4. The 2 k Factorial Designs (Ch.6. Two-Level Factorial Designs)

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1 4. The 2 k Factorial Designs (Ch.6. Two-Level Factorial Designs) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University

2 Introduction to 2 k Factorial Designs Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative) Very widely used in industrial experimentation Form a basic building block for other very useful experimental designs Useful for factor screening 2

3 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery 3

4 The Simplest Case: The and + denote the low and high levels of a factor, respectively Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative, although their treatment in the final model will be different 4

5 Notation of the 2 k Designs A special notation is used to represent the runs. In general, a run is represented by a series of lower case letters. If a letter is present, then the corresponding factor is set at the high level in that run; if it is absent, the factor is run at its low level. For example, run a indicates that factor A is at the high level and factor B is at the low level. The run with both factors at the low level is represented by (). This notation is used throughout the 2k design series. For example, the run in a 24 with A and C at the high level and B and D at the low level is denoted by ac. 5

6 Estimation of Factor Effects A = y - y A ab + a b + () = - 2n 2n = [ ab + a - b - ()] 2n B = y - y B 2n 2n + - A + - B ab + b a + () = - 2n 2n = [ ab + b - a - ()] ab + () a + b AB = - 2n 2n = [ ab + () - a - b] The letters (), a, b, and ab also represent the totals of all n observations taken at these design points. Orthogonal Design 6

7 Contrasts in the 2 2 Recall contrasts C a å i= = c y i. Effect = Contrast/2 i Sum of Square of Contrasts SS c = 2 æ a ö c y i i. ç å è i= ø a 2 å ci n i= SS = C = y + y - y - y A A B A B A B A B = [ ab + a - b - ()] n A é ù ê [ ab + a - b - ()] n ú () = ë û = (4) 4n n ( Contrast) 4 / n 2 2 [ ab a b ] 2 7

8 Sum of Squares of the 2 2 Designs SS SS SS A = B AB = = [ a + ab - b - ( )] 4n [ b + ab - a - ( )] 4n [ ab + ( ) - a - b] 4n The analysis of variance is completed by computing the total sum of squares SST (with 4n- degrees of freedom) as usual, and obtaining the error sum of squares SSE [with 4(n-) degrees of freedom] by subtraction. 8

9 ANOVA of the Chemical Processing The F-test for the model source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important? 9

10 Regression Model Regression model for 2 k Designs y = b + b x + b x + b x x + e o Where x is coded variable of Factor A and x2 is coded variable of Factor B Low lever = - and High level = + Relationship between natural and coded variables x = + - A- ( A + A ) / ( A - A ) / 2 0

11 Regression Model for Chemical Processing Since interaction effect is very small, the regression model employed is y = b + b x + b x + e o 2 2 where x is coded variable of the reactant concentration and x2 is coded variable of the amount of catalyst x Conc- ( Conchigh + Conclow) / 2 = / 2 ( Conchigh -Conclow) Conc- (25+ 5) / 2 Conc-20 = = (25-5) / 2 5 x 2 Catalyst -.5 = 0.5

Regression Model for Chemical Processing Estimating b0, b, b2 of the regression model, using least square method We will return to least square method in response surface method Regression model with

12 Regression Model for Chemical Processing Estimating b0, b, b2 of the regression model, using least square method We will return to least square method in response surface method Regression model with coded factors is yˆ 27.5 æ ö x æ - = + ö + x 2 çè 2 ø çè 2 ø where 27.5 is grand average of all observation, bˆ, bˆ 2 is one-half of the corresponding factor effect estimates Regression model with uncoded factors æ8.33öæconc-20ö æ-5.00öæcatalyst -.5ö yˆ = ç è 2 øèç 5 ø çè 2 øè ç 0.5 ø = Conc-5.00Catalyst 2

13 Residual Analysis of Chemical Processing Residual For example e = y - yˆ e = æ8.33ö æ-5.00ö yˆ = (- ) + (-) ç è 2 ø èç 2 ø 3

14 Review of Analysis Procedure Estimate factor effects Main effects, interaction effects Formulate model 2 2 design example Statistical testing (ANOVA) Refine the model Chemical processing example Regression model estimation By Least Square Method Analyze residuals (graphical) Normal probability plot of residuals Interpret results y = b + b x + b x + b x x + e o y = b + b x + b x + e o 2 2 yˆ = b ˆ + b ˆ x + b ˆ x o 2 2 4

15 The 2 3 Factorial Design 5

16 Factor Effect of the 2 3 Designs 3 factors, each at two levels 8 factor-level combinations 3 main effects: A,B,C 3 two-factor interactions: AB, AC,BC three-factor interaction: ABC 6

17 Factor Effect of the 2 3 Designs Main effect of A A = a + ab + ac + abc- -b-c-bc 4n Main effect of B [ () ] B = b + ab + bc + abc - - a - c - ac 4n [ () ] Main effect of C C = c + ac + bc + abc- -a -b-ab 4n [ () ] 7

18 Factor Effect of the 2 3 Designs Interaction effect of AB AB = éab( Chigh) AB( Clow) ù 2 êë + úû where AB( Clow) = [ ab + ()] - [ a + b] 2 n 2 n AB( Chigh) = [ abc + c] - [ ac + bc] 2n 2n Therefore AB = [ ab + () + abc + c-b-a-bc-ac ] 4n The same approach can be applied for the interaction effect of BC and AC 8

19 Factor Effect of the 2 3 Designs Interaction effect of ABC is defined as the average difference between the AB interaction at the two different level of C ABC = [ AB( C high) - AB( C low) ] 2 ìé ù é ù = ï í ê ( abc + c) - ( ac + ab) - ( ab + () ) - ( a + b) 2 ïê ïîë 2 n 2 n úû êë 2n 2 n úû = [ abc-bc- ac+ c- ab+ b+ a-() ] 4n How to memorize the sign of coefficients? 9

20 Factor Effect of the 2 3 Designs 20

21 Properties of the Table Except for column I, every column has an equal number of + and signs The sum of the product of signs in any two columns is zero Multiplying any column by I leaves that column unchanged (identity element) The product of any two columns yields a column in the table: A B = AB 2 AB BC = AB C = AC Orthogonal design Orthogonality is an important property shared by all factorial designs 2

22 Effects, Sum of Squares, and Contrast The 2 3 Designs Effect = Contrast/4 Sum of squares = n(contrast) 2 /8 Contrast for factor A ContrastA = a + ab + ac + abc- -b-c-bc n Main effect of factor A Sum of Square of factor A [ () ] A = ContrastA / 4 = a + ab + ac + abc-() -b-c-bc 4n [ ] 2 [ ] 2 SS A = n( ContrastA) / 8 = a + ab + ac + abc-() -b-c-bc 8n 22

23 Plasma Etching Process A 2 3 factorial design was used to develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between the electrodes, the gas flow (C 2 F 6 is used as the reactant gas), and the RF power applied to the cathode. Each factor is run at two levels, and the design is replicated twice. The response variable is the etch rate for silicon nitride (Å/m) A = gap, B = Flow, C = Power, y = Etch Rate 23

24 Plasma Etching Process Gap Gas flow Power Wafer Plasma Etching Process Etch rate 24

25 ANOVA Summary Full Model Important effects by A, C, AC, 25

26 The Regression Model with Reduced Factors 26

27 The Regression Model with Reduced Factors 27

28 Cube Plot of Ranges What do the large ranges when gap and power are at the high level tell you? 28

29 The General 2 k Factorial Design Contrast Effect = k- 2 n( Contrast) SS = 2 k 2 29

30 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner of the cube An unreplicated 2 k factorial design is also sometimes called a single replicate of the 2 k These designs are very widely used Risks if there is only one observation at each corner, is there a chance of unusual response observations spoiling the results? 30

31 Spacing of Factor Levels in the Unreplicated 2 k Factorial Designs If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data More aggressive spacing is usually best 3

32 Unreplicated 2 k Factorial Designs Lack of replication causes potential problems in statistical testing Replication admits an estimate of pure error (a better phrase is an internal estimate of error) With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem Pooling high-order interactions to estimate error Normal probability plotting of effects (Daniels, 959) 32

33 Example of an Unreplicated 2 k Design A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product. The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin Experiment was performed in a pilot plant 33

34 The Resin Plant Experiment 34

35 Contrast Constants for the 2 4 Design 35

36 Estimates of the Effects 36

37 ANOVA Summary for the Model as a 2 3 in Factors A, C, and D 37

38 The Regression Model 38

39 Experiments with the larger number of factors The system is usually dominated by the main effects and low-order interactions. Higher interactions are usually negligible. When the number of factors is larger than 3 or 4, a common practice is to run only a single replicate design and then pool the higher order interactions as an estimate of error. Normal probability plot of the effects may be useful If none of the effects is significant, then the estimates will behave like a random sample drawn from a normal distribution with zero mean, and the plotted effects will lie approximately along a straight line. Those effects that do not plot on the line are significant factors. 39

4. Design of Experiments (DOE) (The 2 k Factorial Designs)

4. Design of Experiments (DOE) (The 2 k Factorial Designs) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Example: Golfing How to improve my score in Golfing? Practice!!! Other than