Chapter 10 Analysis of Variance. Chapter 10 - Chapter Outcomes After studying the material in this...

Chapter 10Chapter 10

Analysis of Analysis of VarianceVariance

Chapter 10 - Chapter 10 - Chapter Chapter OutcomesOutcomes

After studying the material in this chapter, you should be able to:•Recognize the applications that call for the use of analysis of variance.•Understand the logic of analysis of variance.•Be aware of several different analysis of variance designs and understand when to use each one.•Perform a single factor hypothesis test using analysis of variance manually and with the aid of Excel or Minitab software.

Chapter 10 - Chapter 10 - Chapter Chapter OutcomesOutcomes

(continued)(continued)

After studying the material in this chapter, you should be able to:•Conduct and interpret post-analysis of variance pairwise comparisons procedures.•Recognize when randomized block analysis of variance is useful and be able to perform the randomized block analysis.•Perform two factor analysis of variance tests with replications using Excel or Minitab and interpret the output.

One-Way Analysis of One-Way Analysis of VarianceVariance

One-way analysis of varianceOne-way analysis of variance is a design in which independent samples are obtained from k levels of a single factor for the purpose of testing whether the k levels have equal means.


A factorfactor refers to a quantity under examination in an experiment as a possible cause of variation in the response variable.


LevelsLevels refer to the categories, measurements, or strata of a factor of interest in the current experiment.


Null hypothesis in an ANOVA experiment:

Alternative hypothesis in an ANOVA experiment:

kH 3210 :

different are means population least twoAt :AH


An experiment is completely randomized if it consists of the independent random selection of observations representing each level of one factor.

COMPLETELY RANDOMIZED DESIGNCOMPLETELY RANDOMIZED DESIGN


An experiment is said to have a balanced design if the factor levels have equal sample sizes.

BALANCED DESIGNBALANCED DESIGN


The ANOVA test is based on three assumptions:

• All populations are normally distributed.

• The population variances are equal.• The observations are independent,

meaning that any one individual value is not dependent on the value of any other observation.


The total variationtotal variation in the data refers to the aggregate dispersion of the individual data values across the various factor levels.

Total Total VariationVariation


The dispersion that exists among the data values within a particular factor level is called the within-within-sample variationsample variation.

Within-Sample Within-Sample VariationVariation


The dispersion among the factor sample means is called the between-between-sample variationsample variation.

Between-Sample Between-Sample VariationVariation


PARTITIONED SUM OF SQUARESPARTITIONED SUM OF SQUARES

where:TSS = Total Sum of SquaresSSB = Sum of Squares

BetweenSSW = Sum of Squares

Within

SSW SSB TSS


Null hypothesis in an ANOVA experiment:

Alternative hypothesis in an ANOVA experiment:

kH 3210 :

different are means population least twoAt :AH


F-Max test for Equal VariancesF-Max test for Equal Variances

where:

s2max = Largest sample

variances2

min = Smallest sample variance

2min

2max

s

sFMax


TOTAL SUM OF SQUARESTOTAL SUM OF SQUARES

where:TSS = Total sum of squares k = Number of populations (levels) ni = Sample size from population i

xij = jth measurement from population i

= Grand Mean (mean of all the data values)

x

k

i

n

jij

j

xxTSS1 1

2)(


SUM OF SQUARES BETWEENSUM OF SQUARES BETWEEN

where:SSB = Sum of Squares Between

Samples k = Number of populations (levels) ni = Sample size from population i

= Sample mean from population i

= Grand Meanx

2

1

)( xxnSSB i

k

ii

ix


SUM OF SQUARES WITHINSUM OF SQUARES WITHIN

SSW = TSS - SSBor

where:SSW = Sum of Squares Within

Samples k = Number of populations ni = Sample size from population i

= Sample mean from population i

xij = jth measurement from population i

2

11

)( iij

n

j

k

i

xxSSWj

ix

One-Way ANOVA TableOne-Way ANOVA Table(Table 10-3)(Table 10-3)

Source ofVariation dfSS MS

BetweenSamples SSB

SSB K-1

WithinSamples

N - kSSW SSW N-k

Total N - 1TSS

F Ratio

k - 1

k = Number of populations

N = Sum of the sample sizes from all populations

df = Degrees of freedom

MSBMSW

One-Way ANOVA TableOne-Way ANOVA Table(Table 10-3)(Table 10-3)

1-kSSB between square Mean MSB

k-NSSW withinsquare Mean MSW

Example of a One-Way Example of a One-Way ANOVA TableANOVA Table

(Table 10-4)(Table 10-4)


BetweenSamples

22 7.333

WithinSamples

28198.88 7.10286

Total 3131220.88220.88

F Ratio

37.333

7.10286

= 1.03244= 1.03244

3333.7322 between square MeanMSB

10286.728

198.88 withinsquare MeanMSW

One-Way Analysis of One-Way Analysis of Variance Variance (Figure 10-5)(Figure 10-5)

F 0

Degrees of freedom:

D1 = k - 1 = 4 - 1 = 3

D2 = N - k = 32 - 4 = 28Rejection Region

95.2F

03244.110286.7

333.7

MSW

MSBF

Since F=1.03244 F= 2. 95, do not reject H0

H0: 1= 2 = 3 = 4

HA: At least two population means are different

= 0.05

The Tukey-Kramer The Tukey-Kramer ProcedureProcedure

The Tukey-KramerTukey-Kramer procedure is a method for testing which populations have different means, after the one-way ANOVA null hypothesis has been rejected.

The Tukey-Kramer The Tukey-Kramer Procedure and One-Way Procedure and One-Way

ANOVAANOVA

The experiment-wide error rateexperiment-wide error rate is the proportion of experiments in which at least one of the set of confidence intervals constructed does not contain the true value of the population parameter being estimated.

The Tukey-Kramer The Tukey-Kramer Procedure and One-Way Procedure and One-Way

ANOVAANOVATUKEY-KRAMER CRITICAL RANGETUKEY-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, respectively.

ji nn

MSWq

11

2Range Critical

Randomized Complete Randomized Complete Block ANOVABlock ANOVA

A treatmenttreatment is a combination of one level of each factor in an experiment associated with each observed value of the response variable.


SUM OF SQUARES PARTITIONING - SUM OF SQUARES PARTITIONING - RANDOMIZED COMPLETE BLOCK DESIGNRANDOMIZED COMPLETE BLOCK DESIGN

where:TSS = Total sum of squaresSSB = Sum of squares between factor

levelsSSBL= Sum of squares between blocksSSW = Sum of squares within levels

SSWSSBLSSBTSS


SUM OF SQUARES FOR BLOCKINGSUM OF SQUARES FOR BLOCKING

where: k = Number of levels for the factor n = Number of blocks = The mean of the jth block = Grand Mean

n

jj xxkSSBL 2)(

jxx


SUM OF SQUARES WITHINSUM OF SQUARES WITHIN

)( SSBLSSBTSSSSW


The randomized block design requires the following assumptions:

• The populations are normally distributed.

• The populations have equal variances.• The observations are independent.

Randomized Block ANOVA Randomized Block ANOVA TableTable

(Table 10-7)(Table 10-7)Source ofVariation dfSS MS

BetweenBlocks SSBL

MSB

MSBL

BetweenSamples

k - 1SSB

MSW

Total N - 1TSS

F Ratio

b- 1

k = Number of levels

b = Number of blocks

df = Degrees of freedom and N = Combined sample size

SamplesWithin SSW (k - 1)(b - 1)

MSBLMSW

MSWMSB


(From Table 10-7)(From Table 10-7)

where:

1k

SSBMSB between square Mean

)1( b

SSBLMSBL blocking square Mean

)1)(1(

bk

SSWMSW withinsquare Mean

Randomized Block ANOVARandomized Block ANOVA(Figure 10-12)(Figure 10-12)

F 0

Degrees of freedom:

D1 = k - 1 = 3 - 1 = 2, D2 = (n - 1)(k - 1)

= (4)(2) = 8 Rejection Region

4589.4F

544.8F

Since F=8.544 > F= 4.4589, reject H0

H0: 1= 2 = 3

HA: At least two population means are different

= 0.05

Fisher’s Least Significant Fisher’s Least Significant Difference (LSD) TestDifference (LSD) Test

FISHER’S LEAST SIGNIFICANT FISHER’S LEAST SIGNIFICANT DIFFERENCE FOR COMPLETE BLOCK DIFFERENCE FOR COMPLETE BLOCK

DESIGNDESIGN

where: t/2 = Upper-tailed value from Student’s t-

distribution for /2 and (k -1)(n - 1) degrees of freedom MSW = Mean square within from ANOVA table

n = Number of blocks k = Number of levels

nMSWtLSD

22/

Two-Factor Analysis of Two-Factor Analysis of VarianceVariance

Two-factor ANOVATwo-factor ANOVA is a technique used to analyze two factors in an analysis of variance framework.

Two-Factor Analysis of Two-Factor Analysis of VarianceVariance(Figure 10-13)(Figure 10-13)

SST

SSA

SSB

SSAB

SSE

Factor A

Factor B

Interaction Between A and BInherent Variation (Error)

Two-Factor Analysis of Two-Factor Analysis of VarianceVariance

The necessary assumptions for the two factor ANOVA are:

• The population values for each combination of pairwise factor levels are normally distributed.

• The variances for each population are equal.

• The samples are independent.• The observations are

independent.


(Table 10-9)(Table 10-9)


Factor A SSA

MSB

MSA

b - 1SSB

MSAB

Total N - 1TSS

F Ratio

a- 1

a = Number of levels of factor A

b = Number of levels of factor B

N = Total number of observations in all cells

AB Interaction SSAB

(a - 1)(b - 1)

MSA

MSE

MSEMSBFactor B

Error SSE N - ab MSE

MSEMSAB


(From Table 10-9)(From Table 10-9)

where:

1a

SSMS A

A A factor square Mean

1b

SSMS B

B B factor square Mean

)1)(1(ninteractio squareMean

ba

SSMS AB

AB

abN

SSEMSE

withinsquare Mean

Two Factor ANOVA Two Factor ANOVA EquationsEquations

a

i

b

j

n

kijk xxTSS

1 1 1

2)(

2

1.. )( xxnbSS

a

iiA

2

1 1..... )( xxxxnSS

a

i

b

jjiijAB

a

i

b

j

n

kijijk xxSSE

1 1 1

2.)(

2

1.. )( xxnaSS

b

jjB

Total Sum of Squares:

Sum of Squares Factor A:

Sum of Squares Factor B:

Sum of Squares Interaction Between A and B:

Sum of Squares Error:

Two Factor ANOVA Two Factor ANOVA EquationsEquations

where:

Mean Grand

nab

x

x

a

i

b

j

n

kijk

1 1 1

A factorof level eachof Mean

nb

x

x

b

j

n

kijk

i1 1

Bfactor of leveleach ofMean 1 1..

na

xx

a

i

n

kijk

j

celleach ofMean 1

.

n

k

ijkij n

xx

a = Number of levels of factor A

b = Number of levels of factor B

n’ = Number of replications in each cell

Differences Between Factor Differences Between Factor Level Mean Values: Level Mean Values:

No InteractionNo Interaction

1 2 3

Factor B Level 1Factor B Level 4Factor B Level 3Factor B Level 2

Factor A Levels

Mean

Resp

on

se

Differences Between Factor Differences Between Factor Level Mean Values: Level Mean Values: Interaction PresentInteraction Present

1 2 3

Factor B Level 1

Factor B Level 4

Factor B Level 3

Factor B Level 2

Factor A Levels

Mean

Resp

on

se

A Caution About A Caution About InteractionInteraction

When conducting hypothesis tests for a two factor ANOVA:

• Test for interaction.• If interaction is present conduct

a one-way ANOVA to test the levels of one of the factors using only one level of the other factor.

• If no interaction is found, test factor A and factor B.

Key TermsKey Terms

• Balanced Design• Between-Sample

Variation• Completely

Randomized Design• Experiment-Wide

Error Rate• Factor

• Levels • One-Way Analysis

of Variance• Total Variation• Treatment• Within-Sample

Variation

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