Chapter 6
The
k
2
Factorial Designs
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2k Factorial Design
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Special case of the general factorial design; k
factors, all at two levels
The two levels are usually called low and high
(they could be either quantitative or qualitative)
Very widely used in industrial experimentation
Form a basic “building block” for other very
useful experimental designs
Particularly useful for factor screening
experiments
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Simple Case 22
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Two factors each having two levels
Completely randomized experiment
The effects model
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Example
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Consider an investigation into the effect of the concentration of the
reactant and the amount of the catalyst on the yield in a chemical process
The reactant concentration is factor A having two levels of 15 and 25
percent
The catalyst is factor B with two levels of 1 and 2 pounds
The data obtained are as follows
A: reactant concentration ; 15 and 25 percent
B: catalyst ; 1 and 2 pounds
y: yield
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“-” and “+” denote the low and high levels of a factor, respectively
Low and high are arbitrary terms
Factors can be quantitative or qualitative, although their treatment in the
final model will be different
Geometrically, the four runs form the corners of a square
Notations for treatment combinations
a : A +, B- / b: A -, B + / ab: A +, B + / (1): A -, B 5
•
•
The averaged effect of a factor: the change in response produced by a
change in the level of that factor averaged over the levels of the other
factor
(1),a, b, ab: total of the response observation at all n replicates taken at
the treatment combination
• Main effect of A (denoted by A):
A= Average of the effect of A at the low level of B and the effect of A at the high
level of B
or
A = Difference in the average response of the combinations at A+ and A-
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•
Main effect of B (denoted by B):
•
Interaction effect AB (denoted by AB):
AB = average difference between the effect of A at the high level of B
and the effect of A at the low level of B
or
AB = the average of the right-to left diagonal treatment combinations
minus the average of the left –to right diagonal treatment combination
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Analysis Procedure for a Factorial Design
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Estimate factor effects
Formulate model
a. With replication, use a full model
b. With an unreplicated design, use normal probability plots
Statistical testing (ANOVA)
Refine the model
Analyze residuals (graphical)
Interpret results
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Estimation of Factor Effects
The effect estimate are:
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A = (90+100-60-80)/(2x3)
= 8.33
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 = (90+60-100-80)/(2X3)
= -5.00
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AB = (90+80-60-100)/(2*3)
= 1.67
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A
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A
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The effect of A is positive, implying that increasing A from the low
level to the high level will increase the yield
The effect of B is negative, suggesting that increasing the amount
of catalyst added to the process will decrease the yield
The interaction effect appears to be small relative to the two main
effects
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ANOVA
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ContrastA=ab+a-b-(1): total effect of A
ContrastB=ab+b-a-(1): total effect of B
ContrastAB=ab+(1)-a-b: total effect of AB
Orthogonal contrasts
Note:
Let C be a contrast
E[C]=0
Var[C]=4*n*2
C2/4n2
~ 2(1)
SS= 2(1)* 2 = C2/4n
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Standard order:
it is convenient to
write down the
treatment
combinations in the
order (1), a,b, ab
Main effect A, B
Interaction effect AB
The total of the entire experiment I
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Regression Model
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For the 22 design, the regression model is
where x1 is a coded variable for the factor A and x2 is a coded
variable for the factor B and
and
Then
The regression coefficients are
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For the chemical process experiment,
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 Response Surface
• The model contains only the main effects.=> The fitted response
surface is plane.
• The yield increases as reactant concentration increases and
catalyst amount decreases. => We use a fitted surface to find a
direction of potential improvement for a process
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 Residuals and Model Adequacy
Residual =observed y value – Fitted value
ex) x1 =-1, x2 =-1 =>
27.5+(8.33/2)*(-1) +(-5.0/2)(-1) = 25.835
e1=28-25.835=2.165 , e2=25-25.835=-0.835, e3= 27-25.835=1.165
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Problem
Temperature (0
C)
500C
1000C
A.
B.
C.
D.
E.
Copper Content (%)
40%
80%
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24
20
22
16
25
12
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State the effects model.
Compute the estimates of the effects in the model.
Construct two factor interaction plots
State and test hypotheses related with the ANOVA table
Give a regression model for the data
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