Factorial Treatments (

1
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).
2
Example 1 of Factors
What factors (characteristics, conditions) will
make a wiring harness for a car last longer?
FACTOR Levels Number of strands 7 or
9. Length of unsoldered, 0, 3, 6, or
12. uninsulated wire (in 0.01 inches)
Diameter of wire (gauge) 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.
3
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.
4
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.
5
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_at_(40,60), P_at_(10,20) lbs/acre.
One-factor-at-a-time approach Fix one factor
then vary the other
Treatment N P Yield Parameter T1 60 10 145 m1 T
2 40 10 125 m2 T3 40 20 160 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 N60 and P20 what
would I expect to find?
E(Y N60,P20) m3(m1 - m2) 160 20
180? E(Y N60,P20) m2(m1 - m2)(m3 - m2)
1252035 180?
6
Interaction and Parallel Lines
We apply the N60, P20 treatment and get the
following
Treatment N P Yield Parameter T1 60 10 145 m1 T
2 40 10 125 m2 T3 40 20 160 m3 T4 60 20 130 m4
Yield
Expected T4
180
170
T3
160
N40
T1
150
140
Observed T4
N60
20
130
120
T2
P
20
10
7
Parallel and Non-Parallel Profiles
Parallel Lines gt the effect of the two factors
is additive (independent). Non-Parallel Lines gt
the effect of the two factors interacts
(dependent).
180
170
160
N40
150
140
N60
20
130
120
P
20
10
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.
8
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.)
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General Data Layout Two Factor (a x b) Factorial
Column Factor (B) Row Factor(A) 1 2 3 b Tota
ls 1 T11 T12 T13 T1b A1 2 T21 T22 T23 T2b A2
3 T31 T32 T33 T3b A3 ... a Ta1 T
a2 Ta3 Tab Aa Totals B1 B2 B3 Bb G
yijk observed response for the kth replicate
(k1,,n) for the treatment defined by the
combination of the ith level of the row factor
and the jth level of the column factor.
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Model
mij mean of the ijth table cell, 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.
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Sums of Squares
12
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.)

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Main Effects
Column Factor (B) Row Factor(A) 1 2 3 b 1 m
11 m12 m13 m1b m1? 2 m21 m22 m13 m1b m2? 3 m
31 m32 m13 m1b m3? ... a ma1 ma2 m
a3 mab ma? Totals m?1 m ?2 m ?3 m ?b m ??
Factor A main effects
Testing is via a set of linear comparisons.
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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.
Reject H0 if F gt F(a-1),ab(n-1),a
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Profile Analysis for Factor A
Mean for level 5 of Factor A
Mean for level 1 of Factor A
m53
180
m11
m51
170
m5 ?
160
m ? ?
m1?
150
m13
140
m52
m12
130
120
5
4
1
3
2
Factor A Levels
Profile of mean of Factor A (main effect of A).
Profile for level 2 of Factor B.
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Insignificant Main Effect for Factor A
m51
180
m11
170
m5 ?
m13
160
m ? ?
m1?
150
140
m52
m12
130
120
5
4
1
3
2
Factor A Levels
Is there strong evidence of a Main Effect for
Factor A?
SSA small (w.r.t. SSE) ? No.
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Significant Main Effect for Factor A
180
170
160
m ? ?
150
140
130
120
5
4
1
3
2
Factor A Levels
Is there strong evidence of a Main Effect for
Factor A?
SSA large (w.r.t. SSE) ? Yes.
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Main Effect Linear Comparisons - Factor A
Column Factor (B) Row Factor(A) 1 2 b3 1 m1
1 m12 m13 m1? 2 m21 m22 m13 m2? 3 m31 m32 m13 m3
? 4 m41 m42 m43 m4? a5 m51 m52 m53 m5? Totals
m?1 m ?2 m ?3 m ? ?
Testing via a set of linear comparisons.
Not mutually orthogonal, but together they
represent a-14 dimensions of comparison.
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Main Effect Linear Comparisons - Factor B
Column Factor (B) Row Factor(A) 1 2 b3 1 m1
1 m12 m13 m1 ? 2 m21 m22 m13 m2
? 3 m31 m32 m13 m3 ? 4 m41 m42 m43 m4
? a5 m51 m52 m53 m5 ? Totals m?1 m ? 2 m ? 3 m
? ?
Testing via a set of linear comparisons.
Not mutually orthogonal, but together they
represent b-12 dimensions of comparison.
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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.
Reject H0 if F gt F(b-1),ab(n-1),a
21
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
5
4
5
1
4
3
1
3
2
2
Factor A Levels
Factor A Levels
Differences between levels of Factor B do not
depend on the level of Factor A.
Differences between levels of Factor B do depend
on the level of Factor A.
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Interaction Linear Comparisons
m51
m41
m21
180
m52
m31
Interaction is lack of consistency in differences
between two levels of Factor B across levels of
Factor A.
m22
m42
m53
160
m11
m32
m43
140
m23
m12
m33
120
m13
5
4
1
3
2
Factor A Levels
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.
23
Model for Interaction
Tests for interaction are based on the abij terms
exclusively.
If all abij terms are equal to zero, then there
is no interaction.
24
Overall Test for Interaction
H0 No interaction, HA Interaction exists.
TS
F gt F(a-1)(b-1),ab(n-1),a
RR
25
Partitioning of Total Sums of Squares
TSS SSR SSE SSA SSB SSAB SSE
ANOVA Table
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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 Tukeys 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.
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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) 1 2 3
4
Variety (B) 1 2 3
49, 39 50, 55 43, 38 53, 48
55, 41 67, 58 53, 42 85, 73
66, 68 85, 92 69, 62 85, 99
This is a completely randomized 3 ? 4 factorial
experiment with factor A at a4 levels, and
factor B at b3 levels. There are t3?412
treatments, each replicated n2 times.
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Example in Minitab
Stat gt ANOVA gt Two-way
A B yield 1 1 49 1 1 39 1 2 55 1 2 41 1 3 66 1 3 6
8 2 1 50 2 1 55 2 2 67 2 2 58 2 3 85 2 3 92 3 1 43
3 1 38 3 2 53 3 2 42 3 3 69 3 3 62 4 1 53 4 1 48
4 2 85 4 2 73 4 3 85 4 3 99
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 gt ANOVA gt Two-way gt additive model
Two-way ANOVA yield versus A, B Source DF
SS MS F P A 3
2227.46 742.49 13.86 0.000 B 2
3996.08 1998.04 37.29 0.000 Error 18
964.42 53.58 Total 23 7187.96 S
7.320 R-Sq 86.58 R-Sq(adj) 82.86
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Analyze Main Effects with Tukeys HSD (MTB)
Stat gt ANOVA gt General Linear Model
Use to get factor or profile plots
MTB will use t4 n6 to compare A main effects,
and t3 n8 to compare B main effects.
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Tukey Analysis of Main Effects (MTB)
A 1 subtracted from Difference SE of
Adjusted A of Means
Difference T-Value P-Value 2 14.833
4.226 3.5100 0.0122 3 -1.833
4.226 -0.4338 0.9719 4 20.833
4.226 4.9297 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 B1 B2 B3
Summary A3 A1 A2 A4
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Compare All Level Means with Tukeys HSD (MTB)
If the interaction had been significant, we would
then compare all level means
MTB will use t4312 n2 to compare all level
combinations of A with B.
32
Tukey Comparison of All Level Means (MTB)
A 1, B 1 subtracted from Difference
SE of Adjusted A B of Means
Difference T-Value P-Value 1 2 4.000
6.503 0.6151 0.9999 1 3 23.000
6.503 3.5367 0.0983 2 1 8.500
6.503 1.3070 0.9623 2 2 18.500
6.503 2.8448 0.2695 2 3 44.500
6.503 6.8428 0.0007 3 1 -3.500
6.503 -0.5382 1.0000 3 2 3.500
6.503 0.5382 1.0000 3 3 21.500
6.503 3.3061 0.1395 4 1 6.500
6.503 0.9995 0.9945 4 2 35.000
6.503 5.3820 0.0055 4 3 48.000
6.503 7.3810 0.0003 ... etc. ... A 4, B
2 subtracted from Difference SE
of Adjusted A B of Means
Difference T-Value P-Value 4 3 13.00
6.503 1.999 0.6882
There are a total of t(t-1)/212(11)/266
pairwise comparisons here! Note that if we
wanted just the comparisons among levels of B
(within each level of A), we should use t3
n2. (Not possible in MTB.)
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Example in R
ANOVA Table Model with interaction gt fruit lt-
read.table("fruit.txt",headerT) gt fruit.lm lt-
lm(yieldfactor(A)factor(B)factor(A)factor(B),d
atafruit) gt anova(fruit.lm)
Df Sum Sq Mean Sq F value Pr(gtF) 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 gt summary(lm(yieldfactor(A)factor
(B),datafruit)) Call lm(formula yield
factor(A) factor(B), data fruit) Coefficients
Estimate Std. Error t value
Pr(gtt) (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

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Profile Plot for the Example (R)
Averaging the 2 reps, in each A,B combination
gives a typical point on the graph
interaction.plot(fruitA,fruitB,fruityield)
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) 1 2 3
4
Variety (B) 1 2 3
49, 39 50, 55 43, 38 53, 48
55, 41 67, 58 53, 42 85, 73
66, 68 85, 92 69, 62 85, 99
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 t3?412 treatments.) The analysis
would therefore proceed as in a 3-way ANOVA.