Designing experiments with several factors
Outlines:
 Factorial Experiments
 Two-factor factorial experiment
 General factorial experiments
 2k factorial designs
Factorial Experiments



Factorial experiment: Powerful technique for analysis experimental results
with more than one factors.
Factorial experiment: investigate all possible combinations of the levels for
the factors.
Main effect: the change in the response produced by a change in the level
of the factor.

Main effect of A (B) : the difference between the average response at the high level
of A (B) and the average response at the low level of A (B)
A
30  40 10  20
20  40 10  30

 20, B 

 10
2
2
2
2
Factorial Experiments

Interaction: the dependency between factors. The difference in response between the
levels of one factor is not the same at all level of the other factors.

At low level of B, the A effect is A=30-10=20

At high level of B, the A effect is A=0-20=-20

When an interaction is large, the corresponding main effects have very little
meaning.
A

30  0 10  20

0
2
2
Very different depends
on the level of B
There is no factor A effect
At different level of B, we can see the different effect of A.
The knowledge of AB
interaction is more useful than
the main effect
Factorial Experiments



The AB Interaction effect can be calculated as the difference in the
diagonal averages.
Ex.1
AB 
20  30 10  40

0
2
2
AB 
20  30 10  0

 20
2
2
Ex. 2
The AB interaction effect in Ex.2 is higher than Ex.1
Factorial Experiments
Ex.1
Low
Average response
30
20
Main effect plots and interaction plots
 Main effect plots: plot the values of average response at different levels
for each factors.
Average response
35
15

high
Low
Factor A

high
Factor B
Interaction plot: plot the response against the levels of each factor.
Ex1
Ex2
Factorial Experiments

Three dimensional surface plot represents main effects and interaction effecs.
Two-factor factorial experiments

The simplest type of factorial experiment

2 factors: A (with a levels) and B (with b levels)

The treatment has n replicates. There are a*b*n observations, run in random
order.
observation
Two-factor factorial experiments

The linear statistical model for each observation:
A,

Statistical Analysis:
B,
AB,
random error
Two-factor factorial experiments

Hypothesis

SS:
Two-factor factorial experiments

Degree of freedom:

MSE:

Test statistics:
Two-factor factorial experiments
Two-factor factorial experiments


Ex.
Aircraft primer paints are applied to aluminum surfaces by two methods: dipping and
spraying. The purpose of the primer is to improve paint adhesion, and some parts can
be primed using either application method. The process engineering group responsible
for this operation is interested in learning whether three different primers differ in their
adhesion properties.
Two-factor factorial experiments
Two-factor factorial experiments
X
Ho
/
Ho
>f0.05,2,12=3.89
>f0.05,1,12=4.75
<f0.05,2,12=3.89
Two-factor factorial experiments


Model Adequacy Checking
Residual = random error Eijk
eijk  yijk  yij.
General Factorial Experiments

Involve more than 2 factors. For example, 3 factors A (a levels), B(b levels), C (c
levels)
General Factorial Experiments

Ex. A mechanical engineer is studying the surface roughness of a part produced in a
metal-cutting operation. Three factors, feed rate (A), depth of cut (B), and tool
angle (C), are of interest. All three factors have been assigned two levels, and two
replicates of a factorial design are run.
General Factorial Experiments
Exercise

The percentage of hardwood concentration in raw pulp, the freeness, and the
cooking time of the pulp are being investigated for their effects on the strength of
paper. The data from a three-factor factorial experiment are shown in the following
table.
(a) Analyze the data using the analysis of variance assuming that all factors are fixed. Use
0.05.
(b) Find P-values for the F-ratios in part (a).
(c) analyze the residuals from this experiment.
Exercise

The quality control department of a fabric finishing plant is studying the effects of several factors on dyeing
for a blended cotton/synthetic cloth used to manufacture shirts. Three operators, three cycle times, and two
temperatures were selected, and three small specimens of cloth were dyed under each set of conditions. The
finished cloth was compared to a standard, and a numerical score was assigned. The results are shown in the
following table.
(a) State and test the appropriate hypotheses using the analysis of variance with 0.05.
(b) Graphically analyze the residuals from this experiment.