ITK-226 Statistika & Rancangan Percobaan Dicky Dermawan [email protected].

3.COMPARING SEVERAL TREATMENTS: ANALYSIS OF VARIANCE

ITK-226 Statistika & Rancangan Percobaan

Dicky [email protected]

http://www.dickydermawan.net78.net/

mailto:[email protected]

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

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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

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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

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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

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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

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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


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


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


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


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


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


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


ANOVA TableExample 3-1


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


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


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


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

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

ITK-226 Statistika & Rancangan Percobaan Dicky Dermawan [email protected].

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