Review
Three subjects measured in four conditions.
Find the sum of squares for
condition differences, SStreatment
A.
B.
C.
D.
84
152
252
336
Condition
Subject
A
B
C
D
Ms
1
59
55
68
58
60
2
71
63
75
63
68
3
68
62
73
65
67
Mi
66
60
72
62
65
Review
Three subjects measured in four conditions.
Find the sum of squares for
individual differences, SSsubject
A.
B.
C.
D.
38
114
152
252
Condition
Subject
A
B
C
D
Ms
1
59
55
68
58
60
2
71
63
75
63
68
3
68
62
73
65
67
Mi
66
60
72
62
65
Review
Three subjects measured in four conditions.
SStreatment = 252
SSsubject = 152
SStotal = 420
dftreatment = 3
dfresidual = 6
Condition
Subject
A
B
C
D
Ms
1
59
55
68
58
60
2
71
63
75
63
68
3
68
62
73
65
67
Mi
66
60
72
62
65
Calculate the F statistic for testing condition differences
A.
B.
C.
D.
1.20
1.88
3.32
31.50
Factorial ANOVA
11/13
Multiple Independent Variables
• Simple (one-way) ANOVA tells whether groups differ
– Compares levels of a single independent variable
• Sometimes we have multiple IVs
– Factors
– Subjects divided in multiple ways
• Training type & testing type
– Not always true independent variables
• Undergrad major & sex
– Some or all can be within-subjects (gets more complicated)
• Memory drug & stimulus type
• Dependent variable measured for all combinations of values
• Factorial ANOVA
– How does each factor affect the outcome?
– Extends ANOVA in same way regression extends correlation
Basic Approach
Testing
Training
Dominant
Dominant
[3,7,5,4,6]
Non-dominant [11,7,10,8,9]
Mean
7
Non-dominant
[14,15,11,13,12]
[10,12,13,11,9]
12
Mean
9
10
9.5
• Calculate sum of squares for each factor
– Variability explained by that factor
– Essentially by averaging all data for each level of that factor
• Separate hypothesis test for each factor
– Convert SS to mean square
– Divide by MSresidual to get F
Interactions
• Effect of one factor may depend on level of another
– Pick any two levels of Factor A, find difference of means,
compare across levels of Factor B
• Testable in same way as main effect of each factor
– SSinteraction, MSinteraction, F, p
• Can have higher-order interactions
– Interaction between Factors ATesting
and B depends on C
• Partitioning
Training variability
Dominant
Non-dominant
– SStotal
= SSA + SSB + SSCM = 5
Dominant
M = 13
+ SSA:B + SSA:C + M
SS=B:C9 + SSA:B:C M = 11
Non-dominant
+
SS
residual
Difference
-4
+2
Example: Memory and Brain Injury
Brain Injury
Delay
None
Occipital
MTL
Mean
Short
78%
65%
73%
72%
Long
66%
53%
37%
61%
52%
Difference
12%
12%
12%
36%
Mean
72%
55%
59%
62%
Testing
Rule formain
an interaction:
effects and interactions:
• Pick any two levels of Factor A (A1, A2) and any two levels of Factor B (B1, B2)
Effect
SS
df
MS
F p
• There’s anDelay
interaction if 6000
M A1,B1 - M1A1,B2 ¹6000
M A2 ,B112.91
- M A2 ,B.0007
2
• Equivalently:
Injury
Delay:Injury
Residual
M A1,B1 - M2A2 ,B1 ¹1580
M A1,B2 -3.40
M A2 ,B.041
3160
2
1920
2
960
25094
54
464.7
2.07 .136
Logic of Sum of Squares
• Total sum of squares: å ( X - M ) 2
• Null hypothesis assumes all data are from same population
– Expected value of ( X - M ) is s2 for each raw score
– No matter how we break up SStotal, every individual square
has expected value s2
– SStreatment, SSinteraction, SSresidual are all sums of numbers
with expected value s2
2
• Every MS has expected value s2
– Average of many numbers that all have expected value s2
– E(MStreatment), E(MSinteraction), E(MSresidual) all equal s2,
according to H0
• If H0 false, then MStreatment and MSinteraction tend to be larger
– F is sensitive to such an increase
Review
A factorial experiment compares men and women on their
memory for different word types, with different distractor tasks.
Factors:
• Sex (male, female)
• Word type (noun, verb, adjective, preposition)
• Second task (speech, manual, none)
How many groups of subjects are there?
A.
B.
C.
D.
2
3
9
24
Men
Speec
h
Manua
l
Non
e
Women
Noun
Noun
Verb
Verb
Adj.
Adj.
Prep.
Prep.
Speec
h
Manua
l
Non
e
Review
A factorial experiment compares people on their memory for different word
types, with different distractor tasks.
Speec
h
Manua
l
Non
e
Mea
n
Nou
n
15
13
17
15
Verb
10
11
15
12
Adj.
9
10
14
11
Is there an interaction? Prep.
8
9
13
10
Group Means:
(ignoring sex)
A.
B.
C.
D.
Yes, because adjectives and prepositions are differentially affected by the
second task
Yes, because the difference between Speech and Manual is different for
nouns than for verbs
No, because the difference between Manual and None is the same for all
word types
Yes, because the overall averages for different word types are different
Review
A factorial experiment compares people on their memory for different
word types, with different distractor tasks.
ANOVA table:
SS
df
Word type
35
2
3
117.33 7.33 .0003
2nd Task
27
3
2
136.50 8.53 .0005
Interaction
27
6
4.50
60
16
96
Find the Fs for the three effects
A.
B.
C.
D.
Residual
0
MS
F
0.28
p
.94
FWord type = 27.08, F2nd task = 30.33, FWord type:2nd task = 0.28
FWord type = 0.37, F2nd task = 0.28, FWord type:2nd task = 0.03
FWord type = 7.33, F2nd task = 8.53, FWord type:2nd task = 0.28
FWord type = 6.52, F2nd task = 3.37, FWord type:2nd task = 0.03