ST 516
Experimental Statistics for Engineers II
Extending a 2k design
The 2k factorial design can detect main effects of factors and
interactions among factors, at the selected levels of the factors.
To predict what will happen at other levels of quantitative factors, we
need a regression equation; e.g. for a model with main effects and
two-factor interactions:
y = β0 +
k
X
j=1
βj x j +
XX
βi,j xi xj + .
i<j
This model can be fitted using the 2k design, with xj the coded
variable for factor j (always ±1).
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The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
But: the model is linear in each xj with the other factors held fixed.
Cannot detect curvature, e.g. if the true dependence is
y = β0 +
k
X
j=1
βj xj +
XX
βi,j xi xj +
i<j
k
X
βj,j xj2 + .
j=1
In this case, extra design points are needed.
E.g. if all factors are quantitative, nC runs at the center point
x1 = x2 = · · · = xk = 0.
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The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
At the original (2k factorial) design points, all xj2 = 1, so
y=
β0 +
k
X
j=1
!
βj,j
+
k
X
βj xj +
j=1
XX
βi,j xi xj + .
i<j
So ȳFP
, the average y at the original design points, estimates
β0 + kj=1 βj,j .
But clearly ȳC , the average y at the center point, estimates β0 .
3 / 13
The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
So we can use ȳF − ȳC to test the hypothesis H0 :
P
βj,j = 0.
This
P will usually detect quadratic terms (but the test has no power if
βj,j = 0).
Test statistic is based on
SSPure quadratic
nF nC (ȳF − ȳC )2
=
,
nF + nC
where nF is the number of points in the original factorial design.
Replicated center values provide nC − 1 degrees of freedom to
estimate σ 2 (“pure error”).
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The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
Example
Filtration rate, with 4 center points (rates 73, 75, 66, 69)
R code
filt2 <- read.table("data/filtration-with-center.txt", header = TRUE)
summary(aov(Rate ~ A * B * C * D + I(A^2) + I(B^2) + I(C^2) + I(D^2),
filt2))
summary(aov(Rate ~ A * C + A * D + I(A^2) + I(C^2) + I(D^2), filt2))
The formulas include all squared variables, but, with this design, only
one can be estimated.
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The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
Full model
Df Sum Sq Mean Sq F value
Pr(>F)
A
1 1870.56 1870.56 115.1115 0.001731
B
1
39.06
39.06
2.4038 0.218821
C
1 390.06 390.06 24.0038 0.016273
D
1 855.56 855.56 52.6500 0.005401
I(A^2)
1
1.51
1.51
0.0931 0.780243
A:B
1
0.06
0.06
0.0038 0.954450
A:C
1 1314.06 1314.06 80.8654 0.002903
B:C
1
22.56
22.56
1.3885 0.323620
A:D
1 1105.56 1105.56 68.0346 0.003731
B:D
1
0.56
0.56
0.0346 0.864273
C:D
1
5.06
5.06
0.3115 0.615686
A:B:C
1
14.06
14.06
0.8654 0.420856
A:B:D
1
68.06
68.06
4.1885 0.133202
A:C:D
1
10.56
10.56
0.6500 0.479099
B:C:D
1
27.56
27.56
1.6962 0.283757
A:B:C:D
1
7.56
7.56
0.4654 0.544069
Residuals
3
48.75
16.25
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
6 / 13
The 2k Factorial Design
**
*
**
**
**
1
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
Reduced model
Df Sum Sq Mean Sq F value
Pr(>F)
A
1 1870.56 1870.56 99.7122 1.830e-07
C
1 390.06 390.06 20.7927 0.0005354
D
1 855.56 855.56 45.6066 1.356e-05
I(A^2)
1
1.51
1.51 0.0806 0.7809238
A:C
1 1314.06 1314.06 70.0474 1.359e-06
A:D
1 1105.56 1105.56 58.9331 3.502e-06
Residuals
13 243.88
18.76
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
7 / 13
The 2k Factorial Design
***
***
***
***
***
1
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
The “Residuals” line in the reduced model may be decomposed into
“Lack of Fit” and “Pure Error”:
summary(aov(Rate ~ A * C + A * D + I(A^2) + I(C^2) + I(D^2) +
factor(A):factor(B):factor(C):factor(D), filt2))
Output
Df
A
1
C
1
D
1
I(A^2)
1
A:C
1
A:D
1
factor(A):factor(B):factor(C):factor(D) 10
Residuals
3
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05
8 / 13
Sum Sq
1870.56
390.06
855.56
1.51
1314.06
1105.56
195.13
48.75
Mean Sq F value
Pr(>F)
1870.56 115.1115 0.001731 **
390.06 24.0038 0.016273 *
855.56 52.6500 0.005401 **
1.51
0.0931 0.780243
1314.06 80.8654 0.002903 **
1105.56 68.0346 0.003731 **
19.51
1.2008 0.494185
16.25
. 0.1
1
The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
The line for factor(A):factor(B):factor(C):factor(D) is Lack of Fit, with 10
d.f. pooled from the omitted terms.
The residuals line is now Pure Error, with the 3 d.f. from the 4 center
point runs.
9 / 13
The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
Central Composite Design
Adding center points allows you to detect pure quadratic terms, but
not to estimate them.
Need to add more points; e.g. axial points:
x1 = ±α, x2 = · · · = xk = 0,
x1 = 0, x2 = ±α, x3 = · · · = xk = 0,
and so on; 2k extra points when unreplicated (usual case).
Choice of α (nF = number of factorial runs):
α = 1 convenient, as it brings in no new levels;
√
α = 4 nF gives desirable property (rotatability).
10 / 13
The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
Center Points with Qualitative Factors
For a qualitative factor (e.g. equipment from two suppliers), x = 0 is
infeasible.
Use two center points, one each at the high and low levels of the
qualitative factor.
Same as axial points with α = 1, but usually replicated.
11 / 13
The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
Coded Variables and Engineering Units
A quantitative factor x ∗ has levels x−∗ and x+∗ .
The coded variable is
x ∗ − 21 x+∗ + x−∗
x=
.
1
(x+∗ − x−∗ )
2
Regression equation in coded variables can be converted to
engineering units for later use, e.g in predicting the response for
arbitrary levels.
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The 2k Factorial Design
Extending a 2k design
ST 516
Experimental Statistics for Engineers II
The same equation results from fitting directly to the original factor
levels.
Coded variable form
is useful for the experimenter:
Gives all effects and interactions;
The t-statistics are equivalent to F -statistics in the ANOVA
table.
Engineering units form
is useful for others:
Does not depend on experimental levels of factors;
Coefficients have a different interpretation: a regression
coefficient represents the effect of changing a factor by 1
(engineering) unit, not the effect of changing from low level to
high level (actually, one-half that effect).
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The 2k Factorial Design
Extending a 2k design