Class 17-2-3 Factorial Design

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.

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.

Example

23 Factorial Design

Introduction

Example

23 Factorial Design

Introduction

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

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

Example

23 Factorial Design

Introduction

Example

23 Factorial Design

Introduction

Example

23 Factorial Design

Introduction

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.

Example

23 Factorial Design

Introduction

Example

23 Factorial Design

Introduction

A Carbonizat ion

B Pressure

C Speed

Main Effects Plot for Fill Hight DeviationFitted Means

Example

23 Factorial Design

Introduction

Residual

6420-2

Fitted Value

l1.00.50.0-0.5-1.0

Residual

16151413121110987654321

Observation Order

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Fill Hight Deviation

Example

23 Factorial Design

Introduction

1-1 1-1

A Carbonization

B Pressure

C Speed

A C arb o n izatio n

B P ressu re

Interaction Plot for Fill Hight DeviationFitted Means

Example

23 Factorial Design

Introduction

1-1 1-1

A Carbonization

B Pressure

C Speed

A C arb o n izatio n

B P ressu re

Interaction Plot for Fill Hight DeviationFitted Means

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?

Example

23 Factorial Design

Introduction

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

Example

23 Factorial Design

Introduction

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

Example

23 Factorial Design

Introduction

Contrasts

What is SST

Example

23 Factorial Design

Introduction

Example

23 Factorial Design

Introduction

Example

23 Factorial Design

Introduction

Cutting Speed

Tool Geometry

Cutting Angle

S p eed

C u ttin g

G eo m etry

To o l

A n g le

C u ttin g

Interaction Plot for Life in hoursFitted Means

Example

23 Factorial Design

Introduction

Cutt ing Speed

Tool Geometry

Cutt ing Angle

Main Effects Plot for Life in hoursFitted Means

Example

23 Factorial Design

Introduction

1050-5-10

Residual

5448423630

Fitted Value

10.07.55.02.50.0-2.5-5.0

Residual

24222018161412108642

Observation OrderR

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Life in hours

Example

23 Factorial Design

Introduction

151050-5-10-15

M ean 5.181041E -15

S tDev 4.581

A D 0.674

P -Va lu e 0.069

Probability Plot of RESI1Normal - 95% CI

Example

23 Factorial Design

Introduction

55504540353025

Scatterplot of RESI1 vs FITS1

Class 17-2-3 Factorial Design

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