Design of Engineering Experimentshaalshraideh/Courses/IE710/Ch03.pdf · 2015-10-04 · Determining...

Design of Engineering Experiments

Hussam Alshraideh

Chapter 3: Analysis of Variance

October 4, 2015

Hussam Alshraideh (JUST) ANOVA October 4, 2015 1 / 44

Overview

1 ANOVAIntroductionFixed effects caseModel Adequacy CheckingComparison of MeansSample Size Determination

2 Other Examples of Single-Factor Experiments

3 The Random Effects Model


ANOVA Introduction

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 thantwo levels of interest, or there are several factors of simultaneousinterest

The analysis of variance (ANOVA) is the appropriate analysis“engine” for these types of experiments

The ANOVA was developed by Fisher in the early 1920s, and initiallyapplied to agricultural experiments.

Used extensively today for industrial experiments.


ANOVA Fixed effects case

An Example (See pg. 66)

An engineer is interested in investigating the relationship between theRF power setting and the etch rate for this tool. The objective of anexperiment like this is to model the relationship between etch rateand RF power, and to specify the power setting that will give adesired target etch rate.

The response variable is etch rate.

She is interested in a particular gas (C2F6) and gap (0.80 cm), andwants to test four levels of RF power: 160W, 180W, 200W, and220W. 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










Does changing the power change the mean etch rate?

Is there an optimum level for power?

We would like to have an objective way to answer these questions

The t-test really doesnt apply here more than two factor levels



The Analysis of Variance (Sec. 3.2, pg. 68)

In general, there will be a levels of the factor, or a treatments, and nreplicates of the experiment, run in random order: a completelyrandomized design (CRD)N = an total runsWe consider the fixed effects case, the random effects case will bediscussed later.Objective is to test hypotheses about the equality of the a treatmentmeans



The Analysis of Variance

The name analysis of variance stems from a partitioning of the totalvariability in the response variable into components that areconsistent with a model for the experiment.

The basic single-factor ANOVA model is:

yij = µ+ τi + εij

{i = 1, 2, · · · , aj = 1, 2, · · · n

where:

µ an overall meanτi= i th treatment effectεij=experimental error ∼ N(0, σ2)



Models for the Data

There are several ways to write a model for the data:

yij = µ+ τi + εij is called the effects model

Let µi = µ+ τi , then

yij = µi + εij is called the means model




Total variability is measured by the total sum os squares:

SST =a∑

i=1

n∑j=1

(yij − y ··)2

The basic ANOVA partitioning is:

a∑i=1

n∑j=1

(yij − y ··)2 =

a∑i=1

n∑j=1

[(y i· − y ··) + (yij − y i·)]2

na∑

i=1

(y i· − y ··)2 +a∑

i=1

n∑j=1

(yij − y i·)2

SST = SSTreatment + SSE




SST = SSTreatment + SSE

A large value of SSTreatments reflects large differences in treatmentmeans

A small value of SSTreatments likely indicates no differences intreatment means

Formal statistical hypotheses are:

H0 : µ1 = µ2 = · · · = µa

H1 : At least one mean is different




While sums of squares cannot be directly compared to test thehypothesis of equal means, mean squares can be compared.

A mean square is a sum of squares divided by its degrees of freedom:

dfTotal = dfTreatment + dfError

an − 1 = (a− 1) + a(n − 1)

MSTreatment =SSTreatment

a− 1, MSE =

SSEa(n − 1)

If the treatment means are equal, the treatment and error meansquares will be (theoretically) equal.

If treatment means differ, the treatment mean square will be largerthan the error mean square.



The Analysis of Variance: Summary

Computing, see text, pp 69

The reference distribution for F0 is the Fa−1,a(n−1) distribution

Reject the null hypothesis (equal treatment means) if

F0 > Fα,a−1,a(n−1)



The Analysis of Variance: Computing formulas

SST =a∑

i=1

n∑j=1

y2ij −

y2··N

SSTreatment =1

n

a∑i=1

y2i· −

y2··N

SSE = SST − SSTreatment



ANOVA table: Example 3.1



ANOVA table: Example 3.1



ANOVA computing using software

Design-Expert,

JMP

Minitab

See pages 102-105 for discussion on summary statistics from thesepackages.


ANOVA Model Adequacy Checking

Model Adequacy Checking in the ANOVA

Checking assumptions is important

NormalityConstant varianceIndependence

Have we fit the right model?

Later we will talk about what to do if some of these assumptions areviolated



Model Adequacy Checking in the ANOVA

Examination of Residuals(sec 3.4):

eij = yij − yij

= yij − y i·

Computer softwaregenerates the residuals

Residual plots are veryuseful

Normal probability plotof residuals



Other Important Residual Plots

Residuals vs order → Independence

Residuals vs fitted values → Constant variance


ANOVA Comparison of Means

Comparison of Means

The analysis of variance tests the hypothesis of equal treatmentmeans

If that hypothesis is rejected, we dont know which specific means aredifferent

Determining which specific means differ following an ANOVA is calledthe multiple comparisons problem

There are lots of ways to do this. → see text, Section 3.5

We will use pairwise t-tests on means sometimes called Fishers LeastSignificant Difference (or Fishers LSD) Method



Graphical Comparison of Means: sliding t-distribution



Fishers’s LSD Method

Controls error rate (α) for each test, but not the rate for the tests as awhole.

H0 : µi = µj

H1 : µi 6= µj

Reject H0 if:

|t0| =

∣∣∣∣∣∣ y i· − y j·√MSE ( 1

ni+ 1

nj)

∣∣∣∣∣∣ > tα/2,N−a

or:

LSD =∣∣y i· − y j·

∣∣ > tα/2,N−a

√MSE (

1

ni+

1

nj)



Tukey’s Method

Based on the studentized range statistic.

Controls the family-wise error rate.



Why Does the ANOVA Work?

We are sampling from Normal populations so:

SSTreatment

σ2∼ χ2

a−1 if H0 is true, andSSEσ2∼ χ2

a(n−1)

so:

F0 =SSTreatment/(a− 1)

SSE/(a(n − 1))=

χ2a−1/(a− 1)

χ2a(n−1)/(a(n − 1))

∼ Fa−1,a(n−1)

Finally,

E [MSTreatment ] = σ2 +

na∑

i=1

τ2i

a− 1, and E [MSE ] = σ2


ANOVA Sample Size Determination

Sample Size Determination

See text section 3.7.

FAQ in designed experiments. Answer depends on lots of things;including

what type of experiment is being contemplated,how it will be conducted,resources, anddesired sensitivity

Sensitivity refers to the difference in means that the experimenterwishes to detect

Generally, increasing the number of replications increases thesensitivity or it makes it easier to detect small differences in means



Sample Size Determination: Fixed Effects Case

Can choose the sample size to detect a specific difference in meansand achieve desired values of type I and type II errors

Type I error : reject H0 when it is true (α)

Type II error : fail to reject H0 when it is false (β)

Power = 1− βOperating characteristic curves plot β against a parameter Φ where

Φ2 =

na∑

i=1

τ2i

aσ2



Sample Size Determination: Fixed Effects Case

The OC curves for the fixed effects model are in the Appendix, TableV

A very common way to use these charts is to define a difference intwo means D of interest, then the minimum value of Φ2 is

Φ2 =nD2

2aσ2

Typically work in term of the ratio of D/σ and try values of n untilthe desired power is achieved

Most statistics software packages will perform power and sample sizecalculations see page 108

There are some other methods discussed in the text



Sample Size Determination: Fixed Effects Case, Table V



Sample Size Determination: Fixed Effects Case, Minitaboutput


Other Examples of Single-Factor Experiments

3.8 Other Examples of Single-Factor Experiments









3.8 Other Examples: Marketing Example



3.8 Other Examples: Marketing Example


The Random Effects Model

3.9 The Random Effects Model

There are a large number of possible levels for the factor(theoretically an infinite number)

The experimenter chooses a of these levels at random

Inference will be to the entire population of levels






3.9 The Random Effects Model: covariance structure



3.9 The Random Effects Model: covariance structure

For a = 3 and n = 2







E (MSTreatment) = σ2 + nσ2τ

E (MSE ) = σ2

ANOVA F-test is identical to the fixed-effects case.

σ2 = MSE

σ2τ =

MSTreatment −MSEn



3.9 The Random Effects Model: Example



Homework, due on Monday October 20th 2014

Solve the following end of chapter problem form the textbook. Solve bothmanually and using Minitab.

Problem 3.7

Problem 3.11

Problem 3.25

Problem 3.31 parts (a) and (b).


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