1. Design of Experiments
An Introduction

2. Experiment Why? To uncover the truth.
Truth: something that is not already known.
We do it very often!
Children learn about hot stove.
Children test parents.
Students test teachers.

3. Experiment How do you get your best listening satisfaction? Manipulate
DENON DRA-685 Stereo Receiver
How do you get your best listening satisfaction?
Manipulate
Volume Knob (low – high)
Tone Knob (Low – High)
Balance Knob (Left – Right)

4. Systematic Experiments
Problem:
How can we have the best sound effect for my Stereo system?
Dependent Variable:
How do we measure? – listening satisfaction
Control (Independent) Variables:
How we set them?
Volume (low – high)
Tone (Low – High)
Balance (Left – Right Speakers)

5. True Experiment A true experiment is a study
Independent variables are manipulated.
Levels (values) of independent variables are assigned at random to the experiment units.
Their effect on dependent variables is determined.

6. Three Steps Experiment Statement of Problem
Choice of Response (dependent variable).
Selection of factors (independent variables) to be varied.
Choices of levels of these factors
Design
Number of observation to be taken
Order of experimentation: Complete randomization.
Mathematical model to describe the experiment.
Hypothesis to be test.
Analysis
Data collection and processing.
Computation of test statistics and preparation of graphics.
Interpretation of results.

7. DOE Investigation Process
Analysis and interpretation of experimental results
Restate problem
Perform Experiment
Design of Experiment

8. Example-one factor
A manufacturer wants to know if any of their four types fabrics, A, B, C, and D resists wear better.
There is a machine for wear testing that fabrics can go though a set length of wearing cycle to measure the fabric weight loss.
Dependent variable: Weight loss of material in grams
Independent variable: Fabric type, 4 levels: A, B, C, D
4 observations for each fabric. A total of 16 samples.
Complete randomization

9. Experiment Data Table Factor Fabric Type Treatment (level #1, A)
Factor Fabric Type
Treatment (level #1, A)
Treatment (level #2, B)
Treatment (level #3, C)
Treatment (level #4, D)
Observation #1
Observation #2
Observation #3
Observation #4

10. Hypothesis We want to see if different fabric makes difference?
Ho: no difference among the 4 fabrics.
H1: At least one fabric is different.

11. Model Description
An observation, Yij, for the ith observation and jth treatment may contain three parts:
Yij = m + tj + eij
m: a common effect for the whole
experiment, often fixed parameter
tj: the effect of the jth treatment,
tj = mj - m
eij: random error present in the ith observation
and jth treatment. NID

12. Analysis ANOVA Ho: t1 = t2 = t3 = t4
Ho: t1 = t2 = t3 = t4
H1: At least one treatment effect  0

13. Experimental Data on Weight Loss
Treatment
Observation A B C D 1
1.93
2.55
2.4
2.33 2
2.38
2.72
2.68 3
2.2
2.75
2.31
2.28 4
2.25
2.7
How do we analyze the data and test if any of these fabric has better wear resistance?
ANOVA

14. 1 2 3 4
Obs #1
Y11
Y12
Y13
Y14
Obs #2
Y21
Y22
Y23
Y24
Obs#3
Y31
Y32
Y33
Y34
Obs #4
Y41
Y42
Y43
Column total
T.1
T.2
T.3
T.4
Number
Mean

15. 1 2 3 4 Obs #1 Y11 Y12 Y13 Y14 Obs #2 Y21 Y22 Y23 Y24 Obs#3 Y31 Y321 2 3 4
Obs #1
Y11
Y12
Y13
Y14
Obs #2
Y21
Y22
Y23
Y24
Obs#3
Y31
Y32
Y33
Y34
Obs #4
Y41
Y42
Y43
Mean
 ~N(0,2)
If 1= 2 = 3= = 4=0, e.1, e.2, e.3,e.4 should also follow  ~N(0,2)

16. If we test against
If the ratio of them is 1,

17. One-way ANOVA Source SS df MS F p Treatment, t 0.5201 3 0.1734 8.53
0.0026
Error, e
0.2438 12
0.0203
Total
0.7639
f-statistic = / = 8.53;
P(F(3,12) > 8.53) =
Ho can be rejected for any  >
There is a significant difference in the wear resistance among the 4 fabric. Factor fabric makes difference. Which (type of) fabric is better? Don’t know!