Two-Level Factorial Designs

Presented by:

Juanito S. Chan, PIE

Standard 23 Design

• Where do the factors (independent variables) appear in this table?

• Where do the responses (dependent variable) appear in this table?

• What do the –1 and +1 mean?

• Should these experimental runs be made in the order they are shown?

Run A B C

1 -1 -1 -1

2 +1 -1 -1

3 -1 +1 -1

4 +1 +1 -1

5 -1 -1 +1

6 +1 -1 +1

7 -1 +1 +1

8 +1 +1 +1

Standard 23 Design

• Factors are A, B, C

• Responses do not appear in this table?

• Choose a high and a low value for each factor.o -1 means set factor to low

level in this run

o +1 means set factor to high level in this run

• Run order should be randomizedo Failure to randomize very

risky for factor C, since it has runs 1-4 at low level and 5-8 at high level

Run A B C

1 -1 -1 -1

2 +1 -1 -1

3 -1 +1 -1

4 +1 +1 -1

5 -1 -1 +1

6 +1 -1 +1

7 -1 +1 +1

8 +1 +1 +1

Maximize Reaction Yield23 Factorial Design

• Objective: maximize reaction yield

• Factors:o A = catalyst weight percent (1,2)o B = reaction time, hours (1,2)o C = temperature, °F (200,250)

• Response: Reaction yield, %

Maximize Reaction Yield

Run Catalyst

Weight %

Reaction Time, hr

Temperature, °F

Yield, %

1 1 1 200 65.3

2 2 1 200 81.3

3 1 2 200 53.3

4 2 2 200 69.9

5 1 1 250 61.8

6 2 1 250 77.4

7 1 2 250 73.9

8 2 2 250 89.9

Now What?

• Calculate effects of each factor and interaction

• Decide which effects are important

• Plan another, multilevel experiment focusing on the important variables

Interactions

Run A B C AB AC BC ABC Y

1 -1 -1 -1 +1 +1 +1 -1 65.3

2 +1 -1 -1 -1 -1 +1 +1 81.3

3 -1 +1 -1 53.3

4 +1 +1 -1 69.9

5 -1 -1 +1 61.8

6 +1 -1 +1 77.4

7 -1 +1 +1 73.9

8 +1 +1 +1 89.9

Note that each factor is tested at each level 4 times.

-1 -1 = +1

Investigating Interactions

• You set the value for each factor in each experiment

• The interactions happen naturallyo You do not set some level of AB interaction; it

happens automatically because of the levels you set for A and B individually

• Interactions are a physical reality of the system, and will happen whether you calculate an effect for them or not

How to Calculate Effects• High Total = sum of all response values

when the factor is at the +1 level• Low Total = sum of all response values

when the factor is at the –1 level• Difference = (High Total) – (Low Total)

o Note that you can also calculate the difference by multiplying each +1 or –1 by the response for its row, then summing all the values in the column. That is what your book says.

• Effect = Difference / (# runs at each level)

Effects

ACat. Wgt. %

BRxn Time

CTemp

AB AC BC ABC

High

Total

Low

Total

Diff

Effect

On Y

Effects

ACat. Wgt. %

BRxn Time

CTemp

AB AC BC ABC

High

Total

318.5 287.0 303.0 286.9 285.9 310.4 286.3

Low

Total

254.3 285.8 269.8 285.9 286.9 262.4 286.5

Diff 64.2 1.2 33.2 1.0 -1.0 48.0 -0.2

Effect

On Y

16.05 .30 8.30 0.25 -0.25 12.00 -0.05

“Scree Plot” to Identify Order of Importance

-2

0

2

4

6

8

10

12

14

16

18

A BC C B AB AC

Factor

Eff

ec

t o

n R

ea

cti

on

Yie

ld

Conclusions

• Increasing catalyst weight % or increasing temperature will increase the yieldo Increasing catalyst is most effective

• Increasing reaction time itself has little effect on yield, but in combination with increased temperature multiplies the effect of temperature

Comparison with OFAT

• OFAT would reveal the effect of catalyst and temperature.

• OFAT would not reveal the time-temperature interaction.

• OFAT would not reveal the lack of time-catalyst and temperature-catalyst interaction.

Adding a Factor• Adding a factor to a full factorial design doubles

the number of experimental runso 3 factors = 23 = 8 runso 4 factors = 24 = 16 runs

• If you are confident that an interaction is unimportant, you can substitute a new factor for that interaction term in the test matrixo 3-way interaction least likely to be importanto Substitution of a factor for an interaction makes an

unsaturated design

Unsaturated Designs and Aliasing

• If a factor replaces an interaction in the design, o You cannot tell the difference between the effect of the

factor and the effect of the interactiono Interaction is an innate property of the system. You do not

control whether or not it happens by deciding whether or not to study it.

o Some or all of the effect you calculate for the new factor could be due to the interaction between other factors.

o You cannot study how the new factor interacts with others