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
+
41.71
W
40.78
40.37
40.21
Temp
40.23
C
38.95
C
Bacteria
39.77
39.19
C
M
E
Inoculation
Estimated Effects

For a k-factor design with n replicates, the
cell means are estimated as
̂i i  Yi i 
1

k
1
k
We can write any effect as a contrast;
interaction contrasts are obtained by
element-wise multiplication of main effect
contrast coefficients.
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
(1)
-1
T
-1
I
-1
BT
+1
BI
+1
TI
+1
BTI
-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:
1
c11Y11.    c22Y22. 
k 1
2

These effects are twice the size of our
usual ANOVA effects.
Estimated Effects

The sum of squares for the estimated
effect can be computed using the sum of
squares formula we learned for contrasts
Effect 2
Effect 2
SS(Effect) 

2
1
2
1 k 1 
c

i
2
 k 1 
n
n 2 
SS(effect)  (Estimated effect) n2
2
k 2
Estimated Effects
Bacteria Example
B effect=(39.19+38.95+40.21+40.78-39.7740.23-40.37-41.71)/4
=-.7375
SSB=(-.7375)2x2=1.088
 The entire ANOVA table for this example
can be constructed in this way

ANOVA
B
T
I
BT
BI
TI
BTI
Error
Total
df
1
1
1
1
1
1
1
0
7
Effect
-.7375
1.2325
.5325
.1925
-.3675
.4225
-.0175
SS
1.0878
3.0381
.5671
.0741
.2701
.3570
.0006
0
5.395
Testing Effects

With replication (n>1)
SSA
k
F
~ F 1,2 n  1
MSE
 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
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

n2
2
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:
so  1.5  med ei
PSE  1.5  med ei
ei  2.5 so
Testing Effects
Though both are consistent estimates,
PSE is more robust
 The following terms are used to test
effects

  1  (1   ) 

ME  t  
, m / 3  PSE
2



  1  (1   )1 / m 

, m / 3  PSE
SME  t  
2



Testing Effects
The df term was developed from a study of
the empirical distribution of PSE2
 ME is a 1- confidence bound for the
absolute value of a single effect
 SME is an exact (since the effects are
independent) simultaneous 1- confidence
bound for all m effects
