77 0.81 0.79 0.71 0.75- Uji Bartlett
cortest.bartlett(r_baru1)## Warning in cortest.bartlett(r_baru1): n not specified, 100 used## $chisq
## [1] 453.2787
##
## $p.value
## [1] 2.64483e-78
##
## $df
## [1] 28- Scree plot penentuan jumlah faktor baru
fa.parallel(Z_new3, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors = 3 and the number of components = NA- Rotasi Varimax Baru
fa_varimax <- fa(Z_new3,
nfactors = 3,
rotate = "varimax",
fm = "pa",
scores = TRUE)## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected. Examine the results carefullyfa_varimax## Factor Analysis using method = pa
## Call: fa(r = Z_new3, nfactors = 3, rotate = "varimax", scores = TRUE,
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA2 PA1 PA3 h2 u2 com
## X1 0.07 0.91 0.28 0.91 0.093 1.2
## X2 1.01 0.06 0.08 1.03 -0.027 1.0
## X3 0.76 0.36 0.24 0.77 0.229 1.6
## X4 0.04 0.19 0.74 0.59 0.411 1.1
## X5 -0.76 -0.19 -0.04 0.61 0.386 1.1
## X6 -0.19 -0.81 -0.26 0.76 0.240 1.3
## X8 0.24 0.63 0.05 0.45 0.547 1.3
## X10 -0.10 -0.11 -0.48 0.26 0.744 1.2
##
## PA2 PA1 PA3
## SS loadings 2.29 2.08 1.00
## Proportion Var 0.29 0.26 0.12
## Cumulative Var 0.29 0.55 0.67
## Proportion Explained 0.43 0.39 0.19
## Cumulative Proportion 0.43 0.81 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 3 factors are sufficient.
##
## df null model = 28 with the objective function = 4.75 with Chi Square = 904.18
## df of the model are 7 and the objective function was 0.17
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.04
##
## The harmonic n.obs is 195 with the empirical chi square 1.91 with prob < 0.96
## The total n.obs was 195 with Likelihood Chi Square = 32.26 with prob < 3.6e-05
##
## Tucker Lewis Index of factoring reliability = 0.883
## RMSEA index = 0.136 and the 90 % confidence intervals are 0.091 0.186
## BIC = -4.65
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA2 PA1 PA3
## Correlation of (regression) scores with factors 1 0.94 0.78
## Multiple R square of scores with factors 1 0.89 0.61
## Minimum correlation of possible factor scores 1 0.78 0.22Hasil rotasi Varimax menunjukkan bahwa terbentuk tiga faktor utama. Faktor pertama terdiri dari variabel X2, X3, X5, dan X6. Faktor kedua terdiri dari X1 dan X8. Faktor ketiga terdiri dari X4, X9, dan X10. Nilai RMSR sebesar 0,02 dan TLI sebesar 0,883 menunjukkan model cukup baik, meskipun nilai RMSEA sebesar 0,136 mengindikasikan model belum sepenuhnya optimal. Variabel X10 memiliki nilai communalities rendah dan X2 memiliki nilai communalities sangat tinggi sehingga kontribusinya terhadap model kurang baik.
6. Perbaikan FA Ketiga
Z_new2 <- Z_new3[, !(colnames(Z_new3) %in% c("X2","X10"))]Variabel X2 dan X10 dihapus dari model karena tidak memiliki factor loading yang signifikan pada ketiga faktor (seluruh loading < 0,30 dan > 1,00). Selain itu, nilai communalitiesnya menunjukkan bahwa variabel X10 tidak mampu dijelaskan oleh faktor yang terbentuk dan untuk X2 menunjukkan relasi yang sangat kuat. Oleh karena itu, x2 dan X10 dianggap tidak representatif dalam struktur faktor dan dikeluarkan dari analisis untuk meningkatkan kualitas model.
- Asumsi Ulang
- MSA
r_baru2 <- cor(Z_new2)
KMO(r_baru2)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = r_baru2)
## Overall MSA = 0.73
## MSA for each item =
## X1 X3 X4 X5 X6 X8
## 0.72 0.69 0.82 0.60 0.75 0.85- Uji Bartlett
cortest.bartlett(r_baru2)## Warning in cortest.bartlett(r_baru2): n not specified, 100 used## $chisq
## [1] 263.1868
##
## $p.value
## [1] 2.368208e-47
##
## $df
## [1] 15- Scree plot penentuan jumlah faktor baru
fa.parallel(Z_new2, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA- Rotasi Varimax Baru
fa_varimax <- fa(Z_new2,
nfactors = 2,
rotate = "varimax",
fm = "pa",
scores = TRUE)## maximum iteration exceededfa_varimax## Factor Analysis using method = pa
## Call: fa(r = Z_new2, nfactors = 2, rotate = "varimax", scores = TRUE,
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## X1 0.94 0.14 0.90 0.095 1.0
## X3 0.40 0.70 0.65 0.350 1.6
## X4 0.38 0.09 0.15 0.846 1.1
## X5 -0.10 -0.89 0.80 0.198 1.0
## X6 -0.84 -0.26 0.77 0.231 1.2
## X8 0.58 0.25 0.40 0.595 1.4
##
## PA1 PA2
## SS loadings 2.24 1.44
## Proportion Var 0.37 0.24
## Cumulative Var 0.37 0.61
## Proportion Explained 0.61 0.39
## Cumulative Proportion 0.61 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## df null model = 15 with the objective function = 2.74 with Chi Square = 523.18
## df of the model are 4 and the objective function was 0.09
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.06
##
## The harmonic n.obs is 195 with the empirical chi square 3.13 with prob < 0.54
## The total n.obs was 195 with Likelihood Chi Square = 17.2 with prob < 0.0018
##
## Tucker Lewis Index of factoring reliability = 0.902
## RMSEA index = 0.13 and the 90 % confidence intervals are 0.071 0.197
## BIC = -3.9
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.96 0.91
## Multiple R square of scores with factors 0.92 0.83
## Minimum correlation of possible factor scores 0.84 0.67Hasil rotasi Varimax menunjukkan bahwa terbentuk dua faktor utama karena pada model 3 faktor muncul ultra-heywood pada X2. Faktor pertama terdiri dari variabel X1, X6, dan X8. Faktor kedua terdiri dari X3 dan X5. Kedua faktor mampu menjelaskan 61% total variansi data. Nilai RMSR sebesar 0,03 dan TLI sebesar 0,902 menunjukkan model cukup baik, meskipun nilai RMSEA sebesar 0,13 mengindikasikan model belum sepenuhnya optimal dan varibael X4 memiliki nilai communalities rendah sehingga kontribusinya terhadap model kurang baik.
7. Perbaikan FA Keempat
Z_new4 <- Z_new2[, !colnames(Z_new2) %in% c("X4")]Variabel X4 dihapus dari model karena memiliki nilai communalities < 0,30 menunjukkan bahwa variabel tersebut tidak mampu dijelaskan oleh faktor yang terbentuk. Oleh karena itu, X4 dianggap tidak representatif dalam struktur faktor dan dikeluarkan dari analisis untuk meningkatkan kualitas model.
- Asumsi Ulang
- MSA
r_baru4 <- cor(Z_new4)
KMO(r_baru4)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = r_baru4)
## Overall MSA = 0.71
## MSA for each item =
## X1 X3 X5 X6 X8
## 0.70 0.69 0.60 0.72 0.85- Uji Bartlett
cortest.bartlett(r_baru4)## Warning in cortest.bartlett(r_baru4): n not specified, 100 used## $chisq
## [1] 244.7261
##
## $p.value
## [1] 6.967857e-47
##
## $df
## [1] 10- Scree plot penentuan jumlah faktor baru
fa.parallel(Z_new4, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors = 2 and the number of components = NA- Rotasi Varimax Baru
fa_varimax <- fa(Z_new4,
nfactors = 2,
rotate = "varimax",
fm = "pa",
scores = TRUE)## maximum iteration exceededfa_varimax## Factor Analysis using method = pa
## Call: fa(r = Z_new4, nfactors = 2, rotate = "varimax", scores = TRUE,
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## X1 0.92 -0.14 0.88 0.12 1.0
## X3 0.39 -0.70 0.64 0.36 1.6
## X5 -0.11 0.89 0.80 0.20 1.0
## X6 -0.84 0.25 0.78 0.22 1.2
## X8 0.61 -0.24 0.43 0.57 1.3
##
## PA1 PA2
## SS loadings 2.10 1.42
## Proportion Var 0.42 0.28
## Cumulative Var 0.42 0.70
## Proportion Explained 0.60 0.40
## Cumulative Proportion 0.60 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## df null model = 10 with the objective function = 2.54 with Chi Square = 485.65
## df of the model are 1 and the objective function was 0.05
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic n.obs is 195 with the empirical chi square 0.94 with prob < 0.33
## The total n.obs was 195 with Likelihood Chi Square = 8.79 with prob < 0.003
##
## Tucker Lewis Index of factoring reliability = 0.835
## RMSEA index = 0.2 and the 90 % confidence intervals are 0.095 0.331
## BIC = 3.51
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.95 0.91
## Multiple R square of scores with factors 0.90 0.83
## Minimum correlation of possible factor scores 0.81 0.65Hasil rotasi Varimax menunjukkan bahwa terbentuk dua faktor utama. Faktor pertama terdiri dari variabel X1, X6, dan X8. Faktor kedua terdiri dari X3 dan X5. Kedua faktor mampu menjelaskan 70% total variansi data. Nilai RMSR sebesar 0,02 dan TLI sebesar 0,835 menunjukkan model cukup baik, meskipun nilai RMSEA sebesar 0,20 tergolong tinggi. Secara keseluruhan, model dua faktor dapat dinilai cukup memadai dan stabil dalam merepresentasikan struktur data.
8. Interpretasi Factor Loading
print(fa_varimax$loadings, cutoff = 0.3)##
## Loadings:
## PA1 PA2
## X1 0.925
## X3 0.388 -0.700
## X5 0.885
## X6 -0.844
## X8 0.608
##
## PA1 PA2
## SS loadings 2.101 1.416
## Proportion Var 0.420 0.283
## Cumulative Var 0.420 0.703loadings <- as.data.frame(unclass(fa_varimax$loadings))
loadings$Variable <- rownames(loadings)
loadings## PA1 PA2 Variable
## X1 0.9246307 -0.1434650 X1
## X3 0.3883844 -0.6999034 X3
## X5 -0.1137521 0.8851337 X5
## X6 -0.8442779 0.2519433 X6
## X8 0.6080635 -0.2414183 X8Hasil rotasi Varimax menunjukkan terbentuk dua faktor utama. Faktor pertama (PA1) dibentuk oleh variabel X1 (0,925), X6 (−0,844), dan X8 (0,608). Faktor kedua (PA2) dibentuk oleh variabel X5 (0,885) dan X3 (−0,700). Kedua faktor mampu menjelaskan 70,3% total variansi data, sehingga model dua faktor dinilai cukup baik dalam merepresentasikan struktur data.
9. Evaluasi Model
fa_varimax$RMSEA## RMSEA lower upper confidence
## 0.19977581 0.09473164 0.33092524 0.90000000fa_varimax$TLI## [1] 0.8351001fa_varimax$rms## [1] 0.02195696fa_varimax$BIC## [1] 3.514678fa_varimax$Vaccounted## PA1 PA2
## SS loadings 2.1012703 1.4156668
## Proportion Var 0.4202541 0.2831334
## Cumulative Var 0.4202541 0.7033874
## Proportion Explained 0.5974717 0.4025283
## Cumulative Proportion 0.5974717 1.0000000summary(fa_varimax)##
## Factor analysis with Call: fa(r = Z_new4, nfactors = 2, rotate = "varimax", scores = TRUE,
## fm = "pa")
##
## Test of the hypothesis that 2 factors are sufficient.
## The degrees of freedom for the model is 1 and the objective function was 0.05
## The number of observations was 195 with Chi Square = 8.79 with prob < 0.003
##
## The root mean square of the residuals (RMSA) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## Tucker Lewis Index of factoring reliability = 0.835
## RMSEA index = 0.2 and the 90 % confidence intervals are 0.095 0.331
## BIC = 3.5110. Faktor Score FA
fa_scores <- fa_varimax$scores
head(fa_scores)## PA1 PA2
## [1,] 0.7793723 0.3066403
## [2,] 0.5125727 -0.2359794
## [3,] -1.5738855 2.0589759
## [4,] 0.5097425 0.2029549
## [5,] 0.5936671 -0.3747409
## [6,] 0.5621896 -0.6708985Visualisasi Factor Analysis
1. Diagram Faktor Analysis
fa.diagram(fa_varimax$loadings)
2. Visualisasi Faktor Analysis Barplot
loadings <- as.data.frame(unclass(fa_varimax$loadings))
loadings$Variable <- rownames(loadings)
load_melt <- melt(loadings, id.vars = "Variable")
ggplot(load_melt, aes(x = Variable, y = value)) +
geom_bar(stat = "identity") +
facet_wrap(~variable) +
coord_flip() +
ggtitle("Factor Loadings (Varimax Rotation)")