Question

A between-subjects factorial design has two levels of factor A and 2 levels of Factor B. Each cell of the design contains n=8 participants. The sums of squares are shown in the table. The alpha level for the experiment is 0.05. Use the provided information and partially completed table to provide the requested values.

Source	SS	df	MS	F	p	n2p	Fcrit
A	12
B	3
AxB	21
Within	84
Total	120

Specifically looking for the df total and F ratio for the AxB interaction.

Answer 1

Concepts

• Degree of freedom for factor A is given by a-1 ,where a is the levels of factor A
• Degree of freedom for factor B is given by b-1 ,where b is the levels of factor B
• Degree of freedom of (A*B) is given by (a-1)*(b-1)
• Degree of freedom for error is given by N -a*b, where N is the total observations
• Degree of freedom, total is equal to N-1
• MS = SS/df for each case
• F = MSeffect/ MSerror

Solution

n=8, a = 2, b = 2.

Since total participants are 8 and A, B has 2 levels each, the total number of observations would be 8*2*2 = 32. So, N =32

Calculating degree of freedom

• df of A = a-1 = 1
• df of B = b-1 = 1
• df of (A*B) = (a-1)*(b-1) = 1*1 = 1
• df of error = N-ab = 32-4 = 28
• df total = N-1 =31

We can check the dftotal should be equal to sum of all other degree of freedom.

dftotal = 1+1+1+28 = 31 = N-1

Calculating MS

MS = SS/df

Calculating using this formula, we get

Source	SS	df	MS
A	12	1	12/1 =12
B	3	1	3/1 = 3
Interaction	21	1	21/1 = 21
Error	84	28	84/28 =3
Total	120	31

Calculating F value

F = MSeffect/ MSerror

Source	SS	df	MS	F
A	12	1	12	12/3 = 4
B	3	1	3	3/3 = 1
Interaction	21	1	21	21/3 = 7
Error	84	28	3
Total	120	31

Observing p-values for the F values for the corresponding degree of freedom from the distribution table

p-value can also be taken from excel function

FDIST(F value, numerator df, denominator df). Now for A, the formula would be FDIST(4,1,28) as F value is 4, numerator degree of freedom is 1, and denominator df is 28

Source	SS	df	MS	F	p-value
A	12	1	12	4	0.05528
B	3	1	3	1	0.326
Interaction	21	1	21	7	0.013
Error	84	28	3
Total	120	31

Calculating effect size (eta-square)

Effect size = SSaffect /SS

Source	SS	df	MS	F	p-value	Eta square
A	12	1	12	4	0.05528	12/120 =0.1
B	3	1	3	1	0.326	3/120= .025
Interaction	21	1	21	7	0.013	21/120 =0.175
Error	84	28	3
Total	120	31

Calculating F-critical

Given that alpha = 0.05

F-critical is characterized by degree of freedom of effect and degree of freedom of error for the given alpha

For A, it is F-crit(df of A, df of Error) = (1,28) = 4.196

For B, it is F-crit(df of B, df of Error) = (1,28) = 4.196

For Interaction,  it is F-crit(df of A*B, df of Error) = (1,28) = 4.196

Populating the full table now

Source	SS	df	MS	F	p-value	Eta square	F-critical
A	12	1	12	4	0.05528	0.1	4.196
B	3	1	3	1	0.326	0.025	4.196
Interaction	21	1	21	7	0.013	0.175	4.196
Error	84	28	3
Total	120	31