©2003 thomson/south-western 1 chapter 11 – analysis of variance slides prepared by jeff heyl,...

©2003 Thomson/South-Western 1

Chapter 11 –Chapter 11 –

Analysis of Analysis of VarianceVariance

Slides prepared by Jeff Heyl, Lincoln UniversitySlides prepared by Jeff Heyl, Lincoln University©2003 South-Western/Thomson Learning™

Introduction toIntroduction to Business StatisticsBusiness Statistics, 6e, 6eKvanli, Pavur, KeelingKvanli, Pavur, Keeling


Analysis of VarianceAnalysis of Variance

Analysis of Variance (ANOVA) Analysis of Variance (ANOVA) determines if a factor has a determines if a factor has a significant effect on the variable significant effect on the variable being measuredbeing measured

Examine variation within samples Examine variation within samples and variation between samplesand variation between samples


Measuring VariationMeasuring Variation

SS(factor)SS(factor) measures between-sample measures between-sample variation variation [SS(between)][SS(between)]

SS(error)SS(error) measures within-sample measures within-sample variation variation [SS(within)][SS(within)]

SS(total)SS(total) measures the total variation in measures the total variation in the sample the sample [SS(factor)] [SS(error)][SS(factor)] [SS(error)]


Determining Sum of SquaresDetermining Sum of Squares

SS(factor) = + -SS(factor) = + -TT22

nn

TT1122

nn11

TT2222

nn22

SS(total) = ∑SS(total) = ∑xx22 - = ∑ - = ∑xx22 - -(∑(∑xx))22

nn

TT22

nn

SS(error) = ∑xSS(error) = ∑x22 - + - + ororTT11

22

nn11

TT2222

nn22

SS(error) = SS(total) - SS(factor)SS(error) = SS(total) - SS(factor)


ANOVA Test for HANOVA Test for Hoo: µ: µ11 = µ = µ22 Versus HVersus Haa: µ: µ11 ≠ µ ≠ µ22

MS(factor) = =MS(factor) = =SS(factor)SS(factor)

df for factordf for factor

SS(factor)SS(factor)

11

MS(error) = =MS(error) = =SS(error)SS(error)

nn11 + + nn22 - 2 - 2

SS(error)SS(error)

df for errordf for error

FF = = = =

estimated population variance based onestimated population variance based onthe variation among the sample meansthe variation among the sample means

estimated population variance based onestimated population variance based onthe variation within each of the samplesthe variation within each of the samples

MS(factor)MS(factor)

MS(error)MS(error)


Defining the Rejection RegionDefining the Rejection Region

Figure 11.1Figure 11.1

FF


p-Values for Battery p-Values for Battery Lifetime ExampleLifetime Example


tt curve with 8 curve with 8 dfdf4.204.20

tt

pp-value-value = 2 (shaded area)= 2 (shaded area)= .0030= .0030

FF curve with 1 and 8 curve with 1 and 8 dfdf17.6417.64

FF

pp-value-value = shaded area= shaded area= .0030= .0030


Dot Array DiagramDot Array Diagram


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Number of cartonsNumber of cartons

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AssumptionsAssumptions

The replicates are obtained The replicates are obtained independently and randomly from independently and randomly from each of the populationseach of the populations

The replicates from each population The replicates from each population follow a (approximate) normal follow a (approximate) normal distributiondistribution

The normal populations all have a The normal populations all have a common variancecommon variance


Deriving the Sum of SquaresDeriving the Sum of Squares

SS(factor) = + + ... + -SS(factor) = + + ... + -TT11

22

nn11

TT2222

nn22

TTkk22

nnkk

TT22

nn

SS(total) = ∑SS(total) = ∑xx22 - -TT22

nn

SS(error) = ∑SS(error) = ∑xx22 - + + ... + - + + ... +TT11

22

nn11

TT2222

nn22

TTkk22

nnkk

= SS(total) - SS(factor)= SS(total) - SS(factor)


The ANOVA TableThe ANOVA Table

SourceSource dfdf SSSS MSMS FF

FactorFactor k - 1k - 1 SS(factor)SS(factor) MS(factor)MS(factor) MS(factor) MS(factor)

ErrorError nn - 2 - 2 SS(error)SS(error) MS(error)MS(error) MS(error)MS(error)

TotalTotal nn - 1 - 1 SS(total)SS(total)

SS(factor)SS(factor)

kk - 1 - 1MS(factor) =MS(factor) =

SS(error)SS(error)

nn - - kkMS(error) =MS(error) =


MS(error)MS(error)FF = =


Test for Equal VariancesTest for Equal Variances

HHoo: : 1122 = = 22

22 = … = = … = kk22

HHaa: at least 2 variances are unequal: at least 2 variances are unequal

reject Hreject Hoo if if HH > > HHTable A.14Table A.14

HH = =maximum maximum ss22

minimum minimum ss22


Confidence Intervals inConfidence Intervals inOne-Factor ANOVAOne-Factor ANOVA

XXii - - tt/2,/2,nn--kksspp to to XXii + + tt/2,/2,nn--kksspp11

nnii

11

nnii

wherewhere

kk = number of populations (levels)= number of populations (levels)nnii = number of replicates in the = number of replicates in the iith sampleth sample

nn = total number of observations= total number of observations

sspp == MS(error)MS(error)


Confidence Intervals inConfidence Intervals inOne-Factor ANOVAOne-Factor ANOVA

The (1 - The (1 - ) • 100% confidence interval for µ) • 100% confidence interval for µii - µ - µjj is is

((XXii - X - Xjj) - ) - tt/2,/2,nn--kksspp + +

to (to (XXii - X - Xjj) + ) + tt/2,/2,nn--kksspp + +

11

nnii

11

nnjj

11

nnii

11

nnjj


Multiple Comparisons Multiple Comparisons ProcedureProcedure

The multiple comparisons procedure compares The multiple comparisons procedure compares all possible pairs of means in such a way that all possible pairs of means in such a way that the probability of making one or more Type 1 the probability of making one or more Type 1 errors is errors is

Tukey’s TestTukey’s Test

Q =Q =maximum (maximum (XXii) - minimum () - minimum (XXii))

MS(error)/MS(error)/nnrr


Multiple Comparisons Multiple Comparisons ProcedureProcedure

1.1. Find QFind Q,,kk,,vv using Table A.16 using Table A.16

4.4. If two means differ by more than D, the conclusion If two means differ by more than D, the conclusion is that the corresponding population means are is that the corresponding population means are unequalunequal

2.2. DetermineDetermine DD = = QQ,,kk,,vv • •MS(error)MS(error)

nnrr

3.3. Place the sample means in order, from smallest Place the sample means in order, from smallest to largestto largest


Plot of Group MeansPlot of Group Means

11 22 33 44 55

GroupGroup

Gro

up

Mea

ns

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up

Mea

ns

2626

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2222

2121


Nylon Breaking StrengthNylon Breaking Strength


One-Factor ANOVA One-Factor ANOVA ProcedureProcedure

1.1. The replicates are obtained The replicates are obtained independently and randomly from each independently and randomly from each of the populationsof the populations

2.2. The observations from each population The observations from each population follow (approximately) a normal follow (approximately) a normal distributiondistribution

3.3. The populations all have a common The populations all have a common variancevariance

RequirementsRequirements


One-Factor ANOVA One-Factor ANOVA ProcedureProcedure

HHoo: µ: µ11 = µ = µ22 = … = µ = … = µkk

HHaa: not all µ’s are equal: not all µ’s are equal

HypothesesHypotheses


FactorFactor k - 1k - 1 SS(factor)SS(factor) MS(factor) MS(factor)

ErrorError n - 2n - 2 SS(error)SS(error) MS(error)MS(error)

TotalTotal n - 1n - 1 SS(total)SS(total)


MS(error)MS(error)

reject reject HHoo if if FF** > > FF,,kk-1,-1,nn--kk


Completely Completely Randomized DesignRandomized Design

Replicates are obtained in a completely Replicates are obtained in a completely random manner from each populationrandom manner from each population

Null hypothesis isNull hypothesis is

HHoo: µ: µ11 = µ = µ22 = ... = µ = ... = µnn


Randomized Block DesignRandomized Block Design

The samples are not independent, the data The samples are not independent, the data are grouped (blocked) by another variableare grouped (blocked) by another variable

The difference between the randomized The difference between the randomized block design and the completely block design and the completely randomized design is that here we use a randomized design is that here we use a blocking strategy rather than independent blocking strategy rather than independent samples to obtain a more precise test for samples to obtain a more precise test for examining differences in the factor level examining differences in the factor level meansmeans



SS(factor) = [SS(factor) = [TT1122 + + TT22

22 + ... + + ... + TTkk22] -] -

11

bbTT22

bkbk

wherewhere

kk= number of factor levels in the design= number of factor levels in the designbb= number of blocks in the design= number of blocks in the designnn= number of observations = = number of observations = bkbkTT11, , TT22, ..., , ..., TTkk represent the totals for the k factor levelsrepresent the totals for the k factor levels

SS11, , SS22, ..., , ..., SSbb are the totals for the are the totals for the bb blocks blocks

TT= = TT11 + + TT22 + ... + + ... + TTkk

= = SS11 + + SS22 + ... + + ... + S Sbb = total of all observations = total of all observations



SS(blocks) = [SS(blocks) = [SS1122 + + SS22

22 + ... + + ... + SSbb22] -] -

11

kkTT22

bkbk

SS(total) = ∑SS(total) = ∑xx22 - - TT22

bkbk

SS(error) + SS(total) - SS(factor) - SS(blocks)SS(error) + SS(total) - SS(factor) - SS(blocks)

df for factordf for factor = = kk - 1 - 1df for blocksdf for blocks = = bb - 1 - 1

df for errordf for error = (= (kk - 1)( - 1)(bb - 1) - 1)df for totaldf for total = = bkbk - 1 - 1




FactorFactor kk - 1 - 1 SS(factor)SS(factor) MS(factor)MS(factor) FF11

BlocksBlocks bb - 1 - 1 SS(blocks)SS(blocks) MS(blocks)MS(blocks) FF22

ErrorError ((kk - 1)( - 1)(bb - 1) - 1) SS(error)SS(error) MS(error)MS(error)

TotalTotal bkbk - 1 - 1 SS(total)SS(total)

FF11 = = MS(factor)MS(factor)

MS(error)MS(error)FF22 = =

MS(blocks)MS(blocks)

MS(error)MS(error)


Factor Hypothesis TestFactor Hypothesis Test

HHoo: µ: µ11 = µ = µ22 = … = µ = … = µkk


reject reject HHoo if if FF** > > FF,,kk-1,(-1,(kk-1)(-1)(bb-1)-1)

FF11 = = MS(factor)MS(factor)

MS(error)MS(error)


Block Hypothesis TestBlock Hypothesis Test

HHoo: µ: µ11 = µ = µ22 = … = µ = … = µbb


reject reject HHoo if if FF** > > FF,,bb-1,(-1,(kk-1)(-1)(bb-1)-1)

FF22 = = MS(blocks)MS(blocks)

MS(error)MS(error)


Hardness Test Data AnalysisHardness Test Data Analysis



Confidence IntervalConfidence IntervalDifference Between Difference Between

Two MeansTwo Means

Randomized BlockRandomized Block (1- (1- ) 100% confidence interval) 100% confidence interval

((XXii - X - Xjj) - ) - tt/2,/2,dfdf • • s • +s • +

to (to (XXii - X - Xjj) + ) + tt/2,df/2,df • • s • +s • +

11

bb11

bb

11

bb11

bb


Dental Claim Data AnalysisDental Claim Data Analysis



Multiple Comparisons Multiple Comparisons Procedure:Procedure:

Randomized BlockRandomized Block

||XXii - - XXjj| > D| > D

DD = = QQ,,kk,(,(kk-1)(-1)(bb-1)-1)MS(error)MS(error)

bb


Machine Choice ExampleMachine Choice Example



Two-Way Factorial DesignTwo-Way Factorial Design

SingleSingle MarriedMarried

MaleMale LowLow HighHigh

FemaleFemale HighHigh LowLow





11 22 ......

bb

11 xx xx

xx

22 xx xx

xx

......

aa xx xx

xx

Factor BFactor B

Factor AFactor A




11 22 ...... BB

TotalsTotals

11

TT11

22

TT22

......

aa

TTaa

SS11 SS22 SSbb

Factor AFactor A

Factor BFactor B

xx, , xx(total =(total = R R1111))



xx, , xx(total =(total = R Raa11))

xx, , xx(total =(total = R Raa22))


xx, , xx(total =(total = R R11bb))

xx, , xx(total =(total = R R11bb))

xx, , xx(total =(total = R Rabab))

TotalsTotals



Factor A: SSA = [Factor A: SSA = [TT1122 + + TT22

22 + ... + + ... + TTaa22] -] -

11

brbrTT22

abrabr

11

rrTT22

abrabrInteraction: SSAB = [∑Interaction: SSAB = [∑RR22] - SSA - SSB -] - SSA - SSB -

Factor B: SSB = [Factor B: SSB = [SS1122 + + SS22

22 + ... + + ... + SSaa22] -] - TT22

abrabr11

arar

TT22

abrabrTotal: SS(total) = ∑Total: SS(total) = ∑xx22 - -



SS(error) = SS(total) - SSA - SSB - SSABSS(error) = SS(total) - SSA - SSB - SSAB

MS(error) =MS(error) =SS(error)SS(error)

abab((rr - 1) - 1)

MSA =MSA =SSASSA

aa - 1 - 1MSB =MSB =

SSBSSB

bb - 1 - 1

MSAB =MSAB =SSABSSAB

((aa - 1)( - 1)(bb - 1) - 1)




Factor AFactor A aa - 1 - 1 SSASSA MSAMSA FF11

Factor BFactor B bb - 1 - 1 SSBSSB MSBMSB FF22

InteractionInteraction ((aa - 1)( - 1)(bb - 1) - 1) SSABSSAB MSABMSAB FF33

ErrorError abab((rr - 1) - 1) SS(error)SS(error) MS(error)MS(error)

TotalTotal abrabr - 1 - 1 SS(total)SS(total)


Hypothesis Test - Factor AHypothesis Test - Factor A

HHoo: Factor A is not significant (µ: Factor A is not significant (µMM = µ = µFF))

HHaa: Factor A is significant (µ: Factor A is significant (µMM ≠ µ ≠ µFF))

reject Hreject Ho,o,AA if if FF11 > > FF,,vv1,1,vv22

FF11 = =MSAMSA



Hypothesis Test - Factor BHypothesis Test - Factor B

HHoo: Factor B is not significant (µ: Factor B is not significant (µ11 = µ = µ22 = µ = µ33))

HHaa: Factor B is significant (not all µ’s are equal): Factor B is significant (not all µ’s are equal)

reject Hreject Ho,o,BB if if FF22 > > FF,,vv1,1,vv22

FF22 = =MSBMSB

MS(error)MS(error)


Hypothesis Test - Hypothesis Test - InteractionInteraction

HHoo: Interaction is not significant : Interaction is not significant

HHaa: Interaction is significant: Interaction is significant

reject Hreject Ho,o,ABAB if if FF22 > > FF,,vv1,1,vv22

FF33 = =MSABMSAB

MS(error)MS(error)


Multiple Comparisons Multiple Comparisons Procedure:Procedure:


DD = = QQ,,kk,,vv • •MS(error)MS(error)

nnrr

vv = df for error = df for errornnrr = number of replicates in each sample = number of replicates in each sample


Interaction EffectInteraction Effect


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|Category 1Category 1




MaleMale

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Employee classificationEmployee classification

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Interaction EffectInteraction Effect


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Gender Factor AnalysisGender Factor Analysis


©2003 thomson/south-western 1 chapter 11 – analysis of variance slides prepared by jeff heyl,...

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