Factorial Treatments (§ 14.3, 14.4, 15.4)
Completely randomized designs - One treatment factor at t levels.
Randomized block design - One treatment factor at t levels, one
block factor at b levels.
Latin square design - One treatment factor at t levels, two block
factors, each at t levels.
Many other blocking structures are available.
Check the literature on Experimental Design.
Now we move on to the situation where the t treatment levels are
defined as combinations of two or more “factors”.
Factor – a controlled variable (e.g. temperature, fertilizer type,
percent sand in concrete mix). Factors can have
several levels (subdivisions).
22-1
Example 1 of Factors
What factors (characteristics, conditions) will make a wiring
harness for a car last longer?
FACTOR
Number of strands
Length of unsoldered,
uninsulated wire
(in 0.01 inches)
Diameter of wire (gauge)
Levels
7 or 9.
0, 3, 6, or 12.
24, 22, or 20
A treatment is a specific combination of levels of the three factors.
T1 = ( 7 strand, 0.06 in, 22 gauge)
Response is the number of stress cycles the harness survives.
22-2
Example 2 of Factors
What is the effect of temperature and pressure on the bonding
strength of a new adhesive?
Factor x1: temperature (any value between 30oF to 100oF)
Factor x2: pressure (any value between 1 and 4 kg/cm2
Factors (temperature, pressure) have continuous levels,
Treatments are combinations of factors at specific levels.
Response is bonding strength - can be determined for any
combination of the two factors.
Response surface above the (x1 by x2) Cartesian surface.
22-3
Other Examples of Factors
•
•
•
•
•
The effect of added Nitrogen, Phosphorus and Potassium on
crop yield.
The effect of replications and duration on added physical
strength in weight lifting.
The effect of age and diet on weight loss achieved.
The effect of years of schooling and gender on Math scores.
The effect of a contaminant dose and body weight on liver
enzyme levels.
Since many of the responses we are interested in are affected by
multiple factors, it is natural to think of treatments as being
constructed as combinations of factor levels.
22-4
One at a Time Approach
Consider a Nitrogen and Phosphorus study on crop yield. Suppose
two levels of each factor were chosen for study: N@(40,60),
P@(10,20) lbs/acre.
One-factor-at-atime approach:
Fix one factor
then vary the
other
Treatment
T1
T2
T3
N
60
40
40
P
10
10
20
Yield
145
125
160
Parameter
m1
m2
m3
H0: m1-m2 = test of N-effect (20 unit difference observed in response).
H0: m2-m3 = test of P-effect (35 unit difference observed in response).
If I examined the yield at N=60 and P=20 what would I expect to find?
E(Y | N=60,P=20) = m3+(m1 - m2) = 160 + 20 = 180?
E(Y | N=60,P=20) = m2+(m1 - m2)+(m3 - m2) = 125+20+35 = 180?
22-5
Interaction and Parallel Lines
We apply the
N=60, P=20
treatment and
get the
following:
Yield
Treatment
T1
T2
T3
T4
N
60
40
40
60
P
10
10
20
20
Yield
145
125
160
130
Parameter
m1
m2
m3
m4
Expected T4
180
170
T3
160
T1
150
140
130
120
Observed T4
20
T2
10
20
P
22-6
Parallel and Non-Parallel Profiles
180
170
160
150
140
130
120
20
10
20
Parallel Lines => the
effect of the two
factors is additive
(independent).
Non-Parallel Lines =>
the effect of the two
factors interacts
P (dependent).
The effect of one factor on the response does not remain the same for different
levels of the second factor. That is, the factors do not act independently of each
other.
Without looking at all combinations of both factors, we
would not be able to determine if the factors interact.
22-7
Factorial Experiment
Factorial Experiment - an experiment in which the response y is
observed at all factor level combinations.
An experiment is not a design. (e.g. one can perform a factorial
experiment in a completely randomized design, or in a
randomized complete block design, or in a Latin square
design.)
Design relates to how the experimental units are arranged,
grouped, selected and how treatments are allocated to units.
Experiment relates to how the treatments are formed. In a
factorial experiment, treatments are formed as combinations of
factor levels.
(E.g. a fractional factorial experiment uses only a fraction
(1/2, 1/3, 1/4, etc.) of all possible factor level
combinations.)
22-8
General Data Layout
Two Factor (a x b) Factorial
Row Factor(A)
1
2
3
…
a
Totals
1
T11
T21
T31
…
Ta1
B1
Column Factor (B)
2
3
…
b
T12 T13 …
T1b
T22 T23 …
T2b
T32 T33 …
T3b
…
… …
…
Ta2 Ta3 …
Tab
B2
B3 …
Bb
Totals
A1
A2
A3
...
Aa
G
n
Tij   yijk  yij
k 1
b
Ai   Tij  yi 
j 1
a
B j   Tij  y j 
i 1
a
b
n
G     yijk  y
i 1
j 1 k 1
yijk= observed response for the kth replicate
(k=1,…,n) for the treatment defined by the
combination of the ith level of the row factor
and the jth level of the column factor.
22-9
Model
mij = mean of the ijth table cell,
yijk  m ij +  ijk
expected value of the response
for the combination for the ith
row factor level and the jth
column factor level.
Overall Test of no treatment differences:
Ho: all mij are equal
Ha: at least two mij differ
Test as in a completely randomized design with a x b treatments.
22-10
Sums of Squares
2
TSS   ( yijk  y )
a
b
n
i 1 j 1 k 1
2
G
2
  yijk 
abn
i 1 j 1 k 1
a
2
SSCells  n ( yij  y )
a
b
i 1 j 1
b
n
1 a b 2 G2
  Tij 
n i 1 j 1
abn
2
SSE   ( yijk  yij )  TSS  SSCells
a
b
n
i 1 j 1 k 1
df total  abn  1
df cells  ab  1
df within  ab(n  1)
SSE
MSE 
 ˆ 2
df within
 SSCells



df
cells 
F
, follows an Fdfcells ,dfwithin
MSE
22-11
After the Overall F test
As with any experiment, if the hypothesis of equal cell means is
rejected, the next step is to determine where the differences are.
In a factorial experiment, there are a number of predefined contrasts
(linear comparisons) that are always of interest.
• Main Effect of Treatment Factor A - Are there differences in the
means of the factor A levels (averaged over the levels of factor B).
• Main Effect of Treatment Factor B - Are there differences in the
means of the factor B levels (averaged over the levels of factor A).
• Interaction Effects of Factor A with Factor B - Are the
differences between the levels of factor A the same for all levels of
factor B? (or equivalently, are the differences among the levels of
factor B the same for all levels of factor A? (Yes  no interaction
present; no  interaction is present.)
22-12
Main Effects
Column Factor (B)
Row Factor(A) 1
2
3
…
b
1
m11
m12
m13
…
m1b m1
2
m21
m22
m13
…
m1b m2
3
m31
m32
m13
…
m1b m3
…
…
…
…
…
…
a
ma1
ma2
ma3
…
mab ma
m1 m 2 m 3
…
m b m 
Totals
Factor A main effects:
H 0 : m1    m a
...
1 b
1
1
m1   m11 +  +  m1b   m1 j
b j 1
b
b
Testing is via a set of linear comparisons.
22-13
Testing for Main Effects: Factor A
There are a levels of Treatment Factor A. This implies that there are a-1
mutually independent linear contrasts that make up the test for main effects for
Treatment Factor A. The Sums of Squares for the main effect for treatment
differences among levels of Factor A is computed as the sum of the individual
contrast sums of squares for any set of a-1 mutually independent linear
comparisons of the a level means. Regardless of the chosen set, this overall
main effect sums of squares will always equal the value of SSA below.
2
a
SSA  nb ( yi  y ) ,
SSA
MSA 
a 1
i 1
H 0 : m1    m a
MSA
F
, follows an Fa 1,ab( n 1)
MSE
Reject H0 if F > F(a-1),ab(n-1),a
22-14
Profile Analysis for Factor A
Mean for level 5
of Factor A
Mean for level 1
of Factor A
m53
180
170
m11
m51
m5 
160
150
140
130
120
m1
m13
m12
m
m52
1
2
Profile for level 2 of Factor B.
3
Factor A Levels
4
5
Profile of mean of Factor A
(main effect of A).
22-15
Insignificant Main Effect for Factor A
180
170
160
150
140
130
120
m11
m13
m1
m51
m5 
m12
m52
1
2
3
Factor A Levels
4
m
5
Is there strong evidence of a Main Effect for Factor A?
SSA small (w.r.t. SSE)  No.
22-16
Significant Main Effect for Factor A
180
170
m
160
150
140
130
120
1
2
3
Factor A Levels
4
5
Is there strong evidence of a Main Effect for Factor A?
SSA large (w.r.t. SSE)  Yes.
22-17
Main Effect Linear Comparisons - Factor A
Column Factor (B)
Row Factor(A)
1
2
b=3
1
m11
m12
m13
m1
2
m21
m22
m13
m2
3
m31
m32
m13
m3
4
m41
m42
m43
m4
a=5
m51
m52
m53
m5
Totals
m1
m 2
m 3
m 
H 0 : m1    m a
1
1
1
 3
 3
 3
E ( yijk )  m ij  m +  i +  j
m1    m11 +   m12 +   m13
Testing via a set of linear comparisons.
L1 : m1  m 5  1   5  0
L2 : m 2  m 5   2   5  0
L3 : m 3  m 5   3   5  0
Not mutually orthogonal, but
together they represent a-1=4
dimensions of comparison.
L4 : m 4  m 5   4   5  0
22-18
Main Effect Linear Comparisons - Factor B
Column Factor (B)
Row Factor(A)
1
2
b=3
1
m11
m12
m13
m1 
2
m21
m22
m13
m2 
3
m31
m32
m13
m3 
4
m41
m42
m43
m4 
a=5
m51
m52
m53
m5 
Totals
m1
m2
m3
m 
H 0 : m 1    m b
L1 : m 1  m 3  1   3  0
L2 : m 2  m 3   2   3  0
1
1
 5
 5
E ( yijk )  m ij  m +  i +  j
1
 5
m 1    m11 +   m 21 +  +   m 51
Testing via a set of linear comparisons.
Not mutually orthogonal, but
together they represent b-1=2
dimensions of comparison.
22-19
Testing for Main Effects: Factor B
There are b levels of Treatment Factor B. This implies that there are b-1
mutually independent linear contrasts that make up the test for main effects for
Treatment Factor B. The Sums of Squares for the main effect for treatment
differences among levels of Factor B is computed as the sum of the individual
contrast sums of squares for any set of b-1 mutually independent linear
comparisons of the b level means. Regardless of the chosen set, this overall
main effect sums of squares will always equal the value of SSB below.
2
SSB  na  ( y j   y ) ,
b
SSB
MSB 
b 1
j 1
H0 : m1    mb
MSB
F
, follows an Fb 1,ab( n 1)
MSE
Reject H0 if F > F(b-1),ab(n-1),a
22-20
Interaction
Two Factors, A and B, are said to interact if the difference in mean
response for two levels of one factor is not constant across levels of
the second factor.
180
180
160
160
140
140
120
120
1
2
3
4
5
Factor A Levels
Differences between levels of Factor B do
not depend on the level of Factor A.
1
2
3
5
4
Factor A Levels
Differences between levels of Factor B do
depend on the level of Factor A.
22-21
Interaction Linear Comparisons
m21
180
Interaction is lack of
consistency in differences
between two levels of
Factor B across levels of
Factor A.
(m11  m12 )  (m51  m52 )  0
(m 21  m 22 )  (m51  m52 )  0
(m31  m32 )  (m51  m52 )  0
(m 41  m 42 )  (m51  m52 )  0
160
140
120
m22
m11
m31
m32
m23
m12
m13
1
m51
m41
m52
m42
m53
m43
m33
4
3
2
Factor A Levels
5
These four linear comparisons tested
simultaneously is equivalent to testing that
the profile line for level 1 of B is parallel to
the profile line for level 2 of B.
Four more similar contrasts would be needed to test the profile line
for level 1 of B to that of level 3 of B.
22-22
Model for Interaction
yijk  m +  i +  j + ij +  ijk
m11  m12  ( m + 1 + 1 + 11 )  ( m + 1 +  2 + 12 )
 ( 1   2 ) + (11  12 )
Tests for interaction are based on the abij terms exclusively.
H 0 : (m11  m12 )  ( m 21  m 22 )  (11  12 )  (21  22 )  0
If all abij terms are equal to zero, then there is no interaction.
22-23
Overall Test for Interaction
SSAB  SSCells  SSA  SSB
2
 n ( yij  yi   y j  + y )
a
b
i 1 j 1
1 a b 2
G2
  Tij  SSA  SSB 
n i 1 j 1
abn
H0: No interaction,
SSAB
TS:
RR:
F
HA: Interaction exists.
(a  1)(b  1) MSAB

MSE
MSE
F > F(a-1)(b-1),ab(n-1),a
22-24
Partitioning of Total Sums of Squares
TSS = SSR + SSE = SSA + SSB + SSAB + SSE
ANOVA Table
Source
df
Between Cells
ab-1
Factor A
a-1
Factor B
b-1
Interaction (a-1)(b-1)
Error(Within Cells)
Total (corrected)
ab(n-1)
abn-1
SS
SSCells
SSA
SSB
SSAB
MS
MSCells
MSA
MSB
MSAB
SSE
TSS
MSE
F
MSCells/MSE
MSA/MSE
MSB/MSE
MSAB/MSE
22-25
Multiple Comparisons in Factorial Experiments
•
•
Methods are the same as in the one-way classification situation i.e.
composition of yardstick. Just need to remember to use:
(i) MSE and df error from the SSE entry in AOV table;
(ii) n is the number of replicates that go into forming the sample
means being compared;
(iii) t in Tukey’s HSD method is # of level means being compared.
Significant interactions can affect how multiple comparisons are
performed.
If Main Effects are significant AND Interactions are NOT significant:
Use multiple comparisons on factor main effects (factor means).
If Interactions ARE significant:
1) Multiple comparisons on main effect level means should NOT
be done as they are meaningless.
2) Should instead perform multiple comparisons among all
factorial means of interest.
22-26
Two Factor Factorial Example: pesticides and fruit trees
(Example 14.6 in Ott & Longnecker, p.896)
An experiment was conducted to determine the effect of 4 different pesticides (factor A) on
the yield of fruit from 3 different varieties of a citrus tree (factor B). 8 trees from each variety
were randomly selected; the 4 pesticides were applied to 2 trees of each variety. Yields
(bushels/tree) obtained were:
Pesticide (A)
Variety (B)
1
2
3
1
49, 39
50, 55
43, 38
53, 48
2
55, 41
67, 58
53, 42
85, 73
3
66, 68
85, 92
69, 62
85, 99
4
This is a completely randomized 3  4 factorial experiment with factor A at a=4 levels, and
factor B at b=3 levels. There are t=34=12 treatments, each replicated n=2 times.
22-27
Example in Minitab
Stat > ANOVA
> Two-way…
Two-way ANOVA: yield versus A, B
Analysis of Variance for yield
Source
DF
SS
MS
F
P
A
3
2227.5 742.5 17.56 0.000
B
2
3996.1 1998.0 47.24 0.000
Interaction
6
456.9
76.2 1.80 0.182
Error
12
507.5
42.3
Total
23
7188.0
Interaction not significant; refit additive model:
Stat > ANOVA > Two-way > additive model
Two-way ANOVA: yield versus A, B
Source
A
B
Error
Total
DF
3
2
18
23
SS
2227.46
3996.08
964.42
7187.96
MS
742.49
1998.04
53.58
F
13.86
37.29
P
0.000
0.000
A
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
B
1
1
2
2
3
3
1
1
2
2
3
3
1
1
2
2
3
3
1
1
2
2
3
3
yield
49
39
55
41
66
68
50
55
67
58
85
92
43
38
53
42
69
62
53
48
85
73
85
99
S = 7.320 R-Sq = 86.58% R-Sq(adj) = 82.86%
22-28
Analyze Main Effects with Tukey’s HSD (MTB)
Stat > ANOVA > General Linear Model
Use to get factor or profile plots
MTB will use t=4
& n=6 to
compare A main
effects, and t=3
& n=8 to
compare B main
effects.
22-29
Tukey Analysis of Main Effects (MTB)
A = 1 subtracted from:
Difference
SE of
A of Means Difference T-Value
2 14.833
4.226
3.5100
3 -1.833
4.226
-0.4338
4 20.833
4.226
4.9297
Adjusted
P-Value
0.0122
0.9719
0.0006
A = 2 subtracted from:
Difference
SE of
Adjusted
A of Means Difference T-Value P-Value
3 -16.67
4.226
-3.944
0.0048
4
6.00
4.226
1.420
0.5038
A = 3 subtracted from:
Difference
SE of
Adjusted
A of Means Difference T-Value P-Value
4
22.67
4.226
5.364
0.0002
All Pairwise Comparisons among Levels of B
B = 1 subtracted from:
Difference
SE of
Adjusted
B of Means Difference T-Value P-Value
2
12.38
3.660
3.381
0.0089
3
31.38
3.660
8.573
0.0000
B = 2 subtracted from:
Difference
SE of
Adjusted
B of Means Difference T-Value P-Value
3
19.00
3.660
5.191
0.0002
Summary:
Summary:
A3 A1 A2 A4
B1 B2 B3
22-30
Compare All Level Means with Tukey’s HSD (MTB)
If the interaction had been significant, we would then compare all level means…
MTB will use
t=4*3=12 & n=2
to compare all
level
combinations of
A with B.
22-31
Tukey Comparison of All Level Means (MTB)
A = 1, B = 1 subtracted from:
Difference
SE of
A B
of Means Difference T-Value
1 2
4.000
6.503
0.6151
1 3
23.000
6.503
3.5367
2 1
8.500
6.503
1.3070
2 2
18.500
6.503
2.8448
2 3
44.500
6.503
6.8428
3 1
-3.500
6.503 -0.5382
3 2
3.500
6.503
0.5382
3 3
21.500
6.503
3.3061
4 1
6.500
6.503
0.9995
4 2
35.000
6.503
5.3820
4 3
48.000
6.503
7.3810
Adjusted
P-Value
0.9999
0.0983
0.9623
0.2695
0.0007
1.0000
1.0000
0.1395
0.9945
0.0055
0.0003
...
etc.
...
A = 4, B = 2 subtracted from:
Difference
SE of
A B
of Means Difference T-Value
4 3
13.00
6.503
1.999
Adjusted
P-Value
0.6882
There are a total of
t(t-1)/2=12(11)/2=66
pairwise comparisons
here!
Note that if we wanted
just the comparisons
among levels of B
(within each level of A),
we should use t=3 &
n=2. (Not possible in
MTB.)
22-32
Example in R
ANOVA Table: Model with interaction
> fruit <- read.table("fruit.txt",header=T)
> fruit.lm <- lm(yield~factor(A)+factor(B)+factor(A)*factor(B),data=fruit)
> anova(fruit.lm)
Df Sum Sq Mean Sq F value
Pr(>F)
factor(A)
3 2227.5
742.5 17.5563 0.0001098 ***
factor(B)
2 3996.1 1998.0 47.2443 2.048e-06 ***
factor(A):factor(B) 6 456.9
76.2 1.8007 0.1816844
Residuals
12 507.5
42.3
Main Effects: Interaction is not significant so fit additive model
> summary(lm(yield~factor(A)+factor(B),data=fruit))
Call: lm(formula = yield ~ factor(A) + factor(B), data = fruit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
38.417
3.660 10.497 4.21e-09 ***
factor(A)2
14.833
4.226
3.510 0.002501 **
factor(A)3
-1.833
4.226 -0.434 0.669577
factor(A)4
20.833
4.226
4.930 0.000108 ***
factor(B)2
12.375
3.660
3.381 0.003327 **
factor(B)3
31.375
3.660
8.573 9.03e-08 ***
mˆ11  38.417

mˆ  38.417 + 14.833
 21

 mˆ 31  38.417  1.833

mˆ12  38.417 + 12.375
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Profile Plot for the Example (R)
interaction.plot(fruit$A,fruit$B,fruit$yield)
Averaging the 2 reps, in
each A,B combination
gives a typical point on
the graph: yij
Level 3, Factor B
Level 2, Factor B
Level 1, Factor B
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Pesticides and Fruit Trees Example in RCBD Layout
Suppose now that the two replicates per treatment in the experiment were
obtained at different locations (Farm 1, Farm 2).
Pesticide (A)
Variety (B)
1
2
3
1
49, 39
50, 55
43, 38
53, 48
2
55, 41
67, 58
53, 42
85, 73
3
66, 68
85, 92
69, 62
85, 99
4
This is now a 3  4 factorial experiment in a randomized complete block design
layout with factor A at 4 levels, factor B at 3 levels, and the location (block) factor at
2 levels. (There are still t=34=12 treatments.)
The analysis would therefore proceed as in a 3-way ANOVA.
22-35