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Material Type: Notes; Professor: Wu; Class: Dsgn & Analy-Experiments; Subject: Industrial & Systems Engr; University: Georgia Institute of Technology-Main Campus; Term: Unknown 1989;
Typology: Study notes
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Instructor : C. F. Jeff Wu
School of Industrial and Systems Engineering
Georgia Institute of Technology
Text book :
Experiments : Planning, Analysis, and Parameter Design
Optimization
(by Wu and Hamada; Wiley, 2000)
1
Sources : Sections 1.1 to 1.5, additional materials (in these notes) on regression
analysis.
Historical perspectives and basic definitions.
Planning and implementation of experiments.
Fisher’s fundamental principles.
Simple linear regression.
Multiple regression, variable selection.
Regression diagnostics.
2
Quality Revolution :
Quality and productivity improvement, variation
reduction, total quality management, Taguchi’s work on robust parameterdesign, Six-sigma movement.
A lot of successful applications in manufacturing (cars, electronics, homeappliances, etc.)
Current Trends and Potential New Areas :
Computer modelling and
experiments, large and complex systems, applications to biotechnology,nanotechnology, material development, etc.
4
Treatment Comparisons :
Purpose is to compare several treatments of a
factor (have 4 rice varieties and would like to see if they are different interms of yield and draught resistence).
Variable Screening :
Have a large number of factors, but only a few are
important. Experiment should identify the important few.
Response Surface Exploration :
After important factors have been
identified, their impact on the system is explored; regression model building.
5
Factor :
variable whose influence upon a response variable is being studied
in the experiment.
Factor Level :
numerical values or settings for a factor.
Trial
(or
run
) : application of a treatment to an experimental unit.
Treatment or level combination :
set of values for all factors in a trial.
Experimental unit :
object to which a treatment is applied.
Randomization :
using a chance mechanism to assign treatments to
experimental units or run order.
7
State the objective of the study.
Choose the response variable
should correspond to the purpose of the
study.
Nominal-the-best, larger-the-better or smaller-the-better
Choose factors and levels.
Use flow chart or cause-and-effect diagram (see Figure 1).
Choose experimental design (i.e., plan).
Perform the experiment (use a planning matrix to determine the set oftreatments and the order to be run).
Analyze data (design should be selected to meet objective so that theanalysis is efficient and easy).
Draw conclusions.
8
To improve a process that often produces underweight soap bars.Obvious choice of response,
y
= soap bar weight.
There are two sub-processes : (i) mixing, which affects soap bar density(=
y
1
), (ii) forming, which affects soap bar dimensions (=
y
2
Even though
y
is a function of
y
1
and
y
2
, better to study
y
1
and
y
2
separately
and identify factors important for each of the two sub-processes.
10
Each treatment is applied to units that are representative of the population(example : measurements of 3 units vs. 3 repeated measurements of 1 unit).
Replication vs Repetition (i.e., repeated measurements).
Enable the estimation of experimental error. Use sample standard deviation.
Decrease variance of estimates and increase the power to detect significantdifferences : for independent
y
i
’s,
Var
1 n
n
i
=
1
y
i
1 n
Var
y
1
11
block
refers to a collection of homogeneous units. Effective blocking : larger
between-block variations than within-block variations.(Examples: hours, batches, lots, street blocks, pairs of twins.)
Run and compare treatments within the same blocks. (Use randomizationwithin blocks.) It can eliminate block-block variation and reduce variabilityof treatment effects estimates.
Block what you can and randomize what you cannot.
Discuss
typing experiment
to demonstrate possible elaboration of the
blocking idea. See next page.
13
To compare two keyboards
and
in terms of typing efficiency. Six
manuscripts 1-6 are given to the same typist.
Several designs (i.e., orders of test sequence) are considered:
always followed by
, why bad ?)
(an improvement, but there are four with
and two with
. Why is
this not desirable? Impact of
learning effect
Balanced randomization
: To mitigate the learning effect, randomly
choose three with
and three with
. (Produce one such plan on
your own).
14
40
45
50
100 90 80 70 60
Temperature
Mortality rate
Figure 2: Scatter Plot of Temperature versus Mortality Rate, Breast Cancer Data.
16
Underlying Model :
y
β
0
β
1
x
ε
ε
σ
2
Coefficients are estimated by minimizing
n
=
1
y
i
β
0
β
1
x
i
2
Least Squares Estimates Estimated Coefficients :
β
1
x
i
x
y
i
y
x
i
x
2
var
β
1
σ
2
x
i
x
2
β
0
y
β
1
x
var
β
0
σ
2
1 n
x
2
x
i
x
2
x
1 n
x
i
y
1 n
y
i
17
To test the null hypothesis
0
β
j
0 against the alternative hypothesis
0
β
j
0, use the test statistic
t
j
β
j
s
d
β
j
The higher the value of
t
, the more significant is the coefficient.
For 2-sided alternatives,
p
-value
Prob
t
d f
t
obs
, df = degrees of
freedom for the
t
-statistic,
t
obs
= observed value of the
t
-statistic. If
p
-value
is very small, then either we have observed something which rarelyhappens, or
0
is not true. In practice, if
p
-value is less then
α
05 or
0
is rejected at level
α
19
α
% confidence interval for
β
j
is given by
β
j
t
n
−
2
,
α 2
s
d
β
j
where
t
n
−
2
,
α^2
is the upper
α
2 point of the
t
distribution with
n
2 degrees of
freedom.If the confidence interval for
β
j
does not contain 0, then
0
is rejected.
20
