14-1 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.

14-2 Factorial Experiments Definition

14-2 Factorial Experiments Figure 14-3 Factorial Experiment, no interaction.

14-2 Factorial Experiments Figure 14-4 Factorial Experiment, with interaction.

14-2 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.

14-2 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.

14-2 Factorial Experiments Figure 14-7 Yield versus reaction time with temperature constant at 155º F.

14-2 Factorial Experiments Figure 14-8 Yield versus temperature with reaction time constant at 1.7 hours.

14-2 Factorial Experiments Figure 14-9 Optimization experiment using the one-factor-at-a-time method.

14-3 Two-Factor Factorial Experiments

The observations may be described by the linear statistical model:

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model

14-3 Two-Factor Factorial Experiments To test H 0 :  i = 0 use the ratio Statistical Analysis of the Fixed-Effects Model To test H 0 :  j = 0 use the ratio To test H 0 : (  ) ij = 0 use the ratio

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Definition

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Example 14-1 Figure Graph of average adhesion force versus primer types for both application methods.

14-3 Two-Factor Factorial Experiments Statistical Analysis of the Fixed-Effects Model Minitab Output for Example 14-1

14-3 Two-Factor Factorial Experiments Model Adequacy Checking

14-3 Two-Factor Factorial Experiments Model Adequacy Checking Figure Normal probability plot of the residuals from Example 14-1

14-3 Two-Factor Factorial Experiments Model Adequacy Checking Figure Plot of residuals versus primer type.

14-3 Two-Factor Factorial Experiments Model Adequacy Checking Figure Plot of residuals versus application method.

14-3 Two-Factor Factorial Experiments Model Adequacy Checking Figure Plot of residuals versus predicted values.

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

14-4 General Factorial Experiments Example 14-2

14-4 General Factorial Experiments Example 14-2

k Factorial Designs Design Figure The 2 2 factorial design.

k Factorial Designs Design The main effect of a factor A is estimated by

k Factorial Designs Design The main effect of a factor B is estimated by

k Factorial Designs Design The AB interaction effect is estimated by

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

k Factorial Designs 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

k Factorial Designs Example 14-3

k Factorial Designs Example 14-3

k Factorial Designs Example 14-3

k Factorial Designs Residual Analysis Figure Normal probability plot of residuals for the epitaxial process experiment.

k Factorial Designs Residual Analysis Figure Plot of residuals versus deposition time.

k Factorial Designs Residual Analysis Figure Plot of residuals versus arsenic flow rate.

k Factorial Designs Residual Analysis Figure The standard deviation of epitaxial layer thickness at the four runs in the 2 2 design.

k Factorial Designs k Design for k  3 Factors Figure The 2 3 design.

Figure Geometric presentation of contrasts corresponding to the main effects and interaction in the 2 3 design. (a) Main effects. (b) Two-factor interactions. (c) Three- factor interaction.

k Factorial Designs k Design for k  3 Factors The main effect of A is estimated by The main effect of B is estimated by

k Factorial Designs k Design for k  3 Factors The main effect of C is estimated by The interaction effect of AB is estimated by

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

k Factorial Designs k Design for k  3 Factors

k Factorial Designs k Design for k  3 Factors

k Factorial Designs k Design for k  3 Factors Contrasts can be used to calculate several quantities:

k Factorial Designs Example 14-4

k Factorial Designs Example 14-4

k Factorial Designs Example 14-4

k Factorial Designs Example 14-4

k Factorial Designs Example 14-4

k Factorial Designs Residual Analysis Figure Normal probability plot of residuals from the surface roughness experiment.

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

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

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

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

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

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

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

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

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

k Factorial Designs Additional Center Points to a 2 k 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 2 k design will provide protection against curvature as well as allow an independent estimate of error to be obtained. Figure illustrates the situation.

k Factorial Designs Additional Center Points to a 2k Design Figure A 2 2 Design with center points.

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

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

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

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

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

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

14-6 Blocking and Confounding in the 2 k Design Figure A 2 3 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 2 k Design General method of constructing blocks employs a defining contrast

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

Figure A 2 4 design in two blocks for Example (a) Geometric view. (b) Assignment of the 16 runs to two blocks.

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

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

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

14-7 Fractional Replication of the 2 k Design One-Half Fraction of the 2 k Design Figure The one-half fractions of the 2 3 design. (a) The principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC

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

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

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

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

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

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

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

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

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

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

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

Example 14-8

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

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

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

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

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

14-7 Fractional Replication of the 2 k Design Example 14-9 Figure 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 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 A contour plot of yield response surface in Figure

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 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 Figure Response surface plots for the first-order model in the Example

14-8 Response Surface Methods and Designs Example Figure Steepest ascent experiment for Example