Visualisasi SST, SSR, dan SSE dalam Analisis Regresi Linier

📘 Pendahuluan

Dalam analisis regresi linier, variasi total dalam data (Total Sum of Squares, SST) dapat dipecah menjadi:

• SSR (Regression Sum of Squares): Variasi yang bisa dijelaskan oleh model.
• SSE (Error Sum of Squares): Variasi yang tidak bisa dijelaskan (galat).

Secara matematis:

\[ SST = SSR + SSE \]

Pemahaman ini sangat penting untuk mengukur seberapa baik model regresi menjelaskan data.

---

🧪 Simulasi Data

# Atur seed agar hasil replikasi tetap
set.seed(123)

# Buat data
x <- 1:10
y <- 2 + 1.5 * x + rnorm(10, mean = 0, sd = 2)  # Model linier + noise

# Buat model regresi
model <- lm(y ~ x)

# Tampilkan ringkasan model
summary(model)

## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.2695 -1.1248 -0.2785  0.7707  3.3628 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   3.0509     1.3346   2.286 0.051577 .  
## x             1.3361     0.2151   6.212 0.000256 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.954 on 8 degrees of freedom
## Multiple R-squared:  0.8283, Adjusted R-squared:  0.8068 
## F-statistic: 38.59 on 1 and 8 DF,  p-value: 0.0002561

# Nilai rata-rata y
y_mean <- mean(y)

# Nilai prediksi dari model
y_hat <- fitted(model)

# Hitung SSE, SSR, SST
SSE <- sum((y - y_hat)^2)
SSR <- sum((y_hat - y_mean)^2)
SST <- sum((y - y_mean)^2)

# Tampilkan hasilnya
data.frame(
  Komponen = c("SSE", "SSR", "SST", "SSE + SSR"),
  Nilai = round(c(SSE, SSR, SST, SSE + SSR), 2)
)

##    Komponen  Nilai
## 1       SSE  30.53
## 2       SSR 147.27
## 3       SST 177.80
## 4 SSE + SSR 177.80

# Buat plot data asli dan model regresi
plot(x, y, pch = 16, col = "blue",
     main = "Visualisasi SST, SSR, dan SSE",
     xlab = "x", ylab = "y")

# Tambahkan garis regresi dan rata-rata y
abline(model, col = "red", lwd = 2)              # Garis regresi
abline(h = y_mean, col = "darkgreen", lty = 2)   # Garis rata-rata

# Tambahkan garis untuk setiap komponen
segments(x, y, x, y_hat, col = "orange", lwd = 2)     # SSE
segments(x, y_hat, x, y_mean, col = "purple", lwd = 2) # SSR
segments(x, y, x, y_mean, col = "grey", lty = 3)       # SST

# Tambahkan legenda
legend("topleft",
       legend = c("Garis Regresi", "Rata-rata y", "SSE", "SSR", "SST"),
       col = c("red", "darkgreen", "orange", "purple", "grey"),
       lty = c(1, 2, 1, 1, 3), lwd = 2)