Chapter 15Chapter 15

Analysis of Analysis of VarianceVariance

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Framework for One-Way Framework for One-Way Analysis of VarianceAnalysis of Variance

Suppose that we have independent samples of n1, n2, . . ., nK observations from K populations. If the population means are denoted by 1, 2, . . ., K, the one-way analysis of variance framework is designed to test the null hypothesis

ji1

210

,pair oneleast at For :

:

ji

K

H

H

Sample Observations from Sample Observations from Independent Random Samples of K Independent Random Samples of K

PopulationsPopulations(Table 15.2)(Table 15.2)

POPULATION

1 2 . . . K

x11

x12

.

.

.x1n1

x21

x22

.

.

.x2n2

. . .. . .

. . .

xK1

xK2

.

.

.xKnK

Sum of Squares Sum of Squares Decomposition for One-Way Decomposition for One-Way

Analysis of VarianceAnalysis of Variance

Suppose that we have independent samples of n1, n2, . . ., nK observations from K populations. Denote by the K group sample means and by x the overall sample mean. We define the following sum of sum of squaressquares:

where xij denotes the jth sample observation in the ith group.Then

jn

jiij

K

i

xxSSW1

2

1

)( :Groups-Within

Kxxx ,,, 21

K

iii xxnSSG

1

2)( :Groups-Between

jn

jij

K

i

xxSST1

2

1

)( :Total

SSGSSWSST

Sum of Squares Decomposition Sum of Squares Decomposition for One-Way Analysis of for One-Way Analysis of

VarianceVariance(Figure 15.2)(Figure 15.2)

Total sum of squares

Within-groupssum of squares

Between-groupssum of squares

Hypothesis Test for One-Way Hypothesis Test for One-Way Analysis of VarianceAnalysis of Variance

Suppose that we have independent samples of n1, n2, . . ., nK observations from K populations. Denote by n the total sample size so that

We define the mean squares as follows:

The null hypothesis to be tested is that the K population means are equal, that is

Kn

SSWMSW

:Groups-Within

1 :Groups-Between

K

SSGMSG

KH 210 :

Knnnn 21

Sum of Squares Sum of Squares Decomposition for Two-Way Decomposition for Two-Way

Analysis of VarianceAnalysis of Variance

Suppose that we have a sample of observations with xij denoting the observation in the ith group and jth block. Suppose that there are K groups and H blocks, for a total of n = KH observations. Denote the group sample means by , the block sample means by and the overall sample mean by x. We define the following sum of squaressum of squares:

H

jj xxKSSB

1

2)( :Blocks-Within

),,2,1( Kixi

K

ii xxHSSG

1

2)( :Groups-Between

H

jij

K

i

xxSST1

2

1

)( :Total

),,2,1( Hjx j

Sum of Squares Sum of Squares Decomposition for Two-Way Decomposition for Two-Way

Analysis of VarianceAnalysis of Variance(continued)(continued)

H

jjiij

K

i

xxxxSSE1

2

1

)( :ERROR

SSESSBSSGSST

Hypothesis Test for Two-Way Hypothesis Test for Two-Way Analysis of VarianceAnalysis of Variance

Suppose that we have a sample observation for each group-block combination in a design containing K groups and H blocks.

Where Gi is the group effect and Bj is the block effect.

Define the following mean squares:

We assume that the error terms ij in the model are independent of one another, are normally distributed, and have the same variance

1 :Groups-Between

K

SSGMSG

1 :Blocks-Between

H

SSBMSB

ijjiij EBGx

)1)(1( :Error

HK

SSEMSE

Hypothesis Test for Two-Way Hypothesis Test for Two-Way Analysis of VarianceAnalysis of Variance

(continued)(continued)

A test of significance level of the null hypothesis H0 that the K population group means are all the same is provided by the decision rule

A test of significance level of the null hypothesis H0 that the H population block means are all the same is provided by the decision rule

Here F v1,v2, is the number exceeded with probability by a random variable following an F distribution with numerator degrees of freedom v1 and denominator degrees of freedom v2

),1)(1(,10 if HReject HKKFMSE

MSG

),1)(1(,10 if HReject HKHFMSE

MSB

General Format of Two-Way General Format of Two-Way Analysis of Variance TableAnalysis of Variance Table

(Table 15.9)(Table 15.9)

Source of Variation

Sums of Squares

Degrees of

Freedom

Mean Squares

F Ratios

Between groups

Between blocks

Error

Total

SSG

SSB

SSE

SST

K – 1

H – 1

(K – 1)(H – 1)

n - 1

1K

SSGMSG

1H

SSBMSB

)1)(1(

HK

SSEMSE

MSE

MSG

MSE

MSB

Sum of Squares Decomposition for Two-Way Sum of Squares Decomposition for Two-Way Analysis of Variance: Several Observations per Analysis of Variance: Several Observations per

CellCell

Suppose that we have a sample of observations on K groups and H blocks, with L observations per cell. Then, we define the following sums of squares and sums of squares and associated degrees of freedomassociated degrees of freedom:

1)( :Blocks-Between1

2

HxxKLSSBH

jj

1)( :Groups-Between1

2

KxxHLSSGK

ii

1)( :Total 2 KHLxxSSTi j l

ijl

)1()( :Error 2 LKHxxSSE iji j l

ijl

)1)(1()( :nInteractio1

2

1

HKxxxxLSSIH

jjiij

K

i

Sum of Squares Decomposition Sum of Squares Decomposition for Two-Way Analysis of for Two-Way Analysis of

Variance with More than One Variance with More than One Observation per CellObservation per Cell

(Figure 15.12)(Figure 15.12)

Total sum of squares

Within-groupssum of squares

Between-groupssum of squares

Interactionsum of squares

Errorsum of squares

Key WordsKey Words

Hypothesis Test for One-Way Analysis of Variance Hypothesis Test for Two-Way Analysis of Variance Interaction Kruskal-Wallis Test One-Way Analysis of Variance Randomized Block Design Sum of Squares Decomposition for One-Way

Analysis of Variance Sum of Squares Decomposition for Two-Way

Analysis of Variance Two-Way Analysis of Variance: Several

Observations per Cell