Chapter 3 Design & Analysis of Experiments 7E 2009 Montgomery

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Chapter 3 Design & Analysis of Experiments 7E 2009 Montgomery

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3.7 Sample Size DeterminationText, Section 3.7, pg. 101

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

type of experiment is being contemplated, how it will be conducted, resources, and desired sensitivity

• Sensitivity refers to the difference in means that the experimenter wishes to detect

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

Chapter 3 Design & Analysis of Experiments 7E 2009 Montgomery

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Sample Size DeterminationFixed Effects Case

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

• Type I error – reject H0 when it is true ( )

• Type II error – fail to reject H0 when it is false ( )

• Power = 1 - • Operating characteristic curves plot against a

parameter where

2

2 12

a

ii

n

a

Chapter 3 Design & Analysis of Experiments 7E 2009 Montgomery

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Sample Size DeterminationFixed Effects Case---use of OC Curves

• The OC curves for the fixed effects model are in the Appendix, Table V

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

• Typically work in term of the ratio of and try values of n until the desired power is achieved

• Most statistics software packages will perform power and sample size calculations – see page 103

• There are some other methods discussed in the text

2 22

22

nD

a

/D

Exemplo• Suponha que no exemplo taxa de gravação vs potência, com

a=4 níveis de potência, deseja-se rejeitar a hipótese nula com probabilidade pelo menos 0,90 se quaisquer dois tratamentos diferem dde pelo menos 75 (u.m.) com nível de significância 0,01.

• Suponha desvio-padrão de 25 (u.m.).• Assim o valor mínimo de 2 é

nn

a

nD125,,1

)25)(4(2

)75(

2 2

2

2

22

n N-a 2

4 12 4,5 2,12

5 16 5,625 2,37

6 20 6,75 2,60

N=na, n=4 no exemplo considerado.

app_A_12a

Sugestão de atividades

• 1,2,4,6,8 a 11, 16, 18 a 20 (quinta edição)• Ler

Design of Engineering Experiments

– The Blocking Principle• Text Reference, Chapter 4

• Blocking and nuisance factors

• The randomized complete block design or the RCBD

• Extension of the ANOVA to the RCBD

• Other blocking scenarios…Latin square designs

The Blocking Principle• Blocking is a technique for dealing with nuisance factors• A nuisance factor is a factor that probably has some

effect on the response, but it’s of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized

• Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units

• Many industrial experiments involve blocking (or should)• Failure to block is a common flaw in designing an

experiment (consequences?)

The Blocking Principle• If the nuisance variable is known and controllable, we

use blocking

• If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance (see Chapter 15) to remove the effect of the nuisance factor from the analysis

• If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment

• Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

The Hardness Testing Example• Text reference, pg 121, 122• We wish to determine whether 4 different tips produce

different (mean) hardness reading on a Rockwell hardness tester

• Gauge & measurement systems capability studies are frequent areas for applying DOX

• Assignment of the tips to an experimental unit; that is, a test coupon

• Structure of a completely randomized experiment• The test coupons are a source of nuisance variability• Alternatively, the experimenter may want to test the

tips across coupons of various hardness levels• The need for blocking

The Hardness Testing Example

• To conduct this experiment as a RCBD, assign all 4 tips to each coupon

• Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips

• Variability between blocks can be large, variability within a block should be relatively small

• In general, a block is a specific level of the nuisance factor• A complete replicate of the basic experiment is conducted

in each block• A block represents a restriction on randomization• All runs within a block are randomized

The Hardness Testing Example• Suppose that we use b = 4 blocks:

• Notice the two-way structure of the experiment• Once again, we are interested in testing the equality of

treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)

Extension of the ANOVA to the RCBD

• Suppose that there are a treatments (factor levels) and b blocks

• A statistical model (effects fixed model) for the RCBD is

• The relevant (fixed effects) hypotheses are

1,2,...,

1, 2,...,ij i j ij

i ay

j b

0 1 2 1: where (1/ ) ( )

b

a i i j ijH b

Extension of the ANOVA to the RCBD

ANOVA partitioning of total variability:

2.. . .. . ..

1 1 1 1

2. . ..

2 2. .. . ..

1 1

2. . ..

1 1

( ) [( ) ( )

( )]

( ) ( )

( )

a b a b

ij i ji j i j

ij i j

a b

i ji j

a b

ij i ji j

T Treatments Blocks E

y y y y y y

y y y y

b y y a y y

y y y y

SS SS SS SS

The degrees of freedom for the sums of squares in

are as follows:

Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means

T Treatments Blocks ESS SS SS SS

Extension of the ANOVA to the RCBD

1 1 1 ( 1)( 1)ab a b a b

ANOVA Display for the RCBD

Manual computing (ugh!)…see Equations (4-9) – (4-12), page 124

Design-Expert analyzes the RCBD

Manual computing:

Vascular Graft Example (pg. 126)

• To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin

• Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures

Vascular Graft Example Design-Expert Output

dado3=read.table("e:\\dox\\graft3.txt",header=T) psi=as.factor(dado3$psi) y=c(dado3$X1,dado3$X2,dado3$X3,dado3$X4,dado3$X5,dado3$X6)graft3=data.frame(block=gl(6,4),psi,y) graft3.aov=aov(y~psi+block,data=graft3) summary(graft3.aov)

Df Sum Sq Mean Sq F value Pr(>F) psi 3 178.171 59.390 8.1071 0.001916 **block 5 192.252 38.450 5.2487 0.005532 **Residuals 15 109.886 7.326 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual Analysis for the Vascular Graft Example

Residual Analysis for the Vascular Graft Example

Residual Analysis for the Vascular Graft Example

• Basic residual plots indicate that normality, constant variance assumptions are satisfied

• No obvious problems with randomization• No patterns in the residuals vs. block• Can also plot residuals versus the pressure

(residuals by factor) • These plots provide more information about the

constant variance assumption, possible outliers