82539897 analysis of variance ppt bec doms

7/29/2019 82539897 Analysis of Variance PPT BEC DOMS

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Analysis of Variance


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Chapter Goals

After completing this chapter, you should be

able to:

Recognize situations in which to use analysis of variance

Understand different analysis of variance designs

Perform a single-factor hypothesis test and interpret results

Conduct and interpret post-analysis of variance pairwise

comparisons procedures

Set up and perform randomized blocks analysis

Analyze two-factor analysis of variance test with replications

results


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Chapter Overview

Analysis of Variance (ANOVA)

F-test

F-testTukey-

Kramer

testFishers Least

Significant

Difference test

One-WayANOVA RandomizedComplete

Block ANOVA

Two-factorANOVA

with replication


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General ANOVA Setting

Investigator controls one or more independent

variables

Called factors (or treatment variables)

Each factor contains two or more levels (or

categories/classifications)

Observe effects on dependent variable

Response to levels of independent variable

Experimental design: the plan used to test

hypothesis


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One-Way Analysis of Variance

Evaluate the difference among the means of three or

more populations

Examples: Accident rates for 1st, 2nd, and 3rd shift

Expected mileage for five brands of tires

Assumptions

Populations are normally distributed

Populations have equal variances

Samples are randomly and independently drawn


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Completely Randomized Design

Experimental units (subjects) are assigned

randomly to treatments

Only one factor or independent variable

With two or more treatment levels

Analyzed by

One-factor analysis of variance (one-way

ANOVA)

Called a Balanced Design if all factor levels

have equal sample size


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Hypotheses of One-Way ANOVA

All population means are equal

i.e., no treatment effect (no variation in means among

groups)

At least one population mean is different

i.e., there is a treatment effect Does not mean that all population means are different (some

pairs may be the same)

3210 :H

tampoptheofallNot:HA


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One-Factor ANOVA

All Means are the same:

The Null Hypothesis is True

(No Treatment Effect)

3210 :H

stharallNot:H iA

321


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One-Factor ANOVA

At least one mean is different:

The Null Hypothesis is NOT true

(Treatment Effect is present)

3210 :H

stharallNot:H iA

21 21

or

(continued)


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Partitioning the Variation

Total variation can be split into two parts:

SST = Total Sum of Squares

SSB = Sum of Squares Between

SSW = Sum of Squares Within

SST = SSB + SSW


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Total Variation = the aggregate dispersion of the individualdata values across the various factor levels (SST)

Within-Sample Variation = dispersion that exists among

the data values within a particular factor level (SSW)

Between-Sample Variation = dispersion among the factor

sample means (SSB)

SST = SSB + SSW

(continued)


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Partition of Total Variation

Variation Due toFactor (SSB)

Variation Due to RandomSampling (SSW)

Total Variation (SST)Commonly referred to as:

Sum of Squares Within

Sum of Squares Error

Sum of Squares Unexplained

Within Groups Variation

Commonly referred to as:

Sum of Squares Between

Sum of Squares Among

Sum of Squares Explained

Among Groups Variation

= +


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Total Sum of Squares

k

i

n

j ij

i

)xx(SST1 1

2

Where:

SST = Total sum of squares

k = number of populations (levels or treatments)

ni = sample size from population i

xij = jth measurement from population i

x = grand mean (mean of all data values)

SST = SSB + SSW


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Total Variation(continued)

Group 1 Group 2 Group 3

Response, X

X

2

12

2

11x(...)xx()xx(SSTkk


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Sum of Squares Between

Where:

SSB = Sum of squares between

k = number of populations

ni = sample size from population i

xi = sample mean from population i


2

1

)xx(nSSB i

k

i

i

SST = SSB + SSW


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Between-Group Variation

Variation Due to

Differences Among Groups

i

j

2

1

)xx(nSSB i

k

i

i

1

k

SSMSB

Mean Square Between =

SSB/degrees of freedom


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Between-Group Variation(continued)


Response, X

X1

X2X

3X

2

22

2

11(n...)xx(n)xx(nSSB kk


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Sum of Squares Within

Where:

SSW = Sum of squares within

k = number of populationsni = sample size from population i

xi = sample mean from population i

xij

= jth measurement from population i

11

)xx(SSW iij

n

j

k

i

j

SST = SSB + SSW


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Within-Group Variation

Summing the variation

within each group and then

adding over all groups

i

kN

SSMSW

Mean Square Within =

SSW/degrees of freedom

11

)xx(SSW iij

n

j

k

i

j


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Within-Group Variation(continued)


Response, X

1X

2X

3X

2

212

2

111x(...)xx()xx(SSW kk


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One-Way ANOVA Table

Source ofVariation

dfSS MS

BetweenSamples SSB MSB =

WithinSamples

N - kSSW MSW =

Total N - 1SST =SSB+SSW

k - 1MSB

MSW

F ratio

k = number of populations

N = sum of the sample sizes from all populations

df = degrees of freedom

SSB

k - 1

SSW

N - k

F =


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One-Factor ANOVAF Test Statistic

Test statistic

MSB is mean squares between variances

MSWis mean squares within variances

Degrees of freedom

df1 = k1 (k = number of populations)

df2 = Nk (N = sum of sample sizes from all populations)

MSWMSBF

H0: 1= 2= = k

HA: At least two population means are different


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Interpreting One-Factor ANOVAF Statistic

The F statistic is the ratio of the betweenestimate of variance and the within estimateof variance

The ratio must always be positive df1 = k-1 will typically be small

df2 =N- k will typically be large

The ratio should be close to 1 ifH0: 1= 2= = k is true

The ratio will be larger than 1 ifH0: 1= 2= = k is false


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One-Factor ANOVAF Test Example

You want to see if three different golf clubs yield different

distances. You randomly select five measurements from trials on

an automated driving machine for each club. At the .05

significance level, is there a difference in mean distance?

Club 1 Club 2 Club 3254 234 200

263 218 222

241 235 197237 227 206

251 216 204


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One-Factor ANOVA Example:Scatter Diagram

270

260

250

240

230

220

210200

190

Distance

1X

2X

3X

X

227.0x

20x226.0x249.2x 321

Club 1 Club 2 Club 3254 234 200

263 218 222

241 235 197237 227 206

251 216 204

Club1 2 3


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One-Factor ANOVA ExampleComputations

Club 1 Club 2 Club 3254 234 200

263 218 222

241 235 197

237 227 206

251 216 204

x1 = 249.2

x2 = 226.0

x3 = 205.8

x = 227.0

n1 = 5

n2 = 5

n3 = 5

N = 15

k = 3

SSB = 5 [ (249.2 227)2 + (226 227)2 + (205.8 227)2 ] = 4716.4

SSW = (254 249.2)2 + (263 249.2)2++ (204 205.8)2 = 1119.6

MSB = 4716.4 / (3-1) = 2358.2

MSW = 1119.6 / (15-3) = 93.325.

93.3

2358.2F


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F= 25.275

One-Factor ANOVA ExampleSolution

H0: 1= 2= 3

HA: i not all equal

= .05

df1= 2 df

2= 12

Test Statistic:

Decision:

Conclusion:

Reject H0 at = 0.05

There is evidence that

at least one i differsfrom the rest

0

= .05

F.05

= 3.885Reject H0Do not

reject H0

25

93.3

2358.

MSW

MSBF

CriticalValue:

F= 3.885


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SUMMARY

Groups Count Sum Average Variance

Club 1 5 1246 249.2 108.2

Club 2 5 1130 226 77.5

Club 3 5 1029 205.8 94.2

ANOVA

Source of

VariationSS df MS F P-value F crit

Between

Groups4716.4 2 2358.2 25.275 4.99E-05 3.885

Within

Groups1119.6 12 93.3

Total 5836.0 14

ANOVA -- Single Factor:Excel Output

EXCEL: tools | data analysis | ANOVA: single factor


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The Tukey-Kramer Procedure

Tells which population means aresignificantly different

e.g.: 1= 23

Done after rejection of equal means in ANOVA

Allows pair-wise comparisons

Compare absolute mean differences with critical

range

x1 = 2 3


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Tukey-Kramer Critical Range

where:

q

= Value from standardized range table

with k and N - k degrees of freedom for

the desired level of

MSW = Mean Square Within

ni and nj = Sample sizes from populations (levels) i and j

ji n

1

n

1

2

MSWqRangeCritical


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The Tukey-Kramer Procedure:Example

1. Compute absolute mean differences:Club 1 Club 2 Club 3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 204 20.205.8226.0xx

43.205.8249.2xx

23.226.0249.2xx

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2. Find the q value from the table in appendix J with k

and N - k degrees of freedom forthe desired level of

3.7q


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The Tukey-Kramer Procedure:Example

5. All of the absolute mean differencesare greater than critical range.Therefore there is a significant

difference between each pair ofmeans at 5% level of significance.

15

1

5

1

2

93.33.77

n

1

n

1

2

MSWqRangeCritical

ji

3. Compute Critical Range:

20.2xx

43.4xx

23.2xx

32

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21

4. Compare:


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Tukey-Kramer in PHStat


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Randomized Complete Block ANOVA

Like One-Way ANOVA, we test for equal population means(for different factor levels, for example)...

...but we want to control for possible variation from a second

factor (with two or more levels)

Used when more than one factor may influence the value ofthe dependent variable, but only one is of key interest

Levels of the secondary factor are called blocks


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Total variation can now be split into three

parts:

SST = Total sum of squares

SSB = Sum of squares between factor levels

SSBL = Sum of squares between blocks

SSW = Sum of squares within levels

SST = SSB + SSBL + SSW


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Sum of Squares for Blocking

Where:

k = number of levels for this factor

b = number of blocks

xj = sample mean from the jth block


2

1)xx(kSSBL j

b

j



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Total variation can now be split into three

parts:

SST and SSB are

computed as they werein One-Way ANOVA


SSW = SST (SSB + SSBL)


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Mean Squares

k

SbetwsquareMeanMSB

b

SblocsquarMeanMSBL

b)(k(

SwithisquareMeanMSW

1

an om ze oc


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an om ze ocTable

Source ofVariation

dfSS MS

BetweenSamples

SSB MSB

Within

Samples (k1)(b-1)SSW MSW

Total N - 1SST

k - 1

MSBL

MSW

F ratio

k = number of populations N = sum of the sample sizes from all populations

b = number of blocks df = degrees of freedom

Between

Blocks

SSBL b - 1 MSBL

MSB

MSW


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Blocking Test Blocking test: df1 = b - 1

df2 = (k1)(b1)

MSBL

MSW

.:H b3b2b10

ambloallNot:HA

F =

Reject H0 if F > F


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Main Factor test: df1 = k - 1

df2 = (k1)(b1)

MSB

MSW

3210 ...:H

ampopallNot:HA

F =

Reject H0 if F > F

Main Factor Test


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FishersLeast Significant Difference Test

To test which population means aresignificantly different

e.g.: 1= 2 3

Done after rejection of equal means inrandomized block ANOVA design

Allows pair-wise comparisons

Compare absolute mean differences with criticalrange

x=1 2 3


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Fishers Least Significant Difference(LSD) Test

where:

t/2 = Upper-tailed value from Students t-distribution

for /2 and (k -1)(n - 1) degrees of freedom

MSW = Mean square within from ANOVA tableb = number of blocks

k = number of levels of the main factor

b

2MSWtLSD /2


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...etc

xx

xx

xx

32

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Fishers Least Significant Difference(LSD) Test

(continued)

b

2MSWtLSD /2

If the absolute mean differenceis greater than LSD then thereis a significant differencebetween that pair of means atthe chosen level of significance.

Compare:LSxxIs ji


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Two-Way ANOVA

Examines the effect of

Two or more factors of interest on the dependent

variable

e.g.: Percent carbonation and line speed on soft drinkbottling process

Interaction between the different levels of these

two factors

e.g.: Does the effect of one particular percentage of

carbonation depend on which level the line speed is

set?


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Two-Way ANOVA

Assumptions

Populations are normally distributed

Populations have equal variances

Independent random samples are drawn

(continued)

T W ANOVA


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Two-Way ANOVASources of Variation

Two Factors of interest: A and B

a = number of levels of factor Ab = number of levels of factor B

N = total number of observations in all cells


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Two-Way ANOVASources of Variation

SST

Total Variation

SSAVariation due to factor A

SSBVariation due to factor B

SSAB

Variation due to interactionbetween A and B

SSEInherent variation (Error)

Degrees of

Freedom:

a 1

b 1

(a 1)(b 1)

N ab

N - 1

SST = SSA + SSB + SSAB + SSE

(continued)


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Two Factor ANOVA Equations

a

i

b

j

n

k

ijk )xx(SST1 1 1

2

1

)xx(nbSSa

i

iA

2

1

)xx(naSSb

j

jB

Total Sum of Squares:

Sum of Squares Factor A:

Sum of Squares Factor B:


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1 1

)xxxx(nSSa

i

b

j

jiijAB

a

i

b

j

n

kijijk )xx(SSE

1 1 1

2

Sum of Squares

Interaction Between

A and B:

Sum of Squares Error:

(continued)


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where:MeGrand

nab

x

x

a

i

b

j

n

k

ijk

1 1 1

facofleveleachofMeannb

x

x

b

j

n

k

ijk

i

1 1

facofleveleachofMeanna

x

x

a

i

n

kijk

j

1 1

ceaofMeann

xx

n

k

ijk

ij

1

a = number of levels of factor A

b = number of levels of factor B

n = number of replications in each cell

(continued)


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Mean Square Calculations

a

SAfactsquarMeanMSA

b

SBfactsquarMeanMSB

b)(a(

SninterasquareMeanMS AAB1

N

SerrosquarMeanMSE

T W ANOVA


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Two-Way ANOVA:The F Test Statistic

F Test for Factor B Main Effect

F Test for Interaction Effect

H0: A1 = A2 = A3=

HA: Not all Ai are equal

H0: factors A and B do not interactto affect the mean response

HA: factors A and B do interact

F Test for Factor A Main Effect

H0: B1 = B2 = B3=

HA: Not all Bi are equal

Reject H0

if F > F MSE

MSF

A

MSE

MSF

B

MSE

MSF

AB

Reject H0

if F > F

Reject H0

if F > F

T W ANOVA


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Two-Way ANOVASummary Table

Source of

Variation

Sum of

Squares

Degrees of

Freedom

Mean

Squares

F

Statistic

Factor A SSA a1MSA

= SSA/(a1)

MSA

MSE

Factor B SSB b1MSB

= SSB/(b1)

MSB

MSE

AB

(Interaction)SS

AB

(a1)(b1)MSAB

= SSAB/ [(a1)(b1)]

MSAB

MSE

Error SSE NabMSE =

SSE/(Nab)

Total SST N1

F t f T W ANOVA


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Features of Two-Way ANOVAFTest

Degrees of freedom always add up

N-1 = (N-ab) + (a-1) + (b-1) + (a-1)(b-1)

Total = error + factor A + factor B + interaction

The denominator of the FTest is always the

same but the numerator is different

The sums of squares always add up

SST = SSE + SSA + SSB + SSAB

Total = error + factor A + factor B + interaction


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Examples:Interaction vs. No Interaction

No interaction:

1 2

Factor B Level 1

Factor B Level 3

Factor B Level 2

Factor A Levels 1 2

Factor B Level 1

Factor B Level 3

Factor B Level 2

Factor A Levels

MeanResponse

Me

anResponse

Interaction is

present:

82539897 analysis of variance ppt bec doms

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