Transcript KNNL-Ch19
Two-Factor Studies with Equal
Replication
KNNL – Chapter 19
Two Factor Studies
• Factor A @ a levels
Factor B @ b levels
ab ≡ # treatments with n replicates per treatment
• Controlled Experiments (CRD) – Randomize abn
experimental units to the ab treatments (n units per trt)
• Observational Studies – Take random samples of n units
from each population/sub-population
• One-Factor-at-a-Time Method – Choose 1 level of one
factor (say A), and compare levels of other factor (B).
Choose best level factor B levels, hold that constant and
compare levels of factor A
Not effective – Poor randomization, logistics, no interaction tests
Better Method – Observe all combinations of factor levels
ANOVA Model Notation – Additive Model
Halo Effect Study: Factor A: Essay Quality(Good,Poor) Factor B: Photo: (Attract,Unatt,None)
A=EQ\B=Pic
j=1: Attract
j=2: Unatt
j=3: None
Row Average
i=1: Good
m11 = 25
m12 = 18
m13 = 20
m1● = 21
i=2: Poor
m21 = 17
m22 = 10
m23 = 12
m2● = 13
Column Average m●1 = 21
m●2 = 14
m●3 = 16
m●● = 17
Additive Effects Model: mij m i j
a
b
1 a b
s.t. i j 0 m
mij
ab i 1 j 1
i 1
j 1
b
1 b
1 b
1
mi mij m i j bm b i j m i
b j 1
b j 1
b
j 1
i mi m
j m j m
m j m j
1 a b
1 a
1 b
m mij mi m j
ab i 1 j 1
a i 1
b j 1
Halo Effect Example:
1 m1 m 21 17 4
1 m1 m 21 17 4
2 m 2 m 13 17 4
2 m2 m 14 17 3
1 2 0
3 m3 m 16 17 1
Mean Score versus - Essay Quality - Additive Model
30
Mean Score
25
20
15
j=1
10
j=2
5
j=3
0
1
2
Essay (1=Good, 2=Poor)
Mean Score versus Picture - Additive Model
30
Mean Score
25
20
15
i=1
i=2
10
5
0
1
2
Picure (1=Attractive, 2=Unattractive, 3=None)
3
ANOVA Model Notation – Interaction Model
Halo Effect Study: Factor A: Essay Quality(Good,Poor) Factor B: Photo: (Attract,Unatt,None)
A=EQ\B=Pic
j=1: Attract
j=2: Unatt
j=3: None
Row Average
i=1: Good
m11 = 23
m12 = 20
m13 = 20
m1● = 21
i=2: Poor
m21 = 19
m22 = 8
m23 = 12
m2● = 13
m●2 = 14
m●3 = 16
m●● = 17
Column Average m●1 = 21
Interaction Model: mij m i j ij
a
s.t.
b
a
b
i 1
i
j 1
j
i 1
ij
j 1
ij
0
ij mij m i j mij m mi m m j m mij mi m j m
Halo Effect Example:
11 23 21 21 17 2
21 19 23 21 17 2
12 20 21 14 17 2
22 8 13 14 17 2
13 20 21 16 17 0
23 12 13 16 17 0
Mean Score versus Essay Quality (A) by Picture - Interaction Model
25
Mean Score
20
15
j=1
10
j=2
5
j=3
0
1
2
Essay (1=Good, 2=Poor)
Mean Score versus Picture (B) by Essay Quality - Interaction Model
25
Mean Score
20
15
i=1
10
i=2
5
0
1
2
Picture (1=Attractive, 2=Unattractive, 3=None)
3
Comments on Interactions
• Some interactions, while present, can be ignored and
analysis of main effects can be conducted. Plots with
“almost” parallel means will be present.
• In some cases, a transformation can be made to
remove an interaction. Typically: logarithmic, square
root, square or reciprocal transformations may work
• In many settings, particular interactions may be
hypothesized, or observed interactions can have
interesting theoretical interpretations
• When factors have ordinal factor levels, we may
observe antagonistic or synergistic interactions
Two Factor ANOVA – Fixed Effects – Cell Means
Fixed Effects - All factor levels of interest are used in the experiment
Cell Means Model:
Yijk mij ijk i 1,.., a; j 1,..., b; k 1,..., n; nT abn
mij mean when Factor A at level i, B at j ijk ~ NID 0, 2
Matrix Form a 2, b 2, n 2 :
Y111
111
m11
111
1 0 0 0
Y
m
1 0 0 0
112
112
11
112
Y121
121
m12
121
0 1 0 0 m11
Y
m
m
0
1
0
0
12 122 12 122
Y = Xβ + ε 122
Y211
211
m21
211
0 0 1 0 m21
Y212
212
m21
212
0 0 1 0 m22
Y
m
0 0 0 1
221
221
22
221
0 0 0 1
Y222
m22
222
222
σ 2 Y = σ 2 ε = σ 2I nT
Two Factor ANOVA – Fixed Effects – Factor Effects
Fixed Effects - All factor levels of interest are used in the experiment
Factor Effects Model:
Yijk m i j ij ijk
1 a b
m mij
ab i 1 j 1
i 1,.., a;
i mi m
j 1,..., b; k 1,..., n; nT abn
j m j m
ij mij mi m j m
m overall mean
i main effect of i th level of A
j main effect of j th level of B
ij
interaction of effect at i th level of A and j th level of B
a
b
a
b
i 1
i
j 1
j
i 1
ij
j 1
Yijk ~ N m i j ij , 2
ij
ijk ~ NID 0, 2
0
independent with
mij m i j ij
Analysis of Variance – Least Squares/ML Estimators
Notation: Observation when A @ i, B @ j , k th replicate: Yijk
Sample mean when A @ i, B @ j :
Sample mean when A @ i :
Sample mean when B @ j :
Overall Mean: Y
Y ij
Yij
1 n
Yijk
n k 1
n
Y i
Y
1 b n
Yijk i
bn j 1 k 1
bn
Y j
Y j
1 a n
Y
ijk bn
an i 1 k 1
Y
1 a b n
Yijk
abn i 1 j 1 k 1
abn
Error Sum of Squares: Q ijk Yijk mij
a
b
n
a
b
n
2
i 1 j 1 k 1
Q
0
mij
2
i 1 j 1 k 1
^
Least squares (and maximum likelihood) estimators: m ij Y ij
^
Fitted values: Y ijk Y ij
^
Residuals: eijk Yijk Y ijk Yijk Y ij
Factor Effects Model Estimators:
^
m
^
^
i Y i Y
Y
^
j Y j Y
^
ij
Y ij Y i Y j Y
Y ijk Y Y i Y Y j Y Y ij Y i Y j Y Y ij
Analysis of Variance – Sums of Squares
Y
Y
Cell Means Model: Yijk Y Yijk Y ij Y ij Y
Y
a
b
n
ijk
i 1 j 1 k 1
Y
a
2
b
n
ijk
i 1 j 1 k 1
Y ij
2
a
b
n
ij
Y
i 1 j 1 k 1
2
SSTO SSE SSTR
df E ab n 1 nT ab
dfTO abn 1 nT 1
Factor Effects Model:
dfTR ab 1
Yijk Y Yijk Y ij Y i Y Y j Y Y ij Y i Y j Y
Y
ijk
i
j
j
Yijk Y ij
k
i
Y
2
i
Y j Y
2
j
k
Y
2
i
k
Y ij Y i Y j Y
k
i
j
k
SSTO SSE SSA SSB SSAB
dfTO abn 1 nT 1
df A a 1
j
df B b 1
df E ab n 1 nT ab
df AB a 1 b 1
2
i
Y
2
Analysis of Variance – Expected Mean Squares
Factor Effects Model:
SSE
i
j
Yijk Y ij
2
df E ab n 1
MSE
k
SSA bn Y i Y
2
df A a 1
MSA
i
a
E MSA 2
bn
i 1
a
2
i
a 1
SSB an Y j Y
2
2
bn mi m
E MSB 2
an
j 1
a 1
MSB
an m j m
b
b 1
2
MSAB
SSB
b 1
2
j 1
b 1
SSAB n Y ij Y i Y j Y
i
SSA
a 1
2
i
2
j
E MSE 2
i 1
df B b 1
b
SSE
ab n 1
2
df AB a 1 b 1
j
SSAB
a 1 b 1
n ij
2
E MSAB 2
i
j
a 1 b 1
2
n mij mi m j m
i
j
a 1 b 1
2
ANOVA Table – F-Tests
Source
df
SS
MS
F*
Factor A
a-1
SSA
MSA=SSA/(a-1)
FA*=MSA/MSE
Factor B
b-1
SSB
MSB=SSB/(b-1)
FB*=MSB/MSE
AB Interaction
(a-1)(b-1)
SSAB
MSAB=SSAB/[(a-1)(b-1)] FAB*=MSAB/MSE
Error
ab(n-1)
SSE
MSE=SSE/[ab(n-1)]
Total
abn-1
SSTO
Testing for Interaction Effects: H 0AB : 11 ... ab 0 mij mi m j m for all (i, j )
*
Test Statistic: FAB
MSAB
MSE
*
Reject H 0 if FAB
F .95; a 1 b 1 , ab n 1
Testing for Factor A Main Effects: H 0A : 1 ... a 0 mi m for all i
Test Statistic: FA*
MSA
MSE
MSB
MSE
No Factor A Level Effects
Reject H 0 if FA* F .95; a 1, ab n 1
Testing for Factor B Main Effects: H 0B : 1 ... b 0 m j m for all j
Test Statistic: FB*
No Interaction
Reject H 0 if FB* F .95; b 1, ab n 1
No Factor B Level Effects
Testing/Modeling Strategy
• Test for Interactions – Determine whether they are
significant or important – If they are:
If the primary interest is the interactions (as is often the case in
behavioral research), describe the interaction in terms of cell
means
If goal is for simplicity of model, attempt simple
transformations on data (log, square, square root, reciprocal)
• If they are not significant or important:
Test for significant Main Effects for Factors A and B
Make post-hoc comparisons among levels of Factors A and B,
noting that the marginal means of levels of A are based on bn
cases and marginal means of levels of B are based on an cases
Factor Effect Contrasts when No Interaction
a
Contrasts among Levels of Factor A : L ci mi
i 1
a
^
c
i 1
i
0
^
Estimator: L ci Y i
Estimated Standard Error: s L
i 1
1 100% CI for L :
a
MSE a 2
ci
bn i 1
L t 1 2 ; ab n 1 s L
^
^
^
Scheffe' Method for Many (or data-driven) tests: L
a 1 F 1 ; a 1, ab n 1 s
Bonferroni Method for g Pre-planned Tests: L t 1 / 2 g ; ab n 1 s L
^
^
^
1
Tukey Method for all Pairs of Factor A Levels: L
q 1 ; a, ab n 1 s L
2
^
Similar Results for Factor B, with a and b being "reversed" in all formulas:
b
L c j m j
j 1
b
c
j 1
j
0
^
b
L cjY j
j 1
^
s L
MSE b 2
cj
an j 1
^
L
Factor Effect Contrasts when Interaction Present
a
b
Contrasts among Cell Means : L cij mij
i 1 j 1
^
a
b
Estimator: L cij Y ij
a
c
i 1 j 1
ij
0
^
Estimated Standard Error: s L
i 1 j 1
1 100% CI for L :
b
MSE a b 2
cij
n i 1 j 1
L t 1 2 ; ab n 1 s L
^
^
^
Scheffe' Method for Many (or data-driven) tests: L
ab 1 F 1 ; ab 1, ab n 1 s
Bonferroni Method for g Pre-planned Tests: L t 1 / 2 g ; ab n 1 s L
^
^
^
1
Tukey Method for all Pairs of Treatment Means: L
q 1 ; ab, ab n 1 s L
2
^
^
L