Chapter 6:
Two-level designs
Petter Mostad
[email protected]
Factorial designs
• We discussed previously various factorial
designs; now, the 2k design
• This design is well suited for factor
screening, i.e., searching among many
factors for those which should be
investigated more closely.
The Simplest Case: The 22
“-” and “+” denote
the low and high
levels of a factor,
respectively
Low and high are
arbitrary terms
Geometrically, the
four runs form the
corners of a square
Factors can be
quantitative or
qualitative, although
their treatment in the
final model will be
different
DOE 6E Montgomery
Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery
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Estimation of Factor Effects
A = y A+ − y A−
ab + a b + (1)
−
2n
2n
= 21n [ab + a − b − (1)]
=
B = yB + − yB −
ab + b a + (1)
−
2n
2n
= 21n [ab + b − a − (1)]
See textbook, pg. 205-206 For
manual calculations
The effect estimates are:
A
= 8.33, B = -5.00, AB = 1.67
Practical interpretation?
=
ab + (1) a + b
AB =
−
2n
2n
= 21n [ab + (1) − a − b]
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Contrasts
• A linear combination
of parameters
• How they can be used
to compute main and
interaction effects
• Orthogonal contrasts
A
B
AB
I
-
-
+
+
-
+
-
+
+
-
-
+
+
+
+
+
Sums of squares
• For two-factor experiments, sums of squares
can be computed easily from contrasts.
• For 22 experiment, the SS is equal to the
contrast squared, divided by 4n, where n is
the number of replications at each setting
• For each sum of square computed this way,
there is only 1 degree of freedom.
Statistical Testing - ANOVA
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Residuals and Diagnostic Checking
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The connection to linear models
and to response surfaces
• One can investigate the same model by
formulating it as a linear model, fitting it
using least squares:
y = β 0 + β1 x1 + β 2 x2 + β 3 x1 x2 + ε
• If the factor levels are choices of values for
a continuous factor, one can investigate
further by fitting a response surface
The 23 Factorial Design
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Effects in The 23 Factorial Design
A = y A+ − y A−
B = yB + − yB−
C = yC + − yC −
etc, etc, ...
Analysis
done via
computer
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An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
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Table of – and + Signs for the 23 Factorial Design (pg. 214)
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Properties of the Table
• 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
• 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
• Orthogonal design
• Orthogonality is an important property shared by all factorial designs
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Estimation of Factor Effects
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ANOVA Summary – Full Model
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Model Summary Statistics (pg. 222)
• Standard error of model coefficients (full
model)
se( βˆ ) = V ( βˆ ) =
σ2
n2
k
=
MS E
2252.56
=
= 11.87
k
n2
2(8)
• Confidence interval on model coefficients
βˆ − tα / 2,df se( βˆ ) ≤ β ≤ βˆ + tα / 2,df se( βˆ )
E
E
DOE 6E Montgomery
Computing confidence intervals
for effects
• The effect is computed as some difference
of averages
• Find an estimate of its standard deviation,
and divide by this
• Use this error estimate in the confidence
interval, together with a t distribution, with
a df equal to the number of degrees of
freedom used in the estimate above.
Example
• Assume now we have a 23 factorial experiment, but with
10 repetitions at each setting (total of 80 experiments).
• A main effect is estimated as an average of 40 numbers
minus an average of 40 numbers.
• Assume that, at each setting of factors, the population of
possible observations has variance 2
• The standard error (estimated standard deviation) of the
main effect can be estimated as
σ 2 / 40 + σ 2 / 40
• The variances
2
must also be estimated from the data
Example cont.
• We have a 23 factorial experiment with each
observation repeated 10 times.
• We get 8 variance estimated, each based on
10 numbers. s−2−− , s−2−+ , s−2+ − , s−2+ + , s+2−− , s+2−+ , s+2+ − , s+2+ +
• The pooled estimate for 2 becomes
2
2
2
2
2
2
2
2
9
s
+
9
s
+
9
s
+
9
s
+
9
s
+
9
s
+
9
s
+
9
s
−−+
−+−
−++
+ −−
+−+
++−
+++
s 2p = − − −
9+9+9+9+9+9+9+9
• It has 9 times 8 equals 72 df.
Example cont.
• If
– The effect estimate is e
– The value such that the t distribution has /2
probability of being above this value is t /2,df
– The number of values averaged over in the
difference used to compute e is n (for us, n=40)
• Then the confidence interval for e is
e ± tα / 2,df s 2p (1 / n + 1 / n)
The full 2k model
• The above is easily generalized to k factors
• One may visualize part of the results in the same
way
• Tables of contrasts can be generalized
• Computations of sums of squares can be done in a
similar way (see textbook for formula)
• ANOVA tables can be done in a similar way
Advantages with factorial
experiments
• Well suited for searching for factors
with influence
• Useful for iterative learning
• Indicate direction for further
experiments
• Simple to analyse and understand
• The ”standard order”
• Inert and active factors
• Difference to ”one-at-a-time”
approaches
”Standardorder”
order
”Standard
+
+
+
+
+
+
+
+
+
+
+
+