Class 17-2-3 Factorial Design

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23 Factorial DesignIndustrial Engineering

Example

23 Factorial Design

Introduction

Suppose that three factors, A, B, and C, each at two levels, are of interest. The design is called a 23 factorial design and the eight treatment combinations can now be displayed geometrically as a cube.

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Example

23 Factorial Design

Introduction

There are seven degrees of freedom between the eight treatment combinations in the 23 design. Three degrees of freedom are associated with the main effects of A, B, and C. Four degrees of freedom are associated with interactions; one each with AB, AC, and BC and one with ABC.

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

Sums of squares for the effects are easily computed, because each effect has a corresponding single-degree-of-freedom contrast. In the 23 design with n replicates, the sum of squares for any effect is

Algebraic Signs for Calculating Effects in the 23 Design

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Example

23 Factorial Design

Introduction

A soft drink bottler is interested in obtaining more uniform fill heights in the bottles produced by his manufacturing process. The filling machine theoretically fills each bottle to the correct target height, but in practice, there is variation around this target, and the bottler would like to understand better the sources of this variability and eventually reduce it. The process engineer can control three variables during the filling process: the percent carbonation (A), the operating pressure in the filler (B), and the bottles produced per minute or the line speed (C). Suppose that only two levels of carbonation are used so that the experiment is a 23 factorial design with two replicates. The data, deviations from the target fill height, are shown in Table 6-4, and the design is shown geometrically

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

The largest effects are for carbonation (A = 3.00), pressure (B = 2.25), speed (C = 1.75) and the carbonation-pressure interaction (AB = 0.75), although the interaction effect does not appear to have as large an impact on fill height deviation as the main effects.

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

1-1

2

1

0

1-1

1-1

2

1

0

A Carbonizat ion

Me

an

B Pressure

C Speed

Main Effects Plot for Fill Hight DeviationFitted Means

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Example

23 Factorial Design

Introduction

10-1

99

90

50

10

1

Residual

Pe

rce

nt

6420-2

1.0

0.5

0.0

-0.5

-1.0

Fitted Value

Re

sid

ua

l1.00.50.0-0.5-1.0

6.0

4.5

3.0

1.5

0.0

Residual

Fre

qu

en

cy

16151413121110987654321

1.0

0.5

0.0

-0.5

-1.0

Observation Order

Re

sid

ua

l

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Fill Hight Deviation

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Example

23 Factorial Design

Introduction

1-1 1-1

4

2

0

4

2

0

A Carbonization

B Pressure

C Speed

-1

1

A C arb o n izatio n

-1

1

B P ressu re

Interaction Plot for Fill Hight DeviationFitted Means

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Example

23 Factorial Design

Introduction

1-1 1-1

4

2

0

4

2

0

A Carbonization

B Pressure

C Speed

-1

1

A C arb o n izatio n

-1

1

B P ressu re

Interaction Plot for Fill Hight DeviationFitted Means

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Example

23 Factorial Design

Introduction

An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results follow:

a. Estimate the factor effectsb. Prepare an analysis of variance table, and determine which factors are

important in explaining yieldc. Plot the residuals versus the predicted yield and on a normal probability

scale. Does the residual analysis appear satisfactory?

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Example

23 Factorial Design

Introduction

N=3 , ɑ = 104 , b = 119 , c = 127, ɑb = 148, ɑc= 113, bc= 164, ɑbc= 127, (-1) = 78

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Example

23 Factorial Design

Introduction

N=3 , ɑ = 104 , b = 119 , c = 127, ɑb = 148, ɑc= 113, bc= 164, ɑbc= 127, (-1) = 78

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Example

23 Factorial Design

Introduction

Contrasts

What is SST

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

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Example

23 Factorial Design

Introduction

50

40

30

1-1

1-1

50

40

30

1-1

50

40

30

Cutting Speed

Tool Geometry

Cutting Angle

-1

1

S p eed

C u ttin g

-1

1

G eo m etry

To o l

-1

1

A n g le

C u ttin g

Interaction Plot for Life in hoursFitted Means

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Example

23 Factorial Design

Introduction

1-1

45.0

42.5

40.0

37.5

35.0

1-1

1-1

45.0

42.5

40.0

37.5

35.0

Cutt ing Speed

Me

an

Tool Geometry

Cutt ing Angle

Main Effects Plot for Life in hoursFitted Means

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Example

23 Factorial Design

Introduction

1050-5-10

99

90

50

10

1

Residual

Pe

rce

nt

5448423630

10

5

0

-5

Fitted Value

Re

sid

ua

l

10.07.55.02.50.0-2.5-5.0

6.0

4.5

3.0

1.5

0.0

Residual

Fre

qu

en

cy

24222018161412108642

10

5

0

-5

Observation OrderR

esid

ua

l

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Life in hours

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Example

23 Factorial Design

Introduction

151050-5-10-15

99

95

90

80

70

60

50

40

30

20

10

5

1

RESI1

Pe

rce

nt

M ean 5.181041E -15

S tDev 4.581

N 24

A D 0.674

P -Va lu e 0.069

Probability Plot of RESI1Normal - 95% CI

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Example

23 Factorial Design

Introduction

55504540353025

55504540353025

12.5

10.0

7.5

5.0

2.5

0.0

-2.5

-5.0

12.5

10.0

7.5

5.0

2.5

0.0

-2.5

-5.0

FITS1

RE

SI1

Scatterplot of RESI1 vs FITS1

Class 17-2-3 Factorial Design

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