doctoral seminar, spring semester 2007 experimental design & analysis two-factor experiments...
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DOCTORAL SEMINAR, SPRING SEMESTER 2007
Experimental Design & Analysis
Two-Factor Experiments
February 20, 2007
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Two-Factor Experiments
Two advantagesEconomyDetection of interaction effects
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Economy
a1 a2 a3
n=30 n=30n=30
b1 b2 b3
n=30 n=30n=30
b1 b2 b3
a1 n=10 n=10 n=10
a2 n=10 n=10 n=10
a3 n=10 n=10 n=10
Compare N for 2 one-factor experiments
with 1 two-factor experiment
N=180 N=90
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Detection of Interactive Effects
Factors may have multiplicative effect, rather than an additive one
Interactions suggest important boundary conditions for hypothesized relationships, giving clues to nature of explanation
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Two-Factor Analysis
Sources of variance when A and B are independent variables A B AxB S/AxB
The model is Yij = μ + αi + βj + (αβ)ij +εij
Overall grand mean
Average effect of α
Average effect of β
Interaction effect of α, β (effect left in data
after subtracting offlower-order effects)
Error term, alsoknown as S/AxB,or randomness
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Two-Factor Analysis
Yijk = μ + αi + βj + (αβ)ij +εijk
We want to test 3 main hypotheses Main effect of A
H0: α1 = α2 = …= αa = 0 vs. H1: at least one α ≠ 0
Main effect of B H0: β1 = β2 = …= βb = 0 vs. H1: at least one β ≠ 0
Interaction effect of AB H0: αβij = 0 for all ij vs. H1: at least one αβ ≠ 0
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Two-Factor Analysis
Sources of variance in two-factor design Total sum of squares: Difference between each score
and grand mean is squared and then summed The deviation of a score from the grand mean can be
divided into 4 independent components 1st component - deviation of row mean from grand mean 2nd component - deviation of column mean from grand
mean 3rd component - deviation of an individual's score from its
corresponding cell mean (only affected by random variation) If we take these 3 components and subtract them from SST
we can find a remaining 4th source of variation, which is interaction effect
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Two-Factor Analysis
Sum of SquaresTotal = (Xijk – X…)2
Sum of SquaresB = an(X.j – X…)2
Sum of SquaresA = bn(Xi. – X…)2
Sum of SquaresS/AxB = n(Xijk – Xij)2
.
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Two-Factor Analysis
Computations in two-way ANOVA involves 4 steps 1. Examining the model for sources of variance when A and B are
independent variables A (with a levels) B (with b levels) AxB (interaction effect of A, B) S/AxB (subjects nested within factors A, B)
2. Determine degrees of freedom A: a-1 B: b-1 AxB: (a-1)(b-1) = ab - a - b +1 S/AxB: ab(n-1) = abn - ab Total: abn - 1
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Two-Factor Analysis
3. Construct formulas for sums of squares using bracket terms [A], [B], [AB], [Y], [T]
Sums and means [A] = ΣAj
2 /bn [A] = bnΣYAj2
[B] = ΣBk2 /an [B] = anΣYBk
2
[AB] = ΣABjk2 /n [AB] = nΣYijk
2
[Y] = ΣYijk2 [Y] = ΣYijk
2
[T] = T2 /abn [T] = abnYT2
Bracket terms SSA = [A] – [T] SSB = [B] – [T] SSAxB = [AB] – [A] – [B] + [T] SSS/AB = [Y] – [T]
See Keppel and Wickens, p. 217-218, for summary table of computational formulas
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Two-Factor Analysis
See Keppel and Wickens, p. 217-218, for summary table of computational formulas
Source SS computation df MS f
A [A]-[T] a-1 SSA/dfA MSA/MSS/AB
B [B]-[T] b-1 SSB/dfB MSB/MSS/AB
AxB [AB]-[A]-[B]+[T] (a-1)(b-1)
= ab-a-b+1
SSAxB
dfAxB
MSAxB/MSS/AB
S/AB [Y]-[AB] ab(n-1)
= abn-ab
SSS/AB
dfS/AB
Total [Y]-[T] abn-1
4. Specify mean squares and F ratios for analysis
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Numerical Example
See Keppel and Wickens, p. 221 Control Drug X Drug Y Control Drug X Drug Y
a1b1 a2b1 a3b1 a1b2 a2b2 a2b2
1 13 9 15 6 14
4 5 16 6 18 7
0 7 18 10 9 6
7 15 13 13 15 13
1-hour deprivation 24-hour deprivation
ABjk 12 40 56 44 48 40
ΣY2 66 468 830 530 666 450
Mean 3 10 14 11 12 10
Std dev 3.16 4.76 3.92 3.92 5.48 4.08
Std error 1.58 2.38 1.96 1.96 2.74 2.04of mean
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Numerical Example
What is the total sum? What are the marginal
sums?
1hour 24hour Sum
Control 12 44 56
Drug X 40 48 88
Drug Y 56 40 96
Sum 108 132 240
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Two-Factor Analysis
[T] = T2/abn = 2402/(3)(2)(4) = 2,400
[A] = ΣAj2/bn = 562 + 882 + 962/(2)(4) = 2,512
[B] = ΣBk2/an = 1082 + 1322/(3)(4) = 2,424
[AB] = ΣABjk2/n = 122 + 402 + … + 482 + 402/4 = 2,680
[Y] = ΣYijk2 = 66 + 468 + 830 + 530 + 666 + 450 = 3,010
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Numerical Example
0
10
20
30
40
50
60
Control Drug X Drug Y
1-hr deprivation
24-hr deprivation
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Main Effects and Interactionsa1
a2 a2a2
a1
a1
b1 b2 b1 b2 b1 b2
a2a2
a2
a1
a1
a1
b1 b2 b1 b2 b1 b2
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What’s the Story?
Excitement ad Nutrition ad
Children
Adults
Cerealrating
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What’s the Story?
“Not easy to use” “Not difficult to use”
10 seconds
45 seconds
Productevaluation
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What’s the Story?
No advertising Advertising
Milk
Soft drink
Grossmargins
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What’s the Story?
Exceededexpectations
Did not meetexpectations
Low expectations
High expectations
Satisfaction
Metexpectations
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What’s the Story?
Think of 2 reasons Think of 10 reasons
Novices
Experts
BMWevaluation
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Ceiling Effect
Effect of Time on Word Memory
02
468
10
1214
15 minutes 25 minutes
Wo
rds
rem
emb
ered
6 year olds
10 year olds
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Ordinal Interactions
Effect of Caffeine, Exercise on Calories Consumed
1500
2000
2500
3000
No exercise Exercise
Cal
ori
es c
on
sum
ed
No caffeine
Caffeine
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Ordinal Interactions
Effect of Caffeine, Exercise on Hunger
1
3
5
7
9
11
No exercise Exercise
Rat
ing
s o
f h
un
ger
No caffeine
Caffeine