Transcript 5장 - Konkuk

Chapter 5
Factorial Experiments
Pages 183-232
Topics
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Principles of factorial experiments
Two factor factorial design
General factorial design
Fitting response surfaces
Blocking in a factorial design
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Principles of Factorial Experiments
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By a factorial experiment we mean that 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.
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Factor effect (Main effect): The change in the
mean response when the factor is changed
from low to high
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Interaction between factors: The average
difference in response between the levels of
one factor at all levels of the other factors
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Suppose that both factors are quantitative.
Then regression model representation of the
two factor factorial experiment is given by
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Interaction twists the plane so that there is
curvature in the response function. That is,
interaction is a form of curvature.
Factorial experiments are the only way to
discover interactions between variables.
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Problem with one
factor at a time
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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.
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Example
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An engineer is designing a battery for use in a
device.
He has three possible choices for the plate
material of a battery.
Once the device is shipped to the customer, it
will be subjected to some extreme variations in
temperature.
However, temperature will affect the battery life
and can be controlled in the lab for testing.
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To test three plate materials, three temperature
levels, 15, 70 and 1250F, are selected which are
consistent with the customer environment.
4 batteries are tested for each combination of
material and temperature.
All 36 tests are run in random order.
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What effects do material type and temperature
have on the life of the battery ?
Is there a choice of material that would give
uniformly long life regardless of temperature?
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Two-Factor Factorial Design
This is a completely randomized design.
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The effects model is
yijk     i   j  ( )ij   ijk
 : overall mean effect
 i  1, 2,..., a

 j  1, 2,..., b
 k  1, 2,..., n
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i : the effect of the ith level of the row factor
j: the effect of the jth level of the column factor
()ij: the effect of the interaction between i and j
ijk: random error component
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Other models (means model, regression models)
can be useful.
The hypotheses are
 for equality of factor A effects
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for equality of factor B effects
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for the interaction
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ANOVA partitioning of total variability
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a
b
n
a
b
2
(
y

y
)

bn
(
y

y
)

an
(
y

y
)
 ijk ...
 i.. ...
 . j. ...
2
i 1 j 1 k 1
2
i 1
a
j 1
b
a
b
n
 n ( yij .  yi..  y. j .  y... ) 2   ( yijk  yij . ) 2
i 1 j 1
i 1 j 1 k 1
SST  SS A  SS B  SS AB  SS E
df breakdown:
abn  1  a  1  b  1  (a  1)(b  1)  ab(n  1)
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Example
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Life에 대한 상호작용 플롯(적합 평균)
Material
Type
1
2
3
150
평균
125
100
75
50
15
70
T e mpe r a tu r e
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Estimating model parameters
The model has 1+a+b+ab parameters.
yijk
 i  1, 2,..., a

    i   j  ( )ij   ijk  j  1, 2,..., b
 k  1, 2,..., n
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Normal equations
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The solution
The fitted value
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Model adequacy checking
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Multiple comparisons
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General Factorial Design
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Basic procedure is similar to the two-factor case;
all abc…kn treatment combinations are run in
random order.
ANOVA identity is also similar:
SST  SS A  SSB 
 SS ABC 
 SS AB  SS AC 
 SS AB
K
 SSE
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Three factor ANOVA Model is
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Example
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A soft drink bottler is interested in obtaining
more uniform fill heights in the bottles
produced by his manufacturing process.
The filling machine theoretically fills each bottle
to the correct target height, but in practice,
there is variation around this target.
The bottler would like to understand the
sources of this variability better and eventually
reduce it.
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The process engineer can control three variables
during the filling process: the percent
carbonation (A), the operating pressure (B) and
the bottles produced per minute or the line
speed (C).
The pressure and speed are easy to control but
the percent carbonation is more difficult to
control during actual manufacturing because it
varies with product temperature.
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However, for purpose of an experiment, the
engineer can control carbonation at three
levels:10, 12 and 14%.
She choose two levels for pressure (25 and 20
psi) and two levels for line speed (200 and 250
bpm).
She decides to run two replicates of a factorial
design in these 3 factors, with all 24 runs taken
in random order.
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The response variable is the average deviation
from the target fill height observed in a
production run of bottles at each set of
conditions.
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Quantitative vs Qualitative Factors
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The basic ANOVA procedure treats every factor
as if it were qualitative.
Sometimes an experiment will involve both
quantitative and qualitative factors.
This can be accounted for in the analysis to
produce regression models for the quantitative
factors at each level (or combination of levels)
of the qualitative factors.
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Example
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The material type is qualitative while the
temperature is quantitative.
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Since the temperature has the three levels, we
can compute a linear and a quadratic
temperature effect to study effects of
temperature on the battery life.
The interaction effect can also subdivided into
the interactions of the linear and quadratic
temperature factor with material type.
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Figure 5.18 Predicted life as a function of
temperature for the three material types
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Blocking in a Factorial Design
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The effects model is
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Example
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An engineer is studying methods for improving
the ability to detect targets on a radar scope.
Two factors are considered to be important: the
amount of background noise (ground clutter)
and the type of filter placed over the screen.
An experiment is designed using three levels of
ground clutter and two filter types.
We will consider these as fixed-type factors.
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The experiment is performed by randomly
selecting a treatment combination and then
introducing a signal representing the target into
the scope.
The intensity of this target is increased until the
operator observes it.
The intensity level at detection is measured as
the response variable.
The operators are considered as blocks.
Four operators are randomly selected.
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Once an operator is chosen, the order in which
the six treatment combinations are run is
randomly determined.
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Example
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Suppose that because of the setup time
required, only six runs can be made per day.
Days become a second randomization restriction.
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The effects model is
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