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Design and Analysis of Experiments. Dr. Tai- Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC. Two-Level Fractional Factorial Designs. Dr. Tai- Yue Wang Department of Industrial and Information Management - PowerPoint PPT Presentation
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Design and Analysis of Experiments
Dr. Tai-Yue Wang Department of Industrial and Information Management
National Cheng Kung UniversityTainan, TAIWAN, ROC
1/33
Two-Level Fractional Factorial Designs
Dr. Tai-Yue Wang Department of Industrial and Information Management
National Cheng Kung UniversityTainan, TAIWAN, ROC
2/33
Outline Introduction The One-Half Fraction of the 2k factorial Design The One-Quarter Fraction of the 2k factorial
Design The General 2k-p Fractional Factorial Design Alias Structures in Fractional Factorials and Other
Designs Resolution III Designs Resolution IV and V Designs Supersaturated Designs
Alias Structures I Fractional Factorials and other Designs
Assuming that we use the following regression equation to fit the experimental results:
where y is an n x 1 vector of the response X1 is an n x p1 matrix β1 is a p1 x 1 vector
Thus the estimated of β1 via LSE is
11βXy
yX)X(Xβ T1
11
T11
Alias Structures I Fractional Factorials and other Designs
Suppose that the true model is
where X2 is an n x p2 matrix with additional variables
β2 is a p2 x 1 vector
2211 βXβXy
Alias Structures Fractional Factorials and other Designs
Thus the expected parameters
The matrix is called alias matrix
The elements of A operating on β2 identify the alias relationships for the parameters in the vector β1
21
22T1
11
T111
AβββXX)X(Xββ
)(E
2T1
11
T1 XX)X(XA
Alias Structures Fractional Factorials and other Designs
Example: 23-1 design with I=ABC
Alias Structures Fractional Factorials and other Designs
Regression model
So, for the four runs 3322110 xβxβxββy
111111111111
1111
and
3
2
1
0
11 Xβ
Alias Structures Fractional Factorials and other Designs
Suppose the true model is
and
322331132112
3322110
xxxxxxxxxy
111111
111111
and 2
23
13
12
Xβ2
Alias Structures Fractional Factorials and other Designs Now try to find A
004040400000
41
004040400000
)4( 144
2T1
11
T1
II
XX)X(XA
Alias Structures Fractional Factorials and other Designs
And
123
132
231
0
23
13
12
3
2
1
0
001010100000
)(
211 AβββE
Alias Structures Fractional Factorials and other Designs
Comparison[A] A+BC [B] B+AC [C] C+AB
123
132
231
0
3
2
1
0
)(
1βE
Resolution III Designs -- Constructing
Resolution III designs are useful for screening (5 – 7 variables in 8 runs, 9 - 15 variables in 16 runs, for example)
A saturated design has k = N – 1 variables
Examples of saturated design:132
III472
III
Resolution III Designs -- Constructing
Case of 472 III
Resolution III Designs -- Constructing
Can be used to generate factors fewer than 7
For example,
472 III
362 III
Resolution III Designs – Fold over
By combining fractional factorial designs that certain signs are switched , one can systematically isolate effects of the potential interest
This type of sequential experiments is called a fold over of the original design
Resolution III Designs – Fold over
For the case of Reversing the sign in factor D - + + - - + + -
472 III
Resolution III Designs – Fold over
Original effects Reversed effects[A]’ A-BD+CE+FG [B]’ B-AD+CF+EG [C]’ C+AE+BF+DG[D]’ D-AB-CG-EF[-D]’ -D+AB+CG+EF[E]’ E+AC+BG-DF [F]’ F+BC+AG-DE [G]’ G-CD+BE+AF
Resolution III Designs – Fold over
Assuming the three-factor and higher interactions are insignificant, one can combine the two fractions
For effect of the factor D½ [D]+1/2[D]’ D
For effects ½ [D]-1/2[D]’ AB+C+EF
Resolution III Designs – Fold over
In general, if we add to a fractional design of resolution III or higher a further fraction with signs of a single factor reversed, the combined design will provide the estimates of the man effect of that factor and its two-factor interactions
This is a single-factor fold over
Resolution III Designs – Fold over
If we add to a fractional design of resolution III a second fraction with signs of all the factors are reversed, the combined design break the alias link between all main effects and their two-factor interaction.
This is a full fold over
Resolution III Designs – example (7—1/9)
Eye focus, Response= time 7 factors Screening experiment
Resolution III Designs – example (7—2/9)
STAT>DOE>Create Factorial Design 2 level fractional (default) Number of factor 7 Choose 1/8 fractional
Resolution III Designs – example (7—3/9)
STAT>DOE>Analyze Factorial Design Only A, B, D are significant
Resolution III Designs – example (7—4/9)
Examining the alias structure
We are not sure if A or BD, B or AD, D or AB are significant!!!!!
Alias Structure (up to order 3)I + A*B*D + A*C*E + A*F*G + B*C*F + B*E*G + C*D*G + D*E*FA + B*D + C*E + F*G + B*C*G + B*E*F + C*D*F + D*E*GB + A*D + C*F + E*G + A*C*G + A*E*F + C*D*E + D*F*GC + A*E + B*F + D*G + A*B*G + A*D*F + B*D*E + E*F*GD + A*B + C*G + E*F + A*C*F + A*E*G + B*C*E + B*F*GE + A*C + B*G + D*F + A*B*F + A*D*G + B*C*D + C*F*GF + A*G + B*C + D*E + A*B*E + A*C*D + B*D*G + C*E*GG + A*F + B*E + C*D + A*B*C + A*D*E + B*D*F + C*E*F
Resolution III Designs – example (7—5/9)
Note that ABD is one of the word in defining relation, do not project into a full 23 factorial in ABD
It does project into two replicates of a 23-1 design.
23-1 is a resolution III design, too Try fold over
472 III
Resolution III Designs – example (7—6/9)
2nd fraction: STAT>DOE>Modify Design
Specify fold all factor OK
Resolution III Designs – example (7—7/9)
Resolution III Designs – example (7—8/9)
Collecting data STAT>DOE>Analyze Factorial Design
Resolution III Designs – example (7—9/9)
Though B, D, BD, and AF are significant, B and D are distinguishable
BD is aliased with CE and FG AF is aliased with CD and BE.
A + B*C*G + B*E*F + C*D*F + D*E*GB + A*C*G + A*E*F + C*D*E + D*F*GC + A*B*G + A*D*F + B*D*E + E*F*GD + A*C*F + A*E*G + B*C*E + B*F*GE + A*B*F + A*D*G + B*C*D + C*F*GF + A*B*E + A*C*D + B*D*G + C*E*GG + A*B*C + A*D*E + B*D*F + C*E*FA*B + C*G + E*F A*C + B*G + D*FA*D + C*F + E*G A*E + B*F + D*GA*F + B*E + C*D A*G + B*C + D*EB*D + C*E + F*G
Resolution III Designs – Fold over
To find the defining relation for a combined design, one can assume that the first fraction has L words and the fold over fraction has U words.
Thus the combined design will have L+U-1 words used as a generators.
Resolution III Designs – Fold over
For example, Generators for the first fraction:
I=ABD, I=ACE, I=BCF, I=ABCG Generators for the second fraction:
I=-ABD, I=-ACE, I=-BCF, I=ABCG We have switched the signs on the generators
with an odd number of letters
472 III
Resolution III Designs – Fold over
The complete defining relations for the combined design are: I=ABCG=BCDE=ACDF=ADEG=BDFG
=ABEF=CEFG
Resolution III Designs – Fold over
Usually the second fraction are different from the first fraction in day, time, shift, material, methods.
This leads to the blocking situation.
Resolution III Designs – Plackett-Burman Designs
For the case of k=N-1 variables in N runs, where N is a multiple of 4, one can use fold over if N is a power of 2.
However, N=12, 20, 24, 28 and 36, The Placket-Burman is of interest.
Because these design cannot be represented as cubes, called non-geometric designs.
Two ways to generate these designs, check example 8.
Resolution III Designs – Plackett-Burman Designs
Upper half: for N=12, 20, 24, and 36 Lower half: for N=28
Resolution III Designs – Plackett-Burman Designs
Example for Upper half: N=12 and k=11Turn into the first column
Resolution III Designs – Plackett-Burman Designs
Shift down one row!
Add “-” sign
Resolution III Designs – Plackett-Burman Designs
Example for Lower Half: N=28 and k=27
X Y Z
Resolution III Designs – Plackett-Burman Designs
N=28 and k=27 Arrange the design into
X Y ZZ X YY Z X- - - - - - - - Add “-” sign to the 28th row
Resolution III Designs – Plackett-Burman Designs
Alias structure Messy and complicated Main effects are partially aliased with every two-
factor interaction not involving itself Non-regular design For the case of N=12
Projected into three replicates of a full 22 design in any two of the original 11 factors
Projected into a full 23 factorial plus a 23-2III
fractional factorial
Resolution III Designs – Plackett-Burman Designs
The resolution II Placket-Burman design has Projectivity 3. It will collapse into a
full factorial in any subset of the three factors.
Resolution III Designs – example (8—1/7)
12 factors If 212-8 fractional is used, all 12 main effects
are aliased with four two-factor interactions. Additional experiments could be required Use 20 run Placket-Burman design Two kinds of designs, one is to follow the
text and Minitab The other is to follow Example 8 in the text.
Resolution III Designs – example (8—2/7)
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19
1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +12 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -13 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +14 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +15 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -16 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -17 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -18 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -19 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +110 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -111 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +112 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -113 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +114 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +115 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +116 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +117 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -118 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -119 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +120 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1
Add “+” sign
Reverse “+” and “-” sign in text
Resolution III Designs – example (8—3/7)
Corrected Table 8.25
Resolution III Designs – example (8—4/7)
Alternate P-B design for N=20
Resolution III Designs – example (8—5/7)
No effect is significant according to traditional analysis.
Resolution III Designs – example (8—6/7)
Use stepwise regressionStepwise Regression: y versus X1, X2, ... Alpha-to-Enter: 0.1 Alpha-to-Remove: 0.15Response is y on 19 predictors, with N = 20Step 1 2 3 4 5 6Constant 200.0 200.0 200.0 200.0 200.0 200.0X2 11.8 11.8 11.8 10.0 10.0 9.9T-Value 2.51 2.78 3.65 4.02 7.30 7.99P-Value 0.022 0.013 0.002 0.001 0.000 0.000X4 9.6 12.0 12.0 12.0 12.1T-Value 2.27 3.64 4.82 8.76 9.78P-Value 0.037 0.002 0.000 0.000 0.000x1x2 -12.0 -12.0 -12.0 -12.5T-Value -3.64 -4.82 -8.76 -9.91P-Value 0.002 0.000 0.000 0.000x1x4 9.0 9.0 9.5T-Value 3.62 6.57 7.54P-Value 0.003 0.000 0.000X1 8.0 8.0T-Value 5.96 6.60P-Value 0.000 0.000X5 2.6T-Value 2.04P-Value 0.062S 21.0 18.9 14.4 10.9 6.00 5.42R-Sq 25.95 43.12 68.89 83.38 95.30 96.44R-Sq(adj) 21.83 36.43 63.05 78.94 93.63 94.80
Resolution III Designs – example (8—7/7)
Fitted model:
4121421 91212108200 xxxxxxxy
Resolution IV and V Designs -- Resolution IV Designs
A 2k-p fractional is of resolution IV if the main effects are clear of two-factor interactions and some two-factor interactions are aliased with each other.
Any 2k-pIV design must contain at least 2k runs.
Resolution IV designs that contain 2k runs are called minimal designs.
Resolution IV designs maybe obtained from resolution III designs by the process of fold over.
Resolution IV and V Designs -- Resolution IV Designs
Resolution IV and V Designs -- Resolution IV Designs
How many runs are needed for different number of factors and resolution.
Resolution IV and V Designs -- Resolution IV Designs
Example: 18 runs for k=9
Resolution IV and V Designs -- Resolution IV Designs
Example: 12 runs for k=6
Resolution IV and V Designs -- Resolution IV Designs
These designs are non-regular designs They are resolution IV with minimum runs No quarantine on orthogonal Useful alternative in screening the main effects If two-factor interaction are proven important,
step-wise regression is used to estimate The price paid for reducing run number is the
complicated alias table
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Fold over in resolution III is to separate the main effect
We can’t use the full fold-over procedure given previously for Resolution III designs – it will result in replicating the runs in the original design.
That is, runs are in different order!!
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Switching the signs in a single column allows all of the two-factor interactions involving that column to be separated.
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Example: 6 factors
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Alias
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Half-Normal
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Since AB is aliased with CE, we do not know whether this is AB or CE or both.
Fold over !! Setting up a new fraction of 26-1
IV and changing sign of factor A
STAT>DOE>Modify design fold over Specify fold just one factorAOK
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Sometimes we can use partial fold over to reduce the run number
In previous example, one can select half of the new fraction.
Here we choose the “-” half of the new fraction because the “-” part has better response in the original 16 runs
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Resolution IV and V Designs –Resolution V Designs
In Resolution V designs, main effects and two-factor interactions do not have other main effect and two-factor interactions as their aliases.
For k=6, 26-1V is required
How about non-regular design? How about k=8? non-regular designs are not orthogonal!! The precision of estimation is higher than the
orthogonal one.
Resolution IV and V Designs –Resolution V Designs
Resolution IV and V Designs –Resolution V Designs