1 1 Slide © 2005 Thomson/South-Western AK/ECON 3480 M & N WINTER 2006 n Power Point Presentation n Professor Ying Kong School of Analytic Studies and Information

1 1 Slide

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

AK/ECON 3480 M & NAK/ECON 3480 M & NWINTER 2006WINTER 2006

Power Point Presentation Power Point Presentation

Professor Ying KongProfessor Ying Kong

School of Analytic Studies and Information School of Analytic Studies and Information TechnologyTechnology

Atkinson Faculty of Liberal and Professional Atkinson Faculty of Liberal and Professional StudiesStudies

York UniversityYork University

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Chapter 13, Part BChapter 13, Part B Analysis of Variance and Experimental Analysis of Variance and Experimental

DesignDesign

Factorial ExperimentsFactorial Experiments

An Introduction to Experimental DesignAn Introduction to Experimental Design Completely Randomized DesignsCompletely Randomized Designs Randomized Block DesignRandomized Block Design

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An Introduction to Experimental DesignAn Introduction to Experimental Design

Statistical studies can be classified as being Statistical studies can be classified as being either experimental or observational.either experimental or observational.

In an In an experimental studyexperimental study, one or more factors are , one or more factors are controlled so that data can be obtained about how controlled so that data can be obtained about how the factors influence the variables of interest.the factors influence the variables of interest.

In an In an observational studyobservational study, no attempt is made , no attempt is made to control the factors.to control the factors.

Cause-and-effect relationshipsCause-and-effect relationships are easier to establish in are easier to establish in experimental studies than in observational studies.experimental studies than in observational studies.

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An Introduction to Experimental DesignAn Introduction to Experimental Design

A A factorfactor is a variable that the experimenter is a variable that the experimenter has selected for investigation.has selected for investigation.

A A treatmenttreatment is a level of a factor. is a level of a factor. Experimental unitsExperimental units are the objects of interest are the objects of interest

in the experiment.in the experiment. A A completely randomized designcompletely randomized design is an is an

experimental design in which the treatments are experimental design in which the treatments are randomly assigned to the experimental units.randomly assigned to the experimental units.

If the experimental units are heterogeneous, If the experimental units are heterogeneous, blocking can be used to form homogeneous blocking can be used to form homogeneous groups, resulting in a groups, resulting in a randomized block designrandomized block design..

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The between-samples estimate of The between-samples estimate of 22 is referred is referredto as the to as the mean square due to treatmentsmean square due to treatments (MSTR).(MSTR).

2

1

( )

MSTR1

k

j jj

n x x

k

2

1

( )

MSTR1

k

j jj

n x x

k

Between-Treatments Estimate of Population VarianceBetween-Treatments Estimate of Population Variance

Completely Randomized DesignCompletely Randomized Design

denominator is thedenominator is thedegrees of degrees of freedomfreedom

associated with associated with SSTRSSTR

numerator is callednumerator is calledthe the sum of squares sum of squares

duedueto treatmentsto treatments

(SSTR)(SSTR)

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The second estimate of The second estimate of 22, the within-samples , the within-samples estimate, is referred to as the estimate, is referred to as the mean square due to errormean square due to error (MSE).(MSE).

Within-Treatments Estimate of Population VarianceWithin-Treatments Estimate of Population Variance


denominator is denominator is thethe

degrees of degrees of freedomfreedom

associated with associated with SSESSE

numerator is numerator is calledcalled

the the sum of sum of squaressquares

due to errordue to error (SSE)(SSE)

MSE

( )n s

n k

j jj

k

T

1 2

1MSE

( )n s

n k

j jj

k

T

1 2

1

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MSTRSSTR

-

k 1MSTR

SSTR-

k 1

MSESSE

-

n kT

MSESSE

-

n kT

MSTRMSE

MSTRMSE

ANOVA TableANOVA Table


Source ofSource ofVariationVariation

Sum ofSum ofSquaresSquares

Degrees ofDegrees ofFreedomFreedom

MeanMeanSquaresSquares FF

TreatmentsTreatments

ErrorError

TotalTotal

kk - 1 - 1

nnTT - 1 - 1

SSTRSSTR

SSESSE

SSTSST

nnT T - - kk

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AutoShine, Inc. is considering marketing a long-AutoShine, Inc. is considering marketing a long-

lasting car wax. Three different waxes (Type 1, Type 2,lasting car wax. Three different waxes (Type 1, Type 2,

and Type 3) have been developed.and Type 3) have been developed.


Example: AutoShine, Inc.Example: AutoShine, Inc.

In order to test the durabilityIn order to test the durability

of these waxes, 5 new cars wereof these waxes, 5 new cars were

waxed with Type 1, 5 with Typewaxed with Type 1, 5 with Type

2, and 5 with Type 3. Each car was then2, and 5 with Type 3. Each car was then

repeatedly run through an automatic carwash repeatedly run through an automatic carwash until theuntil the

wax coating showed signs of deterioration.wax coating showed signs of deterioration.

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The number of times each car went The number of times each car went through thethrough the

carwash is shown on the next slide. carwash is shown on the next slide. AutoShine, Inc.AutoShine, Inc.

must decide which wax to market. Are the must decide which wax to market. Are the threethree

waxes equally effective?waxes equally effective?

Example: AutoShine, Inc.Example: AutoShine, Inc.

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1122334455

27273030292928283131

33332828313130303030

29292828303032323131

Sample MeanSample MeanSample VarianceSample Variance

ObservationObservationWaxWax

Type 1Type 1WaxWax

Type 2Type 2WaxWax

Type 3Type 3

2.52.5 3.3 3.3 2.5 2.529.029.0 30.4 30.4 30.030.0


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HypothesesHypotheses


where: where:

1 1 = mean number of washes for Type 1 wax= mean number of washes for Type 1 wax



HH00: : 11==22==33

HHaa: Not all the means are equal: Not all the means are equal

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Because the sample sizes are all equal:Because the sample sizes are all equal:


MSE = 33.2/(15 - 3) = 2.77MSE = 33.2/(15 - 3) = 2.77

MSTR = 5.2/(3 - 1) = 2.6MSTR = 5.2/(3 - 1) = 2.6

SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2

SSTR = 5(29–29.8)SSTR = 5(29–29.8)22 + 5(30.4–29.8) + 5(30.4–29.8)22 + 5(30–29.8) + 5(30–29.8)22 = 5.2 = 5.2

Mean Square ErrorMean Square Error

Mean Square Between TreatmentsMean Square Between Treatments

1 2 3( )/ 3x x x x 1 2 3( )/ 3x x x x = (29 + 30.4 + 30)/3 = 29.8= (29 + 30.4 + 30)/3 = 29.8

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Rejection RuleRejection Rule


where where FF.05.05 = 3.89 is based on an = 3.89 is based on an FF distribution distributionwith 2 numerator degrees of freedom and 12with 2 numerator degrees of freedom and 12denominator degrees of freedomdenominator degrees of freedom

pp-Value Approach: Reject -Value Approach: Reject HH00 if if pp-value -value << .05 .05

Critical Value Approach: Reject Critical Value Approach: Reject HH00 if if FF >> 3.89 3.89

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Test StatisticTest Statistic


There is insufficient evidence to conclude thatThere is insufficient evidence to conclude thatthe mean number of washes for the three waxthe mean number of washes for the three waxtypes are not all the same.types are not all the same.

ConclusionConclusion

F F = MSTR/MSE = 2.6/2.77 = .939 = MSTR/MSE = 2.6/2.77 = .939

The The pp-value is greater than .10, where -value is greater than .10, where FF = 2.81. = 2.81. (Excel provides a (Excel provides a pp-value of .42.)-value of .42.) Therefore, we cannot reject Therefore, we cannot reject HH00..

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ErrorError

TotalTotal

22

1414

5.25.2

33.233.2

38.438.4

1212


2.602.60

2.772.77

.939.939


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• For a randomized block design the sum of squares For a randomized block design the sum of squares total (SST) is partitioned into three groups: sum total (SST) is partitioned into three groups: sum of squares due to treatments, sum of squares due of squares due to treatments, sum of squares due to blocks, and sum of squares due to error.to blocks, and sum of squares due to error.

ANOVA ProcedureANOVA Procedure

Randomized Block DesignRandomized Block Design

SST = SSTR + SSBL + SSESST = SSTR + SSBL + SSE

• The total degrees of freedom, The total degrees of freedom, nnTT - 1, are - 1, are partitioned such that partitioned such that kk - 1 degrees of freedom - 1 degrees of freedom go to treatments, go to treatments, bb - 1 go to blocks, and ( - 1 go to blocks, and (kk - 1)(- 1)(bb - 1) go to the error term. - 1) go to the error term.

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MSTRSSTR

-

k 1MSTR

SSTR-

k 1

MSTRMSE

MSTRMSE






ErrorError

TotalTotal

kk - 1 - 1

nnTT - 1 - 1

SSTRSSTR

SSESSE

SSTSST



BlocksBlocks SSBLSSBL bb - 1 - 1

((k k – 1)(– 1)(bb – 1) – 1)

SSBL

MSBL-1b

SSBL

MSBL-1b

MSESSE

( )( )k b1 1MSE

SSE ( )( )k b1 1

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Example: Crescent Oil Co.Example: Crescent Oil Co.

Crescent Oil has developed threeCrescent Oil has developed three

new blends of gasoline and mustnew blends of gasoline and must

decide which blend or blends todecide which blend or blends to

produce and distribute. A studyproduce and distribute. A study

of the miles per gallon ratings of theof the miles per gallon ratings of the

three blends is being conducted to determine if thethree blends is being conducted to determine if the

mean ratings are the same for the three blends.mean ratings are the same for the three blends.

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Example: Crescent Oil Co.Example: Crescent Oil Co.

Five automobiles have beenFive automobiles have been

tested using each of the threetested using each of the three

gasoline blends and the milesgasoline blends and the miles

per gallon ratings are shown onper gallon ratings are shown on

the next slide.the next slide.

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29.8 29.8 28.8 28.8 28.4 28.4TreatmentTreatment

MeansMeans

11

22

33

44

55

3131

3030

2929

3333

2626

3030

2929

2929

3131

2525

3030

2929

2828

2929

2626

30.33330.333

29.33329.333

28.66728.667

31.00031.000

25.66725.667

Type of Gasoline (Treatment)Type of Gasoline (Treatment)BlockBlockMeansMeansBlend XBlend X Blend YBlend Y Blend ZBlend Z

AutomobileAutomobile(Block)(Block)

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Mean Square Due to ErrorMean Square Due to Error


MSE = 5.47/[(3 - 1)(5 - 1)] = .68MSE = 5.47/[(3 - 1)(5 - 1)] = .68

SSE = 62 - 5.2 - 51.33 = 5.47SSE = 62 - 5.2 - 51.33 = 5.47

MSBL = 51.33/(5 - 1) = 12.8MSBL = 51.33/(5 - 1) = 12.8SSBL = 3[(30.333 - 29)SSBL = 3[(30.333 - 29)22 + . . . + (25.667 - 29) + . . . + (25.667 - 29)22] = 51.33] = 51.33

MSTR = 5.2/(3 - 1) = 2.6MSTR = 5.2/(3 - 1) = 2.6SSTR = 5[(29.8 - 29)SSTR = 5[(29.8 - 29)22 + (28.8 - 29) + (28.8 - 29)22 + (28.4 - 29) + (28.4 - 29)22] = 5.2] = 5.2

The overall sample mean is 29. Thus,The overall sample mean is 29. Thus,

Mean Square Due to TreatmentsMean Square Due to Treatments

Mean Square Due to BlocksMean Square Due to Blocks

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ErrorError

TotalTotal

22

1414

5.205.20

5.475.47

62.0062.00

88

2.602.60

.68.68

3.823.82



BlocksBlocks 51.3351.33 12.8012.8044

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Rejection RuleRejection Rule


For For = .05, = .05, FF.05.05 = 4.46 = 4.46 (2 d.f. numerator and 8 d.f. denominator)(2 d.f. numerator and 8 d.f. denominator)

pp-Value Approach:-Value Approach: Reject Reject HH00 if if pp-value -value << .05 .05

Critical Value Approach:Critical Value Approach: Reject Reject HH00 if if FF >> 4.46 4.46

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ConclusionConclusion


There is insufficient evidence to conclude thatThere is insufficient evidence to conclude thatthe miles per gallon ratings differ for the threethe miles per gallon ratings differ for the threegasoline blends.gasoline blends.

The The pp-value is greater than .05 (where -value is greater than .05 (where FF = 4.46) and less than .10 (where = 4.46) and less than .10 (where FF = 3.11). = 3.11). (Excel provides a (Excel provides a pp-value of .07). Therefore, -value of .07). Therefore, we cannot reject we cannot reject HH00..

FF = MSTR/MSE = 2.6/.68 = 3.82 = MSTR/MSE = 2.6/.68 = 3.82

Test StatisticTest Statistic

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Factorial ExperimentsFactorial Experiments

In some experiments we want to draw conclusions In some experiments we want to draw conclusions about more than one variable or factor.about more than one variable or factor.

Factorial experimentsFactorial experiments and their corresponding and their corresponding ANOVA computations are valuable designs ANOVA computations are valuable designs when simultaneous conclusions about two or when simultaneous conclusions about two or more factors are required.more factors are required.

For example, for For example, for aa levels of factor A and levels of factor A and bb levels of factor B, the experiment will involve levels of factor B, the experiment will involve collecting data on collecting data on abab treatment combinations. treatment combinations.

The term factorial is used because the The term factorial is used because the experimental conditions include all possible experimental conditions include all possible combinations of the factors.combinations of the factors.

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• The ANOVA procedure for the two-factor factorial The ANOVA procedure for the two-factor factorial experiment is similar to the completely randomized experiment is similar to the completely randomized experiment and the randomized block experiment.experiment and the randomized block experiment.

ANOVA ProcedureANOVA Procedure

SST = SSA + SSB + SSAB SST = SSA + SSB + SSAB + SSE+ SSE

• The total degrees of freedom, The total degrees of freedom, nnTT - 1, are - 1, are partitioned such that (partitioned such that (aa – 1) d.f go to Factor A, – 1) d.f go to Factor A, ((bb – 1) d.f go to Factor B, ( – 1) d.f go to Factor B, (aa – 1)( – 1)(bb – 1) d.f. go to – 1) d.f. go to Interaction, and Interaction, and abab((rr – 1) go to Error. – 1) go to Error.

Two-Factor Factorial ExperimentTwo-Factor Factorial Experiment

• We again partition the sum of squares total We again partition the sum of squares total (SST) into its sources.(SST) into its sources.

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SSA

MSA-1a

SSA

MSA-1a

MSAMSE

MSAMSE





Factor AFactor A

ErrorError

TotalTotal

aa - 1 - 1

nnTT - 1 - 1

SSASSA

SSESSE

SSTSST

Factor BFactor B SSBSSB bb - 1 - 1

abab((rr – 1) – 1)

SSEMSE

( 1)ab r

SSE

MSE( 1)ab r


SSB

MSB-1b

SSB

MSB-1b

InteractionInteraction SSABSSAB ((a a – 1)(– 1)(bb – 1) – 1) SSAB

MSAB( 1)( 1)a b

SSAB

MSAB( 1)( 1)a b

MSBMSE

MSBMSE

MSABMSE

MSABMSE

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Step 3Step 3 Compute the sum of squares for factor B Compute the sum of squares for factor B

2

1 1 1

SST = ( )a b r

ijki j k

x x

2

1 1 1

SST = ( )a b r

ijki j k

x x

2

1

SSA = ( . )a

ii

br x x

2

1

SSA = ( . )a

ii

br x x

2

1

SSB = ( . )b

jj

ar x x

2

1

SSB = ( . )b

jj

ar x x


Step 1Step 1 Compute the total sum of squares Compute the total sum of squares

Step 2Step 2 Compute the sum of squares for factor A Compute the sum of squares for factor A

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Step 4Step 4 Compute the sum of squares for interaction Compute the sum of squares for interaction

2

1 1

SSAB = ( . . )a b

ij i ji j

r x x x x

2

1 1

SSAB = ( . . )a b

ij i ji j

r x x x x


SSE = SST – SSA – SSB - SSABSSE = SST – SSA – SSB - SSAB

Step 5Step 5 Compute the sum of squares due to error Compute the sum of squares due to error

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A survey was conducted of hourly wagesA survey was conducted of hourly wages

for a sample of workers in two industriesfor a sample of workers in two industries

at three locations in Ohio. Part of theat three locations in Ohio. Part of the

purpose of the survey was topurpose of the survey was to

determine if differences existdetermine if differences exist

in both industry type andin both industry type and

location. The sample data are shownlocation. The sample data are shown

on the next slide.on the next slide.

Example: State of Ohio Wage SurveyExample: State of Ohio Wage Survey


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Example: State of Ohio Wage SurveyExample: State of Ohio Wage Survey


IndustrIndustryy

CincinnaCincinnatiti

ClevelandCleveland ColumbuColumbuss

II 12.1012.10 11.8011.80 12.9012.90

II 11.8011.80 11.2011.20 12.7012.70

II 12.1012.10 12.0012.00 12.2012.20

IIII 12.4012.40 12.6012.60 13.0013.00

IIII 12.5012.50 12.0012.00 12.1012.10

IIII 12.0012.00 12.5012.50 12.7012.70

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FactorsFactors


•Each experimental condition is repeated Each experimental condition is repeated 33 times times

•Factor B: Location (3 levels)Factor B: Location (3 levels)•Factor A: Industry Type (2 levels)Factor A: Industry Type (2 levels)

ReplicationsReplications

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Factor AFactor A

ErrorError

TotalTotal

11

1717

.50.50

1.431.43

3.423.42

1212

.50.50

.12.12

4.194.19


Factor BFactor B 1.121.12 .56.5622


InteractionInteraction .37.37 .19.19224.694.69

1.551.55

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Conclusions Using the Conclusions Using the pp-Value Approach-Value Approach


((pp-values were found using Excel)-values were found using Excel)

Interaction is not significant.Interaction is not significant.

•Interaction: Interaction: pp-value = .25 > -value = .25 > = .05 = .05

Mean wages differ by location.Mean wages differ by location.

•Locations: Locations: pp-value = .03 -value = .03 << = .05 = .05

Mean wages do not differ by industry type.Mean wages do not differ by industry type.•Industries: Industries: pp-value = .06 > -value = .06 > = .05 = .05

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Conclusions Using the Critical Value ApproachConclusions Using the Critical Value Approach


Interaction is not significant.Interaction is not significant.

•Interaction: Interaction: FF = 1.55 < = 1.55 < FF = = 3.893.89

Mean wages differ by location.Mean wages differ by location.•Locations: Locations: FF = 4.69 = 4.69 >> FF = 3.89 = 3.89

Mean wages do not differ by industry type.Mean wages do not differ by industry type.•Industries: Industries: FF = 4.19 < = 4.19 < FF = 4.75 = 4.75

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End of Chapter 13, Part BEnd of Chapter 13, Part B

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1 1 Slide © 2005 Thomson/South-Western AK/ECON 3480 M & N WINTER 2006 n Power Point Presentation n Professor Ying Kong School of Analytic Studies and Information