ss total ss between subjects ss within subjects ss a ss as ss b ss bs ss ab ss ab*s - in a two...

SS total

SS between subjects

SS within subjects

SS A SS A*S SS B SS B*S SS A*B SS A*B*S

- In a Two way ANOVA with 2 within subject factors:

Forced Choice Free Response

Subject Ritalin Adderall Caffeine Control Ritalin Adderall Caffeine Control

1 13 11 8 10 8 8 2 2

2 6 6 9 3 8 12 6 6

3 12 12 13 14 12 8 6 6

4 11 14 10 8 13 15 7 9

This is a Repeated Measures design with 2 factors :

1. Question (Forced Choice, Free Response)

2. Drug (Ritalin, Adderal, Caffeine, Control)

Forced Choice Free ResponseSubject Rit Add Caf Con Rit Add Caf Con mean

1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

Msubject 1

M grand

Get the mean of each subject, subtract from the grand mean, square and multiply by the number of scores of each subject

SS_between_subjects = SS subject

SS between_subjects = Level drug x Level question ∑ (M subejct - M grand )2

=4 x 2 x [ ( 7.75-9 )2 + ( 7-9 )2 + ( 10.38-9 )2 + ( 10.38-9 )2 ]

df between_subjects = nsubjects -1 = 4-1 = 3

SS_between_subjects = SS subject


1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

SS_within_subjects

Msubject 1

Subtract each score from its subject mean, square and sum.

SS_within_subjects

SS within _subject = ∑ (X - Msubject )2

= [ (13 - 7.75)2 + (11-7.75)2 + …..+ (6-7)2 + (6-7 )2 +….] df within _subject = total scores -number of subjects

= nsubjects x ( (Level drug x Level question) -1)

= 4 x (2 x 4 -1 ) = 28


1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

SS Drug:

Grand Mean

Subtract mean of each drug level from the grand mean, square and multiply by the number of score in each level of drug. Collapse (mean) across drug ignoring choice.

Mritalin=10.38

SS Drug:

SS Drug = nsubjects x Level Question x ∑ (MDrug - MGrand )2

= 4 x 2 x [ (10.38 - 9)2 + (10.75-9)2 + (7.63-9)2 + (7.25-9 )2 ]

df Drug = Level Drug -1 = 4 - 1 = 3

S Rit Add Caf Con mean1 10.5 9.5 5 6 7.752 7 9 7.5 4.5 7.003 12 10 9.5 10 10.384 12 14.5 8.5 8.5 10.88

mean 10.38 10.75 7.63 7.25 9.00

SS Drug x Subject:

For getting the interaction between subject and Drug we have to calculate SS drug_subject and subtract SS subject and SS drug from it.

SS Drug x Subjetc = SS Drug_subject - SS Drug - SS subject


1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

Subtract the mean of each Drug level for each subject from the grand mean, square and multiply by the number of scores in each drug subject combination

Mritalin-subject 1=10.5

SS Drug_Subject:

S Rit Add Caf Con1 10.5 9.5 5 62 7 9 7.5 4.53 12 10 9.5 104 12 14.5 8.5 8.5

SS Drug_Subject = Level question x ∑ (Mdrug_subject - MGrand )2

= 2 x [ (10.5 - 9)2 + (9.5-9)2 + (5-9)2 + (6-9 )2 + (7-9 )2 + (9-9 )2 + ….]

SS Drug_Subject:

S Rit Add Caf Con1 10.5 9.5 5 62 7 9 7.5 4.53 12 10 9.5 104 12 14.5 8.5 8.5

SS Drug x Subject:

Now we can get the SS interaction between subject and drug. As we have all the other three terms we are all set!

SS Drug x Subjetc = SS Drug_subject - SS Drug - SS subject

df Drug x Subjetc = df Drug X df subject =( level Drug -1) X (nsubject -1)

= (4-1) X (4-1) = 9


1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

SS Question:

Grand Mean

Subtract mean of each Question level from the grand mean, square and multiply by the number of score in each level of quesrtion. Collapse (mean) across question ignoring drug.

Mforced Choice=10

SS Question:

SS Question = nsubjects x Level Drug x ∑ (MQuestion - MGrand )2

= 4 x 4 x [ (10 - 9)2 + (8-9)2]

df Question = Level Question -1 = 2 - 1 = 2

Subject FC FR mean1 10.5 5 7.752 6 8 7.003 12.75 8 10.384 10.75 11 10.88

mean 10 8 9

SS Question x Subject:

For getting the interaction between subject and Question we have to calculate SS Question _subject and subtract SS subject and SS Question from it.

SS Question x Subjetc = SS Question _subject - SS Question - SS subject

Subject FC FR1 10.5 52 6 83 12.75 84 10.75 11


1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

Subtract the mean of each Question level for each subject from the grand mean, square and multiply by the number of scores in each Question subject combination

MforceChoice_subject1=10.5

SS Question_Subject:

Subject FC FR1 10.5 52 6 83 12.75 84 10.75 11

SS Question_Subject = Level Question x ∑ (MQuestion_subject - MGrand )2

= 4 x [ (10.5 - 9)2 + (5-9)2 + (6-9)2 + (8-9 )2 + ….]

SS Question_Subject:

SS Question x Subject:

Now we can get the SS interaction between subject and Question

SS Questionx Subject = SS Question_subject - SS Question- SS subject

df Questionx Subject = df QuestionX df subject =( level Question-1) X (nsubject -1)

= (2-1) X (4-1) = 3

SS Question x Drug:

For getting the interaction between Drug and Question we have to calculate SS Question _Drug and subtract SS Drug and SS Question from it.

SS Question x Drug = SS Question _Drug - SS Question - SS Drug


1 13 11 8 10 8 8 2 2 7.752 6 6 9 3 8 12 6 6 7.003 12 12 13 14 12 8 6 6 10.384 11 14 10 8 13 15 7 9 10.88

mean 10.50 10.75 10.00 8.75 10.25 10.75 5.25 5.75 9

Subtract the mean of each Question Drug Combination from the grand mean, square and multiply by the number of scores in each Question Drug combination.

MforceChoice_aderal=10.75

SS Question_Drug:

SS Question_Subject = n subject x ∑ (MQuestion_drug - MGrand )2

= 4 x [ (10.5 - 9)2 + (10.75-9)2 + (10-9)2 + (8.75-9 )2 + ….]

SS Question_Drug:

SS Question x Drug:

Now we can get the SS interaction between subject and Question

SS Questionx Drug = SS Question_ Drug - SS Question- SS Drug

df Questionx Drug = df QuestionX df Drug =( level Question-1) X (level Drug -1)

= (2-1) X (4-1) = 3

SS Question x Drug x Subject:

For getting the three way interaction we have to subtract SS drug, SS question, SS subject x drug, SS subject x question and SS drug x question from the SS within subjects. As we have all of these SS’s we can calculate the three way interaction.

SS Questionx Drug = SS within_subjects - ( SS Question+ SS Drug + SS Drug x Subject +

SS Question x Subject + SS Drug x Question )

df Questionx Drug = df QuestionX df Drug X df Subject

=( level Question-1) X (level Drug -1) X (n subejct -1)

= (2-1) X (4-1) X (4-1) = 9

After getting all the SS and df We can calculate the Mean Squares. And then we will be able to get the F values for all the effects. Note that you have to use the appropriate error term for each effect:

F Drug = MS Drug / MS Drug x subject

F Question = MS Question / MS Question x subject

F Questionx Drug = MS Question x Drug / MS Question x Drug x subject

ss total ss between subjects ss within subjects ss a ss a*s ss b ss b*s ss a*b ss a*b*s - in a two...

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