Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC This is a basic course blah, blah, blah…

Two-Level Fractional Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC This is a basic course blah, blah, blah…

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

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

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

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

Alias Structures Fractional Factorials and other Designs Suppose the true model is and

Alias Structures Fractional Factorials and other Designs Now try to find A

Alias Structures Fractional Factorials and other Designs

Alias Structures Fractional Factorials and other Designs Comparison [A] A+BC [B] B+AC [C] C+AB

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:

Resolution III Designs -- Constructing Case of

Resolution III Designs -- Constructing Can be used to generate factors fewer than 7 For example,

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  - + + - - + + -

Resolution III Designs – Fold over 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 Original effects

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 D ½ [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*F A + B*D + C*E + F*G + B*C*G + B*E*F + C*D*F + D*E*G B + A*D + C*F + E*G + A*C*G + A*E*F + C*D*E + D*F*G C + A*E + B*F + D*G + A*B*G + A*D*F + B*D*E + E*F*G D + A*B + C*G + E*F + A*C*F + A*E*G + B*C*E + B*F*G E + A*C + B*G + D*F + A*B*F + A*D*G + B*C*D + C*F*G F + A*G + B*C + D*E + A*B*E + A*C*D + B*D*G + C*E*G G + 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

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*G B + A*C*G + A*E*F + C*D*E + D*F*G C + A*B*G + A*D*F + B*D*E + E*F*G D + A*C*F + A*E*G + B*C*E + B*F*G E + A*B*F + A*D*G + B*C*D + C*F*G F + A*B*E + A*C*D + B*D*G + C*E*G G + A*B*C + A*D*E + B*D*F + C*E*F A*B + C*G + E*F A*C + B*G + D*F A*D + C*F + E*G A*E + B*F + D*G A*F + B*E + C*D A*G + B*C + D*E B*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

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=11 Turn 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 Y Z X

Resolution III Designs – Plackett-Burman Designs N=28 and k=27 Arrange the design into X Y Z Z X Y Y 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) Add “+” sign   X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 1 +1 2 -1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 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 regression Stepwise Regression: y versus X1, X2, ... Alpha-to-Enter: 0.1 Alpha-to-Remove: 0.15 Response is y on 19 predictors, with N = 20 Step 1 2 3 4 5 6 Constant 200.0 200.0 200.0 200.0 200.0 200.0 X2 11.8 11.8 11.8 10.0 10.0 9.9 T-Value 2.51 2.78 3.65 4.02 7.30 7.99 P-Value 0.022 0.013 0.002 0.001 0.000 0.000 X4 9.6 12.0 12.0 12.0 12.1 T-Value 2.27 3.64 4.82 8.76 9.78 P-Value 0.037 0.002 0.000 0.000 0.000 x1x2 -12.0 -12.0 -12.0 -12.5 T-Value -3.64 -4.82 -8.76 -9.91 P-Value 0.002 0.000 0.000 0.000 x1x4 9.0 9.0 9.5 T-Value 3.62 6.57 7.54 P-Value 0.003 0.000 0.000 X1 8.0 8.0 T-Value 5.96 6.60 P-Value 0.000 0.000 X5 2.6 T-Value 2.04 P-Value 0.062 S 21.0 18.9 14.4 10.9 6.00 5.42 R-Sq 25.95 43.12 68.89 83.38 95.30 96.44 R-Sq(adj) 21.83 36.43 63.05 78.94 93.63 94.80

Resolution III Designs – example (8—7/7) Fitted model:

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 26-1IV and changing sign of factor A STAT>DOE>Modify design  fold over Specify  fold just one factorAOK

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