Pertemuan 3 Anova

7/30/2019 Pertemuan 3 Anova

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ANOVA

Chap 11-1


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-2

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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-3

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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Chap 11-4

General ANOVA Setting

Investigator controls one or more independent

variables

Called factors (or treatment variables)

Each factor contains two or more levels (orcategories/classifications)

Observe effects on dependent variable

Response to levels of independent variable

Experimental design: the plan used to test

hypothesis


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Chap 11-5

One-Way Analysis of Variance

Evaluate the difference among the means of three

or more populations

Examples: Accident rates for 1

st

, 2

nd

, and 3

rd

shift

Assumptions

Populations are normally distributed

Populations have equal variances

Samples are randomly and independently drawn


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Chap 11-6

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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Chap 11-7

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)

k3210 :H

samethearemeanspopulationtheofallNot:HA


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Chap 11-8

One-Factor ANOVA

All Means are the same:

The Null Hypothesis is True

(No Treatment Effect)

k3210 :H

sametheareallNot:H iA

321


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Chap 11-9

One-Factor ANOVA

At least one mean is different:

The Null Hypothesis is NOT true

(Treatment Effect is present)

k3210 :H

sametheareallNot:H iA

321 321

or

(continued)


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Chap 11-10

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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Chap 11-11


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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Chap 11-12

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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Chap 11-13

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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Chap 11-14

Total Variation(continued)

Group 1 Group 2 Group 3

Response, X

X

22

12

2

11)xx(...)xx()xx(SST

kkn


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Chap 11-15

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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Chap 11-16

Between-Group Variation

Variation Due to

Differences Among Groups

i j

2

1

)xx(nSSB i

k

i

i

1kSSBMSB

Mean Square Between =

SSB/degrees of freedom


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Chap 11-17

Between-Group Variation(continued)


Response, X

X1

X 2X

3X

22

22

2

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


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Chap 11-18

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

2

11

)xx(SSW iij

n

j

k

i

j

SST = SSB + SSW


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Chap 11-19

Within-Group Variation

Summing the variation

within each group and then

adding over all groups

i

kNSSWMSW

Mean Square Within =

SSW/degrees of freedom

2

11

)xx(SSW iij

n

j

k

i

j


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Chap 11-20

Within-Group Variation(continued)


Response, X

1X

2X

3X

22

212

2

111)xx(...)xx()xx(SSW kknk


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Chap 11-21

One-Way ANOVA Table

Source of

VariationdfSS MS

Between

Samples SSB MSB =

Within

SamplesN - 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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Chap 11-22

One-Factor ANOVAF Test Statistic

Test statistic

MSB is mean squares between variances

MSWis mean squares within variances

Degrees of freedom

df1 = k 1 (k = number of populations)

df2 = N k (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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Chap 11-23

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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Chap 11-24

One-Factor ANOVAF Test Example

You want to see if three

different group of woman

yield different distances in 10

seconds. You randomlyselect five measurements

from trials on a sprint. At the

.05 significance level, is there

a difference in meandistance?

group-1group-2 group-3

254 234 200

263 218 222

241 235 197237 227 206

251 216 204


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Chap 11-25

One-Factor ANOVA Example:Scatter Diagram

270

260

250

240

230

220

210200

190

Distance

1X

2X

3X

X

227.0x

205.8x226.0x249.2x 321

group-1group-2 group-3

254 234 200

263 218 222241 235 197

237 227 206

251 216 204

group1 2 3


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Chap 11-26

One-Factor ANOVA ExampleComputations

group-1 group-2 group-3

254 234 200

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) = 93.325.275

93.3

2358.2F


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Chap 11-27

F= 25.275

One-Factor ANOVA ExampleSolution

H0: 1 = 2 = 3

HA: i not all equal

= .05

df1= 2 df2 = 12

Test Statistic:

Decision:

Conclusion:

Reject H0 at = 0.05

There is evidence that

at least one i differs

from the rest

0 = .05

F.05

= 3.885

Reject H0Do notreject H0

25.275

93.3

2358.2

MSW

MSBF

Critical

Value:

F

= 3.885


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Chap 11-28

SUMMARY

Groups Count Sum Average Variance

Club 1 5 1246 249.2 108.2

Club 2 5 1130 226 77.5Club 3 5 1029 205.8 94.2

ANOVA

Source of

VariationSS df MS F P-value F crit

BetweenGroups

4716.4 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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Chap 11-29

The Tukey-Kramer Procedure

Tells which population means are significantlydifferent e.g.: 1 = 23

Done after rejection of equal means in ANOVA Allows pair-wise comparisons

Compare absolute mean differences with criticalrange

x1 = 2 3


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Chap 11-30

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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Chap 11-31

The Tukey-Kramer Procedure:Example

1. Compute absolute meandifferences:group-1 group-2 group-3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 204

20.2205.8226.0xx

43.4205.8249.2xx

23.2226.0249.2xx

32

31

21

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

k and N - k degrees of freedom forthe desired level of

3.77q


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Chap 11-32

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.

16.2855

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

31

21

4. Compare:


33/60

Chap 11-33

Tukey-Kramer in PHStat


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Chap 11-34

Randomized Complete Block ANOVA

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

...but we want to control for possible variation from asecond factor (with two or more levels)

Used when more than one factor may influence thevalue of the dependent variable, but only one is of keyinterest

Levels of the secondary factor are called blocks


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Chap 11-35


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


36/60

Chap 11-36

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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Chap 11-37


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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Chap 11-38

Mean Squares

1k

SSBbetweensquareMeanMSB

1b

SSBLblockingsquareMeanMSBL

)b)(k(

SSWwithinsquareMeanMSW

11


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Chap 11-39

Randomized Block ANOVA Table

Source of

VariationdfSS MS

Between

SamplesSSB 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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Chap 11-40

Blocking Test

Blocking test: df1 = b - 1

df2 = (k 1)(b 1)

MSBL

MSW

...:H b3b2b10

equalaremeansblockallNot:HA

F =

Reject H0 if F > F


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Chap 11-41

Main Factor test: df1 = k - 1

df2 = (k 1)(b 1)

MSB

MSW

k3210 ...:H

equalaremeanspopulationallNot:HA

F =

Reject H0 if F > F

Main Factor Test


42/60

Chap 11-42

FishersLeast Significant Difference Test

To test which population means are significantlydifferent e.g.: 1 = 23 Done after rejection of equal means in randomized

block ANOVA design

Allows pair-wise comparisons Compare absolute mean differences with critical

range

x= 1 2 3


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Chap 11-43

Fishers Least SignificantDifference (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 table

b = number of blocks

k = number of levels of the main factor

b

2MSWtLSD /2

h f


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Chap 11-44

...etc

xx

xx

xx

32

31

21

Fishers Least SignificantDifference (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:?LSDxxIs ji


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RCBD Example

Chap 11-45

A physical therapist wished to compare three

methods for teaching patiens to use a certain

prosthetic device. He felt that there rate oflearning would be different for patients of

different ages and wished to design and

experiment in which the influence of age couldbe taken out


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Chap 11-46

Data:

Three patients in each of five age groups were

selected to participate in the experiment, and onepatient in each age group was randomly assigned

to each of teaching methods. The methods of

instruction constitute our three treatments, and the

five groups are the blocs. The data are shown on

the table.


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Chap 11-47


48/60

Chap 11-48

Two-Way ANOVA

Examines the effect of

Two or more factors of interest on the

dependent variable

e.g.: Percent carbonation and line speed onsoft drink bottling process

Interaction between the different levels of these

two factors

e.g.: Does the effect of one particularpercentage of carbonation depend on which

level the line speed is set?


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Chap 11-49

Two-Way ANOVA

Assumptions

Populations are normally distributed

Populations have equal variances

Independent random samples are

drawn

(continued)

Two Way ANOVA


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Chap 11-50

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

T W ANOVA


51/60

Chap 11-51

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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Chap 11-52

Two Factor ANOVA Equations

a

i

b

j

n

k

ijk )xx(SST1 1 1

2

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:


53/60

Chap 11-53


2

1 1

)xxxx(nSSa

i

b

j

jiijAB

a

i

b

j

n

k

ijijk )xx(SSE1 1 1

2

Sum of Squares

Interaction Between

A and B:

Sum of Squares Error:

(continued)


54/60

Chap 11-54


where:MeanGrand

nab

x

x

a

i

b

j

n

k

ijk

1 1 1

AfactorofleveleachofMeannb

x

x

b

j

n

k

ijk

i

1 1

BfactorofleveleachofMeanna

x

x

a

i

n

k

ijk

j

1 1

celleachofMeann

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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Chap 11-55

Mean Square Calculations

1a

SSAfactorsquareMeanMS AA

1 b

SSBfactorsquareMeanMS BB

)b)(a(

SSninteractiosquareMeanMS ABAB

11

abN

SSEerrorsquareMeanMSE

Two-Way ANOVA:


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Chap 11-56

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 > FMSEMS

F A

MSE

MSF B

MSE

MSF AB

Reject H0

if F > F

Reject H0

if F > F

Two-Way ANOVA


57/60

Chap 11-57

Two-Way ANOVASummary Table

Source of

Variation

Sum of

Squares

Degrees of

Freedom

Mean

Squares

F

Statistic

Factor A SSA a 1MSA

= SSA /(a 1)

MSA

MSE

Factor B SSB b 1MSB

= SSB /(b 1)

MSB

MSE

AB

(Interaction)

SSAB (a 1)(b 1)MSAB

= SSAB / [(a 1)(b 1)]

MSAB

MSE

Error SSE N abMSE =

SSE/(N ab)

Total SST N 1

Features of Two Way ANOVA


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Chap 11-58

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

Examples:


59/60

Chap 11-59

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:


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Chapter Summary

Described one-way analysis of variance The logic of ANOVA

ANOVA assumptions

F test for difference in k means

The Tukey-Kramer procedure for multiple comparisons

Described randomized complete block designs

F test

Fishers least significant difference test for multiplecomparisons

Described two-way analysis of variance

Examined effects of multiple factors and interaction

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Pertemuan 3 Anova