2
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.
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.
2. A factorial experimental allows one to study the interaction between
factors-important in interpretation of combined effects.
Example: A 22 factorial
0
1
Factor A
Factor B
0
1
a 0 b0 a 0 b1
a 1 b0 a 1 b1
3
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.
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.
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.
Stat 402 (Spring 2016): slide set 9
Two-level Factorial Experiments
Stat 402 (Spring 2016): slide set 9
1
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.
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.
2. The treatments consist of all combinations that can be formed from the
different levels of each of the factors chosen for the study.
1. In factorial experiments a number of different factors are investigated
simultaneously.
Advantages and Disadvantages of Factorial Experiments
Last update: March 4, 2016
Stat402 (Spring 2016)
Slide set 9
Stat 402 (Spring 2016): slide set 9
Introduction to Factorial Experiments
Stat 402 (Spring 2016): slide set 9
•
Factor B
0
1
(1)
b
a
ab
k yijk )/n.
The μij may also be denoted by μ(1), μa, μb and μab respectively.
6
These sample means estimates the corresponding population means μij ,
i=1, 2; j=1, 2; respectively.
We may denote these by ȳ(1), ȳa, ȳb and ȳab, respectively.
The treatment means are ȳij , i=1,2; j=1,2 where ȳij = (
0
1
Factor A
Simple Effects of Factor A: μa − μ(1) and μab − μb
Simple Effects of Factor B: μb − μ(1) and μab − μa
The 2 Factorial (contd.)
• between simple effects of A, or equivalently,
• between simple effects of B.
The interaction between factors A and B, denoted by AB, is defined
to be the average difference
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)]
•
•
Definitons:
7
Stat 402 (Spring 2016): slide set 9
2
Stat 402 (Spring 2016): slide set 9
where μij = mean of ij th trt. comb. and ijk ∼ N (0, σ 2).
yijk = μij + ijk , i = 1, 2; j = 1, 2; k = 1, 2 . . . , n
We can write the above model in a more simplified notation as
where i1 = 1, 2; i2 = 1, 2; j = 1, . . . , n
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
yi1,i2,...,ik ,j = μi1,...,ik + i1,...,ik ,j
Here we give the general model for a 2k experiment and assume that a
CRD is the design used. The model is given by:
The Model for 2 factorials
Stat 402 (Spring 2016): slide set 9
5
•
•
k
4
The treatment combinations are: (1), a, b, and ab.
The 2 Factorial
2
(1), a, b, ab, c, ac, bc, and abc.
In this notation, the 8 treatment combinations in a 23 factorial are:
0
1
Factor A
Factor B
0
1
(1)
b
a
ab
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
•
•
:
:
:
:
:
:
:
:
(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
(1)
a
b
ab
Treatment Combination
Factorial Effect
I A B AB
+ +
+ + + - +
+ + + +
10
It is often convenient to put the coefficient of treatments combinations
involved in the effects in a table
A
B
AB
I
The 2 Factorial (contd.)
11
(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
Notes:
Data
•
Stat 402 (Spring 2016): slide set 9
Stat 402 (Spring 2016): slide set 9
2
9
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)].
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 μs; when the estimates are needed we put these subscripts on
the ȳ s, as in the following example.
A
B
AB
I
For ease of notation, we may denote each of these effects and their
estimates by using only the subscripts involved. That is:
•
•
Stat 402 (Spring 2016): slide set 9
The 2 Factorial (contd.)
2
8
 = 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)]
The 2 Factorial (contd.)
2
Main Effect of A
Main Effect of B
Interaction of AB
Overall Mean
It is useful to note the following algebraic relations:
of
of
of
of
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
The mean effect I is defined as 1/4[μab + μa + μb + μ1]
•
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)]
•
12
Stat 402 (Spring 2016): slide set 9
Observed
Total
80
100
60
90
I
+
+
+
+
330
12
27.5
12
9075
AB
+
+
10
6
1.67
12
8.33
k−1
• 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 .
13
4
,
2k (n)
(n). For I,
Stat 402 (Spring 2016): slide set 9
Factorial Effect
A
B
+
+
+
+
50
-30
6
6
8.33
-5
12
12
208.33
75.9
Except for I, the Estimate of Effect =Ê = Contrast/2
Contrast/2k (n)
2 k
SS of Effect
=2(Contrast) /2 (n)
Total SS= yijk - (SS for I)
• The standard error of Ê is given by sE 2k4(n)
•
it is
•
•
Treatment
Combination
(1)
a
b
ab
Contrast
Divisor for Estimate
Estimate of Effect
Divisor for SS
SS of Effect