library(knitr)
library(GGally)
## Warning: package 'GGally' was built under R version 4.4.3
## Loading required package: ggplot2
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library(RVAideMemoire)
## Warning: package 'RVAideMemoire' was built under R version 4.4.3
## *** Package RVAideMemoire v 0.9-83-12 ***
library(car)
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## Loading required package: carData
## Warning: package 'carData' was built under R version 4.4.3
library(CCA)
## Warning: package 'CCA' was built under R version 4.4.3
## Loading required package: fda
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## Loading required package: fds
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library(corrplot)
## corrplot 0.95 loaded
library(candisc)
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library(yacca)
## Warning: package 'yacca' was built under R version 4.4.3
## yacca: Yet Another Canonical Correlation Analysis Package
## Version 1.4-2 created on 2022-03-08.
## copyright (c) 2008, Carter T. Butts, University of California-Irvine
##  For citation information, type citation("yacca").
##  Type help("yacca-package") to get started.
library(CCP)
data_CCA <- as.data.frame(state.x77)
CCA_X <- data_CCA[, c("Income", "Illiteracy", "HS Grad")]
CCA_Y <- data_CCA[, c("Life Exp", "Murder", "Population")]

data_CCA

1 Eksplorasi & Uji Asumsi

1.1 Missing Value

sum(is.na(data_CCA))
## [1] 0

Terbukti bahwa tidak ada missing value

1.2 Multikolinearitas, Linearitas & Singularitas

ggpairs(CCA_X)

det(cor(CCA_X))
## [1] 0.3488921
eigen(cor(CCA_X))$values
## [1] 2.1478324 0.5643164 0.2878512

Variabel dalam set X menunjukkan korelasi sedang hingga kuat, namun tidak ekstrem. Nilai determinan matriks korelasi sebesar 0,3489 (> 0,01) serta seluruh eigenvalue yang bernilai positif dan tidak mendekati nol menunjukkan tidak adanya multikolinearitas maupun singularitas.

ggpairs(CCA_Y)

det(cor(CCA_Y))
## [1] 0.3040795
eigen(cor(CCA_Y))$values
## [1] 1.8797469 0.9499670 0.1702861

Variabel dalam set Y memiliki pola korelasi yang bervariasi. Nilai determinan matriks korelasi sebesar 0,3041 (> 0,01) dan eigenvalue yang seluruhnya lebih besar dari nol mengindikasikan matriks korelasi tidak singular dan bebas dari multikolinearitas.

ggpairs(data_CCA)

1.3 Normalitas Ganda

mqqnorm(CCA_X, main = "Multi-normal Q-Q Plot X")

##     Alaska New Mexico 
##          2         31

Berdasarkan QQ-plot normalitas ganda peubah X, terdapat beberapa titik yang tidak menyebar di sekitar garis lurus, khususnya Alaska dan New Mexico.

mqqnorm(CCA_Y, main = "Multi-normal Q-Q Plot Y")

## California     Hawaii 
##          5         11

Berdasarkan QQ-plot normalitas ganda peubah X, terdapat beberapa titik yang tidak menyebar di sekitar garis lurus, serta dengan beberapa observasi ekstrem, yaitu California dan Hawaii.

mqqnorm(cbind(CCA_X, CCA_Y), main = "Multi-normal Q-Q Plot X dan Y")

##     Alaska California 
##          2          5

Berdasarkan QQ-plot normalitas ganda peubah X dan Y, diperoleh bahwa titik-titik menyebar di sekitar garis diagonal, namun terdapat titik-titik yang tidak menyebar.

1.4 Korelasi antar X & Y

correls_XY <- matcor(CCA_X, CCA_Y)

1.4.1 Korelasi antar X

kable(correls_XY$Xcor)
Income Illiteracy HS Grad
Income 1.0000000 -0.4370752 0.6199323
Illiteracy -0.4370752 1.0000000 -0.6571886
HS Grad 0.6199323 -0.6571886 1.0000000 
corrplot(correls_XY$Xcor, method = "color", addCoef.col ='white', is.cor = T, type = "lower", diag = F)

1.4.2 Korelasi antar Y

kable(correls_XY$Ycor)
Life Exp Murder Population
Life Exp 1.0000000 -0.7808458 -0.0680520
Murder -0.7808458 1.0000000 0.3436428
Population -0.0680520 0.3436428 1.0000000 
corrplot(correls_XY$Ycor, method = "color", addCoef.col ='black', is.cor = T, type = "lower", diag = F)

###Korelasi XY

kable(correls_XY$XYcor)
Income Illiteracy HS Grad Life Exp Murder Population
Income 1.0000000 -0.4370752 0.6199323 0.3402553 -0.2300776 0.2082276
Illiteracy -0.4370752 1.0000000 -0.6571886 -0.5884779 0.7029752 0.1076224
HS Grad 0.6199323 -0.6571886 1.0000000 0.5822162 -0.4879710 -0.0984897
Life Exp 0.3402553 -0.5884779 0.5822162 1.0000000 -0.7808458 -0.0680520
Murder -0.2300776 0.7029752 -0.4879710 -0.7808458 1.0000000 0.3436428
Population 0.2082276 0.1076224 -0.0984897 -0.0680520 0.3436428 1.0000000 
corrplot(correls_XY$XYcor, method = "color", addCoef.col ='white', is.cor = T, type = "lower", diag = F)

img.matcor(correls_XY, type = 2)

#Korelasi Kanonik

cca1 <- cancor(CCA_X, CCA_Y)
summary(cca1)
## 
## Canonical correlation analysis of:
##   3   X  variables:  Income, Illiteracy, HS Grad 
##   with    3   Y  variables:  Life Exp, Murder, Population 
## 
##     CanR CanRSQ  Eigen percent    cum                          scree
## 1 0.7215 0.5206 1.0857  76.646  76.65 ******************************
## 2 0.4157 0.1728 0.2090  14.752  91.40 ******                        
## 3 0.3296 0.1086 0.1219   8.602 100.00 ***                           
## 
## Test of H0: The canonical correlations in the 
## current row and all that follow are zero
## 
##      CanR LR test stat approx F numDF  denDF   Pr(> F)    
## 1 0.72150      0.35350   6.3516     9 107.23 3.527e-07 ***
## 2 0.41575      0.73731   3.7034     4  90.00  0.007745 ** 
## 3 0.32957      0.89138   5.6052     1  46.00  0.022169 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Raw canonical coefficients
## 
##    X  variables: 
##                  Xcan1       Xcan2      Xcan3
## Income      0.00020061  0.00075453 -0.0019243
## Illiteracy  1.52112426 -1.46642271 -0.5343688
## HS Grad    -0.02235599 -0.18555039  0.0247592
## 
##    Y  variables: 
##                  Ycan1       Ycan2       Ycan3
## Life Exp   -9.0472e-02 -1.17227258 -0.47648787
## Murder      2.5371e-01 -0.40642558 -0.10322327
## Population -2.9473e-05  0.00017121 -0.00018497
cc1 <- cc(CCA_X, CCA_Y)
cc1$cor
## [1] 0.7214954 0.4157492 0.3295704
  1. Terdapat hubungan kanonik yang signifikan antara karakteristik sosial-ekonomi (Income, Illiteracy, HS Grad) dan indikator demografis-kesehatan (Life Exp, Murder, Population).
  2. Korelasi kanonik pertama (r = 0,7215) merupakan hubungan paling kuat
  3. Berdasarkan kontribusi keragaman yang dijelaskan oleh fungsi kanonik, fungsi kanonik pertama menjelaskan keragaman total sebesar 76,65%, sedangkan fungsi kanonik kedua dan ketiga masing-masing menjelaskan 14,75% dan 8,60% dari total keragaman.
  4. Hasil uji signifikansi menunjukkan bahwa ketiga korelasi kanonik berbeda nyata dengan nol pada taraf nyata 5%, dengan korelasi kanonik pertama berbeda nyata pada taraf nyata 0,1%.

Dengan demikian, meskipun secara statistik seluruh fungsi kanonik signifikan, fungsi kanonik pertama merupakan yang paling relevan dan dominan dalam menjelaskan hubungan antara gugus variabel X dan gugus variabel Y.

2 Koefisien korelasi kanonik mentah

cc1[3:4]
## $xcoef
##                     [,1]          [,2]         [,3]
## Income      0.0002006131  0.0007545267 -0.001924318
## Illiteracy  1.5211242579 -1.4664227122 -0.534368841
## HS Grad    -0.0223559907 -0.1855503878  0.024759154
## 
## $ycoef
##                     [,1]          [,2]          [,3]
## Life Exp   -9.047223e-02 -1.1722725752 -0.4764878738
## Murder      2.537086e-01 -0.4064255818 -0.1032232656
## Population -2.947339e-05  0.0001712075 -0.0001849682

Koefisien kanonik mentah menunjukkan bahwa pada fungsi kanonik pertama, variabel Illiteracy dan HS Grad merupakan kontributor utama dalam pembentukan peubah kanonik pada set X, sedangkan pada set Y didominasi oleh variabel Murder dan Life Expectancy.

#Menghitung loading kanonik

cc1_comp <- comput(CCA_X, CCA_Y, cc1)

2.1 Korelasi X dengan fungsi kanonik X

cc1_comp$corr.X.xscores
##                  [,1]        [,2]        [,3]
## Income     -0.3939155 -0.07478012 -0.91609963
## Illiteracy  0.9919651 -0.11155406  0.05967344
## HS Grad    -0.7134790 -0.62385139 -0.31899392
corrplot(cc1_comp$corr.X.xscores, method = "color", addCoef.col ='white', is.cor = T)

  • Fungsi kanonik pertama didominasi oleh variabel Illiteracy (0,992) dengan kontribusi berlawanan arah dari HS Grad (−0,713) dan Income (−0,394), sehingga fungsi ini merepresentasikan dimensi pendidikan
  • Fungsi kanonik kedua, variabel HS Grad menunjukkan kontribusi relatif paling besar (−0,624)
  • Fungsi kanonik ketiga dipengaruhi oleh Income (−0,916)

2.2 Korelasi Y dengan fungsi kanonik Y:

cc1_comp$corr.Y.yscores
##                  [,1]         [,2]       [,3]
## Life Exp   -0.8438158 -0.454135838 -0.2858942
## Murder      0.9861909 -0.008892474 -0.1653735
## Population  0.1985285  0.355864876 -0.9132068
corrplot(cc1_comp$corr.Y.yscores, method = "color", addCoef.col ='yellow', is.cor = T)

  • Fungsi kanonik pertama didominasi oleh variabel Murder (0,986) dan Life Expectancy (−0,844)

##Korelasi silang X dengan fungsi kanonik Y:

cc1_comp$corr.X.yscores
##                  [,1]        [,2]       [,3]
## Income     -0.2842082 -0.03108978 -0.3019193
## Illiteracy  0.7156983 -0.04637851  0.0196666
## HS Grad    -0.5147718 -0.25936570 -0.1051309
corrplot(cc1_comp$corr.X.yscores, method = "color", addCoef.col ='white', is.cor = T)

2.3 Korelasi silang Y dengan fungsi kanonik X

cc1_comp$corr.Y.xscores
##                  [,1]         [,2]        [,3]
## Life Exp   -0.6088092 -0.188806603 -0.09422227
## Murder      0.7115322 -0.003697039 -0.05450222
## Population  0.1432374  0.147950531 -0.30096590
corrplot(cc1_comp$corr.Y.xscores, method = "color", addCoef.col ='green', is.cor = T)

3 Uji signifikansi korelasi kanonik

rho <- cc1$cor
n <- dim(CCA_X)[1]
p <- length(CCA_X)
q <- length(CCA_Y)

p.asym(rho, n, p, q, tstat = "Wilks")
## Wilks' Lambda, using F-approximation (Rao's F):
##               stat   approx df1     df2      p.value
## 1 to 3:  0.3534992 6.351574   9 107.235 3.527369e-07
## 2 to 3:  0.7373101 3.703387   4  90.000 7.744693e-03
## 3 to 3:  0.8913834 5.605181   1  46.000 2.216881e-02
p.asym(rho, n, p, q, tstat = "Hotelling")
##  Hotelling-Lawley Trace, using F-approximation:
##               stat   approx df1 df2      p.value
## 1 to 3:  1.4165661 6.715573   9 128 7.600366e-08
## 2 to 3:  0.3308185 3.694140   4 134 6.920293e-03
## 3 to 3:  0.1218518 5.686416   1 140 1.843680e-02
p.asym(rho, n, p, q, tstat = "Pillai")
##  Pillai-Bartlett Trace, using F-approximation:
##               stat   approx df1 df2      p.value
## 1 to 3:  0.8020196 5.594970   9 138 1.422475e-06
## 2 to 3:  0.2814640 3.727265   4 144 6.441474e-03
## 3 to 3:  0.1086166 5.634844   1 150 1.887313e-02
p.asym(rho, n, p, q, tstat = "Roy")
##  Roy's Largest Root, using F-approximation:
##               stat   approx df1 df2      p.value
## 1 to 1:  0.5205556 16.64813   3  46 1.834946e-07
## 
##  F statistic for Roy's Greatest Root is an upper bound.

Hasil pengujian signifikansi menggunakan Wilks’ Lambda, Hotelling–Lawley Trace, Pillai–Bartlett Trace, dan Roy’s Largest Root secara konsisten menunjukkan bahwa korelasi kanonik pertama berbeda nyata dengan nol (p < 0,01). Selain itu, pengujian parsial menunjukkan bahwa korelasi kanonik kedua dan ketiga juga signifikan pada taraf nyata 5%, yang mengindikasikan bahwa terdapat lebih dari satu hubungan kanonik yang bermakna antara gugus variabel X dan gugus variabel Y.

cca1
## 
## Canonical correlation analysis of:
##   3   X  variables:  Income, Illiteracy, HS Grad 
##   with    3   Y  variables:  Life Exp, Murder, Population 
## 
##     CanR CanRSQ  Eigen percent    cum                          scree
## 1 0.7215 0.5206 1.0857  76.646  76.65 ******************************
## 2 0.4157 0.1728 0.2090  14.752  91.40 ******                        
## 3 0.3296 0.1086 0.1219   8.602 100.00 ***                           
## 
## Test of H0: The canonical correlations in the 
## current row and all that follow are zero
## 
##      CanR LR test stat approx F numDF  denDF   Pr(> F)    
## 1 0.72150      0.35350   6.3516     9 107.23 3.527e-07 ***
## 2 0.41575      0.73731   3.7034     4  90.00  0.007745 ** 
## 3 0.32957      0.89138   5.6052     1  46.00  0.022169 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Berdasarkan LR test, diperoleh bahwa korelasi kanonik pertama dan kedua berbeda nyata dengan nol pada taraf nyata 1% (Pr < α, dengan α = 0,01), sedangkan korelasi kanonik ketiga tidak berbeda nyata pada taraf nyata 1% namun signifikan pada taraf nyata 5%. Hal ini menunjukkan bahwa hubungan utama antara gugus variabel X dan gugus variabel Y terutama dijelaskan oleh dua fungsi kanonik pertama, dengan fungsi kanonik pertama sebagai yang paling dominan karena menjelaskan 76,65% dari total keragaman kanonik.

4 Menghitung koefisien korelasi kanonik terstandardisasi

coef_X <- diag(sqrt(diag(cov(CCA_X))))
coef_X %*% cc1$xcoef
##            [,1]       [,2]       [,3]
## [1,]  0.1232707  0.4636340 -1.1824356
## [2,]  0.9271756 -0.8938332 -0.3257155
## [3,] -0.1805693 -1.4986901  0.1999796
coef_Y <- diag(sqrt(diag(cov(CCA_Y))))
coef_Y %*% cc1$ycoef
##            [,1]       [,2]       [,3]
## [1,] -0.1214493 -1.5736511 -0.6396342
## [2,]  0.9365755 -1.5003362 -0.3810528
## [3,] -0.1315837  0.7643545 -0.8257891

5 Peubah Kanonik

$$ U1 = 0.1233X_{1} +0.92717X_2 -0.1806X_3 \ U2 = 0.4636X_{1} -0.8938X_2 -1.4987X_3 \ U3 = -1.1824X_{1} -0.3257X_2 +0.2000X_3 \

$$

\[ V1 = 0.1214X_{1} +0.9366X_2 -0.1316X_3 \\ V2 = -1.5736X_1 -1.5003X_2 + 0.7643X_3 \\ V3 = -0.6396X_1 -0.3810X_2 -0.8258X_3 \]