6BV04

1
6BV04

• Screening Designs

2
Contents

• regression analysis and effects
• 2p-experiments
• blocks
• 2p-k-experiments (fractional factorial
  experiments)
• software
• literature

3
Three factors example
Response deviation filling height
bottles Factors carbon dioxide level
() A pressure (psi) B speed
(bottles/min) C
4
Effects

• How do we determine whether an individual factor
  is of importance?
• Measure the outcome at 2 different settings of
  that factor.
• Scale the settings such that they become the
  values 1 and -1.

5
measurement
-1
1
setting factor A
6
measurement
-1
1
setting factor A
7
measurement
-1
1
setting factor A
8
measurement
-1
1
setting factor A
N.B. effect 2 slope
9
50
measurement
35
-1
1
setting factor A
10
More factors

• We denote factors with capitals
• A, B,
• Each factor only attains two settings
• -1 and 1
• The joint settings of all factors in one
  measurement is called a level combination.

11
More factors
A B
-1 -1
-1 1
1 -1
1 1
Level Combination
12
Notation

• A level combination consists of small letters.
  The small letters denote which factors are set
  at 1 the letters that do not appear are set at
  -1.
• Example ac means A and C at 1, the remaining
  factors at -1
• N.B. (1) means that all factors are set at -1.

13

• An experiment consists of performing
  measurements at different level combinations.
• A run is a measurement at one level combination.
• Suppose that there are 2 factors, A and B.

• We perform 4 measurements with the following
  settings
• A -1 and B -1 (short (1) )
• A 1 and B -1 (short a )
• A -1 and B 1 (short b )
• A 1 and B 1 (short ab )

14
A 22 Experiment with 4 runs
A B yield
(1) -1 -1
b -1 1
a 1 -1
ab 1 1
15
Note
CAPITALS for factors and effects
(A, BC, CDEF)
small letters for level combinations (
settings of the experiments)
(a, bc, cde, (1))
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Graphical display
ab
b
1
B
-1
a
(1)
-1
1
A
17
60
40
1
B
-1
50
35
A
-1
1
18
60
40
1
B
-1
50
35
A
-1
1
2 estimates for effect A
19
60
40
1
B
-1
50
35
A
-1
1
2 estimates for effect A
50 - 35 15
20
60
40
1
B
-1
50
35
A
-1
1
2 estimates for effect A
50 - 35 15
60 - 40 20
21
60
40
1
B
-1
50
35
A
-1
1
2 estimates for effect A
50 - 35 15
60 - 40 20
Which estimate is superior?
22
60
40
1
B
-1
50
35
A
-1
1
2 estimates for effect A
50 - 35 15
60 - 40 20
Combine both estimates ½(50-35) ½(60-40) 17.5
23
60
40
1
B
-1
50
35
A
-1
1
In the same way we estimate the effect B(note
that all 4 measurements are used!)
½(40-35)
½(60-50)
7.5

24
60
40
1
B
-1
50
35
A
-1
1
The interaction effect AB is the difference
between the estimates for the effect A
½(60-40)
½(50-35)
-
2.5
25
Interaction effects

• Cross terms in linear regression models cause
  interaction effects
• Y 3 2 xA 4 xB 7 xA xB
• xA ? xA 1 ?Y?Y 2 7 xB,
• so increase depends on xB. Likewise for xB? xB1
• This explains the notation AB .

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No interaction
55
B low
50
B high
Output
25
20
low
high
Factor A
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Interaction I
55
50
B low
B high
Output
45
20
low
high
Factor A
28
Interaction II
55
50
B low
B high
Output
45
20
low
high
Factor A
29
Interaction III
55
B high
Output
45
B low
20
20
low
high
Factor A
30
Trick to Compute Effects
A B yield
(1) -1 -1 35
b -1 1 40
a 1 -1 50
ab 1 1 60
(coded) measurement settings
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Trick to Compute Effects
A B yield
(1) -1 -1 35
b -1 1 40
a 1 -1 50
ab 1 1 60
Effect estimates
32
Trick to Compute Effects
A B yield
(1) -1 -1 35
b -1 1 40
a 1 -1 50
ab 1 1 60
Effect estimates
Effect A ½(-35 - 40 50 60) 17.5 Effect B
½(-35 40 50 60) 7.5
33
Trick to Compute Effects
A B AB yield
(1) -1 -1 ? 35
b -1 1 ? 40
a 1 -1 ? 50
ab 1 1 ? 60
Effect AB ½(60-40) - ½(50-35) 2.5
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Trick to Compute Effects
A B AB yield
(1) -1 -1 1 35
b -1 1 -1 40
a 1 -1 -1 50
ab 1 1 1 60

AB equals the product of the columns A and B
Effect AB ½(60-40) - ½(50-35) 2.5
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Trick to Compute Effects
I A B AB yield
(1) - - 35
b - - 40
a - - 50
ab 60
Computational rules IA A, IB B, ABAB
etc. This holds true in general (i.e., also for
more factors).
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3 Factors a 23 Design
37
3 Factors a 23 Design
A B C Yield
(1) - - - 5
a - - 2
b - - 7
ab - 1
c - - 7
ac - 6
bc - 9
abc 7
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scheme 23 design
abc7
bc9


ac6
c7


C
effect A
ab1
b7


B
¼(16-28)-3
(1)5
a2


A
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scheme 23 design
abc7
bc9


ac6
c7


C
effect AB
ab1
b7


¼(20-24)-1
B
(1)5
a2


A
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Back to 2 factors Blocking
I A B AB
(1) - -
b - -
a - -
ab
day 1
day 2
Suppose that we cannot perform all measurements
at the same day. We are not interested in the
difference between 2 days, but we must take the
effect of this into account. How do we
accomplish that?
41
Back to 2 factors Blocking
I A B AB day
(1) - - 1
b - - 1
a - - 2
ab 2
hidden block effect
Suppose that we cannot perform all measurements
at the same day. We are not interested in the
difference between 2 days, but we must take the
effect of this into account. How do we
accomplish that?
42
Back to 2 factors Blocking
I A B AB day
(1) - - -
b - - -
a - -
ab
We note that the columns A and day are the
same. Consequence the effect of A and the day
effect cannot be distinguished. This is called
confounding or aliasing).
43
Back to 2 factors Blocking
I A B AB day
(1) - - ?
b - - ?
a - - ?
ab ?
A general guide-line is to confound the day
effect with an interaction of highest possible
order. How can we accomplish that here?
44
Back to 2 factors Blocking
I A B AB day
(1) - -
b - - -
a - - -
ab
Solution day 1 a, b day 2 (1), ab or
interchange the days!
45
Back to 2 factors Blocking
Choose within the days by drawing lots which
experiment must be performed first. In general,
the order of experiments must be determined by
drawing lots. This is called randomisation.
I A B AB day
(1) - -
b - - -
a - - -
ab
Solution day 1 a, b day 2 (1), ab or
interchange the days!
46

• Here is a scheme for 3 factors. Interactions of
  order 3 or higher can be neglected in practice.
  How should we divide the experiments over 2 days?

47
Fractional experiments

• Often the number of parameters is too large to
  allow a complete 2p design (i.e, all 2p
  possible settings -1 and 1 of the p factors).
• By performing only a subset of the 2p experiments
  in a smart way, we can arrange that by
  performing relatively few, it is possible to
  estimate the main effects and (possibly) 2nd
  order interactions.

48
Fractional experiments
I A B AB C AC BC ABC
(1) - - - -
a - - - -
b - - - -
ab - - - -
c - - - -
ac - - - -
bc - - - -
abc
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Fractional experiments
I A B AB C AC BC ABC
(1) - - - -
a - - - -
b - - - -
ab - - - -
c - - - -
ac - - - -
bc - - - -
abc
50
Fractional experiments
I A B AB C AC BC ABC
(1) - - - -
a - - - -
b - - - -
ab - - - -
51
Fractional experiments
I A B AB C AC BC ABC
(1) - - - -
a - - - -
b - - - -
ab - - - -
With this half fraction (only 4 ½8
experiments) we see that a number of columns
are the same (apart from a minus sign)
I -C, A -AC, B -BC, AB -ABC
52
Fractional experiments
I A B AB C AC BC ABC
(1) - - - -
a - - - -
b - - - -
ab - - - -
We say that these factors are confounded or
aliased. In this particular case we have an
ill-chosen fraction, because I and C are
confounded.
I -C, A -AC, B -BC, AB -ABC
53
Fractional experiments Better Choice I
ABC
I A B AB C AC BC ABC
(1) - - - -
a - - - -
b - - - -
ab - - - -
c - - - -
ac - - - -
bc - - - -
abc
54
Fractional experiments Better Choice I
ABC
I A B AB C AC BC ABC
a - - - -
b - - - -
c - - - -
abc
Aliasing structure I ABC, A BC, B AC,
C AB
The other best choice would be I -ABC
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I A B AB C AC BC ABC
a - - - -
b - - - -
c - - - -
abc
In the case of 3 factors further reducing the
number of experiments is not possible in
practice, because this leads to undesired
confounding, e.g. I A BC ABC, B C
AB AC,
56
I A B AB C AC BC ABC
a - - - -
abc
Other quarter fractions also have confounded
main effects, which is unacceptable.
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Further remarks on fractions

• there exist computational rules for aliases.
  E.g., it follows from AC that AB BC. Note
  that I A2 B2 C2 etc. always holds (see the
  next lecture)
• tables and software are available for choosing a
  suitable fraction . The extent of confounding is
  indicated by the resolution. Resolution III is a
  minimal designs with a higher resolution are
  very much preferred.

58
Plackett-Burman designs

• So far we discussed fractional designs for
  screening. This is sensible if one cannot exclude
  the possibility of interactions.
• If one knows based on foreknowledge that there
  are no interactions or if one is for some reason
  is only interested in main effects, than
  Plackett-Burman designs are preferred. They are
  able to detect significant main effects using
  only very few runs. A disadvantage of these
  designs is their complicated aliasing structure.

59
Number of measurements

• For every main or interaction effect that has to
  estimated separately, at least one measurement is
  necessary. If there are k blocks, then this
  requires additional k - 1 measurements. The
  remaining measurements are used for estimation of
  the variance.
• It is important to have sufficient measurements
  for the variance.

60
Choice of design

• After a design has been chosen, the factors A,
  B, must be assigned to the factors of the
  experiment. It is recommended to combine any
  foreknowledge on the factors with the alias
  structure. The individual measurements must be
  performed in a random order.

• never confound two effects that might both be
  significant
• if you know that a certain effect will not be
  significant,you can confound it with an effect
  that might be significant.

61
Centre points and Replications

• If there are not enough measurements to obtain a
  good estimate of the variance, then one can
  perform replications. Another possibility is to
  add centre points .

Centre point

• Adding centre points serves two purposes
• better variance estimate
• allow to test curvature using a lack-of-fit
  test

62
Curvature

• A design in which each factor is only allowed to
  attain the levels -1 and 1, is implicitly
  assuming a linear model. This is because knowing
  only the functions values at -1 and 1, then 1
  and x2 cannot be distinguished. We can
  distinguish them by adding the level 0.
• This is the idea behind adding centre points.

63
Analysis of a Design
A B C Yield
(1) - - - 5
a - - 2
b - - 7
ab - 1
c - - 7
ac - 6
bc - 9
abc 7
64
Analysis of a Design With 2-way Interactions
Analysis Summary ---------------- File name
ltUntitledgt Estimated effects for
Yield --------------------------------------------
-------------------------- average 5.5 /-
0.25 AA -3.0 /- 0.5 BB 1.0 /-
0.5 CC 3.5 /- 0.5 AB -1.0 /-
0.5 AC 1.5 /- 0.5 BC 0.5 /-
0.5 ----------------------------------------------
------------------------ Standard errors are
based on total error with 1 d.f.
65
Analysis of a Design With 2-way Interactions
Analysis of Variance for Yield -------------------
--------------------------------------------------
----------- Source Sum of Squares
Df Mean Square F-Ratio
P-Value ------------------------------------------
-------------------------------------- AA
18.0 1 18.0
36.00 0.1051 BB
2.0 1 2.0 4.00
0.2952 CC 24.5
1 24.5 49.00 0.0903 AB
2.0 1 2.0
4.00 0.2952 AC
4.5 1 4.5 9.00
0.2048 BC 0.5
1 0.5 1.00 0.5000 Total
error 0.5 1
0.5 ----------------------------------------------
---------------------------------- Total (corr.)
52.0 7 R-squared 99.0385
percent R-squared (adjusted for d.f.) 93.2692
percent Standard Error of Est. 0.707107 Mean
absolute error 0.25 Durbin-Watson statistic
2.5 Lag 1 residual autocorrelation -0.375
66
Analysis of a Design Only Main Effects
Analysis Summary ---------------- File name
ltUntitledgt Estimated effects for
Yield --------------------------------------------
-------------------------- average 5.5 /-
0.484123 AA -3.0 /- 0.968246 BB
1.0 /- 0.968246 CC 3.5 /-
0.968246 -----------------------------------------
----------------------------- Standard errors are
based on total error with 4 d.f.
Effect estimates remain the same!
67
Analysis of a Design Only Main Effects
Analysis of Variance for Yield -------------------
--------------------------------------------------
----------- Source Sum of Squares
Df Mean Square F-Ratio
P-Value ------------------------------------------
-------------------------------------- AA
18.0 1 18.0
9.60 0.0363 BB
2.0 1 2.0 1.07
0.3601 CC 24.5
1 24.5 13.07 0.0225 Total
error 7.5 4
1.875 --------------------------------------------
------------------------------------ Total
(corr.) 52.0 7 R-squared
85.5769 percent R-squared (adjusted for d.f.)
74.7596 percent Standard Error of Est.
1.36931 Mean absolute error 0.8125 Durbin-Watson
statistic 2.16667 (P0.3180) Lag 1 residual
autocorrelation -0.125
68
Analysis of a Design with Blocks
Block A B C Yield
(1) 1 - - - 5
ab 1 - 1
ac 1 - 6
bc 1 - 9
a 2 - - 2
b 2 - - 7
c 2 - - 7
abc 2 7
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Analysis of a Design with Blocks With 2-way
Interactions
Analysis of Variance for Yield -------------------
--------------------------------------------------
----------- Source Sum of Squares
Df Mean Square F-Ratio
P-Value ------------------------------------------
-------------------------------------- AA
18.0 1
18.0 BB 2.0 1
2.0 CC
24.5 1 24.5 AB
2.0 1 2.0 AC
4.5 1 4.5 BC
0.5 1
0.5 blocks 0.5 1
0.5 Total error 0.0
0 -------------------------------------------
------------------------------------- Total
(corr.) 52.0 7 R-squared
100.0 percent R-squared (adjusted for d.f.)
100.0 percent
Saturated design 0 df for the error term ? no
testing possible
70
Analysis of a Design with Blocks Only Main
Effects
Analysis of Variance for Yield -------------------
--------------------------------------------------
----------- Source Sum of Squares
Df Mean Square F-Ratio
P-Value ------------------------------------------
-------------------------------------- AA
18.0 1 18.0
7.71 0.0691 BB
2.0 1 2.0 0.86
0.4228 CC 24.5
1 24.5 10.50 0.0478 blocks
0.5 1 0.5
0.21 0.6749 Total error
7.0 3 2.33333 ---------------------
--------------------------------------------------
--------- Total (corr.) 52.0
7 R-squared 86.5385 percent R-squared
(adjusted for d.f.) 76.4423 percent Standard
Error of Est. 1.52753 Mean absolute error
0.75 Durbin-Watson statistic 3.21429
(P0.0478) Lag 1 residual autocorrelation
-0.642857
71
Analysis of a Fractional Design (I -ABC)
A B C Yield
(1) - - - 5
ac - 6
bc - 9
ab - 1
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Analysis of a Fractional Design (I -ABC)
Analysis of Variance for Yield -------------------
--------------------------------------------------
----------- Source Sum of Squares
Df Mean Square F-Ratio
P-Value ------------------------------------------
-------------------------------------- AA-BC
12.25 1
12.25 BB-AC 0.25 1
0.25 CC-AB
20.25 1 20.25 Total error
0.0 0 ----------------------------
--------------------------------------------------
-- Total (corr.) 32.75
3 R-squared 100.0 percent R-squared (adjusted
for d.f.) 0.0 percent
Estimated effects for Yield ----------------------
------------------------------------------------ a
verage 5.25 AA-BC -3.5 BB-AC
-0.5 CC-AB 4.5 ------------------------------
---------------------------------------- No
degrees of freedom left to estimate standard
errors.
73
Analysis of a Design with Centre Points
A B Yield
(1) - - 5
a - 6
b - 9
ab 1
0 0 8
0 0 8
0 0 7
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Analysis of a Design with Centre Points
Analysis of Variance for Yield -------------------
--------------------------------------------------
----------- Source Sum of Squares
Df Mean Square F-Ratio
P-Value ------------------------------------------
-------------------------------------- AA
12.25 1 12.25
36.75 0.0261 BB
0.25 1 0.25 0.75
0.4778 AB 20.25
1 20.25 60.75
0.0161 Lack-of-fit 10.0119
1 10.0119 30.04 0.0317 Pure error
0.666667 2
0.333333 -----------------------------------------
--------------------------------------- Total
(corr.) 43.4286 6 R-squared
75.4112 percent R-squared (adjusted for d.f.)
50.8224 percent Standard Error of Est.
0.57735 Mean absolute error 1.18367 Durbin-Watso
n statistic 0.801839 (P0.1157) Lag 1 residual
autocorrelation 0.524964
P-Value lt 0.05 ? Lack-of-fit!
75
Software

• Statgraphics menu Special -gt Experimental
  Design
• StatLab http//www.win.tue.nl/statlab2/
• Design Wizard (illustrates blocks and fractions)
  http//www.win.tue.nl/statlab2/designApplet.html
  
• Box (simple optimization illustration)
  http//www.win.tue.nl/marko/box/box.html

76
Literature

• J. Trygg and S. Wold, Introduction to
  Experimental Design What is it? Why and Where
  is it Useful?, homepage of chemometrics,
  editorial August 2002 www.acc.umu.se/tnkjtg/Chem
  ometrics/editorial/aug2002.html
• Introduction from moresteam.com
  www.moresteam.com/toolbox/t408.cfm
• V. Czitrom, One-Factor-at-a-Time Versus Designed
  Experiments, American Statistician 53 (1999),
  126-131
• Thumbnail Handbook for Factorial DOE, StatEase