design and analysis of experiments

Design and Analysis of Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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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


Example: 23-1 design with I=ABC


Regression model

So, for the four runs 3322110 xβxβxββy

111111111111

1111

and

3

2

1

0

11 Xβ


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


And

123

132

231

0

23

13

12

3

2

1

0

001010100000

)(

211 AβββE


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


Case of 472 III


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


For the case of Reversing the sign in factor D - + + - - + + -

472 III


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


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


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


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


STAT>DOE>Create Factorial Design 2 level fractional (default) Number of factor 7 Choose 1/8 fractional


STAT>DOE>Analyze Factorial Design Only A, B, D are significant


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


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


2nd fraction: STAT>DOE>Modify Design

Specify fold all factor OK


Collecting data STAT>DOE>Analyze Factorial Design


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


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.


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


The complete defining relations for the combined design are: I=ABCG=BCDE=ACDF=ADEG=BDFG

=ABEF=CEFG


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.


Upper half: for N=12, 20, 24, and 36 Lower half: for N=28


Example for Upper half: N=12 and k=11Turn into the first column


Shift down one row!

Add “-” sign


Example for Lower Half: N=28 and k=27

X Y Z


N=28 and k=27 Arrange the design into

X Y ZZ X YY Z X- - - - - - - - Add “-” sign to the 28th row


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


The resolution II Placket-Burman design has Projectivity 3. It will collapse into a

full factorial in any subset of the three factors.


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.


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


Corrected Table 8.25


Alternate P-B design for N=20


No effect is significant according to traditional analysis.


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


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.


How many runs are needed for different number of factors and resolution.


Example: 18 runs for k=9


Example: 12 runs for k=6


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!!


Switching the signs in a single column allows all of the two-factor interactions involving that column to be separated.


Example: 6 factors


Alias


Half-Normal


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


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 –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

design and analysis of experiments

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factor d d

designs fold overby

designs fold overfor

designs fold overassuming

d ab c ef resolution

d dfor effects d

alias relationships