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…

Factorial 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…

Outline Basic Definition and Principles The Advantages of Factorials The Two Factors Factorial Design The General Factorial Design Fitting Response Curve and Surfaces Blocking in Factorial Design

Basic Definitions and Principles Factorial Design—all of the possible combinations of factors’ level are investigated When factors are arranged in factorial design, they are said to be crossed Main effects – the effects of a factor is defined to be changed Interaction Effect – The effect that the difference in response between the levels of one factor is not the same at all levels of the other factors.

Basic Definitions and Principles Factorial Design without interaction

Basic Definitions and Principles Factorial Design with interaction

Basic Definitions and Principles Average response – the average value at one factor’s level Average response increase – the average value change for a factor from low level to high level No Interaction:

Basic Definitions and Principles With Interaction:

Basic Definitions and Principles Another way to look at interaction: When factors are quantitative In the above fitted regression model, factors are coded in (-1, +1) for low and high levels This is a least square estimates

Basic Definitions and Principles Since the interaction is small, we can ignore it. Next figure shows the response surface plot

Basic Definitions and Principles The case with interaction

Advantages of Factorial design Efficiency Necessary if interaction effects are presented The effects of a factor can be estimated at several levels of the other factors

The Two-factor Factorial Design Two factors a levels of factor A, b levels of factor B n replicates In total, nab combinations or experiments

The Two-factor Factorial Design – An example Two factors, each with three levels and four replicates 32 factorial design

The Two-factor Factorial Design – An example Questions to be answered: What effects do material type and temperature have on the life the battery Is there a choice of material that would give uniformly long life regardless of temperature?

The Two-factor Factorial Design Statistical (effects) model: Means model Regression model 𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 12 𝑥 12 +𝜖

The Two-factor Factorial Design Hypothesis Row effects: Column effects: Interaction:

The Two-factor Factorial Design -- Statistical Analysis

The Two-factor Factorial Design -- Statistical Analysis Mean square: A: B: Interaction:

The Two-factor Factorial Design -- Statistical Analysis Mean square: Error:

The Two-factor Factorial Design -- Statistical Analysis ANOVA table

The Two-factor Factorial Design -- Statistical Analysis Example—Batery-life.mtw

The Two-factor Factorial Design -- Statistical Analysis STATANOVAGeneral Linear Model Fit General Linear Model

The Two-factor Factorial Design -- Statistical Analysis Model…. Select Temp and Material then click Add OK, OK

The Two-factor Factorial Design -- Statistical Analysis Example General Linear Model: Life versus Temp, Material Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Temp Fixed 3 15, 70, 125 Material Fixed 3 1, 2, 3 Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Temp 2 39119 19559.4 28.97 0.000 Material 2 10684 5341.9 7.91 0.002 Temp*Material 4 9614 2403.4 3.56 0.019 Error 27 18231 675.2 Total 35 77647

The Two-factor Factorial Design -- Statistical Analysis Example General Linear Model: Life versus Temp, Material Model Summary S R-sq R-sq(adj) R-sq(pred) 25.9849 76.52% 69.56% 58.26% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 105.53 4.33 24.37 0.000 Temp 15 39.31 6.12 6.42 0.000 1.33 70 2.06 6.12 0.34 0.740 1.33 Material 1 -22.36 6.12 -3.65 0.001 1.33 2 2.81 6.12 0.46 0.651 1.33 Temp*Material 15 1 12.28 8.66 1.42 0.168 1.78 15 2 8.11 8.66 0.94 0.357 1.78 70 1 -27.97 8.66 -3.23 0.003 1.78 70 2 9.36 8.66 1.08 0.289 1.78

The Two-factor Factorial Design -- Statistical Analysis Example General Linear Model: Life versus Temp, Material Regression Equation Life = 105.53 + 39.31 Temp_15 + 2.06 Temp_70 - 41.36 Temp_125 - 22.36 Material_1 + 2.81 Material_2 + 19.56 Material_3 + 12.28 Temp*Material_15 1 + 8.11 Temp*Material_15 2 - 20.39 Temp*Material_15 3 - 27.97 Temp*Material_70 1 + 9.36 Temp*Material_70 2 + 18.61 Temp*Material_70 3 + 15.69 Temp*Material_125 1 - 17.47 Temp*Material_125 2 + 1.78 Temp*Material_125 3 Fits and Diagnostics for Unusual Observations Obs Life Fit Resid Std Resid 2 74.0 134.8 -60.8 -2.70 R 8 180.0 134.8 45.2 2.01 R R Large residual

The Two-factor Factorial Design -- Statistical Analysis Example

The Two-factor Factorial Design -- Statistical Analysis Example

The Two-factor Factorial Design -- Statistical Analysis Example STATANOVA--GLM

The Two-factor Factorial Design -- Statistical Analysis Example STATANOVA--GLM

The Two-factor Factorial Design -- Statistical Analysis Estimating the model parameters

The Two-factor Factorial Design -- Statistical Analysis Choice of sample size Row effects Column effects Interaction effects D:difference, :standard deviation

The Two-factor Factorial Design -- Statistical Analysis

The Two-factor Factorial Design -- Statistical Analysis Appendix Chart V For n=4, giving D=40 on temperature, v1=2, v2=27, Φ 2 =1.28n. β =0.06 n Φ2 Φ υ1 υ2 β 2 2.56 1.6 9 0.45 3 3.84 1.96 18 0.18 4 5.12 2.26 27 0.06

The Two-factor Factorial Design -- Statistical Analysis – example with no interaction Analysis of Variance for Life, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Material 2 10684 10684 5342 5.95 0.007 Temp 2 39119 39119 19559 21.78 0.000 Error 31 27845 27845 898 Total 35 77647 S = 29.9702 R-Sq = 64.14% R-Sq(adj) = 59.51%

The Two-factor Factorial Design – One observation per cell Single replicate The effect model

The Two-factor Factorial Design – One observation per cell ANOVA table

The Two-factor Factorial Design -- One observation per cell The error variance is not estimable unless interaction effect is zero Needs Tuckey’s method to test if the interaction exists. Check page 183 for details.

The General Factorial Design In general, there will be abc…n total observations if there are n replicates of the complete experiment. There are a levels for factor A, b levels of factor B, c levels of factor C,..so on. We must have at least two replicate (n≧2) to include all the possible interactions in model.

The General Factorial Design If all the factors are fixed, we may easily formulate and test hypotheses about the main effects and interaction effects using ANOVA. For example, the three factor analysis of variance model:

The General Factorial Design ANOVA.

The General Factorial Design where

The General Factorial Design --example Three factors: pressure, percent of carbonation, and line speed.

The General Factorial Design --example STATANOVAGeneral Linear Model Fit General Linear Model Model…three factors, up to 3 level

The General Factorial Design --example General Linear Model: Deviation versus Speed, Carbonation, Pressure Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Speed Fixed 2 200, 250 Carbonation Fixed 3 10, 12, 14 Pressure Fixed 2 25, 30 Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Speed 1 22.042 22.042 31.12 0.000 Carbonation 2 252.750 126.375 178.41 0.000 Pressure 1 45.375 45.375 64.06 0.000 Speed*Carbonation 2 0.583 0.292 0.41 0.671 Speed*Pressure 1 1.042 1.042 1.47 0.249 Carbonation*Pressure 2 5.250 2.625 3.71 0.056 Speed*Carbonation*Pressure 2 1.083 0.542 0.76 0.487 Error 12 8.500 0.708 Total 23 336.625

The General Factorial Design --example ANOVA

Fitting Response Curve and Surfaces When factors are quantitative, one can fit a response curve (surface) to the levels of the factor so the experimenter can relate the response to the factors. These surface could be linear or quadratic. Linear regression model is generally used

Fitting Response Curve and Surfaces -- example Battery life data Factor temperature is quantitative

Fitting Response Curve and Surfaces -- example Example STATANOVA—GLM Response  Life Factors: Material Covariates  temp Model … interactions, 2, add terms, 2 , add cross, Add temp, material, temp*temp, material*temp, material*temp*temp

Fitting Response Curve and Surfaces -- example General Linear Model: Life versus Temp, Material Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Material Fixed 3 1, 2, 3 Analysis of Variance Source DF Seq SS Seq MS F-Value P-Value Temp 1 39042.7 39042.7 57.82 0.000 Material 2 10683.7 5341.9 7.91 0.002 Temp*Temp 1 76.1 76.1 0.11 0.740 Temp*Material 2 2315.1 1157.5 1.71 0.199 Temp*Temp*Material 2 7298.7 3649.3 5.40 0.011 Error 27 18230.7 675.2 Total 35 77647.0

Fitting Response Curve and Surfaces -- example coding method: -1, 0, +1 General Linear Model: Life versus Material Factor Type Levels Values Material fixed 3 1, 2, 3 Analysis of Variance for Life, using Sequential SS for Tests Source DF Seq SS Adj SS Seq MS F P Temp 1 39042.7 1239.2 39042.7 57.82 0.000 Material 2 10683.7 1147.9 5341.9 7.91 0.002 Temp*Temp 1 76.1 76.1 76.1 0.11 0.740 Material*Temp 2 2315.1 7170.7 1157.5 1.71 0.199 Material*Temp*Temp 2 7298.7 7298.7 3649.3 5.40 0.011 Error 27 18230.8 18230.8 675.2 Total 35 77647.0 S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56% Term Coef SE Coef T P Constant 153.92 11.87 12.96 0.000 Temp -0.5906 0.4360 -1.35 0.187 Temp*Temp -0.001019 0.003037 -0.34 0.740 Temp*Material 1 -1.9108 0.6166 -3.10 0.005 2 0.4173 0.6166 0.68 0.504 Temp*Temp*Material 1 0.013871 0.004295 3.23 0.003 2 -0.004642 0.004295 -1.08 0.289 Two kinds of coding methods: 1, 0, -1 0, 1, -1

Fitting Response Curve and Surfaces -- example Model Summary S R-sq R-sq(adj) R-sq(pred) 25.9849 76.52% 69.56% 58.26% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 153.9 11.9 12.96 0.000 Temp -0.591 0.436 -1.35 0.187 20.44 Material 1 15.5 16.8 0.92 0.365 10.02 2 5.7 16.8 0.34 0.737 10.02 Temp*Temp -0.00102 0.00304 -0.34 0.740 20.44 Temp*Material 1 -1.911 0.617 -3.10 0.005 93.46 2 0.417 0.617 0.68 0.504 93.46 Temp*Temp*Material 1 0.01387 0.00430 3.23 0.003 58.62 2 -0.00464 0.00430 -1.08 0.289 58.62 Regression Equation 1 Life = 169.4 - 2.501 Temp + 0.01285 Temp*Temp 2 Life = 159.6 - 0.173 Temp - 0.00566 Temp*Temp 3 Life = 132.8 + 0.903 Temp - 0.01025 Temp*Temp

Fitting Response Curve and Surfaces -- example Final regression equation:

Fitting Response Curve and Surfaces – example –32 factorial design Tool life Factors: cutting speed, total angle Data are coded

Fitting Response Curve and Surfaces – example –32 factorial design

Fitting Response Curve and Surfaces – example –32 factorial design STATRegressionRegression Fit Regression Model Response  Life Continuous predictors: Speed, Angle Model…select: Speed, Angle  interaction, 2,Add Terms, 2, Add cross, Add Speed, Angle, Speed*Speed, Angle*Angle, Speed*Angle, Speed*Speed*Angle, Speed*Angle*Angle Speed*Speed*Angle*Angle

Fitting Response Curve and Surfaces – example –32 factorial design Results for: Tool-Life.MTW Regression Analysis: Life versus Speed, Angle Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 8 111.000 13.875 9.61 0.001 Speed 1 3.353 3.353 2.32 0.162 Angle 1 5.090 5.090 3.52 0.093 Speed*Speed 1 3.549 3.549 2.46 0.151 Angle*Angle 1 7.336 7.336 5.08 0.051 Speed*Angle 1 5.197 5.197 3.60 0.090 Speed*Speed*Angle 1 5.527 5.527 3.83 0.082 Speed*Angle*Angle 1 7.544 7.544 5.22 0.048 Speed*Speed*Angle*Angle 1 8.000 8.000 5.54 0.043 Error 9 13.000 1.444 Total 17 124.000

Fitting Response Curve and Surfaces – example –32 factorial design Analysis of Variance Source DF SS MS F P Regression 8 111.000 13.875 9.61 0.001 Residual Error 9 13.000 1.444 Total 17 124.000 Source DF Seq SS Speed 1 21.333 Angle 1 8.333 Angle*Angle 1 16.000 Speed*Speed 1 4.000 Angle*Speed 1 8.000 Angle*Speed*Speed 1 42.667 Angle*Angle*Speed 1 2.667 Angle*Angle*Speed*Speed 1 8.000

Fitting Response Curve and Surfaces – example –32 factorial design StatRegressionregressioncontour plot

Fitting Response Curve and Surfaces – example –32 factorial design StatRegressionregressionsurface plot

Blocking in a Factorial Design We may have a nuisance factor presented in a factorial design Original two factor factorial model: Two factor factorial design with a block factor model:

Blocking in a Factorial Design

Blocking in a Factorial Design – example—target-detection.mtw Response: intensity level Factors: Ground cutter and filter type Block factor: Operator

Blocking in a Factorial Design -- example Results for: Target-Detection.MTW General Linear Model: Intensity versus Blocks, Filter, Clutter Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Blocks Fixed 4 1, 2, 3, 4 Filter Fixed 2 1, 2 Clutter Fixed 3 High, Low, Medium Analysis of Variance Source DF Seq SS Seq MS F-Value P-Value Blocks 3 402.17 134.06 12.09 0.000 Filter 1 1066.67 1066.67 96.19 0.000 Clutter 2 335.58 167.79 15.13 0.000 Filter*Clutter 2 77.08 38.54 3.48 0.058 Error 15 166.33 11.09 Total 23 2047.83 Model Summary S R-sq R-sq(adj) R-sq(pred) 3.33000 91.88% 87.55% 79.21%

Blocking in a Factorial Design -- example Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 94.917 0.680 139.64 0.000 Blocks 1 0.42 1.18 0.35 0.728 1.50 2 1.58 1.18 1.34 0.199 1.50 3 4.58 1.18 3.89 0.001 1.50 Filter 1 6.667 0.680 9.81 0.000 1.00 Clutter High 4.333 0.961 4.51 0.000 1.33 Low -4.792 0.961 -4.98 0.000 1.33 Filter*Clutter 1 High 2.083 0.961 2.17 0.047 1.33 1 Low -2.292 0.961 -2.38 0.031 1.33 Regression Equation Intensity = 94.917 + 0.42 Blocks_1 + 1.58 Blocks_2 + 4.58 Blocks_3 - 6.58 Blocks_4 + 6.667 Filter_1 - 6.667 Filter_2 + 4.333 Clutter_High - 4.792 Clutter_Low + 0.458 Clutter_Medium + 2.083 Filter*Clutter_1 High - 2.292 Filter*Clutter_1 Low + 0.208 Filter*Clutter_1 Medium - 2.083 Filter*Clutter_2 High + 2.292 Filter*Clutter_2 Low - 0.208 Filter*Clutter_2 Medium