Two-Way ANOVA without Replication

When we conduct experiments which involve two factors, and it is not possible to obtain repeated measures for a given set of experimental conditions, a two-way analysis of variance without replication may be used.


	···	Col_j	···	Mean
···	···	···	···	···
Row_i	···	X_ij	···	R_i
···	···	···	···	···
Mean	···	C_j	···	M

where

X_ij:the measure for i^th row and j^th column
R_i: the mean for i^th row
C_j: the mean for j^th column
M: the overall mean

Hypotheses for Rows

H₀: Row means are equal.
H_a: At least one row mean is different.

Hypotheses for Columns

H₀: Column means are equal.
H_a: At least one column mean is different.

Significance level = α

Degrees of Freedom

DF_r = r - 1
DF_c = c - 1
DF_e = (r - 1)(c - 1)
DF_t = rc - 1 or N - 1

where

r: number of rows
c: number of columns
N: number of observations

Sum of Squares

The 4 SS values have the following relation.

Mean Squares

F-values

Critical values

F(DF_r, DF_e, α) for rows
F(DF_c, DF_e, α) for columns

ANOVA Table

Source	DF	SS	MS	F
Row	DF_r	SS_r	MS_r	F_r
Column	DF_c	SS_c	MS_c	F_c
Error	DF_e	SS_e	MS_e
Total	DF_t	SS_t

If the F_r is greater than the F(DF_r, DF_e, α), you can reject the first null hypothesis. If the F_c is greater than the F(DF_c, DF_e, α), you can reject the second null hypothesis.

Example

Determine at the 0.05 significance level whether the rows have the different means and whether the columns have the different means.

  ( 45 88 59 )
  ( 64 78 68 )
  ( 72 96 57 )
  ( 67 70 52 )

	Col 1	Col 2	Col 3	Mean
Row 1	45	88	59	R₁ = 64
Row 2	64	78	68	R₂ = 70
Row 3	72	96	57	R₃ = 75
Row 4	67	70	52	R₄ = 63
Mean	C₁ = 62	C₂ = 83	C₃ = 59	M = 68

Degrees of Freedom

DF_r = 4 - 1 = 3
DF_c = 3 - 1 = 2
DF_e = (4 - 1)(3 - 1) = 6
DF_t = (4)(3) - 1 = 11

Sum of Squares

SS_r = 3Σ(R_i - 68)² = 282
SS_c = 4Σ(C_j - 68)² = 1368
SS_t = ΣΣ(X_ij - 68)² = 2308
SS_e = SS_t - SS_r - SS_c = 658

Mean Squares

MS_r = 282 / 3 = 94
MS_c = 1368 / 2 = 684
MS_e = 658 / 6 = 109.6667

F-Values

F_r = 94 / 109.6667 = 0.8571
F_c = 684 / 109.6667 = 6.2371

ANOVA Table

Source	DF	SS	MS	F
Row	3	282	94	0.8571
Column	2	1368	684	6.2371
Error	6	658	109.67
Total	11	2308

Test for Row Means

Critical value:
F(3, 6, 0.05) = 4.7571
Test statistic: F_r = 0.8571
P(>F_r) = 0.5122
Fail the reject the null hypothesis.
∴ The alternative hypothesis cannot be supported.

Test for Column Means

Critical value:
F(2, 6, 0.05) = 5.1433
Test statistic: F_c = 6.2371
P(>F_c) = 0.0343
Reject the null hypothesis.
∴ At least one column mean is different.