Chapter 3: Screening Designs 2 Chapter 3: Screening Designs 3.1 Fractional Factorial Designs 3.2 Blocking with Screening Designs

Chapter 3: Screening Designs 2 Chapter 3: Screening Designs 3.1 Fractional Factorial Designs 3.2 Blocking with Screening Designs

Objectives Understand screening designs. Distinguish between important and significant factors using a fractional factorial design. Change the aliasing structure of a fractional factorial design. Generate and analyze a fractional factorial screening design.

Screening Designs Concentration Catalyst Pressure and Concentration Temperature Catalyst and Temperature Pressure Temperature and Pressure Pressure and Catalyst Catalyst and Concentration

Two-Level Full Factorial Designs The 23 design requires 8 runs.

Two-Level Fractional Factorial Designs The 23-1 design requires 4 runs.

3.01 Quiz Match the types of fractional factorial designs on the left with the number of necessary runs on the right. 1. 23-1 2. 26-2 3. 26-3 A. 4 runs B. 16 runs C. 8 runs Answer: 1-A, 2-B, 3-C

3.01 Quiz – Correct Answer Match the types of fractional factorial designs on the left with the number of necessary runs on the right. 1. 23-1 2. 26-2 3. 26-3 1-A, 2-B, 3-C A. 4 runs B. 16 runs C. 8 runs Answer: 1-A, 2-B, 3-C

Principles of Fractional Factorial Designs The Pareto principle states that there might be a lot of effects, but very few are important. The sparsity of effects principle states that usually the more important effects are main effects and low-order interactions. The projection property states that every fractional factorial contains full factorials in fewer factors. These designs can be used in sequential experimentation; that is, additional design points can be added to these designs to resolve difficulties or unanswered questions.

22 Full Factorial Design Treatment I A B AB −1 −1 +1 −1 −1 +1 +1 −1

Confounding or Aliasing Suppose you want to include another factor in the experiment, but cannot afford additional runs. You can use the levels of the AB interaction to set the levels of a third factor, C. This means that you cannot separate the effect of C from the effect of AB. Two effects are confounded (or aliased) if it is impossible to estimate each effect separately. Treatment I A B C −1 −1 +1 −1 −1 +1 +1 −1 +1 +1

23-1 Fractional Factorial Design Treatment I A B C AB AC BC ABC +1 +1 +1 +1 −1 +1 −1 −1 −1 +1 −1 −1 −1 +1 Suggested interaction: Explain that the coding in the columns shows where the aliasing is. Give a specific example of a column that is aliased for this design, such as A=BC. Then have students text in the other two columns’ aliasing structure. Answers are: B=AC and C=AB.

Resolution Fractional factorial designs are classified according to their resolution. For resolution 3, main effects are not aliased with other main effects. However, some main effects are aliased with one or more two-factor interactions. For resolution 4, main effects are not aliased with either other main effects or two-factor interactions. However, two-factor interactions can be aliased with other two-factor interactions. For resolution 5, main effects and two-factor interactions are not aliased with other main effects or two-factor interactions.

Plackett-Burman Designs are an alternative to two-level fractional factorial designs for screening use run sizes that are a multiple of 4 rather than a power of 2 have main effects that are orthogonal and two-factor interactions that are only partially confounded are generally resolution 3 designs have good projection properties.

3.02 Multiple Choice Poll With which of the following types of screening designs are you most familiar? Full factorial designs Fractional factorial designs Plackett-Burman designs Other None of these Answers vary.

Important versus Significant Factors Screening studies test many potential effects for significance. You want to separate the vital few from the trivial many. Often, screening tools are necessary to determine which effects are important in explaining variability in the response.

Screening Tools Scaled estimates Prediction profiler Half normal plot Pareto plot Interaction plot Screening platform

Screening Platform Primarily intended for two-level designs in cases with many potential effects but relatively few active effects. Works best with orthogonal effects, but orthogonality is not required. Handles saturated and supersaturated cases. Provides information and tools to decide about the terms in the final model. Provides a bridge to Fit Model for detailed analysis with the final model. Not suitable for all designs.

Screening Platform Contrasts function as parameter estimates. Tests with a t-ratio based on Lenth’s pseudo-standard error (PSE). Provides an individual and a simultaneous p-value for each contrast. Selects any contrast with a p-value less than 0.1. Flags any contrast with a p-value less than 0.05. Includes a half-normal plot for visual determination. Indicates any exact aliases (confounded effects).

p-Values by Simulation The Lenth PSE is used instead of the SE for t-ratio. These ratios do not have a t distribution. An empirical sampling distribution for t-ratios is made by simulation under the null hypothesis (all effects are equal to zero).

Filtration Time Example

Factors of Interest Name Values Temperature cold / hot Presence of Recycled Materials device / no device Water Supply Source 80 / 160 Filter Cloth Type new / old Raw Material Origin on site / other Caustic Soda Rate 5 / 10 Hold-Up Time fast / slow

Two-Level Fractional Factorial Screening Design This demonstration illustrates the concepts discussed previously. This demonstration illustrates the design of a two-level fractional factorial screening design with seven factors. Suggested Interactions: 1. In the Fit Model window, ask students to text in why they think JMP only put the main effects in the Construct Model Effects box. Answer: There are no interactions in the model because they are aliased with the main effects (the design has resolution III). 2. During the demo, ask students to use their seat indicators to answer the following true/false question: There are no p-values accompanying the analysis because there are no degrees of freedom for error. Answer: True – there are seven main effects to be estimated and only eight runs in the experiment. Note: After the foldover design is analyzed, it might be a good idea to refine the model. The final model contains only Water Supply, Caustic Soda, and their interaction. To help lead students into the exercise for this section, show them how to generate the Prediction Expression. Once you have refined the model, go to the Hot Spot next to Response Filtration Time (min) and select Estimates -> Show Prediction Expression. Explain that then, if you were to pick values for your factors, you could plug them in to get an average predicted response for those factor settings.

3.03 Quiz Match the tool on the left with its interpretation on the right. Prediction Profiler Scaled estimates Pareto plot Normal plot Interaction plot deviations from the overall pattern indicate important effects a scale-invariant reference identifies if the effect of one factor depends on the level of another indicates an important effect with long bars changes the level of one variable at a time to see the effect on the response Answer is 1-E, 2-B, 3-D, 4-A, 5-C.

3.03 Quiz – Correct Answer Match the tool on the left with its interpretation on the right. 1-E, 2-B, 3-D, 4-A, 5-C Prediction Profiler Scaled estimates Pareto plot Normal plot Interaction plot deviations from the overall pattern indicate important effects a scale-invariant reference identifies if the effect of one factor depends on the level of another indicates an important effect with long bars changes the level of one variable at a time to see the effect on the response Answer is 1-E, 2-B, 3-D, 4-A, 5-C.

Exercise This exercise reinforces the concepts discussed previously.

3.04 Quiz In the exercise on etch rate, 3 factors, each at two levels, were examined in a full factorial design with 1 replicate. Such a design required 16 runs. The final model equation for etch rate is shown below. The model only contains two of the three factors. In future experiments, how many runs would be necessary to run a new full factorial design with 1 replicate? Answer: 8 runs. This is a 22 factorial design with one replicate, so the number of necessary runs is 2*(22)=8.

3.04 Quiz – Correct Answer In the exercise on etch rate, 3 factors, each at two levels, were examined in a full factorial design with 1 replicate. Such a design required 16 runs. The final model equation for etch rate is shown below. The model only contains two of the three factors. In future experiments, how many runs would be necessary to run a new full factorial design with 1 replicate? 8 runs. This is a 22 factorial design with one replicate, so the number of necessary runs is 2*(22)=8. Answer: 8 runs. This is a 22 factorial design with one replicate, so the number of necessary runs is 2*(22)=8.

Chapter 3: Screening Designs 2 Chapter 3: Screening Designs 3.1 Fractional Factorial Designs 3.2 Blocking with Screening Designs

Objectives Understand blocking in a screening experiment. Generate and analyze a screening design with blocking.

Blocking Blocks are groups of experimental units that are formed such that units within blocks are as homogeneous as possible. Blocking is a statistical technique designed to identify and control variation among groups of experimental units. Blocking is a restriction on randomization. Suggested Interaction: Use seat indicators to answer this true/false question: If a blocking factor is not found to be helpful to the analysis, it is customary to remove that factor from the model. Answer: False – a blocking factor is never removed from a model as it was a restriction on randomization.

Two Factor, Two-Level Full Factorial Design Treatment I A B AB=Block −1 −1 +1 −1 −1 +1 +1 −1 +1 +1

Three Factor, Two-Level Factorial Design Treatment I A B C AB=Block AC=Block BC=Block −1 −1 −1 +1 −1 −1 −1 +1 −1 +1 −1 −1 +1 +1 +1 −1 −1 +1 −1 +1 +1 +1 −1 +1 +1 +1

Aliasing of Effects with a Blocking Factor The aliasing structure of the design indicates that each block is aliased with an interaction. The block effect cannot be estimated separately.

Concentration (continuous) Process Rate Concentration (continuous) 10 & 12 Catalyst (continuous) Temperature (continuous) 10 & 15 220 & 240 Pressure (continuous) 50 & 80

Aliasing of Effects with a Blocking Factor Suppose the design generated by JMP confounds a two-way interaction of interest with a block. JMP enables you to change the aliasing structure of a design.

Generating and Analyzing a Blocked Full Factorial Screening Design This demonstration illustrates the concepts discussed previously. This demonstration illustrates the design of a two-level factorial screening design with four factors of interest and a blocking factor. Suggested Interactions: 1. During the demo, at the point that you choose a design, note that the resolution of the design is 4 and that some 2-factor interactions can be estimated. Ask students to use seat indicators to answer this true/false question: The selected design will allow us to estimate the main effects. Answer: True – a resolution 4 design means that main effects are aliased with neither main effects nor second order interactions. As long as we can assume that the third order interactions are insignificant, the main effects can be estimated. 2. After changing the generating rules, note the first block is aliased as follows: Block1 = Catalyst*Pressure. Have students text in what is aliased with the second block. Answer: Block2 = Catalyst*Temperature*Concentration. 3. In the Fit Model window, ask students why JMP did not include the Pressure*Catalyst interaction in the Construct Model Effects box. Answer: JMP automatically omits that interaction since it is aliased with the first block. To estimate the block effect, you must assume this two-way interaction is negligible and omit the interaction from the model.

Exercise This exercise reinforces the concepts discussed previously. Exercises 2 & 3

3.05 Quiz The Prediction Profiler output from Exercise 3 is below. Which factor is the most important? How did you determine that? Answer: The most important factor is Post Height; it has the steepest slope, meaning changes in Post Height result in a larger change in the response (Pull Strength) as compared to changes in the other factors.

3.05 Quiz – Correct Answer The Prediction Profiler output from Exercise 3 is below. Which factor is the most important? How did you determine that? The most important factor is Post Height; it has the steepest slope, meaning changes in Post Height result in a larger change in the response (Pull Strength) as compared to changes in the other factors. Answer: The most important factor is Post Height; it has the steepest slope, meaning changes in Post Height result in a larger change in the response (Pull Strength) as compared to changes in the other factors.