Chapter 7
Design of Engineering
Experiments Part I
7-1
The Strategy of Experimentation
Every experiment involves a sequence of activities:
1.
Conjecture – the original hypothesis that motivates the experiment.
2.
Experiment – the test performed to investigate the conjecture.
3.
Analysis – the statistical analysis of the data from the experiment.
4.
Conclusion – what has been learned about the original conjecture from the
experiment. Often the experiment will lead to a revised conjecture, and a new
experiment, and so forth.
•
“Best-guess” experiments
– Select (guess) a level combination of factors based on experience and technical
or theoretical knowledge
– Used a lot in practice and often giving a good result
– Disadvantages: no guarantee for an optimal (or satisfactory ) solution in a
reasonable time
•
One-factor-at-a-time (OFAT) experiments
– Selecting a baseline set of levels for each factor and then successively varying
each factor over its range with the other factors held at the baseline level
– Disadvantage: not considering any interaction between factors (“Interaction”:
failure of one factor to produce the same effect on the response at different level
of another factor)
•
Statistically designed experiments
– Based on Fisher’s factorial concept
– Factors are varied together, instead of one at a time
=> correct approach to dealing with several factors
3
Example
Objective: lowering golf score
Response variable: score (per round)
Possible Factors
1.
2.
3.
4.
5.
6.
7.
8.
Type of driver (oversized or regular-sized)
Type of ball used (balata or three piece)
Walking or riding in a golf cart
Drinking while playing (water or beer)
Playing time (morning or afternoon)
Weather temperature (cool or hot)
Type of golf shoe
Windy or calm day
Factors for
experiment
Ignored by
experience
4
OFAT
• Baseline: oversized driver, balata ball, walking and water drinking (O, B,
W, W)
• Result suggests that
- the combination of regular sized driver, riding, drinking water is
optimal (R, -, R, W)
- the ball type is not an important factor
Figure 1-2 Results of the one-factor-at-a-time strategy for the golf experiment
5
Figure 1-3 Interaction between type of driver and type of beverage
for the golf experiment
6
Factorial Design
• In a factorial experiment, all
possible combinations of factor
levels are tested
• Suppose that only two factors
are of interested: type of ball
and type of driver
7
8
7-2


Factorial Experiments
In a factorial experiment all possible combinations of the
levels of the factors are investigated in each complete
replicate of the experiment.
If there are two factors A and B with a levels of factor A and
b levels of factor B, each replicate contains all ab treatments.

Main effect (Factor effect): The change in the mean
response when the factor is changed from low to high
11

Interaction between factors: The average difference in one
factor’s effect at all levels of the other factor
52
40
50
30
40
20
20
12
12



Factorial experiments are the only way to discover interactions between
variables.
A significant interaction can mask the significance of main effects.
When interaction is present, the main effects of the factors involved in
the interaction may not have much meaning.
Problem with One
Factor at a Time
14
7-3





2k Factorial Design
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
7-3.1 22 Design

Two factors each having two levels
Completely randomized experiment

The effects model

Example
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
17





“-” 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 18
• 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-
2
19
•
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
20
Estimation of Factor Effects
The effect estimate are:

A = (90+100-60-80)/(2x3)

A

A



= 8.33

= (90+60-100-80)/(2X3)
= -5.00

AB = (90+80-60-100)/(2*3)
= 1.67
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
22
ANOVA



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
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
23
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
24
Regression Model

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
25
For the chemical process experiment,
26
 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
27
 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= 27-5.835=1.165
28
Problem
Temperature
(0C)
500C
1000C
Copper Content (%)
40%
80%
17
24
20
22
16
25
12
23
A. State the effects model.
B. Compute the estimates of the effects in the model.
C. Construct two factor interaction plots
D. State and test hypotheses related with the ANOVA table
E. Give a regression model for the data
29
7-4
2k Design for k  3 Factors
Consider a 23 Design
Label
(1)
a
b
ab
c
ac
bc
abc




Except for column I, every column has an equal number of + and – signs
The sum of the product of signs in any two columns is zero; the columns in the
table are orthogonal.
Multiplying any column by I leaves that column unchanged (identity element)
The product of any two columns yields a column in the table:
A  B  AB
AB  BC  AB 2C  AC
Estimation of Factor Effects
(1) Main effects
(2) Interaction Effects
Example





A 23 factorial design was used to develop a nitride etch process on a single wafer
plasma etching tool
The gap between the electrodes is factor A having two levels of 0.80 and 1.20 cm
The gas flow is factor B with two levels of 125 and 200 SCCM
The RF power applied to the cathode is factor C with two levels of 275 and 325 W
The data obtained are as follows
35
Main Effect Estimate
A= Average of the effects of A at the low level of B, at the high level of B,
at the low level of C and at the high level of C
36
or
A = Difference in the average response of the combinations at A+ and A-
Interaction Effect Estimate
AB = One-half of the difference between the average A effects at the two levels of B
ABC = the average difference between the AB interactions at the two different
levels of C
37
38

The regression model for predicting etch rate is
, where x1 and x3 are the coded variables representing A and C, respectively
39
Figure 6.7 Response surface and contour plot of etch rate for Example 6-1.
40
2k Factorial Design
41
7-5




Single Replicate of a 2k Design
Resources are usually limited and only allow a single replicate.
These are 2k factorial designs with one observation at each corner of the “cube” (also
called a “single replicate” of the 2k )
This is widely used in screening experiments when there are relatively many factors
under consideration.
Risks…if there is only one observation at each corner, there is a chance of unusual
response observations spoiling the results .


Lack of replication causes potential problems in statistical testing;
•
With only one replicate, there’s no internal estimate of error (or”pure error”).
•
With no replication, fitting the full model results in zero degrees of freedom for
error => F- test cannot be applied.
Potential solutions to this problem:
•
pooling high-order interactions to estimate error (i.e. assuming that certain high
order interactions are negligible and combining their mean squares to estimate
the error). => Most systems are dominated by some of the main effect and low
order interactions. The three-factor and higher-order interactions are usually
negligible.
•
How to select the significant effects? => Normal probability plotting of effects
(Daniels, 1959) :
•
The negligible effects ~ N(0, 2)
•
Significant effects will have nonzero means and will not lie along the
straight line in the normal probability plot.
43
Example
A chemical product is produced in a pressure vessel
 A factorial experiment is carried out in the pilot plant to study the
factors that influence the filtration rate of this product
 Factor A: temperature
 Factor B: pressure
 Factor C: concentration of formaldehyde
 Factor D: stirring rate
 The data obtained are as follows

44
45
46
47
48
• The main effects of A, C and D are positive and if we considered only
these main effects, A+, C+, D+ will be optimal.
• However, there are significant interactions AC and AD !!
• From AC interaction, effect A is large when C= -, and small when
C=+ with the best results with C-, A+.
• From ASD interaction, A+, D+ is the best.
• Thus, A=+, C= -, D=+ is the best combination.
49
Design Projection

In the previous example, B (pressure) is not significant and all interactions involving
B are negligible

We may discard B from the experiment so that the design becomes a 23 factorial in
A,C and D in two replicates

In general, if we have a single replicate of a 2k design, and if h (h<k) factors are
negligible, then the original data correspond to a full two-level factorial in the
remaining k-h factors (2k-h factorial design) with 2h replicates.
50