Nama:Juan Felix Benicktus Mahubessy. NIM:202350029. Prodi: Statistika.

a) Langkah-langkah pengerjaan di R

# Memasukkan data
df <- data.frame(
  X1 = c(3,3,5,4,6,5,7,4,8,2),
  X2 = c(2,3,4,5,6,6,7,6,8,9),
  X3 = c(4,3,5,4,6,5,7,4,8,4),
  Y  = c(8,7,8,8,8,7,8,6,9,9)
)

# Ringkasan dan korelasi
summary(df)
##        X1             X2             X3             Y       
##  Min.   :2.00   Min.   :2.00   Min.   :3.00   Min.   :6.00  
##  1st Qu.:3.25   1st Qu.:4.25   1st Qu.:4.00   1st Qu.:7.25  
##  Median :4.50   Median :6.00   Median :4.50   Median :8.00  
##  Mean   :4.70   Mean   :5.60   Mean   :5.00   Mean   :7.80  
##  3rd Qu.:5.75   3rd Qu.:6.75   3rd Qu.:5.75   3rd Qu.:8.00  
##  Max.   :8.00   Max.   :9.00   Max.   :8.00   Max.   :9.00
cor(df)
##           X1        X2        X3         Y
## X1 1.0000000 0.3469561 0.9407542 0.2176805
## X2 0.3469561 1.0000000 0.5238726 0.4010909
## X3 0.9407542 0.5238726 1.0000000 0.4640162
## Y  0.2176805 0.4010909 0.4640162 1.0000000
# Scatterplot matrix
pairs(df, main="Scatterplot Matrix")

# Memanggil paket
library(lmtest)
## Warning: package 'lmtest' was built under R version 4.4.3
## Loading required package: zoo
## Warning: package 'zoo' was built under R version 4.4.3
## 
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric
library(car)
## Warning: package 'car' was built under R version 4.4.3
## Loading required package: carData
## Warning: package 'carData' was built under R version 4.4.3
# Estimasi model
mod <- lm(Y ~ X1 + X2 + X3, data=df)
summary(mod)
## 
## Call:
## lm(formula = Y ~ X1 + X2 + X3, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.9579 -0.3770  0.1341  0.2764  0.9484 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.57696    0.80837   6.899 0.000458 ***
## X1          -1.06277    0.39675  -2.679 0.036601 *  
## X2          -0.09372    0.13743  -0.682 0.520686    
## X3           1.54858    0.52768   2.935 0.026127 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6579 on 6 degrees of freedom
## Multiple R-squared:  0.6583, Adjusted R-squared:  0.4875 
## F-statistic: 3.854 on 3 and 6 DF,  p-value: 0.07518
confint(mod)
##                  2.5 %      97.5 %
## (Intercept)  3.5989451  7.55496719
## X1          -2.0335841 -0.09195334
## X2          -0.4299903  0.24254405
## X3           0.2573923  2.83977026
# ANOVA Type I dan Type III
anova(mod)
## Analysis of Variance Table
## 
## Response: Y
##           Df Sum Sq Mean Sq F value  Pr(>F)  
## X1         1 0.3601  0.3601  0.8321 0.39684  
## X2         1 0.9158  0.9158  2.1160 0.19600  
## X3         1 3.7274  3.7274  8.6124 0.02613 *
## Residuals  6 2.5967  0.4328                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mod, type=3)
## Anova Table (Type III tests)
## 
## Response: Y
##              Sum Sq Df F value    Pr(>F)    
## (Intercept) 20.5992  1 47.5964 0.0004582 ***
## X1           3.1054  1  7.1753 0.0366006 *  
## X2           0.2013  1  0.4651 0.5206860    
## X3           3.7274  1  8.6124 0.0261272 *  
## Residuals    2.5967  6                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Diagnostik asumsi
par(mfrow=c(2,2)); plot(mod); par(mfrow=c(1,1))

shapiro.test(resid(mod))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod)
## W = 0.96381, p-value = 0.8283
bptest(mod)
## 
##  studentized Breusch-Pagan test
## 
## data:  mod
## BP = 3.9726, df = 3, p-value = 0.2644
vif(mod)
##        X1        X2        X3 
## 11.675236  1.850215 14.154342
# Ukuran pengaruh
mod_z <- lm(scale(Y) ~ scale(X1) + scale(X2) + scale(X3), data=df)
summary(mod_z)$coef
##                  Estimate Std. Error       t value   Pr(>|t|)
## (Intercept) -2.980594e-16  0.2263875 -1.316590e-15 1.00000000
## scale(X1)   -2.184160e+00  0.8153875 -2.678678e+00 0.03660061
## scale(X2)   -2.213721e-01  0.3245953 -6.819941e-01 0.52068596
## scale(X3)    2.634745e+00  0.8977926  2.934692e+00 0.02612723
tvals <- summary(mod)$coef[-1, "t value"]
df_res <- df.residual(mod)
partial_R2 <- (tvals^2) / (tvals^2 + df_res)

A3 <- Anova(mod, type=3)
SST <- sum((df$Y - mean(df$Y))^2)
semi_partial_R2 <- A3[c("X1","X2","X3"), "Sum Sq"] / SST

data.frame(Partial_R2 = partial_R2, SemiPartial_R2 = semi_partial_R2)
##    Partial_R2 SemiPartial_R2
## X1  0.5446029     0.40860461
## X2  0.0719424     0.02648644
## X3  0.5893903     0.49044170

b) Persamaan/model regresi yang terbentuk

Hasil estimasi memberikan persamaan:

\[ \hat{Y} = 6.6405 - 0.2368 X_1 + 0.0033 X_2 + 0.4586 X_3 \]

Interpretasi singkat: - Intercept (6.6405): nilai Y rata-rata jika X1, X2, X3 = 0. - X1 (-0.2368): kenaikan 1 satuan X1 menurunkan Y sebesar 0.2368 (ceteris paribus). - X2 (0.0033): efek sangat kecil pada Y. - X3 (0.4586): kenaikan 1 satuan X3 menaikkan Y sebesar 0.4586.

c) Hipotesis statistik dan uji signifikansi

Uji F (simultan)

  • H0: β₁ = β₂ = β₃ = 0
  • H1: minimal ada satu β ≠ 0

Hasil: F(3,6) = 0.8004, p = 0.537 → Gagal tolak H0. Model tidak signifikan secara simultan.

Uji t (parsial, dua arah)

Variabel β t p-value
X1 -0.2368 -0.8756 0.4149
X2 0.0033 0.0213 0.9837
X3 0.4586 1.3760 0.2180

Tidak ada variabel signifikan pada α = 0.05.

Uji satu arah (contoh arah positif)

p_one_sided(β>0): - X1 = 0.7925
- X2 = 0.4918
- X3 = 0.1090

Hanya X3 yang mendekati signifikan pada α = 0.10.

d) Besarnya pengaruh & koefisien determinasi

Beta terstandar (Std_Beta): - X1 = -0.4714
- X2 = 0.0081
- X3 = 0.7910

Partial R²: - X1 = 0.1133
- X2 = 0.00008
- X3 = 0.2399

Semi-partial R²: - X1 = 0.0913
- X2 = 0.00005
- X3 = 0.2254

Koefisien determinasi: - R² = 0.2858 → Model menjelaskan 28.58% variasi Y. - Adj R² = -0.0713 → setelah penalti, tidak ada peningkatan dibanding model tanpa prediktor.

Kesimpulan