Question

Below is an ANOVA summary table for a 3x5 ANOVA reflecting the data of 12 subjects. Some of the df are incorrect, and/or it may be missing degrees of freedom.  Use the blank space below to fix the incorrect and/or provide the missing degrees of freedom (do not do more than this). What type of ANOVA design does this table represent?

Source	SS	Df	MS	F
Subjects		11
A		2
Error A		21
B		4
Error B		11
AxB		8
Total		179

Answer 1

Here, this table represent spli-plot design ANOVA with two two factors.

ANOVA table for split-plot design is -

Source of variation	SS	df	MS	F-value
Replicate	Ryy	r-1	Ry	Fr
A	Ayy	p-1	Ay	Fa
Error A	Eyya	(r-1)(p-1)	Eya	Fea
B	Byy	q-1	By	Fb
Error B	Eyyb	p(r-1)(q-1)	Eyb	-
A$\times$B	(AB)yy	(p-1)(q-1)	(AB)y	Fab
Total	Tyy	rpq - 1	-	-

Here, we have 12 subjects.

Hence, r = 12

Given ANOVA table for split-plot design is -

Source of variation	SS	df	MS	F-value
Replicate	Ryy	r-1 = 11	Ry	Fr
A	Ayy	p-1 = 2	Ay	Fa
Error A	Eyya	(r-1)(p-1) = 21	Eya	Fea
B	Byy	q-1 = 4	By	Fb
Error B	Eyyb	p(r-1)(q-1) = 11	Eyb	-
A$\times$B	(AB)yy	(p-1)(q-1) = 8	(AB)y	Fab
Total	Tyy	rpq - 1 = 179	-	-

From the table, we have -

r - 1 = 11 $\Rightarrow$ r = 12

p - 1 = 2  $\Rightarrow$ p = 3

So, (r-1)(p-1) = (11)(2) = 22

q - 1 = 4 $\Rightarrow$ q = 5

So, p(r-1)(q-1) = (3)(11)(4) = 132

(p-1)(q-1) = (2)(4) = 8

rpq - 1 = (12)(3)(5) - 1 = 180 - 1 = 179

Hence, the corrected ANOVA table is as follows -

Source of variation	SS	df	MS	F-value
Replicate	Ryy	r-1 = 11	Ry	Fr
A	Ayy	p-1 = 2	Ay	Fa
Error A	Eyya	(r-1)(p-1) = 22	Eya	Fea
B	Byy	q-1 = 4	By	Fb
Error B	Eyyb	p(r-1)(q-1) = 132	Eyb	-
A$\times$B	(AB)yy	(p-1)(q-1) = 8	(AB)y	Fab
Total	Tyy	rpq - 1 = 179	-	-