1 1 slide © 2008 thomson south-western. all rights reserved chapter 13 experimental design and...

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© 2008 Thomson South-Western. All Rights Reserved© 2008 Thomson South-Western. All Rights Reserved

Chapter 13Chapter 13 Experimental Design and Analysis of Experimental Design and Analysis of

Variance Variance Introduction to Experimental Introduction to Experimental

DesignDesign and Analysis of Variance and Analysis of Variance Analysis of Variance Analysis of Variance and the Completely Randomized Designand the Completely Randomized Design

Multiple Comparison Multiple Comparison ProceduresProcedures

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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 , one or more factors are controlled so that data can be obtained are controlled so that data can be obtained about how the factors influence the variables about how the factors influence the variables of interest.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 are easier to establish in experimental studies than in establish in experimental studies than in observational studies.observational studies.

An Introduction to Experimental DesignAn Introduction to Experimental Designand Analysis of Varianceand Analysis of Variance

Analysis of variance (ANOVA) can be used to Analysis of variance (ANOVA) can be used to analyze the data obtained from experimental or analyze the data obtained from experimental or observational studies.observational studies.

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An Introduction to Experimental DesignAn Introduction to Experimental Designand Analysis of Varianceand Analysis of Variance

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

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Analysis of Variance: A Conceptual Analysis of Variance: A Conceptual OverviewOverview

Analysis of VarianceAnalysis of Variance (ANOVA) can be used to test (ANOVA) can be used to test for the equality of three or more population means.for the equality of three or more population means. Analysis of VarianceAnalysis of Variance (ANOVA) can be used to test (ANOVA) can be used to test for the equality of three or more population means.for the equality of three or more population means.

Data obtained from observational or experimentalData obtained from observational or experimental studies can be used for the analysis.studies can be used for the analysis. Data obtained from observational or experimentalData obtained from observational or experimental studies can be used for the analysis.studies can be used for the analysis.

We want to use the sample results to test theWe want to use the sample results to test the following hypotheses:following hypotheses: We want to use the sample results to test theWe want to use the sample results to test the following hypotheses:following hypotheses:

HH00: : 11==22==33==. . . . . . = = kk

HHaa: Not all population means are equal: Not all population means are equal

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HH00: : 11==22==33==. . . . . . = = kk


If If HH00 is rejected, we cannot conclude that is rejected, we cannot conclude that allall population means are different.population means are different. If If HH00 is rejected, we cannot conclude that is rejected, we cannot conclude that allall population means are different.population means are different.

Rejecting Rejecting HH00 means that at least two population means that at least two population means have different values.means have different values. Rejecting Rejecting HH00 means that at least two population means that at least two population means have different values.means have different values.


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For each population, the response (dependent)For each population, the response (dependent) variable is normally distributed.variable is normally distributed. For each population, the response (dependent)For each population, the response (dependent) variable is normally distributed.variable is normally distributed.

The variance of the response variable, denoted The variance of the response variable, denoted 22,, is the same for all of the populations.is the same for all of the populations. The variance of the response variable, denoted The variance of the response variable, denoted 22,, is the same for all of the populations.is the same for all of the populations.

The observations must be independent.The observations must be independent. The observations must be independent.The observations must be independent.

Assumptions for Analysis of Assumptions for Analysis of VarianceVariance


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Sampling Distribution of Given Sampling Distribution of Given HH00 is True is Truexx

1x1x 3x3x2x2x

Sample means are close togetherSample means are close together because there is onlybecause there is only

one sampling distributionone sampling distribution when when HH00 is true. is true.

22x n

2

2x n


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Sampling Distribution of Given Sampling Distribution of Given HH00 is False is Falsexx

33 1x1x 2x2x3x3x 11 22

Sample means come fromSample means come fromdifferent sampling distributionsdifferent sampling distributionsand are not as close togetherand are not as close together

when when HH00 is false. is false.


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Analysis of VarianceAnalysis of Variance

Between-Treatments Estimate of Population Between-Treatments Estimate of Population VarianceVariance Within-Treatments Estimate of Population Within-Treatments Estimate of Population VarianceVariance Comparing the Variance Estimates: The Comparing the Variance Estimates: The F F TestTest ANOVA TableANOVA Table

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2

1

( )

MSTR1

k

j jj

n x x

k

2

1

( )

MSTR1

k

j jj

n x x

k

Between-Treatments EstimateBetween-Treatments Estimateof Population Variance of Population Variance 22

Denominator is theDenominator is thedegrees of freedomdegrees of freedom

associated with associated with SSTRSSTR

Numerator is calledNumerator is calledthe the sum of squares sum of squares

duedueto treatmentsto treatments

(SSTR)(SSTR)

The estimate of The estimate of 22 based on the variation of based on the variation of thethe

sample means is called the sample means is called the mean square due mean square due toto

treatmentstreatments and is denoted by and is denoted by MSTRMSTR..

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The estimate of The estimate of 22 based on the variation of based on the variation of the sample observations within each sample is the sample observations within each sample is called the called the mean square errormean square error and is denoted and is denoted by by MSEMSE..

Within-Treatments EstimateWithin-Treatments Estimateof Population Variance of Population Variance 22

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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Comparing the Variance Estimates: The Comparing the Variance Estimates: The FF TestTest

If the null hypothesis is true and the ANOVAIf the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution ofassumptions are valid, the sampling distribution of MSTR/MSE is an MSTR/MSE is an FF distribution with MSTR d.f. distribution with MSTR d.f. equal to equal to kk - 1 and MSE d.f. equal to - 1 and MSE d.f. equal to nnTT - - kk..

If the means of the If the means of the kk populations are not equal, the populations are not equal, the value of MSTR/MSE will be inflated because MSTRvalue of MSTR/MSE will be inflated because MSTR overestimates overestimates 22.. Hence, we will reject Hence, we will reject HH00 if the resulting value of if the resulting value of MSTR/MSE appears to be too large to have beenMSTR/MSE appears to be too large to have been selected at random from the appropriate selected at random from the appropriate FF distribution.distribution.

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Sampling Distribution of Sampling Distribution of MSTR/MSEMSTR/MSE

Do Not Reject H0Do Not Reject H0

Reject H0Reject H0

MSTR/MSEMSTR/MSE

Critical ValueCritical ValueFF

Sampling DistributionSampling Distributionof MSTR/MSEof MSTR/MSE

Comparing the Variance Estimates: The Comparing the Variance Estimates: The FF TestTest

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MSTRSSTR

-

k 1MSTR

SSTR-

k 1

MSESSE

-

n kT

MSESSE

-

n kT

MSTRMSE

MSTRMSE

Source ofSource ofVariationVariation

Sum ofSum ofSquaresSquares

Degrees ofDegrees ofFreedomFreedom

MeanMeanSquareSquare FF

TreatmentsTreatments

ErrorError

TotalTotal

kk - 1 - 1

nnTT - 1 - 1

SSTRSSTR

SSESSE

SSTSST

nnT T - - kk

SST is SST is partitionedpartitioned

into SSTR and into SSTR and SSE.SSE.

SST’s degrees of SST’s degrees of freedomfreedom

(d.f.) are partitioned (d.f.) are partitioned intointo

SSTR’s d.f. and SSE’s SSTR’s d.f. and SSE’s d.f.d.f.

ANOVA TableANOVA Table

pp--ValueValue

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SST divided by its degrees of freedom SST divided by its degrees of freedom nnTT – 1 is the – 1 is the overall sample variance that would be obtained if weoverall sample variance that would be obtained if we treated the entire set of observations as one data set.treated the entire set of observations as one data set.

SST divided by its degrees of freedom SST divided by its degrees of freedom nnTT – 1 is the – 1 is the overall sample variance that would be obtained if weoverall sample variance that would be obtained if we treated the entire set of observations as one data set.treated the entire set of observations as one data set.

With the entire data set as one sample, the formulaWith the entire data set as one sample, the formula for computing the total sum of squares, SST, is:for computing the total sum of squares, SST, is: With the entire data set as one sample, the formulaWith the entire data set as one sample, the formula for computing the total sum of squares, SST, is:for computing the total sum of squares, SST, is:

2

1 1

SST ( ) SSTR SSEjnk

ijj i

x x

2

1 1

SST ( ) SSTR SSEjnk

ijj i

x x

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ANOVA can be viewed as the process of partitioningANOVA can be viewed as the process of partitioning the total sum of squares and the degrees of freedomthe total sum of squares and the degrees of freedom into their corresponding sources: treatments and error.into their corresponding sources: treatments and error.

ANOVA can be viewed as the process of partitioningANOVA can be viewed as the process of partitioning the total sum of squares and the degrees of freedomthe total sum of squares and the degrees of freedom into their corresponding sources: treatments and error.into their corresponding sources: treatments and error.

Dividing the sum of squares by the appropriateDividing the sum of squares by the appropriate degrees of freedom provides the variance estimatesdegrees of freedom provides the variance estimates and the and the FF value used to test the hypothesis of equal value used to test the hypothesis of equal population means.population means.

Dividing the sum of squares by the appropriateDividing the sum of squares by the appropriate degrees of freedom provides the variance estimatesdegrees of freedom provides the variance estimates and the and the FF value used to test the hypothesis of equal value used to test the hypothesis of equal population means.population means.

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Test for the Equality of Test for the Equality of kk Population Population MeansMeans

FF = MSTR/MSE = MSTR/MSE

HH00: : 11==22==33==. . . . . . = = kk


HypotheseHypothesess

Test Test StatisticStatistic

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Test for the Equality of Test for the Equality of kk Population Population MeansMeans

Rejection Rejection RuleRule

where the value of where the value of FF is based on an is based on anFF distribution with distribution with kk - 1 numerator d.f. - 1 numerator d.f.and and nnTT - - kk denominator d.f. denominator d.f.

Reject Reject HH00 if if pp-value -value << pp-value Approach:-value Approach:

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

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Multiple Comparison ProceduresMultiple Comparison Procedures

Suppose that analysis of variance has Suppose that analysis of variance has provided statistical evidence to reject the null provided statistical evidence to reject the null hypothesis of equal population means.hypothesis of equal population means.

Fisher’s least significant difference (LSD) Fisher’s least significant difference (LSD) procedure can be used to determine where procedure can be used to determine where the differences occur.the differences occur.

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Fisher’s LSD ProcedureFisher’s LSD Procedure

1 1MSE( )

i j

i j

x xt

n n

1 1MSE( )

i j

i j

x xt

n n

Test StatisticTest Statistic

HypothesesHypotheses

0 : i jH 0 : i jH : a i jH : a i jH

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Fisher’s LSD ProcedureFisher’s LSD Procedure

where the value of where the value of ttaa/2 /2 is based on ais based on a

tt distribution with distribution with nnTT - - kk degrees of freedom. degrees of freedom.

Rejection RuleRejection Rule

Reject Reject HH00 if if pp-value -value <<

pp-value Approach:-value Approach:

Critical Value Approach:Critical Value Approach:

Reject Reject HH00 if if tt < - < -ttaa/2 /2 or or tt > > ttaa/2 /2

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Test StatisticTest Statistic

Fisher’s LSD ProcedureFisher’s LSD ProcedureBased on the Test Statistic Based on the Test Statistic xxii - - xxjj

__ __

/ 21 1LSD MSE( )

i jt n n / 2

1 1LSD MSE( )i j

t n n wherewhere

i jx xi jx x

Reject Reject HH00 if > LSD if > LSDi jx xi jx x

HypothesesHypotheses

Rejection Rejection RuleRule

0 : i jH 0 : i jH : a i jH : a i jH

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The experiment-wise Type I error rate gets larger for The experiment-wise Type I error rate gets larger for problems with more populations (larger problems with more populations (larger kk).).

Type I Error RatesType I Error Rates

EWEW = 1 – (1 – = 1 – (1 – ))((k k – 1)!– 1)!

The The comparison-wise Type I error ratecomparison-wise Type I error rate indicates the level of significance associated indicates the level of significance associated with a single pairwise comparison.with a single pairwise comparison.

The The experiment-wise Type I error rateexperiment-wise Type I error rate EWEW is is the probability of making a Type I error on at the probability of making a Type I error on at least one of the (least one of the (kk – 1)! pairwise comparisons. – 1)! pairwise comparisons.

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