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Chapter 15. Analysis of Variance. Framework for One-Way Analysis of Variance. - PowerPoint PPT Presentation
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Chapter 15Chapter 15
Analysis of Analysis of VarianceVariance
©
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