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Two-level Factorial Designs
Bacteria Example:– Response: Bill length– Factors:
B: Bacteria (Myco, Control) T: Room Temp (Warm, Cold) I: Inoculation (Eggs, Chicks)
Yandell, B. (2002) Practical Data Analysis for Designed Experiments, Chapman & Hall, London
Two-level Factorial Designs
Bacteria
Temp. Egg Chick
Control
Cold 39.77 40.23
Myco Cold 39.19 38.95
Control
Warm 40.37 41.71
Myco Warm 40.21 40.78
Cube Plot
+
Temp
Inoculation
W
C
CE
C
39.77
40.78
40.21
41.71
38.95
40.23
39.19
40.37
Bacteria
M
Estimated Effects
For a k-factor design with n replicates, the cell means are estimated as
We can write any effect as a contrast; interaction contrasts are obtained by element-wise multiplication of main effect contrast coefficients.
kk iiii Y 11
Estimated Effects
The resulting contrasts are mutually orthogonal.
The contrasts (up to a scaling constant) can be summarized as a table of ±1’s.
Orthogonal Contrast Coefficients
Run
B
T
I
BT
BI
TI
BTI
(1) -1 -1 -1 +1 +1 +1 -1
b +1 -1 -1 -1 -1 +1 +1
t -1 +1 -1 -1 +1 -1 +1
bt +1 +1 -1 +1 -1 -1 -1
i -1 -1 +1 +1 -1 -1 +1
bi +1 -1 +1 -1 +1 -1 -1
ti -1 +1 +1 -1 -1 +1 -1
bti +1 +1 +1 +1 +1 +1 +1
Estimated Effects
If we code contrast coefficients as ±1, the estimated effects are:
These effects are twice the size of our usual ANOVA effects.
.2222.111112
1 YcYc
k
Estimated Effects
The sum of squares for the estimated effect can be computed using the sum of squares formula we learned for contrasts
2
1
2
2
2
21
21
Effect1Effect
SS(Effect)
kk
in
cn
22 2effect) (EstimatedSS(effect) kn
Estimated Effects
Bacteria ExampleB effect=(39.19+38.95+40.21+40.78-
39.77-40.23-40.37-41.71)/4 =-.7375SSB=(-.7375)2x2=1.088 The entire ANOVA table for this
example can be constructed in this way
ANOVA df Effect SS
B 1 -.7375 1.0878
T 1 1.2325 3.0381
I 1 .5325 .5671
BT 1 .1925 .0741
BI 1 -.3675 .2701
TI 1 .4225 .3570
BTI 1 -.0175 .0006
Error 0 0
Total 7 5.395
Testing Effects
With replication (n>1)
Without replication (k large)– Claim higher-order interactions are
negligible and pool them– For k=6, if 3-way (and higher)
interactions are negligible, 42 d.f. would be available for error
FSSA
MSE~ F 1,2k n 1
Testing Effects
Without replication--Normal Probability Plots– If none of the effects is significant, the
effects are orthogonal normal random variables with mean 0 and variance
2
n2 k 2
Testing Effects
Because the effects are normal, they are also independent
IID normal effects can be “tested” using a normal probability plot (Minitab Example)
Yandell uses a half-normal plot You can pool values on the line as
error and construct an ANOVA table
Testing Effects
Lenth (1989) developed a more formal test of effects.
Denote the effects by ei, i=1,…,m. We say that the ei’s are iid N(0,t2),
where t is their common standard error.
Testing Effects
Lenth develops two estimates of the common standard error, t, of the ci’s:
ise
io
emedPSE
emeds
oi 5.25.1
5.1
Testing Effects
Though both are consistent estimates, PSE is more robust
The following terms are used to test effects
PSEmtSME
PSEmtME
m
3/,2
)1(1
3/,2
)1(1
/1
Testing Effects
The df term was developed from a study of the empirical distribution of PSE2
ME is a 1- a confidence bound for the absolute value of a single effect
SME is an exact (since the effects are independent) simultaneous 1-a confidence bound for all m effects