1 Pendahuluan

Dokumen ini merangkum praktikum pengantar analisis multivariat menggunakan R, meliputi:

Operasi dasar R dan struktur data.
Statistik deskriptif multivariat.
Pengecekan normalitas multivariat.
Pengujian hipotesis vektor mean (Z² dan T² Hotelling).
MANOVA dan contrast.
Studi kasus program remedial.

2 Persiapan Package

pkgs <- c("MVN", "DescTools", "readr", "ggplot2")
to_install <- pkgs[!pkgs %in% installed.packages()[, "Package"]]
if (length(to_install) > 0) install.packages(to_install)
invisible(lapply(pkgs, library, character.only = TRUE))

3 1. Pengenalan Dasar R

3.1 1.1 Operasi Aritmetika

2 + 3

## [1] 5

10 - 4

## [1] 6

3 * 7

## [1] 21

15 / 4

## [1] 3.75

10 / 3

## [1] 3.333333

10 %/% 3

## [1] 3

10 %% 3

## [1] 1

2^5

## [1] 32

sqrt(16)

## [1] 4

log(exp(1))

## [1] 1

log(100, 10)

## [1] 2

exp(1)

## [1] 2.718282

Pembahasan singkat:

Operator %/% memberi hasil pembagian bulat.
Operator %% memberi sisa bagi.
Fungsi log(100, 10) berbeda dengan log(100) karena basis ditentukan eksplisit.

3.2 1.2 Assignment dan Vektor

x <- 10
y <- 3.5
z <- x + y
nama <- "Salman"
cat("Halo,", nama, "! Nilai z =", z, "\n")

## Halo, Salman ! Nilai z = 13.5

v <- c(2, 5, 1, 3, 4)
v + 10

## [1] 12 15 11 13 14

v * 2

## [1]  4 10  2  6  8

v^2

## [1]  4 25  1  9 16

length(v)

## [1] 5

sum(v)

## [1] 15

mean(v)

## [1] 3

var(v)

## [1] 2.5

sd(v)

## [1] 1.581139

min(v); max(v)

## [1] 1

## [1] 5

Interpretasi:

Operasi pada vektor dilakukan elemen demi elemen.
var(v) menggunakan variansi sampel (pembagi n-1), sesuai kebutuhan inferensi statistik.

3.3 1.3 Matriks dan Data Frame

A <- matrix(c(1,2,3, 4,5,6, 7,8,9), nrow = 3, ncol = 3)
A

##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9

t(A)

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6
## [3,]    7    8    9

B <- matrix(c(1,0,0, 0,2,0, 0,0,3), nrow = 3)
A %*% B

##      [,1] [,2] [,3]
## [1,]    1    8   21
## [2,]    2   10   24
## [3,]    3   12   27

C <- matrix(c(2,1, 1,3), nrow = 2)
C

##      [,1] [,2]
## [1,]    2    1
## [2,]    1    3

solve(C)

##      [,1] [,2]
## [1,]  0.6 -0.2
## [2,] -0.2  0.4

det(C)

## [1] 5

mahasiswa <- data.frame(
  Nama   = c("Andi", "Budi", "Cici", "Dani", "Edo"),
  Nilai1 = c(75, 82, 90, 68, 45),
  Nilai2 = c(80, 78, 95, 72, 89),
  Nilai3 = c(70, 85, 88, 84, 95)
)
str(mahasiswa)

## 'data.frame':    5 obs. of  4 variables:
##  $ Nama  : chr  "Andi" "Budi" "Cici" "Dani" ...
##  $ Nilai1: num  75 82 90 68 45
##  $ Nilai2: num  80 78 95 72 89
##  $ Nilai3: num  70 85 88 84 95

head(mahasiswa)

Pembahasan:

* pada matriks berarti perkalian elemen, sedangkan %*% adalah perkalian matriks.
solve(C) hanya valid jika determinan matriks tidak nol.

4 2. Statistik Deskriptif Multivariat

4.1 2.1 Data Contoh

data_ujian <- data.frame(
  Aljabar   = c(75, 82, 90, 68, 77, 85, 71, 93, 66, 80),
  Kalkulus  = c(70, 78, 88, 65, 73, 83, 69, 91, 63, 76),
  Statistik = c(72, 80, 92, 67, 75, 81, 70, 89, 64, 78),
  Komputer  = c(68, 75, 85, 62, 70, 79, 65, 87, 60, 73)
)
data_ujian

4.2 2.2 Vektor Mean, Kovariansi, dan Korelasi

x_bar <- colMeans(data_ujian)
S <- cov(data_ujian)
R <- cor(data_ujian)

x_bar

##   Aljabar  Kalkulus Statistik  Komputer 
##      78.7      75.6      76.8      72.4

round(S, 3)

##           Aljabar Kalkulus Statistik Komputer
## Aljabar    81.789   84.978    80.600   83.133
## Kalkulus   84.978   89.378    84.133   86.956
## Statistik  80.600   84.133    82.400   82.533
## Komputer   83.133   86.956    82.533   84.933

round(R, 4)

##           Aljabar Kalkulus Statistik Komputer
## Aljabar    1.0000   0.9939    0.9818   0.9974
## Kalkulus   0.9939   1.0000    0.9804   0.9980
## Statistik  0.9818   0.9804    1.0000   0.9866
## Komputer   0.9974   0.9980    0.9866   1.0000

pairs(
  data_ujian,
  main = "Scatterplot Matrix Skor Ujian",
  pch = 19,
  col = "steelblue"
)

Interpretasi:

Vektor mean merangkum pusat data tiap variabel.
Kovariansi positif antarmata kuliah menunjukkan kecenderungan skor bergerak searah.
Korelasi tinggi menandakan asosiasi linear yang kuat antarvariabel.

4.3 2.3 Kombinasi Linier

c_vec <- c(0.4, 0.3, 0.2, 0.1)
EY <- t(c_vec) %*% x_bar
VarY <- t(c_vec) %*% S %*% c_vec
EY

##       [,1]
## [1,] 76.76

VarY

##          [,1]
## [1,] 83.83156

Pembahasan: kombinasi linier berguna untuk menyusun indeks performa gabungan dari beberapa komponen nilai.

5 3. Normalitas Multivariat

5.1 3.1 Jarak Mahalanobis dan Q-Q Plot Chi-square

X <- as.matrix(data_ujian)
n <- nrow(X)
p <- ncol(X)
d2 <- mahalanobis(X, center = colMeans(X), cov = cov(X))
head(d2)

## [1] 5.191184 2.075978 7.873216 1.974719 1.633245 3.117544

d2_sorted <- sort(d2)
probs <- (seq_len(n) - 0.5) / n
chi2_quant <- qchisq(probs, df = p)

plot(
  chi2_quant, d2_sorted,
  xlab = expression(paste("Kuantil ", chi[p]^2)),
  ylab = expression(paste("Jarak Mahalanobis Terurut ", d[(j)]^2)),
  main = paste0("Chi-square Q-Q Plot (p = ", p, ")"),
  pch = 19, col = "darkorange"
)
abline(0, 1, col = "tomato", lty = 2, lwd = 2)

5.2 3.2 Uji Mardia dan Henze-Zirkler

hasil_mardia <- mvn(data = data_ujian, mvnTest = "mardia", multivariatePlot = "qq")

hasil_hz <- mvn(data = data_ujian, mvnTest = "hz", multivariatePlot = "qq")

hasil_mardia$multivariateNormality

hasil_hz$multivariateNormality

Interpretasi umum:

Jika p-value > 0.05, data tidak menolak asumsi normal multivariat.
Asumsi ini penting sebelum menjalankan prosedur inferensi seperti Hotelling T² atau MANOVA.

6 4. Pengujian Vektor Mean

6.1 4.1 Uji Z² (Sigma Diketahui)

Sigma_pop <- diag(c(70, 65, 75, 70))
mu_0 <- c(76, 74, 75, 71)

n <- nrow(data_ujian)
p <- ncol(data_ujian)
x_bar <- colMeans(data_ujian)
diff <- x_bar - mu_0

Z2 <- n * t(diff) %*% solve(Sigma_pop) %*% diff
p_val_z2 <- 1 - pchisq(Z2, df = p)

data.frame(
  Statistik = c("Z2", "p-value"),
  Nilai = c(round(Z2, 4), round(p_val_z2, 4))
)

6.2 4.2 Uji T² Hotelling (Satu Sampel)

S <- cov(data_ujian)
T2 <- n * t(diff) %*% solve(S) %*% diff
F_stat <- ((n - p) / (p * (n - 1))) * T2
p_val_t2 <- 1 - pf(F_stat, df1 = p, df2 = n - p)

data.frame(
  Statistik = c("T2", "F", "p-value"),
  Nilai = c(round(T2, 4), round(F_stat, 4), round(p_val_t2, 4))
)

HotellingsT2Test(as.matrix(data_ujian), mu = mu_0)

## 
##  Hotelling's one sample T2-test
## 
## data:  as.matrix(data_ujian)
## T.2 = 12.735, df1 = 4, df2 = 6, p-value = 0.00431
## alternative hypothesis: true location is not equal to c(76,74,75,71)

Interpretasi:

H0: vektor mean populasi sama dengan vektor acuan mu_0.
Tolak H0 bila p-value < 0.05.

6.3 4.3 Uji T² Hotelling Dua Sampel Tidak Berpasangan

Grup1 <- data.frame(
  X1 = c(52, 48, 55, 60, 47, 53, 58, 50, 45, 56),
  X2 = c(30, 28, 32, 35, 27, 31, 34, 29, 26, 33)
)
Grup2 <- data.frame(
  X1 = c(58, 62, 55, 65, 60, 57, 63, 59, 61, 64, 56, 60),
  X2 = c(34, 37, 33, 40, 36, 35, 38, 34, 36, 39, 33, 37)
)

hotelling_two_sample <- function(X1, X2) {
  X1 <- as.matrix(X1); X2 <- as.matrix(X2)
  n1 <- nrow(X1); n2 <- nrow(X2); p <- ncol(X1)
  xbar1 <- colMeans(X1); xbar2 <- colMeans(X2)
  S1 <- cov(X1); S2 <- cov(X2)
  Sp <- ((n1 - 1) * S1 + (n2 - 1) * S2) / (n1 + n2 - 2)
  d <- xbar1 - xbar2
  T2 <- (n1 * n2 / (n1 + n2)) * t(d) %*% solve(Sp) %*% d
  F_stat <- ((n1 + n2 - p - 1) / (p * (n1 + n2 - 2))) * T2
  p_val <- 1 - pf(F_stat, p, n1 + n2 - p - 1)
  c(T2 = T2, F = F_stat, p_value = p_val)
}

round(hotelling_two_sample(Grup1, Grup2), 4)

##      T2       F p_value 
## 26.3814 12.5312  0.0003

6.4 4.4 Uji T² Hotelling Berpasangan

Sebelum <- data.frame(
  Tinggi = c(165,170,158,172,163,168,175,160,167,173,162,170,164,166,171),
  Berat  = c(62,68,55,72,60,65,73,58,64,70,61,67,63,66,69)
)
Sesudah <- data.frame(
  Tinggi = Sebelum$Tinggi + c(1,2,1,1,0,2,1,1,2,1,1,2,0,1,1),
  Berat  = Sebelum$Berat  + c(-2,-3,-1,-3,-2,-2,-3,-1,-2,-3,-1,-2,-2,-2,-3)
)

D <- Sesudah - Sebelum
d_bar <- colMeans(D)
S_D <- cov(D)
n_d <- nrow(D)
p_d <- ncol(D)

T2_pair <- n_d * t(d_bar) %*% solve(S_D) %*% d_bar
F_pair <- ((n_d - p_d) / (p_d * (n_d - 1))) * T2_pair
pval_pair <- 1 - pf(F_pair, df1 = p_d, df2 = n_d - p_d)

data.frame(
  Statistik = c("T2", "F", "p-value"),
  Nilai = c(round(T2_pair, 4), round(F_pair, 4), round(pval_pair, 4))
)

7 5. MANOVA Satu Arah

7.1 5.1 Simulasi Data dan Fit Model

set.seed(789)
n_per <- 10
Kelompok <- factor(rep(c("A", "B", "C"), each = n_per))

Y1 <- c(rnorm(n_per, 50, 5), rnorm(n_per, 55, 5), rnorm(n_per, 60, 5))
Y2 <- c(rnorm(n_per, 30, 4), rnorm(n_per, 33, 4), rnorm(n_per, 36, 4))
Y3 <- c(rnorm(n_per, 20, 3), rnorm(n_per, 22, 3), rnorm(n_per, 24, 3))

data_manova <- data.frame(Y1, Y2, Y3, Kelompok)
fit_manova <- manova(cbind(Y1, Y2, Y3) ~ Kelompok, data = data_manova)

7.2 5.2 Ringkasan Uji MANOVA

summary(fit_manova, test = "Wilks")

##           Df  Wilks approx F num Df den Df    Pr(>F)    
## Kelompok   2 0.2834   7.3205      6     50 1.197e-05 ***
## Residuals 27                                            
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(fit_manova, test = "Pillai")

##           Df  Pillai approx F num Df den Df    Pr(>F)    
## Kelompok   2 0.75051   5.2057      6     52 0.0002936 ***
## Residuals 27                                             
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(fit_manova, test = "Hotelling")

##           Df Hotelling-Lawley approx F num Df den Df    Pr(>F)    
## Kelompok   2            2.409   9.6358      6     48 5.975e-07 ***
## Residuals 27                                                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(fit_manova, test = "Roy")

##           Df    Roy approx F num Df den Df    Pr(>F)    
## Kelompok   2 2.3582   20.438      3     26 5.151e-07 ***
## Residuals 27                                            
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary.aov(fit_manova)

##  Response Y1 :
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## Kelompok     2 567.35 283.673  21.163 2.96e-06 ***
## Residuals   27 361.91  13.404                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##  Response Y2 :
##             Df Sum Sq Mean Sq F value    Pr(>F)    
## Kelompok     2 365.83 182.916   10.22 0.0004958 ***
## Residuals   27 483.22  17.897                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##  Response Y3 :
##             Df  Sum Sq Mean Sq F value  Pr(>F)  
## Kelompok     2  70.441  35.220  3.6418 0.03979 *
## Residuals   27 261.123   9.671                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Pembahasan:

Jika MANOVA signifikan, ada perbedaan multivariat antar kelompok.
ANOVA lanjutan membantu melihat variabel respon mana yang paling berkontribusi.

8 6. Studi Kasus Program Remedial

8.1 6.1 Data Sebelum-Sesudah dan Eksplorasi

set.seed(2024)
n_sc <- 20
Sebelum_SC <- data.frame(
  Aljabar = round(rnorm(n_sc, mean = 58, sd = 8)),
  Kalkulus = round(rnorm(n_sc, mean = 55, sd = 9)),
  Statistika = round(rnorm(n_sc, mean = 60, sd = 7))
)
Sesudah_SC <- data.frame(
  Aljabar = round(Sebelum_SC$Aljabar + rnorm(n_sc, mean = 9, sd = 5)),
  Kalkulus = round(Sebelum_SC$Kalkulus + rnorm(n_sc, mean = 8, sd = 6)),
  Statistika = round(Sebelum_SC$Statistika + rnorm(n_sc, mean = 7, sd = 4))
)

colMeans(Sebelum_SC)

##    Aljabar   Kalkulus Statistika 
##      55.65      54.15      60.20

colMeans(Sesudah_SC)

##    Aljabar   Kalkulus Statistika 
##      65.30      60.90      67.65

pairs(
  Sebelum_SC,
  main = "Scatterplot Matrix Nilai Sebelum Remedial",
  pch = 19,
  col = "steelblue"
)

8.2 6.2 Uji T² Hotelling pada Selisih

D_SC <- Sesudah_SC - Sebelum_SC
d_bar_SC <- colMeans(D_SC)
S_D_SC <- cov(D_SC)
n_SC <- nrow(D_SC)
p_SC <- ncol(D_SC)

T2_SC <- n_SC * t(d_bar_SC) %*% solve(S_D_SC) %*% d_bar_SC
F_SC <- ((n_SC - p_SC) / (p_SC * (n_SC - 1))) * T2_SC
pval_SC <- 1 - pf(F_SC, df1 = p_SC, df2 = n_SC - p_SC)

data.frame(
  Statistik = c("T2", "F", "p-value"),
  Nilai = c(round(T2_SC, 4), round(F_SC, 4), round(pval_SC, 4))
)

HotellingsT2Test(as.matrix(D_SC), mu = rep(0, p_SC))

## 
##  Hotelling's one sample T2-test
## 
## data:  as.matrix(D_SC)
## T.2 = 40.157, df1 = 3, df2 = 17, p-value = 6.225e-08
## alternative hypothesis: true location is not equal to c(0,0,0)

Interpretasi:

H0: tidak ada peningkatan rata-rata multivariat (vektor selisih = 0).
Jika p-value kecil, program remedial efektif secara simultan pada semua mata kuliah.

9 7. Kesimpulan

Kesimpulan praktikum (isi sesuai hasil knit):

R mendukung komputasi multivariat dari tahap dasar hingga inferensi.
Data contoh menunjukkan alur lengkap: deskriptif -> asumsi -> uji hipotesis.
Uji T² Hotelling dan MANOVA memberikan keputusan berbasis banyak variabel sekaligus.
Pada studi kasus remedial, keputusan akhir didasarkan pada p-value uji multivariat.

10 Lampiran (Opsional): Latihan Dasar

x <- c(3,7,2,9,4,6,1,8,5,10)
n <- length(x)
mean_manual <- sum(x) / n
var_manual <- sum((x - mean_manual)^2) / (n - 1)
sd_manual <- sqrt(var_manual)

data.frame(
  mean_builtin = mean(x),
  mean_manual = mean_manual,
  var_builtin = var(x),
  var_manual = var_manual,
  sd_builtin = sd(x),
  sd_manual = sd_manual
)

Praktikum 1 - Metode Statistika Lanjut

03 April 2026

1 Pendahuluan

2 Persiapan Package

3 1. Pengenalan Dasar R

3.1 1.1 Operasi Aritmetika

3.2 1.2 Assignment dan Vektor

3.3 1.3 Matriks dan Data Frame

4 2. Statistik Deskriptif Multivariat

4.1 2.1 Data Contoh

4.2 2.2 Vektor Mean, Kovariansi, dan Korelasi

4.3 2.3 Kombinasi Linier

5 3. Normalitas Multivariat

5.1 3.1 Jarak Mahalanobis dan Q-Q Plot Chi-square

5.2 3.2 Uji Mardia dan Henze-Zirkler

6 4. Pengujian Vektor Mean

6.1 4.1 Uji Z² (Sigma Diketahui)

6.2 4.2 Uji T² Hotelling (Satu Sampel)

6.3 4.3 Uji T² Hotelling Dua Sampel Tidak Berpasangan

6.4 4.4 Uji T² Hotelling Berpasangan

7 5. MANOVA Satu Arah

7.1 5.1 Simulasi Data dan Fit Model

7.2 5.2 Ringkasan Uji MANOVA

8 6. Studi Kasus Program Remedial

8.1 6.1 Data Sebelum-Sesudah dan Eksplorasi

8.2 6.2 Uji T² Hotelling pada Selisih

9 7. Kesimpulan

10 Lampiran (Opsional): Latihan Dasar