ENM 310 Design of Experiments and Regression Analysis Chapter 3 Ilgın ACAR Spring 2019

Introduction An experiment is a test or series of tests. The design of an experiment plays a major role in the eventual solution of the problem. In a factorial experimental design, experimental trials (or runs) are performed at all combinations of the factor levels. The analysis of variance (ANOVA) will be used as one of the primary tools for statistical data analysis.

Factorial Experiments Definition

Factorial Experiments Figure 14-3 Factorial Experiment, no interaction.

Factorial Experiments Figure 14-4 Factorial Experiment, with interaction.

Factorial Experiments Figure 14-5 Three-dimensional surface plot of the data from Table 14-1, showing main effects of the two factors A and B.

Factorial Experiments Figure 14-6 Three-dimensional surface plot of the data from Table 14-2, showing main effects of the A and B interaction.

Factorial Experiments Figure 14-7 Yield versus reaction time with temperature constant at 155º F.

Factorial Experiments Figure 14-8 Yield versus temperature with reaction time constant at 1.7 hours.

Factorial Experiments Figure 14-9 Optimization experiment using the one-factor-at-a-time method.

Two-Factor Factorial Experiments

Two-Factor Factorial Experiments The observations may be described by the linear statistical model: where μ is the overall mean effect, τi is the effect of the ith level of factor A, β j is the effect of the jth level of factor B, (τβ)ij is the effect of the interaction between A and B, and eijk is a random error component having a normal distribution with mean 0 and variance σ2.

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model To test H0: i = 0 use the ratio To test H0: j = 0 use the ratio To test H0: ()ij = 0 use the ratio

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Definition

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 Figure 14-10 Graph of average adhesion force versus primer types for both application methods.

Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Minitab Output for Example 14-1

Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking

Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-11 Normal probability plot of the residuals from Example 14-1 This plot has tails that do not fall exactly along a straight line passing through the center of the plot, indicating some potential problems with the normality assumption, but the deviation from normality does not appear severe.

Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking There is some indication that primer type 3 results in slightly lower variability in adhesion force than the other two primers. Figure 14-12 Plot of residuals versus primer type.

Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-13 Plot of residuals versus application method.

Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking The graph of residuals versus fitted values in does not reveal any unusual or diagnostic pattern. Figure 14-14 Plot of residuals versus predicted values.

Example As an example of a factorial design involving two factors, an engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature. The only design parameter that he can select at this point is the plate material for the battery, and he has three possible choices. When the device is manufactured and is shipped to the field, the engineer has no control over the temperature extremes that the device will encounter, and he knows from experience that temperature will probably affect the effective battery life. However, temperature can be controlled in the product development laboratory for the purposes of a test.

14-4: General Factorial Experiments Model for a three-factor factorial experiment

14-4: General Factorial Experiments Example 14-2

Example 14-2

14-4: General Factorial Experiments Example 14-2

14-5: 2k Factorial Designs 14-5.1 22 Design Figure 14-15 The 22 factorial design.

14-5: 2k Factorial Designs 14-5.1 22 Design The main effect of a factor A is estimated by

14-5: 2k Factorial Designs 14-5.1 22 Design The main effect of a factor B is estimated by

14-5: 2k Factorial Designs 14-5.1 22 Design The AB interaction effect is estimated by

14-5: 2k Factorial Designs 14-5.1 22 Design The quantities in brackets in Equations 14-11, 14-12, and 14-13 are called contrasts. For example, the A contrast is ContrastA = a + ab – b – (1)

14-5: 2k Factorial Designs 14-5.1 22 Design Contrasts are used in calculating both the effect estimates and the sums of squares for A, B, and the AB interaction. The sums of squares formulas are

14-5: 2k Factorial Designs Example 14-3

14-5: 2k Factorial Designs Example 14-3

14-5: 2k Factorial Designs Example 14-3

14-5: 2k Factorial Designs Residual Analysis Figure 14-16 Normal probability plot of residuals for the epitaxial process experiment.

14-5: 2k Factorial Designs Residual Analysis Figure 14-17 Plot of residuals versus deposition time.

14-5: 2k Factorial Designs Residual Analysis Figure 14-18 Plot of residuals versus arsenic flow rate.

14-5: 2k Factorial Designs Residual Analysis Figure 14-19 The standard deviation of epitaxial layer thickness at the four runs in the 22 design.

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Figure 14-20 The 23 design.

Figure 14-21 Geometric presentation of contrasts corresponding to the main effects and interaction in the 23 design. (a) Main effects. (b) Two-factor interactions. (c) Three-factor interaction.

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors The main effect of A is estimated by The main effect of B is estimated by

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors The main effect of C is estimated by The interaction effect of AB is estimated by

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Other two-factor interactions effects estimated by The three-factor interaction effect, ABC, is estimated by

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors

14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Contrasts can be used to calculate several quantities:

14-5: 2k Factorial Designs Example 14-4

14-5: 2k Factorial Designs Example 14-4

14-5: 2k Factorial Designs Example 14-4

14-5: 2k Factorial Designs Example 14-4

14-5: 2k Factorial Designs Example 14-4

Example 14-4

14-5: 2k Factorial Designs Residual Analysis Figure 14-22 Normal probability plot of residuals from the surface roughness experiment.

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5 Figure 14-23 Normal probability plot of effects from the plasma etch experiment.

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5 Figure 14-24 AD (Gap-Power) interaction from the plasma etch experiment.

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5

14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design Example 14-5 Figure 14-25 Normal probability plot of residuals from the plasma etch experiment.

14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design A potential concern in the use of two-level factorial designs is the assumption of the linearity in the factor effect. Adding center points to the 2k design will provide protection against curvature as well as allow an independent estimate of error to be obtained. Figure 14-26 illustrates the situation.

14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Figure 14-26 A 22 Design with center points.

A single-degree-of-freedom sum of squares for curvature is given by: 14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design A single-degree-of-freedom sum of squares for curvature is given by:

14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6 Figure 14-27 The 22 Design with five center points for Example 14-6.

14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6

14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6

14-5: 2k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6

14-6: Blocking and Confounding in the 2k Design Figure 14-28 A 22 design in two blocks. (a) Geometric view. (b) Assignment of the four runs to two blocks.

14-6: Blocking and Confounding in the 2k Design Figure 14-29 A 23 design in two blocks with ABC confounded. (a) Geometric view. (b) Assignment of the eight runs to two blocks.

14-6: Blocking and Confounding in the 2k Design General method of constructing blocks employs a defining contrast

14-6: Blocking and Confounding in the 2k Design Example 14-7

14-6: Blocking and Confounding in the 2k Design Example 14-7

Example 14-7 Figure 14-30 A 24 design in two blocks for Example 14-7. (a) Geometric view. (b) Assignment of the 16 runs to two blocks.

14-6: Blocking and Confounding in the 2k Design Example 14-7 Figure 14-31 Normal probability plot of the effects from Minitab, Example 14-7.

14-6: Blocking and Confounding in the 2k Design Example 14-7

14-7: Fractional Replication of the 2k Design 14-7.1 One-Half Fraction of the 2k Design

14-7: Fractional Replication of the 2k Design 14-7.1 One-Half Fraction of the 2k Design Figure 14-32 The one-half fractions of the 23 design. (a) The principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC

14-7: Fractional Replication of the 2k Design Example 14-8

14-7: Fractional Replication of the 2k Design Example 14-8 Figure 14-33 The 24-1 design for the experiment of Example 14-8.

14-7: Fractional Replication of the 2k Design Example 14-8

14-7: Fractional Replication of the 2k Design Example 14-8

14-7: Fractional Replication of the 2k Design Example 14-8

14-7: Fractional Replication of the 2k Design Example 14-8 Figure 14-34 Normal probability plot of the effects from Minitab, Example 14-8.

14-7: Fractional Replication of the 2k Design Projection of the 2k-1 Design Figure 14-35 Projection of a 23-1 design into three 22 designs.

14-7: Fractional Replication of the 2k Design Projection of the 2k-1 Design Figure 14-36 The 22 design obtained by dropping factors B and C from the plasma etch experiment in Example 14-8.

14-7: Fractional Replication of the 2k Design Design Resolution

14-7: Fractional Replication of the 2k Design 14-7.2 Smaller Fractions: The 2k-p Fractional Factorial

14-7: Fractional Replication of the 2k Design Example 14-9

Example 14-8

14-7: Fractional Replication of the 2k Design Example 14-9

14-7: Fractional Replication of the 2k Design Example 14-9 Figure 14-37 Normal probability plot of effects for Example 14-9.

14-7: Fractional Replication of the 2k Design Example 14-9 Figure 14-38 Plot of AB (mold temperature-screw speed) interaction for Example 14-9.

14-7: Fractional Replication of the 2k Design Example 14-9 Figure 14-39 Normal probability plot of residuals for Example 14-9.

14-7: Fractional Replication of the 2k Design Example 14-9 Figure 14-40 Residuals versus holding time (C) for Example 14-9.

14-7: Fractional Replication of the 2k Design Example 14-9 Figure 14-41 Average shrinkage and range of shrinkage in factors A, B, and C for Example 14-9.

14-8: Response Surface Methods and Designs Response surface methodology, or RSM , is a collection of mathematical and statistical techniques that are useful for modeling and analysis in applications where a response of interest is influenced by several variables and the objective is to optimize this response.

14-8: Response Surface Methods and Designs Figure 14-42 A three-dimensional response surface showing the expected yield as a function of temperature and feed concentration.

14-8: Response Surface Methods and Designs Figure 14-43 A contour plot of yield response surface in Figure 14-42.

14-8: Response Surface Methods and Designs The first-order model The second-order model

14-8: Response Surface Methods and Designs Method of Steepest Ascent

14-8: Response Surface Methods and Designs Method of Steepest Ascent Figure 14-44 First-order response surface and path of steepest ascent.

14-8: Response Surface Methods and Designs Example 14-11

14-8: Response Surface Methods and Designs Example 14-11 Figure 14-45 Response surface plots for the first-order model in the Example 14-11.

14-8: Response Surface Methods and Designs Example 14-11 Figure 14-46 Steepest ascent experiment for Example 14-11.

Important Terms & Concepts of Chapter 14 Analysis of variance (ANOVA) Blocking & nuisance factors Center points Central composite design Confounding Contrast Defining relation Design matrix Factorial experiment Fractional factorial design Generator Interaction Main effect Normal probability plot of factor effects Optimization experiment Orthogonal design Regression model Residual analysis Resolution Response surface Screening experiment Steepest ascent (or descent) 2k factorial design Two-level factorial design Chapter 14 Summary