1

Chapter 3: Screening Designs

3.1 Fractional Factorial Designs

3.2 Blocking with Screening Designs

2

Chapter 3: Screening Designs

3.1 Fractional Factorial Designs3.1 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.

3

Screening Designs

4

Catalyst and ConcentrationCatalyst and Concentration

ConcentrationConcentration

Press

ure a

nd

Press

ure a

nd

Conce

ntrati

on

Conce

ntrati

on

Temperature

Temperature

Catalyst and Catalyst and

TemperatureTemperature

Pressure

Pressure

Temperature and Pressure

Temperature and Pressure

Pressure and Catalyst

Pressure and Catalyst

PressurePressure

TemperatureTemperature

Temperature and PressureTemperature and PressureC

atal

yst

Cat

alys

t

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

5

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

6

7

3.01 QuizMatch 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

8

A. 4 runs

B. 16 runs

C. 8 runs

3.01 Quiz – Correct AnswerMatch 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

9

A. 4 runs

B. 16 runs

C. 8 runs

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.

10

22 Full Factorial Design

Treatment I A B AB

−1 −1 +1 −1 −1 +1

−1 +1 +1 −1 +1 −1

+1 −1 +1 +1 −1 −1

+1 +1 +1 +1 +1 +1

11

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.

12

Treatment I A B C

−1 −1 +1 −1 −1 +1

−1 +1 +1 −1 +1 −1

+1 −1 +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 +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 +1 −1 −1 +1

13

ResolutionFractional 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.

14

Plackett-Burman DesignsPlackett-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.

15

16

3.02 Multiple Choice PollWith which of the following types of screening designs are you most familiar?

a. Full factorial designs

b. Fractional factorial designs

c. Plackett-Burman designs

d. Other

e. None of these

17

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.

18

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

19

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.

20

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).

21

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).

22

Filtration Time Example

23

Factors of InterestName 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

24

Two-Level Fractional Factorial Screening Design

This demonstration illustrates the concepts discussed previously.

25

26

27

3.03 QuizMatch the tool on the left with its interpretation on the right.

28

1. Prediction Profiler

2. Scaled estimates

3. Pareto plot

4. Normal plot

5. Interaction plot

A. deviations from the overall pattern indicate important effects

B. a scale-invariant referenceC. identifies if the effect of one

factor depends on the level of another

D. indicates an important effect with long bars

E. changes the level of one variable at a time to see the effect on the response

3.03 Quiz – Correct AnswerMatch the tool on the left with its interpretation on the right. 1-E, 2-B, 3-D, 4-A, 5-C

29

1. Prediction Profiler

2. Scaled estimates

3. Pareto plot

4. Normal plot

5. Interaction plot

A. deviations from the overall pattern indicate important effects

B. a scale-invariant referenceC. identifies if the effect of one

factor depends on the level of another

D. indicates an important effect with long bars

E. changes the level of one variable at a time to see the effect on the response

Exercise

This exercise reinforces the concepts discussed previously.

30

31

3.04 QuizIn 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?

32

3.04 Quiz – Correct AnswerIn 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.

33

34

Chapter 3: Screening Designs

3.1 Fractional Factorial Designs

3.2 Blocking with Screening Designs3.2 Blocking with Screening Designs

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

blocking.

35

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.

36

Two Factor, Two-Level Full Factorial Design

37

Treatment I A B AB=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

Three Factor, Two-Level Factorial Design

38

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 −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 +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 +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.

39

Process Rate

40

Concentration (continuous)

10 & 12

Catalyst (continuous)

10 & 15

Temperature (continuous)

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.

41

Generating and Analyzing a Blocked Full Factorial Screening Design

This demonstration illustrates the concepts discussed previously.

42

43

Exercise

This exercise reinforces the concepts discussed previously.

44

45

3.05 QuizThe Prediction Profiler output from Exercise 3 is below. Which factor is the most important? How did you determine that?

46

3.05 Quiz – Correct AnswerThe 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.

47