itk-226 statistika & rancangan percobaan dicky dermawan dickydermawan@gmail.com
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3.COMPARING SEVERAL TREATMENTS: ANALYSIS OF VARIANCE
ITK-226 Statistika & Rancangan Percobaan
Dicky Dermawanwww.dickydermawan.net78.netdickydermawan@gmail.com
DOX 6E Montgomery 2
Introduction to DOX
An experiment is a test or a series of tests Experiments are used widely in the
engineering world Process characterization & optimization Evaluation of material properties Product design & development Component & system tolerance determination
“All experiments are designed experiments, some are poorly designed, some are well-designed”
DOX 6E Montgomery 3
Engineering Experiments
Reduce time to design/develop new products & processes
Improve performance of existing processes
Improve reliability and performance of products
Achieve product & process robustness
Evaluation of materials, design alternatives, setting component & system tolerances, etc.
DOX 6E Montgomery 4
Four Eras in the History of DOX The agricultural origins, 1918 – 1940s
R. A. Fisher & his co-workers Profound impact on agricultural science Factorial designs, ANOVA
The first industrial era, 1951 – late 1970s Box & Wilson, response surfaces Applications in the chemical & process industries
The second industrial era, late 1970s – 1990 Quality improvement initiatives in many
companies Taguchi and robust parameter design, process
robustness The modern era, beginning circa 1990
DOX 6E Montgomery 5
The Basic Principles of DOX Randomization
Running the trials in an experiment in random order
Notion of balancing out effects of “lurking” variables
Replication Sample size (improving precision of effect
estimation, estimation of error or background noise)
Replication versus repeat measurements? (see page 13)
Blocking Dealing with nuisance factors
DOX 6E Montgomery 6
What If There Are More Than Two Factor Levels? The t-test does not directly apply There are lots of practical situations where
there are either more than two levels of interest, or there are several factors of simultaneous interest
The analysis of variance (ANOVA) is the appropriate analysis “engine” for these types of experiments – Chapter 3, textbook
The ANOVA was developed by Fisher in the early 1920s, and initially applied to agricultural experiments
Used extensively today for industrial experiments
DOX 6E Montgomery 7
An Example (See pg. 60) An engineer is interested in investigating the
relationship between the RF power setting and the etch rate for this tool. The objective of an experiment like this is to model the relationship between etch rate and RF power, and to specify the power setting that will give a desired target etch rate.
The response variable is etch rate. She is interested in a particular gas (C2F6) and gap
(0.80 cm), and wants to test four levels of RF power: 160W, 180W, 200W, and 220W. She decided to test five wafers at each level of RF power.
The experimenter chooses 4 levels of RF power 160W, 180W, 200W, and 220W
The experiment is replicated 5 times – runs made in random order
DOX 6E Montgomery 8
The Analysis of Variance (Sec. 3-2, pg. 63)
In general, there will be a levels of the factor, or a treatments, and n replicates of the experiment, run in random order…a completely randomized design (CRD)
N = an total runs We consider the fixed effects case…the random effects
case will be discussed later Objective is to test hypotheses about the equality of the a
treatment means
DOX 6E Montgomery 9
The Analysis of Variance The name “analysis of variance” stems
from a partitioning of the total variability in the response variable into components that are consistent with a model for the experiment
The basic single-factor ANOVA model is
2
1,2,...,,
1, 2,...,
an overall mean, treatment effect,
experimental error, (0, )
ij i ij
i
ij
i ay
j n
ith
NID
DOX 6E Montgomery 10
Models for the Data
There are several ways to write a model for the data:
is called the effects model
Let , then
is called the means model
Regression models can also be employed
ij i ij
i i
ij i ij
y
y
DOX 6E Montgomery 11
The Analysis of Variance
Total variability is measured by the total sum of squares:
The basic ANOVA partitioning is:
2..
1 1
( )a n
T iji j
SS y y
2 2.. . .. .
1 1 1 1
2 2. .. .
1 1 1
( ) [( ) ( )]
( ) ( )
a n a n
ij i ij ii j i j
a a n
i ij ii i j
T Treatments E
y y y y y y
n y y y y
SS SS SS
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The Analysis of Variance
A large value of SSTreatments reflects large differences in treatment means
A small value of SSTreatments likely indicates no differences in treatment means
Formal statistical hypotheses are:
T Treatments ESS SS SS
0 1 2
1
:
: At least one mean is different aH
H
DOX 6E Montgomery 13
The Analysis of Variance While sums of squares cannot be directly
compared to test the hypothesis of equal means, mean squares can be compared.
A mean square is a sum of squares divided by its degrees of freedom:
If the treatment means are equal, the treatment and error mean squares will be (theoretically) equal.
If treatment means differ, the treatment mean square will be larger than the error mean square.
1 1 ( 1)
,1 ( 1)
Total Treatments Error
Treatments ETreatments E
df df df
an a a n
SS SSMS MS
a a n
DOX 6E Montgomery 14
The Analysis of Variance is Summarized in a Table
Computing…see text, pp 66-70 The reference distribution for F0 is the Fa-1, a(n-1)
distribution Reject the null hypothesis (equal treatment
means) if 0 , 1, ( 1)a a nF F
DOX 6E Montgomery 15
An Example (See pg. 60) An engineer is interested in investigating the
relationship between the RF power setting and the etch rate for this tool. The objective of an experiment like this is to model the relationship between etch rate and RF power, and to specify the power setting that will give a desired target etch rate.
The response variable is etch rate. She is interested in a particular gas (C2F6) and gap
(0.80 cm), and wants to test four levels of RF power: 160W, 180W, 200W, and 220W. She decided to test five wafers at each level of RF power.
The experimenter chooses 4 levels of RF power 160W, 180W, 200W, and 220W
The experiment is replicated 5 times – runs made in random order
DOX 6E Montgomery 16
An Example (See pg. 62)
Does changing the power change the mean etch rate?
Is there an optimum level for power?
Power (W) Etch Rate (A/min)160 575 542 530 539 570180 565 593 590 579 610200 600 651 610 637 629220 725 700 715 685 710
15 5 20 7 8
2 10 1 11 13
16 12 18 6 4
19 14 3 9 17
DOX 6E Montgomery 17
ANOVA TableExample 3-1
DOX 6E Montgomery 18
The Reference Distribution:
A product development engineer is interesed in investigating the tensile strength of a new synthetic fiber that will be used to make clothe for men’s shirt.
The engineer knows from previous experience that the strength is affected by the weight percent of cotton used in the blend of materials for the fiber. Furthermore, she also knows that cotton content should range between 10% - 40% if the final product is to have other quality characteristics that are desired (such as the ability to take a permanent-press finishing treatment).
The engineer decided to test speciments at 5 levels of cotton weight percent: 15%, 20%, 25%, 30%, and 35%. She also decided to test 5 speciments at each level of cotton weight.
Exercise: Analysis of Variance
Exercise: Analysis of Variance (cont)5 levels = 5 treatments : 15%, 20%, 25%, 30%, and 35%.
5 speciments at each level : 5 replicates
Exercise: Analysis of Variance (cont)
Randomization
Exercise: Analysis of Variance (cont)Summary of Experimental Results
a. Draw the Box & Whisker Plot
b. Make the appropriate analysis of variance
DOX 6E Montgomery 23
Model Adequacy Checking in the ANOVAText reference, Section 3-4, pg. 75 Checking assumptions is
important Normality Constant variance Independence Have we fit the right model? Later we will talk about what to do if
some of these assumptions are violated
DOX 6E Montgomery 24
Model Adequacy Checking in the ANOVA
Examination of residuals (see text, Sec. 3-4, pg. 75)
Design-Expert generates the residuals
Residual plots are very useful
Normal probability plot of residuals
.
ˆij ij ij
ij i
e y y
y y
DOX 6E Montgomery 25
Other Important Residual Plots
Statistical Test for Equality of Variance:Bartlett’s Test
1. Tukey’s Test
2. Fisher Least Significant Difference (LSD) Method
3. Duncan’s Multiple Range Test
4. Neuman-Keul’s Test
We have concluded that at least one of the treatment means is different, but…..Which one(s) ?
To use the Fisher LSD procedure, we simply compare the observed difference between each pair of averages to the corresponding LSD. If the absolute difference > LSD, we conclude that the population means mi and mj differ.
jiEaN,2/ n
1
n
1MStLSD
Unbalanced Data
Unbalanced Data
Randomized Complete Block DesignIs one of the most widely used experimental design.
b,....,2,1j
a,....,2,1iy ijijij
b,....,2,1j
a,....,2,1iy ijjiij
The models:
The ANOVA
RBCD
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