1 Deskripsi Data

Data berasal dari desain faktorial 2 × 2 dengan:

Simbol	Keterangan
Faktor A	2 level: `l1`, `l2`
Faktor B	2 level: `f1`, `f2`
n	4 observasi per sel
p	3 variabel dependen: `X1`, `X2`, `X3`

2 Input Data

data <- data.frame(
  A = factor(rep(c("l1","l1","l2","l2"), each = 4)),
  B = factor(rep(c("f1","f2","f1","f2"), each = 4)),
  X1 = c(1.3, 1.2, 0.7, 1.3,   # A=l1, B=f1
         1.3, 1.4, 1.5, 1.7,   # A=l1, B=f2
         1.3, 1.5, 1.5, 1.8,   # A=l2, B=f1
         1.3, 1.4, 1.5, 1.6),  # A=l2, B=f2
  X2 = c(1.2, 1.1, 1.9, 2.1,
         1.3, 1.9, 1.2, 2.1,
         1.2, 1.7, 1.5, 2.1,
         1.7, 1.8, 1.9, 2.0),
  X3 = c(1.0, 1.3, 1.4, 1.5,
         0.2, 0.7, 0.9, 0.9,
         1.3, 2.1, 0.5, 1.3,
         1.9, 1.6, 1.8, 2.7)
)

knitr::kable(data, caption = "Data Two-Way MANOVA", align = "c")

Data Two-Way MANOVA
A	B	X1	X2	X3
l1	f1	1.3	1.2	1.0
l1	f1	1.2	1.1	1.3
l1	f1	0.7	1.9	1.4
l1	f1	1.3	2.1	1.5
l1	f2	1.3	1.3	0.2
l1	f2	1.4	1.9	0.7
l1	f2	1.5	1.2	0.9
l1	f2	1.7	2.1	0.9
l2	f1	1.3	1.2	1.3
l2	f1	1.5	1.7	2.1
l2	f1	1.5	1.5	0.5
l2	f1	1.8	2.1	1.3
l2	f2	1.3	1.7	1.9
l2	f2	1.4	1.8	1.6
l2	f2	1.5	1.9	1.8
l2	f2	1.6	2.0	2.7

3 Helper Function

# Fungsi menghitung SSCP matrix (sum of squares and cross products)
sscp <- function(mat) {
  centered <- scale(mat, scale = FALSE)
  t(centered) %*% centered
}

# Fungsi untuk membulatkan hanya kolom numerik dari hasil aggregate
round_agg <- function(df_agg, digits = 4) {
  num_cols <- sapply(df_agg, is.numeric)
  df_agg[, num_cols] <- round(df_agg[, num_cols], digits)
  return(df_agg)
}

4 Grand Mean & Marginal Means

Y       <- as.matrix(data[, c("X1","X2","X3")])
n_cell  <- 4   # observasi per sel
p       <- 3   # jumlah variabel dependen
g       <- 2   # level Faktor A
b       <- 2   # level Faktor B

grand_mean <- colMeans(Y)

Grand Mean:

Grand Mean (x̄)
	X1	X2	X3
round(grand_mean, 4)	1.3938	1.6687	1.3188

Marginal Means Faktor A:

Marginal Means Faktor A
A	X1	X2	X3
l1	1.3000	1.6000	0.9875
l2	1.4875	1.7375	1.6500

Marginal Means Faktor B:

Marginal Means Faktor B
B	X1	X2	X3
f1	1.3250	1.6000	1.3000
f2	1.4625	1.7375	1.3375

Cell Means (A × B):

Cell Means (kombinasi A × B)
A	B	X1	X2	X3
l1	f1	1.125	1.575	1.300
l2	f1	1.525	1.625	1.300
l1	f2	1.475	1.625	0.675
l2	f2	1.450	1.850	2.000

5 SSP Total

\[ \text{SSP}_{\text{total}} = \sum_{\ell}\sum_{k}\sum_{r} (\mathbf{x}_{\ell kr} - \bar{\mathbf{x}})(\mathbf{x}_{\ell kr} - \bar{\mathbf{x}})^T \]

SSP_total <- sscp(Y)

SSP Total
	X1	X2	X3
X1	0.9094	0.3669	0.1619
X2	0.3669	1.9944	1.3294
X3	0.1619	1.3294	5.9644

6 SSP Faktor A

\[ \text{SSP}_{\text{fak1}} = \sum_{\ell=1}^{g} bn\,(\bar{\mathbf{x}}_{\ell.} - \bar{\mathbf{x}})(\bar{\mathbf{x}}_{\ell.} - \bar{\mathbf{x}})^T \]

SSP_A <- matrix(0, p, p)
for (la in levels(data$A)) {
  idx  <- data$A == la
  ni   <- sum(idx)
  diff <- colMeans(Y[idx, ]) - grand_mean
  SSP_A <- SSP_A + ni * outer(diff, diff)
}

SSP Faktor A
	X1	X2	X3
X1	0.1406	0.1031	0.4969
X2	0.1031	0.0756	0.3644
X3	0.4969	0.3644	1.7556

7 SSP Faktor B

\[ \text{SSP}_{\text{fak2}} = \sum_{k=1}^{b} gn\,(\bar{\mathbf{x}}_{.k} - \bar{\mathbf{x}})(\bar{\mathbf{x}}_{.k} - \bar{\mathbf{x}})^T \]

SSP_B <- matrix(0, p, p)
for (lb in levels(data$B)) {
  idx  <- data$B == lb
  ni   <- sum(idx)
  diff <- colMeans(Y[idx, ]) - grand_mean
  SSP_B <- SSP_B + ni * outer(diff, diff)
}

SSP Faktor B
	X1	X2	X3
X1	0.0756	0.0756	0.0206
X2	0.0756	0.0756	0.0206
X3	0.0206	0.0206	0.0056

8 SSP Error (Within Cells)

\[ \text{SSP}_{\text{res}} = \sum_{\ell=1}^{g}\sum_{k=1}^{b}\sum_{r=1}^{n} (\mathbf{x}_{\ell kr} - \bar{\mathbf{x}}_{\ell k})(\mathbf{x}_{\ell kr} - \bar{\mathbf{x}}_{\ell k})^T \]

SSP_E <- matrix(0, p, p)
for (la in levels(data$A)) {
  for (lb in levels(data$B)) {
    idx   <- data$A == la & data$B == lb
    Ycell <- Y[idx, ]
    SSP_E <- SSP_E + sscp(Ycell)
  }
}

SSP Error (Residual)
	X1	X2	X3
X1	0.5125	0.2625	0.2075
X2	0.2625	1.8125	0.7125
X3	0.2075	0.7125	2.4475

9 SSP Interaksi (A × B)

\[ \text{SSP}_{\text{int}} = \text{SSP}_{\text{total}} - \text{SSP}_{\text{fak1}} - \text{SSP}_{\text{fak2}} - \text{SSP}_{\text{res}} \]

SSP_AB <- SSP_total - SSP_A - SSP_B - SSP_E

SSP Interaksi (A × B)
	X1	X2	X3
X1	0.1806	-0.0744	-0.5631
X2	-0.0744	0.0306	0.2319
X3	-0.5631	0.2319	1.7556

Verifikasi Dekomposisi:

check     <- SSP_A + SSP_B + SSP_AB + SSP_E
max_selisih <- max(abs(check - SSP_total))
cat("Max selisih (harus ~0):", max_selisih, "\n")

Max selisih (harus ~0): 8.326673e-17

10 Degrees of Freedom

\[ gbn - 1 = (g-1) + (b-1) + (g-1)(b-1) + gb(n-1) \]

df_A  <- g - 1              # 1
df_B  <- b - 1              # 1
df_AB <- (g - 1) * (b - 1) # 1
df_E  <- g * b * (n_cell - 1) # 12
df_T  <- g * b * n_cell - 1   # 15

Degrees of Freedom
Sumber	Formula	df
Faktor A	g − 1	1
Faktor B	b − 1	1
Interaksi A×B	(g−1)(b−1)	1
Error	gb(n−1)	12
Total	gbn − 1	15

11 Uji Wilks’ Lambda

11.1 Formula

\[ \Lambda^* = \frac{|\text{SSP}_{\text{res}}|}{|\text{SSP}_H + \text{SSP}_{\text{res}}|} \]

\[ F = \left(\frac{1 - \Lambda^*}{\Lambda^*}\right) \times \frac{(gb(n-1) - p + 1)/2}{(|df_H - p| + 1)/2} \]

11.2 Fungsi Wilks

wilks_test <- function(SSP_H, SSP_E, df_h, g, b, n, p, label) {

  lam <- det(SSP_E) / det(SSP_H + SSP_E)

  num_term <- (g * b * (n - 1) - p + 1) / 2
  den_term <- (abs(df_h - p) + 1) / 2
  F_stat   <- ((1 - lam) / lam) * (num_term / den_term)
  v1       <- abs(df_h - p) + 1
  v2       <- g * b * (n - 1) - p + 1
  p_val    <- pf(F_stat, v1, v2, lower.tail = FALSE)
  sig      <- ifelse(p_val < 0.001, "***",
               ifelse(p_val < 0.01,  "**",
                ifelse(p_val < 0.05,  "*", "ns")))

  list(label  = label,
       lambda = lam,
       F      = F_stat,
       v1     = v1,
       v2     = v2,
       pval   = p_val,
       sig    = sig)
}

11.3 Hasil Pengujian

res_A  <- wilks_test(SSP_A,  SSP_E, df_A,  g, b, n_cell, p, "Faktor A")
res_B  <- wilks_test(SSP_B,  SSP_E, df_B,  g, b, n_cell, p, "Faktor B")
res_AB <- wilks_test(SSP_AB, SSP_E, df_AB, g, b, n_cell, p, "Interaksi A×B")

Tabel MANOVA Uji Wilks’ Lambda
Sumber	df	Lambda	F_hitung	v1	v2	p_value	Sig
Faktor A	1	0.5297	2.9598	3	10	0.084179	ns
Faktor B	1	0.8609	0.5388	3	10	0.666364	ns
Interaksi A×B	1	0.4346	4.3359	3	10	0.033497	*
Error	12	NA	NA	NA	NA	NA
Total	15	NA	NA	NA	NA	NA

Keterangan signifikansi: *** p < 0.001 | ** p < 0.01 | * p < 0.05 | ns tidak signifikan

12 Interpretasi Hasil

Faktor A: tidak signifikan (p = 0.0842, Λ = 0.5297)
Faktor B: tidak signifikan (p = 0.6664, Λ = 0.8609)
Interaksi A×B: signifikan (p = 0.0335, Λ = 0.4346)

13 Konfirmasi dengan Built-in `manova()` R

fit <- manova(cbind(X1, X2, X3) ~ A * B, data = data)
summary(fit, test = "Wilks")

          Df   Wilks approx F num Df den Df  Pr(>F)  
A          1 0.52968   2.9598      3     10 0.08418 .
B          1 0.86085   0.5388      3     10 0.66636  
A:B        1 0.43464   4.3359      3     10 0.03350 *
Residuals 12                                         
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

14 Ringkasan

Ringkasan Analisis
Komponen	Nilai
Jumlah observasi	16
Faktor A (level)	2
Faktor B (level)	2
Variabel dependen (p)	3
Observasi per sel (n)	4
df Error	12

Two-Way MANOVA Analisis Soal Papan tulis

Sasmita Kusuma Jati

2026-04-12

1 Deskripsi Data

2 Input Data

3 Helper Function

4 Grand Mean & Marginal Means

5 SSP Total

6 SSP Faktor A

7 SSP Faktor B

8 SSP Error (Within Cells)

9 SSP Interaksi (A × B)

10 Degrees of Freedom

11 Uji Wilks’ Lambda

11.1 Formula

11.2 Fungsi Wilks

11.3 Hasil Pengujian

12 Interpretasi Hasil

13 Konfirmasi dengan Built-in `manova()` R

14 Ringkasan

A	B	X1	X2	X3
l1	f1	1.3	1.2	1.0
l1	f1	1.2	1.1	1.3
l1	f1	0.7	1.9	1.4
l1	f1	1.3	2.1	1.5
l1	f2	1.3	1.3	0.2
l1	f2	1.4	1.9	0.7
l1	f2	1.5	1.2	0.9
l1	f2	1.7	2.1	0.9
l2	f1	1.3	1.2	1.3
l2	f1	1.5	1.7	2.1
l2	f1	1.5	1.5	0.5
l2	f1	1.8	2.1	1.3
l2	f2	1.3	1.7	1.9
l2	f2	1.4	1.8	1.6
l2	f2	1.5	1.9	1.8
l2	f2	1.6	2.0	2.7

A	B	X1	X2	X3
l1	f1	1.3	1.2	1.0
l1	f1	1.2	1.1	1.3
l1	f1	0.7	1.9	1.4
l1	f1	1.3	2.1	1.5
l1	f2	1.3	1.3	0.2
l1	f2	1.4	1.9	0.7
l1	f2	1.5	1.2	0.9
l1	f2	1.7	2.1	0.9
l2	f1	1.3	1.2	1.3
l2	f1	1.5	1.7	2.1
l2	f1	1.5	1.5	0.5
l2	f1	1.8	2.1	1.3
l2	f2	1.3	1.7	1.9
l2	f2	1.4	1.8	1.6
l2	f2	1.5	1.9	1.8
l2	f2	1.6	2.0	2.7

Two-Way MANOVA Analisis Soal Papan tulis

Sasmita Kusuma Jati

2026-04-12

1 Deskripsi Data

2 Input Data

3 Helper Function

4 Grand Mean & Marginal Means

5 SSP Total

6 SSP Faktor A

7 SSP Faktor B

8 SSP Error (Within Cells)

9 SSP Interaksi (A × B)

10 Degrees of Freedom

11 Uji Wilks’ Lambda

11.1 Formula

11.2 Fungsi Wilks

11.3 Hasil Pengujian

12 Interpretasi Hasil

13 Konfirmasi dengan Built-in manova() R

14 Ringkasan

13 Konfirmasi dengan Built-in `manova()` R

A	B	X1	X2	X3
l1	f1	1.3	1.2	1.0
l1	f1	1.2	1.1	1.3
l1	f1	0.7	1.9	1.4
l1	f1	1.3	2.1	1.5
l1	f2	1.3	1.3	0.2
l1	f2	1.4	1.9	0.7
l1	f2	1.5	1.2	0.9
l1	f2	1.7	2.1	0.9
l2	f1	1.3	1.2	1.3
l2	f1	1.5	1.7	2.1
l2	f1	1.5	1.5	0.5
l2	f1	1.8	2.1	1.3
l2	f2	1.3	1.7	1.9
l2	f2	1.4	1.8	1.6
l2	f2	1.5	1.9	1.8
l2	f2	1.6	2.0	2.7