Slide set 9
Stat402 (Spring 2016)
Last update: March 4, 2016
Stat 402 (Spring 2016): slide set 9
Introduction to Factorial Experiments
1. In factorial experiments a number of different factors are investigated
simultaneously.
2. The treatments consist of all combinations that can be formed from the
different levels of each of the factors chosen for the study.
3. Example: Study effects of 3 factors - light, oxygen, temperature on a
species of marine life. We may consider two levels of light, say, l0-low ,
l1-high, two levels of oxygen and two levels of temperature, etc.
4. Each treatment combination involves one level each of light, oxygen
and temperature- a factorial treatment. Thus there are 2 × 2 × 2 = 8
treatment combinations in this experiment.
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Stat 402 (Spring 2016): slide set 9
Advantages and Disadvantages of Factorial Experiments
1. The factorial arrangement of factors allows an experimenter to obtain
a broad picture of the effects much more efficiently than conducting
separate experiments for each of the factors involved.
2. A factorial experimental allows one to study the interaction between
factors-important in interpretation of combined effects.
3. The number of treatment combinations for factorials is large when it
is necessary to study several factors at different levels. For example, 7
factors at 3 levels each produce 2187 treatment combinations.
4. For large experiments it is difficult to find a set of homogeneous units and
hence the experimental error is likely to be higher than for comparable
single factor experiments.
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Stat 402 (Spring 2016): slide set 9
Two-level Factorial Experiments
The 2k factorials: The 2k factorial experiment involves k factors each at
2 levels. For example a 23 factorial has 3 factors at 2 levels each.
Notation Uppercase letters A, B, C etc., denote the factors and the
levels of each factor are denoted by subscripted lower case letters. For
example, a0, a1, a2 denote 3 levels of factor A.
Example: A 22 factorial
Factor A
0
1
Factor B
0
1
a0b0 a0b1
a1b0 a1b1
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Stat 402 (Spring 2016): slide set 9
Example (Cont’d)
An alternatively notation can be used to denote treatment combinations
in the 2k factorials. Denote the presence of the high level by including a
letter and the low level by excluding the letter. For example, the above
arrangement is equivalent to
Factor A
0
1
Factor B
0
1
(1)
b
a
ab
In this notation, the 8 treatment combinations in a 23 factorial are:
(1), a, b, ab, c, ac, bc, and abc.
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Stat 402 (Spring 2016): slide set 9
The Model for 2k factorials
•
Here we give the general model for a 2k experiment and assume that a
CRD is the design used. The model is given by:
yi1,i2,...,ik ,j = µi1,...,ik + i1,...,ik ,j
•
where i1 = 1, 2; i2 = 1, 2; ik = 1, 2 and j = 1, 2 . . . , n; yi1,i2,...,ik ,j
denotes the response from the j-th experimental unit receiving the
treatment combination (ai1 bi2 , . . . , kik ) and µi1,dots,ık denote the
expected mean response from applying this treatment combination. We
shall discuss the 22 and the 23 factorials as special cases of this model.
Model for a 22 factorial:
yi1,i2,j = µi1,i2 + i1,i2, j
where i1 = 1, 2; i2 = 1, 2; j = 1, . . . , n
We can write the above model in a more simplified notation as
yijk = µij + ijk , i = 1, 2; j = 1, 2; k = 1, 2 . . . , n
where µij = mean of ij th trt. comb. and ijk ∼ N (0, σ 2).
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Stat 402 (Spring 2016): slide set 9
The 22 Factorial
•
The treatment combinations are: (1), a, b, and ab.
Factor A
0
1
Factor B
0
1
(1)
b
a
ab
P
The treatment means are ȳij , i=1,2; j=1,2 where ȳij = ( k yijk )/n.
We may denote these by ȳ(1), ȳa, ȳb and ȳab, respectively.
These sample means estimates the corresponding population means µij ,
i=1, 2; j=1, 2; respectively.
The µij may also be denoted by µ(1), µa, µb and µab respectively.
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Stat 402 (Spring 2016): slide set 9
The 22 Factorial (contd.)
Definitons:
•
•
Simple Effects of Factor A: µa − µ(1) and µab − µb
Simple Effects of Factor B: µb − µ(1) and µab − µa
The main effect of a factor is the average of its simple effects. Thus:
• Main effect of Factor A:
1/2[(µa − µ(1)) + (µab − µb)] = 1/2[µab + µa − µb − µ(1)]
• Main Effects of Factor B:
1/2[(µb − µ(1)) + (µab − µa)] = 1/2[µab + µb − µa − µ(1)]
The interaction between factors A and B, denoted by AB, is defined
to be the average difference
• between simple effects of A, or equivalently,
• between simple effects of B.
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Stat 402 (Spring 2016): slide set 9
•
Thus interaction AB is:
1/2[(µab − µb) − (µa − µ(1))] = 1/2[µab + µa − µb − µ(1)]
or
1/2[(µab − µa) − (µb − µ(1))] = 1/2[µab − µa + µb − µ(1)]
•
The mean effect I is defined as 1/4[µab + µa + µb + µ1]
•
All of these effects may be estimated by substituting the respective
estimates of µ(1), µa, µb and µab in the above expressions. Thus
Estimate
Estimate
Estimate
Estimate
of
of
of
of
Main Effect of A
Main Effect of B
Interaction of AB
Overall Mean
:
:
:
:
 = 1/2[ȳab + ȳa − ȳb − ȳ(1)]
B̂ = 1/2[ȳab + ȳb − ȳa − ȳ(1)]
ˆ = 1/2[ȳab − ȳa − ȳb + ȳ(1)]
AB
Iˆ = 1/4[ȳab + ȳa + ȳb + ȳ(1)]
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Stat 402 (Spring 2016): slide set 9
The 22 Factorial (contd.)
•
For ease of notation, we may denote each of these effects and their
estimates by using only the subscripts involved. That is:
A
B
AB
I
:
:
:
:
1/2[ab+a-b-(1)]
1/2[ab+b-a-(1)]
1/2[ab-a-b+(1)]
1/4[ab+a+b+(1)]
•
That is, when the definition of effects are needed we use these subscripts
on the µ0s; when the estimates are needed we put these subscripts on
the ȳ 0s, as in the following example.
•
The effect A is defined as A = 1/2[µab + µa − µb − µ(1)] and its estimate
 is  = 1/2[ȳab + ȳa − ȳb − ȳ(1)].
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Stat 402 (Spring 2016): slide set 9
•
The 22 Factorial (contd.)
It is useful to note the following algebraic relations:
A
B
AB
I
•
:
:
:
:
(a-1)(b+1)=ab+a-b-1
(a+1)(b-1)=ab-a+b-1
(a-1)(b-1)=ab-a-b+1
(a+1)(b+1)=ab+a+b+1
It is often convenient to put the coefficient of treatments combinations
involved in the effects in a table
Treatment Combination
(1)
a
b
ab
Factorial Effect
I A B AB
+ +
+ + + - +
+ + + +
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Stat 402 (Spring 2016): slide set 9
The 22 Factorial (contd.)
•
Notes:
(a) Each of the effects A, B and AB are contrasts of observations of the
treatment combinations.
(b) These contrasts are orthogonal to each other; hence corresponding
estimated effects are independent of each other.
(c) The contrast for AB may be obtained by multiplying the corresponding
coefficients of the A and B contrasts.
(d) See worked example in Section 6.2 of Montgomery
Data
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Stat 402 (Spring 2016): slide set 9
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Stat 402 (Spring 2016): slide set 9
Treatment
Combination
(1)
a
b
ab
Contrast
Divisor for Estimate
Estimate of Effect
Divisor for SS
SS of Effect
•
it is
•
•
Observed
Total
80
100
60
90
I
+
+
+
+
330
12
27.5
12
9075
Factorial Effect
A
B
+
+
+
+
50
-30
6
6
8.33
-5
12
12
208.33
75.9
AB
+
+
10
6
1.67
12
8.33
k−1
Except for I, the Estimate of Effect =Ê = Contrast/2
Contrast/2k (n)
2 k
SS of Effect
=
(Contrast)
/2 (n)
P 2
Total SS= yijk - (SS for I)
q
• The standard error of Ê is given by sE 2k4(n)
(n). For I,
• The a 100(1 − α)% CI for an effect E is given by Ê ± t α2 ,ν sE ·
where ν = d.f. for M SE .
q
4
2k (n)
,
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