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 ChoiceFree Response SubjectRitalinAdderallCaffeineControlRitalinAdderallCaffeineControl This is a Repeated Measures design with 2 factors : 1. Question (Forced Choice, Free Response) 2. Drug (Ritalin, Adderal, Caffeine, Control)

M subject 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 [ ( ) 2 + ( 7-9 ) 2 + ( ) 2 + ( ) 2 ] df between_subjects = n subjects -1 = 4-1 = 3 SS_between_subjects = SS subject

SS_within_subjects M subject 1 Subtract each score from its subject mean, square and sum.

SS_within_subjects SS within _subject = ∑ ( X - M subject ) 2 = [ ( ) 2 + ( ) 2 + …..+ (6-7) 2 + (6-7 ) 2 +….] df within _subject = total scores -number of subjects = n subjects x ( (Level drug x Level question ) -1) = 4 x (2 x 4 -1 ) = 28

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. M ritalin =10.38

SS Drug: SS Drug = n subjects x Level Question x ∑ ( M Drug - M Grand ) 2 = 4 x 2 x [ ( ) 2 + ( ) 2 + (7.63-9) 2 + ( ) 2 ] df Drug = Level Drug -1 = = 3

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

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 M ritalin-subject 1 =10.5 SS Drug_Subject:

SS Drug_Subject = Level question x ∑ ( M drug_subject - M Grand ) 2 = 2 x [ ( ) 2 + (9.5-9) 2 + (5-9) 2 + (6-9 ) 2 + (7-9 ) 2 + (9-9 ) 2 + ….] SS Drug_Subject:

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 ( n subject -1) = (4-1) X (4-1) = 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. M forced Choice =10

SS Question: SS Question = n subjects x Level Drug x ∑ ( M Question - M Grand ) 2 = 4 x 4 x [ (10 - 9) 2 + (8-9) 2 ] df Question = Level Question -1 = = 2

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

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 M forceChoice_subject1 =10.5 SS Question_Subject:

SS Question_Subject = Level Question x ∑ ( M Question_subject - M Grand ) 2 = 4 x [ ( ) 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 Question X df subject =( level Question -1) X ( n subject -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

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. M forceChoice_aderal =10.75 SS Question_Drug:

SS Question_Subject = n subject x ∑ ( M Question_drug - M Grand ) 2 = 4 x [ ( ) 2 + ( ) 2 + (10-9) 2 + ( ) 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 Question X 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 Question X 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