Introduction to Design of
Experiments
Reference: book by D. C. Montgomery
1
The Blocking Principle
• Blocking is a technique for dealing with nuisance factors
• A nuisance factor is a factor that probably has some effect
on the response, but it’s of no interest to the
experimenter…however, the variability it transmits to the
response needs to be minimized
• Typical nuisance factors include batches of raw material,
operators, pieces of test equipment, time (shifts, days, etc.),
different experimental units
• Many industrial experiments involve blocking (or should)
• Failure to block is a common flaw in designing an
experiment (consequences?)
2
3
Vascular Graft Example
• To conduct this experiment as a RCBD, assign all 4
pressures to each of the 6 batches of resin
• Each batch of resin is called a “block”; that is, it’s a
more homogenous experimental unit on which to test
the extrusion pressures
4
5
Vascular Graft Example
Design-Expert Output
6
Factorial Experiments
• 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
7
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 30
A y A y A 21
2 2
30 52 20 40
B yB yB 11
2 2
52 20 30 40
AB 1
2 2
8
The Case of Interaction:
50 12 20 40
A y A y A 1
2 2
40 12 20 50
B yB yB 9
2 2
12 20 40 50
AB 29
2 2
9
Example 5.1 The Battery Life Experiment
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)?
10
The General Two-Factor
Factorial Experiment
a levels of factor A; b levels of factor B; n replicates
11
ANOVA Table – Fixed Effects Case
12
Output
13
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:
SST SS A SS B SS AB SS AC
SS ABC SS AB K SS E
14
Design of Engineering Experiments
Two-Level Factorial Designs
• Special case of the general factorial design; k factors,
all at two levels
• The two levels are usually called low and high (they
could be either quantitative or qualitative)
• Very widely used in industrial experimentation
• Form a basic “building block” for other very useful
experimental designs (DNA)
• Special (short-cut) methods for analysis
15
The Simplest Case: The 22
“-” and “+” denote the low and high levels of a factor, respectively
• Low and high are arbitrary terms
• Geometrically, the four runs form the corners of a square
• Factors can be quantitative or qualitative, although their treatment in the final
model will be different
16
Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery
17
Estimation of Factor Effects
A y A y A
ab a b (1)
2n 2n
21n [ab a b (1)]
B yB yB
ab b a (1)
2n 2n
21n [ab b a (1)]
ab (1) a b
AB
2n 2n
21n [ab (1) a b]
18
Statistical Testing - ANOVA
The F-test for the “model” source is testing the significance of the
overall model; that is, is either A, B, or AB or some combination of
these effects important?
19
The 23 Factorial Design
20
An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
21
Table of – and + Signs for the 23 Factorial Design (pg. 218)
22
Properties of the Table
• Except for column I, every column has an equal number of + and –
signs
• The sum of the product of signs in any two columns is zero
• Multiplying any column by I leaves that column unchanged (identity
element)
• The product of any two columns yields a column in the table:
A B AB
AB BC AB 2C AC
• Orthogonal design
• Orthogonality is an important property shared by all factorial designs
23
Estimation of Factor Effects
24
ANOVA Summary – Full Model
25
The General 2k Factorial Design
• There will be k main effects, and
k
2 two-factor interactions
k
3 three-factor interactions
1 k factor interaction
26
Design Projection: ANOVA Summary for
the Model as a 23 in Factors A, C, and D
27
