Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC 1/33

Two-Level Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC 2/33

Outline Introduction The 2 2 Design The 2 3 Design The general 2 k Design A single Replicate of the 2 k design Additional Examples of Unreplicated 2 k Designs 2 k Designs are Optimal Designs The additional of center Point to the 2 k Design

Introduction Special case of general factorial designs k factors each with two levels Factors maybe qualitative or quantitative A complete replicate of such design is 2 k factorial design Assumed factors are fixed, the design are completely randomized, and normality Used as factor screening experiments Response between levels is assumed linear

The 2 2 Design FactorTreatment Combination Replication ABIIIIIIIV --A low, B low A high, B low A low, B high A high, B high

The 2 2 Design “-” 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

Estimate factor effects Formulate model With replication, use full model With an unreplicated design, use normal probability plots Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results The 2 2 Design

Standard order  Yates’s order Effects(1)abab A+1+1 B +1 AB+1 +1 Effects A, B, AB are orthogonal contrasts with one degree of freedom Thus 2 k designs are orthogonal designs

The 2 2 Design ANOVA table

The 2 2 Design Algebraic sign for calculating effects in 2 2 design

The 2 2 Design Regression model x 1 and x 2 are code variable in this case Where con and catalyst are natural variables

The 2 2 Design Regression model Factorial Fit: Yield versus Conc., Catalyst Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant Conc Catalyst Conc.*Catalyst S = PRESS = 70.5 R-Sq = 90.30% R-Sq(pred) = 78.17% R-Sq(adj) = 86.66% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Pure Error Total

The 2 2 Design Regression model

The 2 2 Design Regression model

The 2 2 Design Regression model Estimated Coefficients for Yield using data in uncoded units Term Coef Constant Conc Catalyst Conc.*Catalyst Estimated Coefficients for Yield using data in uncoded units Term Coef Constant Conc Catalyst Regression model (without interaction)

The 2 2 Design Response surface

The 2 2 Design Response surface (note: the axis of catalyst is reversed with the one from textbook)

The 2 3 Design 3 factors, each at two level. Eight combinations

The 2 3 Design Design matrix Or geometric notation

The 2 3 Design Algebraic sign

22 The 2 3 Design -- 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)

23 The 2 3 Design -- Properties of the Table The product of any two columns yields a column in the table: Orthogonal design Orthogonality is an important property shared by all factorial designs

The 2 3 Design -- example Nitride etch process Gap, gas flow, and RF power

The 2 3 Design -- example Nitride etch process Gap, gas flow, and RF power

The 2 3 Design -- example Estimated Effects and Coefficients for Etch Rate (coded units) Term Effect Coef SE Coef T P Constant Gap Gas Flow Power Gap*Gas Flow Gap*Power Gas Flow*Power Gap*Gas Flow*Power S = PRESS = R-Sq = 96.61% R-Sq(pred) = 86.44% R-Sq(adj) = 93.64% Analysis of Variance for Etch Rate (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Way Interactions Residual Error Pure Error Total Full model

The 2 3 Design -- example Factorial Fit: Etch Rate versus Gap, Power Estimated Effects and Coefficients for Etch Rate (coded units) Term Effect Coef SE Coef T P Constant Gap Power Gap*Power S = PRESS = R-Sq = 96.08% R-Sq(pred) = 93.02% R-Sq(adj) = 95.09% Analysis of Variance for Etch Rate (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Pure Error Total Reduced model

28 R 2 and adjusted R 2 R 2 for prediction (based on PRESS) The 2 3 Design – example -- Model Summary Statistics for Reduced Model

The 2 3 Design -- example

31 The Regression Model

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

33 The General 2 k Factorial Design There will be k main effects, and

34 The General 2 k Factorial Design Statistical Analysis

35 The General 2 k Factorial Design Statistical Analysis

36 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? Modeling “noise”?

37 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 Unreplicated 2 k Factorial Designs

38 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, 1959) Unreplicated 2 k Factorial Designs

39 A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate Experiment was performed in a pilot plant Unreplicated 2 k Factorial Designs -- example

40 Unreplicated 2 k Factorial Designs -- example

41 Unreplicated 2 k Factorial Designs -- example

42 Unreplicated 2 k Factorial Designs – example –full model

43 Unreplicated 2 k Factorial Designs -- example –full model

44 Unreplicated 2 k Factorial Designs -- example –full model

45 Unreplicated 2 k Factorial Designs -- example –reduced model Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant Temperature Conc Stir Rate Temperature*Conc Temperature*Stir Rate S = PRESS = R-Sq = 96.60% R-Sq(pred) = 91.28% R-Sq(adj) = 94.89% Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Lack of Fit Pure Error Total

46 Unreplicated 2 k Factorial Designs -- example –reduced model

47 Unreplicated 2 k Factorial Designs -- example –reduced model

48 Unreplicated 2 k Factorial Designs -- example –reduced model

49 Unreplicated 2 k Factorial Designs -- example –Design projection Since factor B is negligible, the experiment can be interpreted as a 2 3 factorial design with factors A, C, D. 2 replicates

50 Unreplicated 2 k Factorial Designs -- example –Design projection

51 Unreplicated 2 k Factorial Designs -- example –Design projection Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant Temperature Conc Stir Rate Temperature*Conc Temperature*Stir Rate Conc.*Stir Rate Temperature*Conc.*Stir Rate S = PRESS = 718 R-Sq = 96.87% R-Sq(pred) = 87.47% R-Sq(adj) = 94.13% Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Way Interactions Residual Error Pure Error Total

52 Dealing with Outliers Replace with an estimate Make the highest-order interaction zero In this case, estimate cd such that ABCD = 0 Analyze only the data you have Now the design isn’t orthogonal Consequences?

53 Duplicate Measurements on the Response Four wafers are stacked in the furnace Four factors: temperature, time, gas flow, and pressure. Response: thickness Treated as duplicate not replicate Use average as the response

54 Duplicate Measurements on the Response

55 Duplicate Measurements on the Response Stat  DOE  Factorial  Pre- process Response for Analyze

56 Duplicate Measurements on the Response Stat  DOE  Factorial  Analyze Factorial Design

57 Duplicate Measurements on the Response Factorial Fit: average versus Temperature, Time, Pressure Estimated Effects and Coefficients for average (coded units) Term Effect Coef SE Coef T P Constant Temperature Time Pressure Temperature*Time Temperature*Pressure S = PRESS = R-Sq = 98.39% R-Sq(pred) = 95.88% R-Sq(adj) = 97.59% Analysis of Variance for average (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Lack of Fit Pure Error Total

58 Duplicate Measurements on the Response

59 Duplicate Measurements on the Response

60 The 2 k design and design optimality The model parameter estimates in a 2 k design (and the effect estimates) are least squares estimates. For example, for a 2 2 design the model is

61 The four observations from a 2 2 design The 2 k design and design optimality In matrix form:

62 The matrix is diagonal – consequences of an orthogonal design The regression coefficient estimates are exactly half of the ‘usual” effect estimates The “usual” contrasts The 2 k design and design optimality

63 The 2 k design and design optimality The matrix X’X has interesting and useful properties: Minimum possible value for a four-run design Maximum possible value for a four-run design Notice that these results depend on both the design that you have chosen and the model

The 2 k design and design optimality The 2 2 design is called D-optimal design In fact, all 2 k design is D-optimal design for fitting first order model with interaction. Consider the variance of the predicted response in the 2 2 design:

The 2 k design and design optimality

The 2 2 design is called G-optimal design In fact, all 2 k design is G-optimal design for fitting first order model with interaction. Minimize the maximum prediction variance

The 2 k design and design optimality The 2 2 design is called I-optimal design In fact, all 2 k design is I-optimal design for fitting first order model with interaction. Smallest possible value of the average prediction variance

The 2 k design and design optimality The Minitab provide the function on “Select Optimal Design” when you have a full factorial design and are trying to reduce the it to a partial design or “fractional design”. It only provide the “D-optimal design” One needs to have a full factorial design first and the choose the number of data points to be allowed to use.

69 These results give us some assurance that these designs are “good” designs in some general ways Factorial designs typically share some (most) of these properties There are excellent computer routines for finding optimal designs The 2 k design and design optimality

70 Addition of Center Points to a 2 k Designs Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: Quadratic effects

71 Addition of Center Points to a 2 k Designs When adding center points, we assume that the k factors are quantitative. Example on 2 2 design

72 Addition of Center Points to a 2 k Designs Five point: (-,-),(-,+),(+,-),(+,+), and (0,0). n F =4 and n C =4 Let be the average of the four runs at the four factorial points and let be the average of n C run at the center point.

73 Addition of Center Points to a 2 k Designs If the difference of is small, the center points lie on or near the plane passing through factorial points and there is no quadratic effects. The hypotheses are:

74 Addition of Center Points to a 2 k Designs Test statistics: with one degree of freedom

75 Addition of Center Points to a 2 k Designs -- example In example 6.2, it is a 2 4 factorial. By adding center points x1=x2=x3=x4=0, four additional responses (filtration rates) are : 73, 75, 66,69. So =70.75 and =70.06.

76 Addition of Center Points to a 2 k Designs -- example Term Effect Coef SE Coef T P Constant Temperature Pressure Conc Stir Rate Temperature*Pressure Temperature*Conc Temperature*Stir Rate Pressure*Conc Pressure*Stir Rate Conc.*Stir Rate Temperature*Pressure*Conc Temperature*Pressure*Stir Rate Temperature*Conc.*Stir Rate Pressure*Conc.*Stir Rate Temperature*Pressure*Conc.*Stir Rate Ct Pt

77 Addition of Center Points to a 2 k Designs -- example Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Way Interactions Way Interactions Curvature Residual Error Pure Error Total

78 Addition of Center Points to a 2 k Designs If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model

79 Addition of Center Points to a 2 k Designs

80 Addition of Center Points to a 2 k Designs Use current operating conditions as the center point Check for “abnormal” conditions during the time the experiment was conducted Check for time trends Use center points as the first few runs when there is little or no information available about the magnitude of error

81 Center Points and Qualitative Factors