Perbandingan Dua Vektor Nilai Tengah Sampel Berpasangan

Input data

library(MASS)
set.seed(105)

n_daring <- 20
n_luring <- 20

mu_daring <- c(60, 55)
mu_luring <- c(65, 60)
Sigma <- matrix(c(200,50,50,180), 2)

data_daring <- mvrnorm(n_daring, mu = mu_daring, Sigma = Sigma)
data_luring <- mvrnorm(n_luring, mu = mu_luring, Sigma = Sigma)

dataset <- data.frame(
  Daring_Teori = data_daring[,1],
  Daring_Praktik = data_daring[,2],
  Luring_Teori = data_luring[,1],
  Luring_Praktik = data_luring[,2]
)
dataset

##    Daring_Teori Daring_Praktik Luring_Teori Luring_Praktik
## 1      69.10082       75.57203     82.92667       66.54077
## 2      64.76782       60.59846     71.48184       50.42095
## 3      75.19226       33.19184     42.15755       61.56826
## 4      33.12650       56.08377     67.27187       51.00631
## 5      67.66235       42.76799     86.96436       49.35985
## 6      54.53408       51.76298     49.26123       48.99604
## 7      60.63178       48.37937     68.14539       65.54106
## 8      65.31747       68.40494     51.38198       49.66886
## 9      55.39224       51.69868     64.79917       34.83578
## 10     64.87338       57.19761     66.79023       83.93738
## 11     81.13363       56.79122     63.67320       66.45155
## 12     70.54292       51.37868     58.92118       67.62025
## 13     60.81948       57.31139     71.99837       53.20297
## 14     57.48305       41.12359     76.67869       51.84916
## 15     55.38135       58.12605     52.98789       43.60255
## 16     54.59488       63.15775     65.60093       73.75354
## 17     70.54695       78.95870     90.02358       65.05747
## 18     59.33022       74.05079     58.61602       78.22222
## 19     71.87210       53.65545     48.33564       48.26555
## 20     65.01429       63.53392     79.94754       65.62850

Memisahkan Data dari Populasi 1 dan 2

data_daring <- dataset[,1:2]
data_luring <- dataset[,3:4]

Menghitung vektor rataan dan matriks kovarians

xbar1 = apply(data_daring, 2, mean)
xbar1

##   Daring_Teori Daring_Praktik 
##       62.86588       57.18726

xbar2 = apply(data_luring, 2, mean)
xbar2

##   Luring_Teori Luring_Praktik 
##       65.89817       58.77645

cov_m1 = cov(data_daring)
cov_m1

##                Daring_Teori Daring_Praktik
## Daring_Teori    102.6125993     -0.5359109
## Daring_Praktik   -0.5359109    133.5326911

cov_m2 = cov(data_luring)
cov_m2

##                Luring_Teori Luring_Praktik
## Luring_Teori      175.77439       24.41551
## Luring_Praktik     24.41551      155.96229

n1 = nrow(data_daring)
n2 = nrow(data_luring)

# S Gabungan
s_gab = ((n1-1)*cov_m1 + (n2-1)*cov_m2)/(n1+n2-2)
s_gab

##                Daring_Teori Daring_Praktik
## Daring_Teori       139.1935        11.9398
## Daring_Praktik      11.9398       144.7475

Uji T2 Hotelling Dua Populasi Sampel Berpasangan

library(Hotelling)

## Loading required package: corpcor

t2_homogen = hotelling.test(data_daring, data_luring, var.equal=TRUE)
t2_homogen

## Test stat:  0.78348 
## Numerator df:  2 
## Denominator df:  37 
## P-value:  0.6855

Selang Kepercayaan Simultan

T.ci = function(mu1, mu2, S_gab, n1, n2, avec=rep(1,length(mu1)), level=0.95){
  p = length(mu1)
  mu = mu1 - mu2
  cval = qf(level, p, n1+n2-p-1) * p * (n1+n2-2) / (n1+n2-p-1)
  zhat = crossprod(avec, mu)
  zvar = crossprod(avec, S_gab %*% avec) * (1/n1 + 1/n2)
  const = sqrt(cval * zvar)
  c(lower = zhat - const, upper = zhat + const)
}

# Teori
T.ci(xbar1, xbar2, s_gab, n1, n2, avec=c(1,0), level=0.95)

##      lower      upper 
## -12.674698   6.610122

# Praktik
T.ci(xbar1, xbar2, s_gab, n1, n2, avec=c(0,1), level=0.95)

##     lower     upper 
## -11.42209   8.24371

Selang Kepercayaan Bonferroni

bon = function(mu1, mu2 ,S, n1, n2, alpha, k){
  p = length(mu1)
  mu = mu1 - mu2
  lower = mu[k] - sqrt((S[k,k]) *(1/n1+1/n2))* abs(qt(alpha/(2*p), df=n1+n2-2))
  upper = mu[k] + sqrt((S[k,k]) *(1/n1+1/n2))* abs(qt(alpha/(2*p), df=n1+n2-2))
  ci = c(lower = lower, upper = upper)
  names(ci)= c("lower","upper")
  ci
}

# Teori
bon(xbar1, xbar2, s_gab, n1, n2, 0.05, 1)

##      lower      upper 
## -11.739085   5.674509

# Praktik
bon(xbar1, xbar2, s_gab, n1, n2, 0.05, 2)

##      lower      upper 
## -10.467994   7.289614

Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Sama

Input data

library(MASS)
set.seed(105)

n1 <- 20
n2 <- 20
mu1 <- c(60, 55)
mu2 <- c(65, 60)
Sigma <- matrix(c(200,50,50,180), 2)  # sama untuk dua populasi

data1 <- mvrnorm(n1, mu = mu1, Sigma = Sigma)
data2 <- mvrnorm(n2, mu = mu2, Sigma = Sigma)

dataset1 <- data.frame(
  Pop1_X1 = data1[,1],
  Pop1_X2 = data1[,2],
  Pop2_X1 = data2[,1],
  Pop2_X2 = data2[,2]
)
dataset1

##     Pop1_X1  Pop1_X2  Pop2_X1  Pop2_X2
## 1  69.10082 75.57203 82.92667 66.54077
## 2  64.76782 60.59846 71.48184 50.42095
## 3  75.19226 33.19184 42.15755 61.56826
## 4  33.12650 56.08377 67.27187 51.00631
## 5  67.66235 42.76799 86.96436 49.35985
## 6  54.53408 51.76298 49.26123 48.99604
## 7  60.63178 48.37937 68.14539 65.54106
## 8  65.31747 68.40494 51.38198 49.66886
## 9  55.39224 51.69868 64.79917 34.83578
## 10 64.87338 57.19761 66.79023 83.93738
## 11 81.13363 56.79122 63.67320 66.45155
## 12 70.54292 51.37868 58.92118 67.62025
## 13 60.81948 57.31139 71.99837 53.20297
## 14 57.48305 41.12359 76.67869 51.84916
## 15 55.38135 58.12605 52.98789 43.60255
## 16 54.59488 63.15775 65.60093 73.75354
## 17 70.54695 78.95870 90.02358 65.05747
## 18 59.33022 74.05079 58.61602 78.22222
## 19 71.87210 53.65545 48.33564 48.26555
## 20 65.01429 63.53392 79.94754 65.62850

Memisahkan data dari Populasi 1 dan 2

pop1 <- dataset1[,1:2]
pop2 <- dataset1[,3:4]

Menghitung vektor rataan dan matriks kovarians

xbar1 <- colMeans(pop1); xbar1

##  Pop1_X1  Pop1_X2 
## 62.86588 57.18726

xbar2 <- colMeans(pop2); xbar2

##  Pop2_X1  Pop2_X2 
## 65.89817 58.77645

cov1 <- cov(pop1); cov1

##             Pop1_X1     Pop1_X2
## Pop1_X1 102.6125993  -0.5359109
## Pop1_X2  -0.5359109 133.5326911

cov2 <- cov(pop2); cov2

##           Pop2_X1   Pop2_X2
## Pop2_X1 175.77439  24.41551
## Pop2_X2  24.41551 155.96229

n1 <- nrow(pop1)
n2 <- nrow(pop2)

# S Gabungan
s_gab <- ((n1-1)*cov1 + (n2-1)*cov2)/(n1+n2-2)
s_gab

##          Pop1_X1  Pop1_X2
## Pop1_X1 139.1935  11.9398
## Pop1_X2  11.9398 144.7475

Uji T2 Hotelling Dua Populasi Ragam Sama

library(Hotelling)
t2_homogen <- hotelling.test(pop1, pop2, var.equal=TRUE)
t2_homogen

## Test stat:  0.78348 
## Numerator df:  2 
## Denominator df:  37 
## P-value:  0.6855

Selang Kepercayaan Simultan

T.ci = function(mu1, mu2, S_gab, n1, n2, avec=rep(1,length(mu1)), level=0.95){
  p = length(mu1)
  mu = mu1 - mu2
  cval = qf(level, p, n1+n2-p-1) * p * (n1+n2-2) / (n1+n2-p-1)
  zhat = crossprod(avec, mu)
  zvar = crossprod(avec, S_gab %*% avec) * (1/n1 + 1/n2)
  const = sqrt(cval * zvar)
  c(lower = zhat - const, upper = zhat + const)
}

# X1
T.ci(xbar1, xbar2, s_gab, n1, n2, avec=c(1,0), level=0.95)

##      lower      upper 
## -12.674698   6.610122

# X2
T.ci(xbar1, xbar2, s_gab, n1, n2, avec=c(0,1), level=0.95)

##     lower     upper 
## -11.42209   8.24371

Selang Kepercayaan Bonferroni

bon = function(mu1, mu2 ,S, n1, n2, alpha, k){
  p = length(mu1)
  mu = mu1 - mu2
  lower = mu[k] - sqrt((S[k,k]) *(1/n1+1/n2))* abs(qt(alpha/(2*p), df=n1+n2-2))
  upper = mu[k] + sqrt((S[k,k]) *(1/n1+1/n2))* abs(qt(alpha/(2*p), df=n1+n2-2))
  ci = c(lower = lower, upper = upper)
  names(ci)= c("lower","upper")
  ci
}

# X1
bon(xbar1, xbar2, s_gab, n1, n2, 0.05, 1)

##      lower      upper 
## -11.739085   5.674509

# X2
bon(xbar1, xbar2, s_gab, n1, n2, 0.05, 2)

##      lower      upper 
## -10.467994   7.289614

Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Tidak Sama

Input data

set.seed(105)

n1 <- 20
n2 <- 20   # samain jumlah sampel biar bisa digabung
mu1 <- c(50, 55)
mu2 <- c(60, 65)

Sigma1 <- matrix(c(150,20,20,100), 2)   # beda
Sigma2 <- matrix(c(250,80,80,220), 2)

data1 <- mvrnorm(n1, mu = mu1, Sigma = Sigma1)
data2 <- mvrnorm(n2, mu = mu2, Sigma = Sigma2)

dataset2 <- data.frame(
  Pop1_X1 = data1[,1],
  Pop1_X2 = data1[,2],
  Pop2_X1 = data2[,1],
  Pop2_X2 = data2[,2]
)
dataset2

##     Pop1_X1  Pop1_X2  Pop2_X1  Pop2_X2
## 1  62.54998 68.19037 80.11398 73.07736
## 2  55.15814 57.94200 66.79697 54.90775
## 3  55.59097 33.99569 34.76487 65.49644
## 4  29.85452 63.42660 62.15782 55.32208
## 5  52.48262 43.46163 83.90131 54.57329
## 6  44.95678 54.06473 42.13129 52.17617
## 7  48.66959 49.74852 63.70417 71.20380
## 8  57.71161 63.76817 44.50679 53.02175
## 9  45.59196 53.77289 58.77526 37.57334
## 10 54.30818 55.30634 62.93611 91.17438
## 11 66.56574 50.39876 58.78757 71.95845
## 12 57.02914 49.24514 53.57135 72.97956
## 13 51.25561 56.53941 67.47983 57.96613
## 14 44.28970 45.07897 72.60928 56.73954
## 15 47.34182 58.70079 46.04367 46.49781
## 16 48.11993 62.77852 61.21334 80.01615
## 17 64.57640 70.37655 87.91460 71.83793
## 18 54.70163 69.78655 53.65568 84.51404
## 19 58.66300 50.61395 41.07712 51.33121
## 20 56.14859 60.12156 76.77832 71.92536

Memisahkan data dari populasi 1 dan 2

pop1 <- dataset2[,1:2]
pop2 <- dataset2[,3:4]

Menghitung vektor rataan dan matriks kovarians

xbar1 <- colMeans(pop1); xbar1

##  Pop1_X1  Pop1_X2 
## 52.77829 55.86586

xbar2 <- colMeans(pop2); xbar2

##  Pop2_X1  Pop2_X2 
## 60.94597 63.71463

cov1 <- cov(pop1); cov1

##           Pop1_X1   Pop1_X2
## Pop1_X1 69.142296  5.653242
## Pop1_X2  5.653242 86.828349

cov2 <- cov(pop2); cov2

##           Pop2_X1   Pop2_X2
## Pop2_X1 217.99157  46.60925
## Pop2_X2  46.60925 188.44134

Uji T2 Hotelling Dua Populasi Ragam Tidak Sama

t2_heterogen <- hotelling.test(pop1, pop2, var.equal=FALSE)
t2_heterogen

## Test stat:  7.6928 
## Numerator df:  2 
## Denominator df:  31.5259166090171 
## P-value:  0.03473

Tugas Data Bangkitan

Agustin Salsabilla

2025-09-30

Perbandingan Dua Vektor Nilai Tengah Sampel Berpasangan

Input data

Memisahkan Data dari Populasi 1 dan 2

Menghitung vektor rataan dan matriks kovarians

Uji T2 Hotelling Dua Populasi Sampel Berpasangan

Selang Kepercayaan Simultan

Selang Kepercayaan Bonferroni

Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Sama

Input data

Memisahkan data dari Populasi 1 dan 2

Menghitung vektor rataan dan matriks kovarians

Uji T2 Hotelling Dua Populasi Ragam Sama

Selang Kepercayaan Simultan

Selang Kepercayaan Bonferroni

Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Tidak Sama

Input data

Memisahkan data dari populasi 1 dan 2

Menghitung vektor rataan dan matriks kovarians

Uji T2 Hotelling Dua Populasi Ragam Tidak Sama