factorial designs: identifying and understanding interactions lawrence r. gordon
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FACTORIAL DESIGNS:Identifying and Understanding
Interactions
Lawrence R. Gordon
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BUILDING-BLOCK EXAMPLE, REVISITED “Effects of timing and amount of reward on problem
solving”
Nomenclature– 1st IV (A) has two levels of reward timing– 2nd IV (B) has four levels of reward amount– AxB = 2 x 4 = 8 cells (“conditions,”
treatment combinations”), with different Ss in each
– “a 2x4 between-Ss factorial design”
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Layout / Nomenclature
DV: # ofAnagrams Solved
REWARD
$1 +0
$1 +.50
$1 +1.00
$1 +1.50
Imm 9.2 11.0 16.0 24.8 15.25
DELAY Del 9.0 11.2 12.4 14.8 11.85
9.10 11.10 14.20 19.80 13.55
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BUILDING-BLOCK EXAMPLE, cont’d.. Analysis
– Descriptives: means, sds, ns• In cells
• Marginals -- for each DV
– Graph of cell means– Inferential: “Two-way ANOVA, Between-Ss”
• Summary table
• Main effects (each IV ignoring other): A, B
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“ANOVA SUMMARY TABLE”
Source SS DF MS F p
Delay 115.60 1 115.60 7.64 .0094
Reward 652.90 3 217.63 14.38 .0000
D R 167.00 3 55.67 3.68 .0221
Error 484.00 32 15.14
Total 1419.90 39
Significant effects: Delay main effect, Reward main effect,and Delay by Reward interaction effect: “F(3,32)=3.68, p<.05”
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INTERPRETATION: “Reward” Descriptive Statistics
ANAGRAMS * DELAY
ANAGRAMS
15.2500 20 7.2321
11.8500 20 4.0429
13.5500 40 6.0339
DELAYImmediate`
Delayed
Total
Mean N Std. Deviation
ANAGRAMS * REWARD
ANAGRAMS
9.1000 10 3.2472
11.1000 10 2.2336
14.2000 10 3.8528
19.8000 10 7.4057
13.5500 40 6.0339
REWARD+0
+.50
+1.00
+1.50
Total
Mean N Std. Deviation
MEAN ANAGRAMS SOLVED
REWARD
+1.50+1.00+.50+0
Me
an
s
30
20
10
0
DELAY
Immediate`
Delayed
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BUILDING-BLOCK EXAMPLE, cont’d.. Analysis
– Descriptives: means, sds, ns• In cells
• Marginals -- for each DV
– Graph of cell means– Inferential: “Two-way ANOVA, Between-Ss”
• Summary table
• Main effects (each IV ignoring other): A, B
• Interaction: A x B or AB -- is significant; what does this mean? First, let’s quickly review a study without a significant interaction!
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NO INTERACTION EXAMPLEReview: Rosenzweig & Tryon (1950) Rats running a maze:
– 3 strains: maze dull, mixed, maze bright– 2 rearing environments: basic, enriched– a “P”E design (ok, “R”E)
Results– Both main effects significant– Interaction is not– Q: “What does this mean?”– A: “Let me tell you…”
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NO INTERACTION EXAMPLE
REARING X STRAIN OF RATS
Rozenzweig & Tryon
Rearing Enivronment
ImpovrdEnriched
Mea
n Tr
ials
to L
earn
Maz
e
12
10
8
6
4
2
Maze Strain Type
Bright
Mixed
Dull
REARING X STRAIN OF RATS
Rozenzweig & Tryon
Rearing Enivronment
ImpovrdEnriched
Mean
Trial
s to L
earn
Maze
12
10
8
6
4
2
Maze Strain Type
Bright
Mixed
Dull
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NO INTERACTION EXAMPLE
REARING X STRAIN OF RATS
Rozenzweig & Tryon
Maze Strain Type
DullMixedBright
Mea
n Tr
ials
to L
earn
Maz
e
12
10
8
6
4
2
Rearing Enivronment
Enriched
Impovrd
REARING X STRAIN OF RATS
Rozenzweig & Tryon
Maze Strain Type
DullMixedBright
Mean
Tria
ls to
Lea
rn M
aze
12
10
8
6
4
2
Rearing Enivronment
Enriched
Impovrd
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BUT…Replicate and Extend
Cooper & Zubeck (1958), studied “genotype - environment interaction” (PxE again -- oops, “R” by E)
“R” -- maze-bright vs. maze-dull rats E -- Restricted, Intermediate, Stimulating What happened? “IT DEPENDS…” --
there were marked performance differences only in the Intermediate environment
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REARING X STRAIN OF RATS
Rozenzweig & Tryon
Rearing Enivronment
ImpovrdEnriched
Mean
Trial
s to L
earn
Maze
12
10
8
6
4
2
Maze Strain Type
Bright
Mixed
Dull
Cooper & Zubeck (1958)
110
120
130
140
150
160
170
180
Rearing Environment
# of
Err
ors
Dull
Bright
INTERACTION OR NOT? What did they look at?
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INTERACTIONS: our last “new” concept Graphs of an interaction: (overhead)
– No interaction --- parallel line segments– Interaction --- non-parallel line segments– No lines perfectly so, must use statistical test
What is the null hypothesis? How is interaction measured?
Testing after finding an interaction is different than when only main effects are significant.
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YES, INTERACTION, EXAMPLES Q: “What do these mean?” A: “It depends…”
“Blunder” (Aronson et al., 1966)
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Aronson et al. (1966)The effect of a pratfall on increasing interpersonal attractiveness. Ps heard audiotape of student said to be a
candidate for the “College Quiz Bowl.” An interview asked difficult questions.
Four tapes:– Candidate “nearly perfect,” no blunder– Candidate “nearly perfect,” blunder (coffee
spill)– Candidate “average,” no blunder– Candidate “average,” blunder
Asked to rate “liking” of the candidate
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Aronson (1966) continued…
Blunder (A)
Type of Person (B)Average Superior Main effect
(A)
None 17.8 20.8 19.30
YES -2.5 30.2 13.85
Main effect (B)
7.65 25.50 16.58 n = 10/cell N = 40 total
ANOVA table
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Aronson (1966) continued…
Results:Source df F p
Blunder 1 1.67 >.05 ns
Person 1 17.72 < .05 *
B x P 1 12.28 < .05 *
Error 36 --
Total 39 F.05(1,36) = 4.17
Graph of the interaction
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Aronson (1966) continued…
Blunder (A)
Type of Person (B)Average Superior Main effect
(A)
None 17.8 20.8 19.30 n=10/cell
YES -2.5 30.2 13.85 N=40 total
Main effect (B)
7.65 25.50 16.58
Graph of the interaction
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Aronson et al., Person x Blunder Interaction
-5
0
5
10
15
20
25
30
35
No Yes
Blunder
Rat
ing
of L
ikin
g
Superior personAverage person
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YES, INTERACTION, EXAMPLES Q: “What do these mean?” A: “It depends…”
“Blunder” (Aronson et al., 1966) “Stroop (1935),” reconstrued
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Stroop (1935), reconsideredRef. Goodwin, Box 7.1, p. 219
Did two experiments:– RCNb vs RCNd (no
difference)
– NC vs NCWd (“Stroop effect”)
Could consider as two factors:– Control vs. Different
– Read color vs. Name color
0
20
40
60
80
100
120
Same Diff
Sec
to
Rea
d/N
ame
100
Item
s
ReadName
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YES, INTERACTION, EXAMPLES Q: “What do these mean?” A: “It depends…”
“Blunder” (Aronson et al., 1966) “Stroop (1935),” reconstrued “Underwater” (Godden &Baddeley,
1975)
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Godden & Baddeley, 1975: Encoding Specificity
Interested in the match between the conditions of encoding and the conditions of retrieval on recall
Four conditions:– Learn on land -- recall on land
– Learn on land -- recall under water
– Learn under water -- recall on land
– Learn under water -- recall under water All divers eventually participated in all four conditions,
making this a repeated-measures factorial design. DV is number of words recalled per list A reference: Goodwin, pp. 254-255. Graph
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Godden & Baddeley (1975):Encoding Retrieval Interaction
0
2
4
6
8
10
12
14
16
Land Water
Where They Recalled
Mea
n #
Wor
ds R
ecal
led
LandWater
Where They Learned
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Further Example
“Dr. Jones” in-class experiment (done Fall 1999)– Written scenarios varied two factors:
• Gender of “Dr. Jones”: He vs. She
• Time teaching since PhD: “since that time,” 10, or 30 yrs.
– DV was a “Teaching Evaluation” scale (8 items)
– Design: 2 x 3 Between-Ss randomized experiment
Summary: “The main effects of Sex and Time were not significant; there was a significant Sex By Time interaction, F(2,96)=3.86, p=.024.”
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Dr. Jones Experiment F99
1. Dr. Jones' sex...
Dependent Variable: Student Rating Scale (6r)
5.103 .094 4.916 5.290
4.966 .094 4.779 5.152
Dr. Jones' sex...'her'
'him'
Mean Std. Error Lower Bound Upper Bound
95% Confidence Interval
2. Professional psychologist how long?
Dependent Variable: Student Rating Scale (6r)
5.201 .115 4.972 5.429
4.882 .113 4.657 5.107
5.020 .117 4.788 5.253
Professionalpsychologist how long?Unspecified
10 years
30 years
Mean Std. Error Lower Bound Upper Bound
95% Confidence Interval
Tests of Between-Subjects Effects
Dependent Variable: Student Rating Scale (6r)
a
.481 1 .481 1.072 .303
1.753 2 .876 1.952 .148
3.466 2 1.733 3.860 .024
43.100 96 .449
48.722 101
Source
SEX
TIME
SEX * TIME
Error
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
a.
Main effects (I.e., on marginal means)
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Dr. Jones Experiment F99
Interaction effect (…but what’s it mean?)
Means of Student Rating Scale (6r)
Dr. Jones' sex...
'him''her'
Ra
tin
g S
ca
le M
ea
ns
5.4
5.2
5.0
4.8
4.6
Professional psychol
Unspecified
10 years
30 years
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Further Example
Summary: “The main effects of Sex and Time were not significant; there was, however, a significant Sex By Time interaction, F(2,96) = 3.86, p =.024. Although there was no sex difference in attributed teaching performance at 10 yrs post-PhD, there was a sex difference at 30 yrs post-PhD, with females seen as improving over the 10 yr mark, and males seen as declining under the 10 yr mark. The vague “since that time” control was better than the ten-yr result for both, but had a nonsignificant sex difference.”
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Wrapup
NO INTERACTION: main effects are unqualified; generalizes from one factor over the other(s) [often the goal of a PE design]. “Let me tell you…”
INTERACTION: main effects ignored or qualified; does not generalize [especially if a PE design]. “It depends…” This may lead to theory revision if not already predicted.
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EXTENSIONS FROM TWO-LEVEL DESIGNS, next?
To more than 2 groups or levels of a single factor (multiple-level)– Previously covered
To more than one factor (IVs) (“factorial” designs)– Today: interactions & examples
Is there another extension from the simple 2-level experiment? – YES -- to multiple simultaneous DVs!
– Will we study? NO - quite advanced!