Koefisien Regresi

Pertemuan 6

Deden Istiawan

Itesa Muhammadiyah
ASE 4335 - Ganjil

2025-04-23

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Variables Independent dan Dependent

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Persamaan Model Regresi

  • Model statistik untuk menjelaskan pengaruh antara satu variabel independen (X) dan satu variabel dependen (Y).

  • Persamaan model:

    \[ Y = \beta_0 + \beta_1 X_1 \]

Tujuan Estimasi OLS

Mencari nilai \(\hat{\beta}_0\) dan \(\hat{\beta}_1\) yang meminimalkan jumlah kuadrat selisih (residual) antara nilai aktual dan prediksi:

\[ \min \sum_{i=1}^n (Y_i - \hat{Y}_i)^2 \]

Error

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Error

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Rumus Koefisien Regresi

  • Slope (kemiringan):

\[ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]

  • Intercept:

\[ \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} \]

Contoh Kasus

Misal data:

X (Jam Belajar) Y (Nilai Ujian)
2 65
3 70
5 75
7 85
9 95

Langkah 1: Hitung Rataan

  • \(\bar{X} = \frac{2 + 3 + 5 + 7 + 9}{5} = 5.2\)
  • \(\bar{Y} = \frac{65 + 70 + 75 + 85 + 95}{5} = 78\)

Langkah 2: Hitung \(\hat{\beta}_1\)

Gunakan:

\[ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]

Hitung:

X Y \(X_i - \bar{X}\) \(Y_i - \bar{Y}\) \((X_i - \bar{X})(Y_i - \bar{Y})\) \((X_i - \bar{X})^2\)
2 65 -3.2 -13 41.6 10.24
3 70 -2.2 -8 17.6 4.84
5 75 -0.2 -3 0.6 0.04
7 85 1.8 7 12.6 3.24
9 95 3.8 17 64.6 14.44

Hitung \(\hat{\beta}_1\)

\[ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]

  • \(\sum (X_i - \bar{X})(Y_i - \bar{Y}) = 137\)
  • \(\sum (X_i - \bar{X})^2 = 32.8\)

\[ \hat{\beta}_1 = \frac{137}{32.8} = 4.18 \]

Langkah 3: Hitung \(\hat{\beta}_0\)

\[ \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} = 78 - (4.18)(5.2) = 56.26 \]

Persamaan Regresi

\[ \hat{Y} = 56.26 + 4.18 X \]

Misal ingin prediksi nilai ujian jika belajar 6 jam:

\[ \hat{Y} = 56.26 + 4.18(6) = 81.34 \]

Dataset

data <- data.frame(
  X = c(2, 3, 5, 7, 9),
  Y = c(65, 70, 75, 85, 95)
)

Executable Code

Scatter Plot

library(ggplot2)

ggplot(data, aes(x = X, y = Y)) +
  geom_point(size = 3, color = "darkblue") +
  labs(title = "Jam Belajar vs Nilai Ujian", x = "Jam Belajar", y = "Nilai Ujian")

model <- lm(Y ~ X, data = data)
summary(model)

Call:
lm(formula = Y ~ X, data = data)

Residuals:
      1       2       3       4       5 
 0.3659  1.1890 -2.1646 -0.5183  1.1280 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  56.2805     1.6293   34.54 5.33e-05 ***
X             4.1768     0.2811   14.86 0.000661 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.61 on 3 degrees of freedom
Multiple R-squared:  0.9866,    Adjusted R-squared:  0.9821 
F-statistic: 220.8 on 1 and 3 DF,  p-value: 0.0006613

Visualisasi Regresi

ggplot(data, aes(x = X, y = Y)) +
  geom_point(size = 3, color = "darkred") +
  geom_smooth(method = "lm", se = FALSE, color = "steelblue") +
  labs(title = "Garis Regresi Linier", x = "Jam Belajar", y = "Nilai Ujian") +
  theme_minimal()