5 factorial design

Factorial Experiments

Dr. Jitesh J. Thakkar

INDIAN INSTITUTE OF TECHNOLOGY (IIT) KHARAGPUR

KHARAGPUR – 721 3021Dr. Jitesh Thakkar, IIT Kharagpur

Contents

• General principles of factorial experiments

• The two-factor factorial with fixed effects

• The ANOVA for factorials

• Extensions to more than two factors

• Quantitative and qualitative factors –

response curves and surfaces

Source

D. C. Montgomery, Design and analysis of experiments, 7th edition

Design & Analysis of Experiments 8E 2012 Montgomery

4

Basic Definition• Many experiments involve the study of the effects

of two or more factors.• Factorial designs most efficient for this type of

experiment.• By a factorial design, we mean that in each

complete trial or replicate of the experiment all possible combinations of the levels of the factors are investigated

• If there are a levels of factor A and b levels of factor B, each replicate contains all ab treatment combinations.


5

Some Basic Definitions

Definition of a factor effect: The change in the mean response when the factor is changed from low to high

40 52 20 3021

2 230 52 20 40

112 2

52 20 30 401

2 2

A A

B B

A y y

B y y

AB


6

The Case of Interaction:

50 12 20 401

2 240 12 20 50

92 2

12 20 40 5029

2 2

A A

B B

A y y

B y y

AB


7

Regression Model & The Associated Response Surface

0 1 1 2 2 12 1 2

1 2 1 2 1 2

The least squares fit is

ˆ 35.5 10.5 5.5 0.5 35.5 10.5 5.5

y x x x x

y x x x x x x


8

The Effect of Interaction on the Response Surface

Suppose that we add an interaction term to the model:

1 2 1 2ˆ 35.5 10.5 5.5 8y x x x x

Interaction is actually a form of curvature


9

Select Insights• When an interaction is large, the corresponding main effects have little

practical meaning.• For the experiment in Figure 5.2, we would estimate the main effect of

A to be • A=(50+12)/2 – (20+40)/2 = 1• This is very small and we are tempted to conclude that there is no

effect due to A.• When we examine the effects of A at different levels of factor B, we

see that this is not the case• Factor A has an effect, but it depends on the level of factor B. • Knowledge of AB interaction is more useful than knowledge of the

main effect.• A significant interaction will often mask the significance of main

effects.

One factor at a time experiment


11

Insight• If a factorial experiment had been performed, an additional

treatment combination, A+ B+ would have been taken • Two estimates of the effect A can be made A+B- - A-B- and A+B+- A-

B+

• Two estimates of the B can be made similarly• Two estimates of each main effect could be averaged to produce

average main effects that are just as precise as those from the single-factor experiment, but only four total observations are required

• Relative efficiency of the factorial design to the one-factor-at-a-time experiment is 6/4=1.5

• Relative efficiency will increase as the number of factors increases• Factorial designs are necessary when interactions may be present

to avoid misleading conclusions.

Relative efficiency of a factorial design to a one-factor-at-a-time experiment (two factor levels)


13

Example 5.1 The Battery Life ExperimentText reference pg. 187

A = Material type; B = Temperature (A quantitative variable)

1. What effects do material type & temperature have on life?

2. Is there a choice of material that would give long life regardless of temperature (a robust product)?


14

The General Two-Factor Factorial Experiment

a levels of factor A; b levels of factor B; n replicates

This is a completely randomized design


15

Statistical (effects) model:

1,2,...,

( ) 1, 2,...,

1, 2,...,ijk i j ij ijk

i a

y j b

k n

Other models (means model, regression models) can be useful


16

Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 189

2 2 2... .. ... . . ...

1 1 1 1 1

2 2. .. . . ... .

1 1 1 1 1

( ) ( ) ( )

( ) ( )

a b n a b

ijk i ji j k i j

a b a b n

ij i j ijk iji j i j k

y y bn y y an y y

n y y y y y y

breakdown:

1 1 1 ( 1)( 1) ( 1)

T A B AB ESS SS SS SS SS

df

abn a b a b ab n


17

ANOVA Table – Fixed Effects Case


18


22

Multiple Comparison

• When ANOVA indicates that row or column means are differ, it is usually of interest to make comparisons between the individual row or column means to discover specific differences

• Turkey’s test• When interaction is significant, comparisons between the

means of one factor (e.g. A) may be obscured by the AB interaction

• Fix factor B at a specific level and apply Turkey’s test toe means of factor A at that level

Chapter 5


23

Design-Expert Output – Example 5.1


25Chapter 5

JMP output – Example 5.1


27

RESIDUAL ANALYSIS

Chapter 5

Choice of Sample Size

The Assumption of No Interaction in a Two-Factor Model


35

One Observation Per Cell

Chapter 5


39

The General Factorial Design

Chapter 5


40Chapter 5

Factorials with More Than Two Factors

• Basic procedure is similar to the two-factor case; all abc…kn treatment combinations are run in random order

• ANOVA identity is also similar:

• Complete three-factor example in text, Example 5.5

T A B AB AC

ABC AB K E

SS SS SS SS SS

SS SS SS


45Chapter 5

Fitting Response Curves and Surfaces

• The basic ANOVA procedure treats every factor as if it were qualitative

• Sometimes an experiment will involve both quantitative and qualitative factors, such as in Example 5.1

• This can be accounted for in the analysis to produce regression models for the quantitative factors at each level (or combination of levels) of the qualitative factors

• These response curves and/or response surfaces are often a considerable aid in practical interpretation of the results


46Chapter 5

Quantitative and Qualitative Factors

Candidate model terms from Design- Expert: Intercept

A B B2

AB B3

AB2

A = Material type

B = Linear effect of Temperature

B2 = Quadratic effect of Temperature

AB = Material type – TempLinear

AB2 = Material type - TempQuad

B3 = Cubic effect of Temperature (Aliased)


47Chapter 5

Quantitative and Qualitative Factors


48

Insights• The P-value indicate that A2 and AB are not significant

whereas the A2B term is significant • Often we think about removing non-significant terms or

factors from a model, but in this case, removing A2 and AB and retaining A2B will result in a model that is not hierarchical

• The hierarchical principle indicates that if a model contains a high-order term (such as A2B), it should also contain all of the lower order terms that compose it (in this case A2 and AB)

• Hierarchy promotes a type of consistency in the model

Chapter 5


49Chapter 5

Regression Model Summary of Results


50Chapter 5


51Chapter 5


52Chapter 5


53Chapter 5


54Chapter 5


55Chapter 5


56

Blocking in a Factorial Design

Chapter 5

5 factorial design

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